Mechanics

Mechanics. The subject of mechanics may be divided into two parts: (1) theoretical or abstract mechanics, and (2) applied mechanics.

1\. Theoretical Mechanics

Historically theoretical mechanics began with the study of practical contrivances such as the lever, and the name mechanics (Gr. τὰ μηχανικά), which might more properly be restricted to the theory of mechanisms, and which was indeed used in this narrower sense by Newton, has clung to it, although the subject has long attained a far wider scope. In recent times it has been proposed to adopt the term dynamics (from Gr. δύναμις force,) as including the whole science of the action of force on bodies, whether at rest or in motion. The subject is usually expounded under the two divisions of statics and kinetics, the former dealing with the conditions of rest or equilibrium and the latter with the phenomena of motion as affected by force. To this latter division the old name of dynamics (in a restricted sense) is still often applied. The mere geometrical description and analysis of various types of motion, apart from the consideration of the forces concerned, belongs to kinematics. This is sometimes discussed as a separate theory, but for our present purposes it is more convenient to introduce kinematical motions as they are required. We follow also the traditional practice of dealing first with statics and then with kinetics. This is, in the main, the historical order of development, and for purposes of exposition it has many advantages. The laws of equilibrium are, it is true, necessarily included as a particular case under those of motion; but there is no real inconvenience in formulating as the basis of statics a few provisional postulates which are afterwards seen to be comprehended in a more general scheme.

The whole subject rests ultimately on the Newtonian laws of motion and on some natural extensions of them. As these laws are discussed under a separate heading (Motion, Laws of), it is here only necessary to indicate the standpoint from which the present article is written. It is a purely empirical one. Guided by experience, we are able to frame rules which enable us to say with more or less accuracy what will be the consequences, or what were the antecedents, of a given state of things. These rules are sometimes dignified by the name of “laws of nature,” but they have relation to our present state of knowledge and to the degree of skill with which we have succeeded in giving more or less compact expression to it. They are therefore liable to be modified from time to time, or to be superseded by more convenient or more comprehensive modes of statement. Again, we do not aim at anything so hopeless, or indeed so useless, as a complete description of any phenomenon. Some features are naturally more important or more interesting to us than others; by their relative simplicity and evident constancy they have the first hold on our attention, whilst those which are apparently accidental and vary from one occasion to another arc ignored, or postponed for later examination. It follows that for the purposes of such description as is possible some process of abstraction is inevitable if our statements are to be simple and definite. Thus in studying the flight of a stone through the air we replace the body in imagination by a mathematical point endowed with a mass-coefficient. The size and shape, the complicated spinning motion which it is seen to execute, the internal strains and vibrations which doubtless take place, are all sacrificed in the mental picture in order that attention may be concentrated on those features of the phenomenon which are in the first place most interesting to us. At a later stage in our subject the conception of the ideal rigid body is introduced; this enables us to fill in some details which were previously wanting, but others are still omitted. Again, the conception of a force as concentrated in a mathematical line is as unreal as that of a mass concentrated in a point, but it is a convenient fiction for our purpose, owing to the simplicity which it lends to our statements.

The laws which are to be imposed on these ideal representations are in the first instance largely at our choice. Any scheme of abstract dynamics constructed in this way, provided it be self-consistent, is mathematically legitimate; but from the physical point of view we require that it should help us to picture the sequence of phenomena as they actually occur. Its success or failure in this respect can only be judged a posteriori by comparison of the results to which it leads with the facts. It is to be noticed, moreover, that all available tests apply only to the scheme as a whole; owing to the complexity of phenomena we cannot submit any one of its postulates to verification apart from the rest.

It is from this point of view that the question of relativity of motion, which is often felt to be a stumbling-block on the very threshold of the subject, is to be judged. By “motion” we mean of necessity motion relative to some frame of reference which is conventionally spoken of as “fixed.” In the earlier stages of our subject this may be any rigid, or apparently rigid, structure fixed relatively to the earth. If we meet with phenomena which do not fit easily into this view, we have the alternatives either to modify our assumed laws of motion, or to call to our aid adventitious forces, or to examine whether the discrepancy can be reconciled by the simpler expedient of a new basis of reference. It is hardly necessary to say that the latter procedure has hitherto been found to be adequate. As a first step we adopt a system of rectangular axes whose origin is fixed in the earth, but whose directions are fixed by relation to the stars; in the planetary theory the origin is transferred to the sun, and afterwards to the mass-centre of the solar system; and so on. At each step there is a gain in accuracy and comprehensiveness; and the conviction is cherished that some system of rectangular axes exists with respect to which the Newtonian scheme holds with all imaginable accuracy.

A similar account might be given of the conception of time as a measurable quantity, but the remarks which it is necessary to make under this head will find a place later.

The following synopsis shows the scheme on which the treatment is based:—

Part 1.—Statics.  
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1. | Statics of a particle.  
2. | Statics of a system of particles.  
3. | Plane kinematics of a rigid body.  
4. | Plane statics.  
5. | Graphical statics.  
6. | Theory of frames.  
7. | Three-dimensional kinematics of a rigid body.  
8. | Three-dimensional statics.  
9. | Work.  
10. | Statics of inextensible chains.  
11. | Theory of mass-systems.  
Part 2.—Kinetics.  
12. | Rectilinear motion.  
13. | General motion of a particle.  
14. | Central forces. Hodograph.  
15. | Kinetics of a system of discrete particles.  
16. | Kinetics of a rigid body. Fundamental principles.  
17. | Two-dimensional problems.  
18. | Equations of motion in three dimensions.  
19. | Free motion of a solid.  
20. | Motion of a solid of revolution.  
21. | Moving axes of reference.  
22. | Equations of motion in generalized co-ordinates.  
23. | Stability of equilibrium. Theory of vibrations.  
  
Part I.—Statics

§ 1. Statics of a Particle.—By a particle is meant a body whose position can for the purpose in hand be sufficiently specified by a mathematical point. It need not be “infinitely small,” or even small compared with ordinary standards; thus in astronomy such vast bodies as the sun, the earth, and the other planets can for many purposes be treated merely as points endowed with mass.

A force is conceived as an effort having a certain direction and a certain magnitude. It is therefore adequately represented, for mathematical purposes, by a straight line AB drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force. In other words, a force is mathematically of the nature of a “vector” (see Vector Analysis, Quaternions). In most questions of pure statics we are concerned only with the ratios of the various forces which enter into the problem, so that it is indifferent what unit of force is adopted. For many purposes a gravitational system of measurement is most natural; thus we speak of a force of so many pounds or so many kilogrammes. The “absolute” system of measurement will be referred to below in Part II., Kinetics. It is to be remembered that all “force” is of the nature of a push or a pull, and that according to the accepted terminology of modern mechanics such phrases as “force of inertia,” “accelerating force,” “moving force,” once classical, are proscribed. This rigorous limitation of the meaning of the word is of comparatively recent origin, and it is perhaps to be regretted that some more technical term has not been devised, but the convention must now be regarded as established.  
  
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Fig. 1.  
  
The fundamental postulate of this part of our subject is that the two forces acting on a particle may be compounded by the “parallelogram rule.” Thus, if the two forces P,Q be represented by the lines OA, OB, they can be replaced by a single force R represented by the diagonal OC of the parallelogram determined by OA, OB. This is of course a physical assumption whose propriety is justified solely by experience. We shall see later that it is implied in Newton’s statement of his Second Law of motion. In modern language, forces are compounded by “vector-addition”; thus, if we draw in succession vectors HK>, KL> to represent P, Q, the force R is represented by the vector HL> which is the “geometric sum” of HK>, KL>.

By successive applications of the above rule any number of forces acting on a particle may be replaced by a single force which is the vector-sum of the given forces: this single force is called the resultant. Thus if AB>, BC>, CD> ..., HK> be vectors representing the given forces, the resultant will be given by AK>. It will be understood that the figure ABCD ... K need not be confined to one plane.  
  
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Fig. 2.  
  
If, in particular, the point K coincides with A, so that the resultant vanishes, the given system of forces is said to be in equilibrium—i.e. the particle could remain permanently at rest under its action. This is the proposition known as the polygon of forces. In the particular case of three forces it reduces to the triangle of forces, viz. “If three forces acting on a particle are represented as to magnitude and direction by the sides of a triangle taken in order, they are in equilibrium.”

A sort of converse proposition is frequently useful, viz. if three forces acting on a particle be in equilibrium, and any triangle be constructed whose sides are respectively parallel to the forces, the magnitudes of the forces will be to one another as the corresponding sides of the triangle. This follows from the fact that all such triangles are necessarily similar.  
  
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Fig. 3.  
  
As a simple example of the geometrical method of treating statical problems we may consider the equilibrium of a particle on a “rough” inclined plane. The usual empirical law of sliding friction is that the mutual action between two plane surfaces in contact, or between a particle and a curve or surface, cannot make with the normal an angle exceeding a certain limit λ called the angle of friction. If the conditions of equilibrium require an obliquity greater than this, sliding will take place. The precise value of λ will vary with the nature and condition of the surfaces in contact. In the case of a body simply resting on an inclined plane, the reaction must of course be vertical, for equilibrium, and the slope α of the plane must therefore not exceed λ. For this reason λ is also known as the angle of repose. If α > λ, a force P must be applied in order to maintain equilibrium; let θ be the inclination of P to the plane, as shown in the left-hand diagram. The relations between this force P, the gravity W of the body, and the reaction S of the plane are then determined by a triangle of forces HKL. Since the inclination of S to the normal cannot exceed λ on either side, the value of P must lie between two limits which are represented by L1H, L2H, in the right-hand diagram. Denoting these limits by P1, P2, we have

P1/W = L1H/HK = sin (α − λ)/cos (θ \+ λ),  
P2/W = L2H/HK = sin (α \+ λ)/cos (θ − λ).

It appears, moreover, that if θ be varied P will be least when L1H is at right angles to KL1, in which case P1 = W sin (α − λ), corresponding to θ = −λ.  
  
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Just as two or more forces can be combined into a single resultant, so a single force may be resolved into components acting in assigned directions. Thus a force can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig. 1. If, as is usually most convenient, the two assigned directions are at right angles, the two components of a force P will be P cos θ, P sin θ, where θ is the inclination of P to the direction of the former component. This leads to formulae for the analytical reduction of a system of coplanar forces acting on a particle. Adopting rectangular axes Ox, Oy, in the plane of the forces, and distinguishing the various forces of the system by suffixes, we can replace the system by two forces X, Y, in the direction of co-ordinate axes; viz.—

X = P1 cos θ1 \+ P2 cos θ2 \+ ... = Σ (P cos θ),  
Y = P1 sin θ1 \+ P2 sin θ2 \+ ... = Σ (P sin θ).

(1)

These two forces X, Y, may be combined into a single resultant R making an angle φ with Ox, provided

X = R cos φ,  Y = R sin φ,

(2)

whence

R2 = X2 \+ Y2, tan φ = Y/X.

(3)

For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words, the sum of the components of the system must be zero for each of two perpendicular directions in the plane.  
  
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Fig. 5.  
  
A similar procedure applies to a three-dimensional system. Thus if, O being the origin, OH> represent any force P of the system, the planes drawn through H parallel to the co-ordinate planes will enclose with the latter a parallelepiped, and it is evident that OH> is the geometric sum of OA>, AN>, NH>, or OA>, OB>, OC>, in the figure. Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are the “direction-ratios” of OH>. The whole system can be reduced in this way to three forces

X = Σ (Pl),  Y = Σ (Pm),  Z = Σ (Pn),

(4)

acting along the co-ordinate axes. These can again be combined into a single resultant R acting in the direction (λ, μ, ν), provided

X = Rλ,  Y = Rμ,  Z = Rν.

(5)

If the axes are rectangular, the direction-ratios become direction-cosines, so that λ2 \+ μ2 \+ ν2 = 1, whence

R2 = X2 \+ Y2 \+ Z2.

(6)

The conditions of equilibrium are X = 0, Y = 0, Z = 0.

§ 2. Statics of a System of Particles.—We assume that the mutual forces between the pairs of particles, whatever their nature, are subject to the “Law of Action and Reaction” (Newton’s Third Law); i.e. the force exerted by a particle A on a particle B, and the force exerted by B on A, are equal and opposite in the line AB. The problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate. For if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 3n Cartesian (or other) co-ordinates of the particles, which are to be found. If the system be subject to frictionless constraints, e.g. if some of the particles be constrained to lie on smooth surfaces, or if pairs of particles be connected by inextensible strings, then for each geometrical relation thus introduced we have an unknown reaction (e.g. the pressure of the smooth surface, or the tension of the string), so that the problem is still determinate.  
  
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Fig. 6.  
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Fig. 7.  
Fig. 8.  
  
The case of the funicular polygon will be of use to us later. A number of particles attached at various points of a string are acted on by given extraneous forces P1, P2, P3 ... respectively. The relation between the three forces acting on any particle, viz. the extraneous force and the tensions in the two adjacent portions of the string can be exhibited by means of a triangle of forces; and if the successive triangles be drawn to the same scale they can be fitted together so as to constitute a single force-diagram, as shown in fig. 6. This diagram consists of a polygon whose successive sides represent the given forces P1, P2, P3 ..., and of a series of lines connecting the vertices with a point O. These latter lines measure the tensions in the successive portions of string. As a special, but very important case, the forces P1, P2, P3 ... may be parallel, e.g. they may be the weights of the several particles. The polygon of forces is then made up of segments of a vertical line. We note that the tensions have now the same horizontal projection (represented by the dotted line in fig. 7). It is further of interest to note that if the weights be all equal, and at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical. To prove this statement, let A, B, C, D ... be successive vertices, and let H, K ... be the middle points of AC, BD ...; then BH, CK ... will be vertical by the hypothesis, and since the geometric sum of BA>, BC> is represented by 2BH>, the tension in BA: tension in BC: weight at B

as BA : BC : 2BH.

The tensions in the successive portions of the string are therefore proportional to the respective lengths, and the lines BH, CK ... are all equal. Hence AD, BC are parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through A, B, C, D. The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ... must be coincident.

§ 3. Plane Kinematics of a Rigid Body.—The ideal rigid body is one in which the distance between any two points is invariable. For the present we confine ourselves to the consideration of displacements in two dimensions, so that the body is adequately represented by a thin lamina or plate.  
  
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Fig. 9.  
  
The position of a lamina movable in its own plane is determinate when we know the positions of any two points A, B of it. Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the invariability of the length AB, it is plain that virtually three independent elements are required and suffice to specify the position of the lamina. For instance, the lamina may in general be fixed by connecting any three points of it by rigid links to three fixed points in its plane. The three independent elements may be chosen in a variety of ways (e.g. they may be the lengths of the three links in the above example). They may be called (in a generalized sense) the co-ordinates of the lamina. The lamina when perfectly free to move in its own plane is said to have three degrees of freedom.  
  
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Fig. 10.  
Fig. 11.  
  
By a theorem due to M. Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J. For suppose that in consequence of the displacement a point of the lamina is brought from A to B, whilst the point of the lamina which was originally at B is brought to C. Since AB, BC, are two different positions of the same line in the lamina they are equal, and it is evident that the rotation could have been effected by a rotation about J, the centre of the circle ABC, through an angle AJB. As a special case the three points A, B, C may be in a straight line; J is then at infinity and the displacement is equivalent to a pure translation, since every point of the lamina is now displaced parallel to AB through a space equal to AB.

Next, consider any continuous motion of the lamina. The latter may be brought from any one of its positions to a neighbouring one by a rotation about the proper centre. The limiting position J of this centre, when the two positions are taken infinitely close to one another, is called the instantaneous centre. If P, P′ be consecutive positions of the same point, and δθ the corresponding angle of rotation, then ultimately PP′ is at right angles to JP and equal to JP·δθ. The instantaneous centre will have a certain locus in space, and a certain locus in the lamina. These two loci are called pole-curves or centrodes, and are sometimes distinguished as the space-centrode and the body-centrode, respectively. In the continuous motion in question the latter curve rolls without slipping on the former (M. Chasles). Consider in fact any series of successive positions 1, 2, 3... of the lamina (fig. 11); and let J12, J23, J34... be the positions in space of the centres of the rotations by which the lamina can be brought from the first position to the second, from the second to the third, and so on. Further, in the position 1, let J12, J′23, J′34 ... be the points of the lamina which have become the successive centres of rotation. The given series of positions will be assumed in succession if we imagine the lamina to rotate first about J12 until J′23 comes into coincidence with J23, then about J23 until J′34 comes into coincidence with J34, and so on. This is equivalent to imagining the polygon J12 J′23 J′34 ..., supposed fixed in the lamina, to roll on the polygon J12 J23 J34 ..., which is supposed fixed in space. By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated. The particular case where both centrodes are circles is specially important in mechanism.  
  
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Fig. 12.  
Fig. 13.  
  
The theory may be illustrated by the case of “three-bar motion.” Let ABCD be any quadrilateral formed of jointed links. If, AB being held fixed, the quadrilateral be slightly deformed, it is obvious that the instantaneous centre J will be at the intersection of the straight lines AD, BC, since the displacements of the points D, C are necessarily at right angles to AD, BC, respectively. Hence these displacements are proportional to JD, JC, and therefore to DD′ CC′, where C′D′ is any line drawn parallel to CD, meeting BC, AD in C′, D′, respectively. The determination of the centrodes in three-bar motion is in general complicated, but in one case, that of the “crossed parallelogram” (fig. 13), they assume simple forms. We then have AB = DC and AD = BC, and from the symmetries of the figure it is plain that

AJ + JB = CJ + JD = AD.

Hence the locus of J relative to AB, and the locus relative to CD are equal ellipses of which A, B and C, D are respectively the foci. It may be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but admits of infinitesimal displacement, whenever the directions of the three links are concurrent (or parallel).

The matter may of course be treated analytically, but we shall only require the formula for infinitely small displacements. If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are λ, μ, and if the axes are simultaneously turned through an angle ε, the co-ordinates of a point of the lamina, relative to the original axes, are changed from x, y to λ \+ x cos ε − y sin ε, μ \+ x sin ε \+ y cos ε, or λ \+ x − yε, μ \+ xε \+ y, ultimately. Hence the component displacements are ultimately

δx = λ − yε, δy = μ \+ xε

(1)

If we equate these to zero we get the co-ordinates of the instantaneous centre.

§ 4. Plane Statics.—The statics of a rigid body rests on the following two assumptions:—

(i) A force may be supposed to be applied indifferently at any point in its line of action. In other words, a force is of the nature of a “bound” or “localized” vector; it is regarded as resident in a certain line, but has no special reference to any particular point of the line.

(ii) Two forces in intersecting lines may be replaced by a force which is their geometric sum, acting through the intersection. The theory of parallel forces is included as a limiting case. For if O, A, B be any three points, and m, n any scalar quantities, we have in vectors

m · OA> \+ n · OB> = (m + n) OC>,

(1)

provided

m · CA> \+ n · CB> = 0.

(2)

Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided

m = P/OA, n = Q/OB;

also

R = P·OC/OA + Q·OC/OB,

(3)

and

P · AC : Q·CB = OA : OB.

(4)

These formulae give a means of constructing the resultant by means of any transversal AB cutting the lines of action. If we now imagine the point O to recede to infinity, the forces P, Q and the resultant R are parallel, and we have

R = P + Q,  P·AC = Q·CB.

(5)  
  
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Fig. 14.  
  
When P, Q have opposite signs the point C divides AB externally on the side of the greater force. The investigation fails when P + Q = 0, since it leads to an infinitely small resultant acting in an infinitely distant line. A combination of two equal, parallel, but oppositely directed forces cannot in fact be replaced by anything simpler, and must therefore be recognized as an independent entity in statics. It was called by L. Poinsot, who first systematically investigated its properties, a couple.

We now restrict ourselves for the present to the systems of forces in one plane. By successive applications of (ii) any such coplanar system can in general be reduced to a single resultant acting in a definite line. As exceptional cases the system may reduce to a couple, or it may be in equilibrium.  
  
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Fig. 15.  
Fig. 16.  
  
The moment of a force about a point O is the product of the force into the perpendicular drawn to its line of action from O, this perpendicular being reckoned positive or negative according as O lies on one side or other of the line of action. If we mark off a segment AB along the line of action so as to represent the force completely, the moment is represented as to magnitude by twice the area of the triangle OAB, and the usual convention as to sign is that the area is to be reckoned positive or negative according as the letters O, A, B, occur in “counter-clockwise” or “clockwise” order.

The sum of the moments of two forces about any point O is equal to the moment of their resultant (P. Varignon, 1687). Let AB, AC (fig. 16) represent the two forces, AD their resultant; we have to prove that the sum of the triangles OAB, OAC is equal to the triangle OAD, regard being had to signs. Since the side OA is common, we have to prove that the sum of the perpendiculars from B and C on OA is equal to the perpendicular from D on OA, these perpendiculars being reckoned positive or negative according as they lie to the right or left of AO. Regarded as a statement concerning the orthogonal projections of the vectors AB> and AC> (or BD), and of their sum AD>, on a line perpendicular to AO, this is obvious.

It is now evident that in the process of reduction of a coplanar system no change is made at any stage either in the sum of the projections of the forces on any line or in the sum of their moments about any point. It follows that the single resultant to which the system in general reduces is uniquely determinate, i.e. it acts in a definite line and has a definite magnitude and sense. Again it is necessary and sufficient for equilibrium that the sum of the projections of the forces on each of two perpendicular directions should vanish, and (moreover) that the sum of the moments about some one point should be zero. The fact that three independent conditions must hold for equilibrium is important. The conditions may of course be expressed in different (but equivalent) forms; e.g. the sum of the moments of the forces about each of the three points which are not collinear must be zero.  
  
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Fig. 17.  
  
The particular case of three forces is of interest. If they are not all parallel they must be concurrent, and their vector-sum must be zero. Thus three forces acting perpendicular to the sides of a triangle at the middle points will be in equilibrium provided they are proportional to the respective sides, and act all inwards or all outwards. This result is easily extended to the case of a polygon of any number of sides; it has an important application in hydrostatics.

Again, suppose we have a bar AB resting with its ends on two smooth inclined planes which face each other. Let G be the centre of gravity (§ 11), and let AG = a, GB = b. Let α, β be the inclinations of the planes, and θ the angle which the bar makes with the vertical. The position of equilibrium is determined by the consideration that the reactions at A and B, which are by hypothesis normal to the planes, must meet at a point J on the vertical through G. Hence

JG/a = sin (θ − α) / sin α,  JG/b = sin (θ \+ β) / sin β,

whence

cot θ = | a cot α − b cot β | .  
---|---|---  
a + b  
  
(6)

If the bar is uniform we have a = b, and

cot θ = 1⁄2 (cot α − cot β).

(7)

The problem of a rod suspended by strings attached to two points of it is virtually identical, the tensions of the strings taking the place of the reactions of the planes.  
  
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Fig. 18.  
  
Just as a system of forces is in general equivalent to a single force, so a given force can conversely be replaced by combinations of other forces, in various ways. For instance, a given force (and consequently a system of forces) can be replaced in one and only one way by three forces acting in three assigned straight lines, provided these lines be not concurrent or parallel. Thus if the three lines form a triangle ABC, and if the given force F meet BC in H, then F can be resolved into two components acting in HA, BC, respectively. And the force in HA can be resolved into two components acting in BC, CA, respectively. A simple graphical construction is indicated in fig. 19, where the dotted lines are parallel. As an example, any system of forces acting on the lamina in fig. 9 is balanced by three determinate tensions (or thrusts) in the three links, provided the directions of the latter are not concurrent.  
  
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Fig. 19.  
  
If P, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish. Hence in “trilinear” co-ordinates, with ABC as fundamental triangle, its equation is Pα \+ Qβ \+ Rγ = 0. If P : Q : R = a : b : c, where a, b, c are the lengths of the sides, this becomes the “line at infinity,” and the forces reduce to a couple.  
  
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Fig. 20.  
  
The sum of the moments of the two forces of a couple is the same about any point in the plane. Thus in the figure the sum of the moments about O is P·OA − P·OB or P·AB, which is independent of the position of O. This sum is called the moment of the couple; it must of course have the proper sign attributed to it. It easily follows that any two couples of the same moment are equivalent, and that any number of couples can be replaced by a single couple whose moment is the sum of their moments. Since a couple is for our purposes sufficiently represented by its moment, it has been proposed to substitute the name torque (or twisting effort), as free from the suggestion of any special pair of forces.

A system of forces represented completely by the sides of a plane polygon taken in order is equivalent to a couple whose moment is represented by twice the area of the polygon; this is proved by taking moments about any point. If the polygon intersects itself, care must be taken to attribute to the different parts of the area their proper signs.  
  
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Fig. 21.  
  
Again, any coplanar system of forces can be replaced by a single force R acting at any assigned point O, together with a couple G. The force R is the geometric sum of the given forces, and the moment (G) of the couple is equal to the sum of the moments of the given forces about O. The value of G will in general vary with the position of O, and will vanish when O lies on the line of action of the single resultant.  
  
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Fig. 22.  
  
The formal analytical reduction of a system of coplanar forces is as follows. Let (x1, y1), (x2, y2), ... be the rectangular co-ordinates of any points A1, A2, ... on the lines of action of the respective forces. The force at A1 may be replaced by its components X1, Y1, parallel to the co-ordinate axes; that at A2 by its components X2, Y2, and so on. Introducing at O two equal and opposite forces ±X1 in Ox, we see that X1 at A1 may be replaced by an equal and parallel force at O together with a couple −y1X1. Similarly the force Y1 at A1 may be replaced by a force Y1 at O together with a couple x1Y1. The forces X1, Y1, at O can thus be transferred to O provided we introduce a couple x1Y1 − y1X1. Treating the remaining forces in the same way we get a force X1 \+ X2 \+ ... or Σ(X) along Ox, a force Y1 \+ Y2 \+ ... or Σ(Y) along Oy, and a couple (x1Y1 − y1X1) + (x2Y2 − y2X2) + ... or Σ(xY − yX). The three conditions of equilibrium are therefore

Σ(X) = 0,  Σ(Y) = 0,  Σ(xY − yX) = 0.

(8)

If O′ be a point whose co-ordinates are (ξ, η), the moment of the couple when the forces are transferred to O′ as a new origin will be Σ{(x − ξ) Y − (y − η) X}. This vanishes, i.e. the system reduces to a single resultant through O′, provided

−ξ·Σ(Y) + η·Σ(X) + Σ(xY − yX) = 0.

(9)

If ξ, η be regarded as current co-ordinates, this is the equation of the line of action of the single resultant to which the system is in general reducible.

If the forces are all parallel, making say an angle θ with Ox, we may write X1 = P1 cos θ, Y1 = P1 sin θ, X2 = P2 cos θ, Y2 = P2 sin θ, .... The equation (9) then becomes

{Σ(xP) − ξ·Σ(P)} sin θ − {Σ(yP) − η·Σ(P)} cos θ = 0.

(10)

If the forces P1, P2, ... be turned in the same sense through the same angle about the respective points A1, A2, ... so as to remain parallel, the value of θ is alone altered, and the resultant Σ(P) passes always through the point

x = | Σ(xP) | ,  y = | Σ(yP) | ,  
---|---|---|---|---  
Σ(P) | Σ(P)  
  
(11)

which is determined solely by the configuration of the points A1, A2, ... and by the ratios P1 : P2 : ... of the forces acting at them respectively. This point is called the centre of the given system of parallel forces; it is finite and determinate unless Σ(P) = 0. A geometrical proof of this theorem, which is not restricted to a two-dimensional system, is given later (§ 11). It contains the theory of the centre of gravity as ordinarily understood. For if we have an assemblage of particles whose mutual distances are small compared with the dimensions of the earth, the forces of gravity on them constitute a system of sensibly parallel forces, sensibly proportional to the respective masses. If now the assemblage be brought into any other position relative to the earth, without alteration of the mutual distances, this is equivalent to a rotation of the directions of the forces relatively to the assemblage, the ratios of the forces remaining unaltered. Hence there is a certain point, fixed relatively to the assemblage, through which the resultant of gravitational action always passes; this resultant is moreover equal to the sum of the forces on the several particles.  
  
---  
Fig. 23.  
  
The theorem that any coplanar system of forces can be reduced to a force acting through any assigned point, together with a couple, has an important illustration in the theory of the distribution of shearing stress and bending moment in a horizontal beam, or other structure, subject to vertical extraneous forces. If we consider any vertical section P, the forces exerted across the section by the portion of the structure on one side on the portion on the other may be reduced to a vertical force F at P and a couple M. The force measures the shearing stress, and the couple the bending moment at P; we will reckon these quantities positive when the senses are as indicated in the figure.

If the remaining forces acting on the portion of the structure on either side of P are known, then resolving vertically we find F, and taking moments about P we find M. Again if PQ be any segment of the beam which is free from load, Q lying to the right of P, we find

FP = FQ,  MP − MQ = −F·PQ;

(12)

hence F is constant between the loads, whilst M decreases as we travel to the right, with a constant gradient −F. If PQ be a short segment containing an isolated load W, we have

FQ − FP = −W, MQ = MP;

(13)  
  
---  
Fig. 24.  
  
hence F is discontinuous at a concentrated load, diminishing by an amount equal to the load as we pass the loaded point to the right, whilst M is continuous. Accordingly the graph of F for any system of isolated loads will consist of a series of horizontal lines, whilst that of M will be a continuous polygon.

To pass to the case of continuous loads, let x be measured horizontally along the beam to the right. The load on an element δx of the beam may be represented by wδx, where w is in general a function of x. The equations (12) are now replaced by

δF = −wδx,  δM = −Fδx,

whence

FQ − FP = − ∫QP w dx,  MQ − MP = − ∫QP F dx.

(14)  
  
---  
Fig. 25.  
  
The latter relation shows that the bending moment varies as the area cut off by the ordinate in the graph of F. In the case of uniform load we have

F = −wx + A,  M = 1⁄2wx2 − Ax + B,

(15)

where the arbitrary constants A,B are to be determined by the conditions of the special problem, e.g. the conditions at the ends of the beam. The graph of F is a straight line; that of M is a parabola with vertical axis. In all cases the graphs due to different distributions of load may be superposed. The figure shows the case of a uniform heavy beam supported at its ends.  
  
---  
Fig. 26.  
  
§ 5. Graphical Statics.—A graphical method of reducing a plane system of forces was introduced by C. Culmann (1864). It involves the construction of two figures, a force-diagram and a funicular polygon. The force-diagram is constructed by placing end to end a series of vectors representing the given forces in magnitude and direction, and joining the vertices of the polygon thus formed to an arbitrary pole O. The funicular or link polygon has its vertices on the lines of action of the given forces, and its sides respectively parallel to the lines drawn from O in the force-diagram; in particular, the two sides meeting in any vertex are respectively parallel to the lines drawn from O to the ends of that side of the force-polygon which represents the corresponding force. The relations will be understood from the annexed diagram, where corresponding lines in the force-diagram (to the right) and the funicular (to the left) are numbered similarly. The sides of the force-polygon may in the first instance be arranged in any order; the force-diagram can then be completed in a doubly infinite number of ways, owing to the arbitrary position of O; and for each force-diagram a simply infinite number of funiculars can be drawn. The two diagrams being supposed constructed, it is seen that each of the given systems of forces can be replaced by two components acting in the sides of the funicular which meet at the corresponding vertex, and that the magnitudes of these components will be given by the corresponding triangle of forces in the force-diagram; thus the force 1 in the figure is equivalent to two forces represented by 01 and 12\. When this process of replacement is complete, each terminated side of the funicular is the seat of two forces which neutralize one another, and there remain only two uncompensated forces, viz., those resident in the first and last sides of the funicular. If these sides intersect, the resultant acts through the intersection, and its magnitude and direction are given by the line joining the first and last sides of the force-polygon (see fig. 26, where the resultant of the four given forces is denoted by R). As a special case it may happen that the force-polygon is closed, i.e. its first and last points coincide; the first and last sides of the funicular will then be parallel (unless they coincide), and the two uncompensated forces form a couple. If, however, the first and last sides of the funicular coincide, the two outstanding forces neutralize one another, and we have equilibrium. Hence the necessary and sufficient conditions of equilibrium are that the force-polygon and the funicular should both be closed. This is illustrated by fig. 26 if we imagine the force R, reversed, to be included in the system of given forces.

It is evident that a system of jointed bars having the shape of the funicular polygon would be in equilibrium under the action of the given forces, supposed applied to the joints; moreover any bar in which the stress is of the nature of a tension (as distinguished from a thrust) might be replaced by a string. This is the origin of the names “link-polygon” and “funicular” (cf. § 2).

If funiculars be drawn for two positions O, O′ of the pole in the force-diagram, their corresponding sides will intersect on a straight line parallel to OO′. This is essentially a theorem of projective geometry, but the following statical proof is interesting. Let AB (fig. 27) be any side of the force-polygon, and construct the corresponding portions of the two diagrams, first with O and then with O′ as pole. The force corresponding to AB may be replaced by the two components marked x, y; and a force corresponding to BA may be represented by the two components marked x′, y′. Hence the forces x, y, x′, y′ are in equilibrium. Now x, x′ have a resultant through H, represented in magnitude and direction by OO′, whilst y, y′ have a resultant through K represented in magnitude and direction by O′O. Hence HK must be parallel to OO′. This theorem enables us, when one funicular has been drawn, to construct any other without further reference to the force-diagram.  
  
---  
Fig. 27.  
  
The complete figures obtained by drawing first the force-diagrams of a system of forces in equilibrium with two distinct poles O, O′, and secondly the corresponding funiculars, have various interesting relations. In the first place, each of these figures may be conceived as an orthogonal projection of a closed plane-faced polyhedron. As regards the former figure this is evident at once; viz. the polyhedron consists of two pyramids with vertices represented by O, O′, and a common base whose perimeter is represented by the force-polygon (only one of these is shown in fig. 28). As regards the funicular diagram, let LM be the line on which the pairs of corresponding sides of the two polygons meet, and through it draw any two planes ω, ω′. Through the vertices A, B, C, ... and A′, B′, C′, ... of the two funiculars draw normals to the plane of the diagram, to meet ω and ω′ respectively. The points thus obtained are evidently the vertices of a polyhedron with plane faces.  
  
---  
Fig. 28.  
---  
Fig. 29.  
  
To every line in either of the original figures corresponds of course a parallel line in the other; moreover, it is seen that concurrent lines in either figure correspond to lines forming a closed polygon in the other. Two plane figures so related are called reciprocal, since the properties of the first figure in relation to the second are the same as those of the second with respect to the first. A still simpler instance of reciprocal figures is supplied by the case of concurrent forces in equilibrium (fig. 29). The theory of these reciprocal figures was first studied by J. Clerk Maxwell, who showed amongst other things that a reciprocal can always be drawn to any figure which is the orthogonal projection of a plane-faced polyhedron. If in fact we take the pole of each face of such a polyhedron with respect to a paraboloid of revolution, these poles will be the vertices of a second polyhedron whose edges are the “conjugate lines” of those of the former. If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel. Another proof will be indicated later (§ 8) in connexion with the properties of the linear complex. It is convenient to have a notation which shall put in evidence the reciprocal character. For this purpose we may designate the points in one figure by letters A, B, C, ... and the corresponding polygons in the other figure by the same letters; a line joining two points A, B in one figure will then correspond to the side common to the two polygons A, B in the other. This notation was employed by R. H. Bow in connexion with the theory of frames (§ 6, and see also Applied Mechanics below) where reciprocal diagrams are frequently of use (cf. Diagram).

When the given forces are all parallel, the force-polygon consists of a series of segments of a straight line. This case has important practical applications; for instance we may use the method to find the pressures on the supports of a beam loaded in any given manner. Thus if AB, BC, CD represent the given loads, in the force-diagram, we construct the sides corresponding to OA, OB, OC, OD in the funicular; we then draw the closing line of the funicular polygon, and a parallel OE to it in the force diagram. The segments DE, EA then represent the upward pressures of the two supports on the beam, which pressures together with the given loads constitute a system of forces in equilibrium. The pressures of the beam on the supports are of course represented by ED, AE. The two diagrams are portions of reciprocal figures, so that Bow’s notation is applicable.  
  
---  
Fig. 30.  
---  
Fig. 31.  
  
A graphical method can also be applied to find the moment of a force, or of a system of forces, about any assigned point P. Let F be a force represented by AB in the force-diagram. Draw a parallel through P to meet the sides of the funicular which correspond to OA, OB in the points H, K. If R be the intersection of these sides, the triangles OAB, RHK are similar, and if the perpendiculars OM, RN be drawn we have

HK·OM = AB·RN = F·RN,

which is the moment of F about P. If the given forces are all parallel (say vertical) OM is the same for all, and the moments of the several forces about P are represented on a certain scale by the lengths intercepted by the successive pairs of sides on the vertical through P. Moreover, the moments are compounded by adding (geometrically) the corresponding lengths HK. Hence if a system of vertical forces be in equilibrium, so that the funicular polygon is closed, the length which this polygon intercepts on the vertical through any point P gives the sum of the moments about P of all the forces on one side of this vertical. For instance, in the case of a beam in equilibrium under any given loads and the reactions at the supports, we get a graphical representation of the distribution of bending moment over the beam. The construction in fig. 30 can easily be adjusted so that the closing line shall be horizontal; and the figure then becomes identical with the bending-moment diagram of § 4. If we wish to study the effects of a movable load, or system of loads, in different positions on the beam, it is only necessary to shift the lines of action of the pressures of the supports relatively to the funicular, keeping them at the same, distance apart; the only change is then in the position of the closing line of the funicular. It may be remarked that since this line joins homologous points of two “similar” rows it will envelope a parabola.

The “centre” (§ 4) of a system of parallel forces of given magnitudes, acting at given points, is easily determined graphically. We have only to construct the line of action of the resultant for each of two arbitrary directions of the forces; the intersection of the two lines gives the point required. The construction is neatest if the two arbitrary directions are taken at right angles to one another.

§ 6. Theory of Frames.—A frame is a structure made up of pieces, or members, each of which has two joints connecting it with other members. In a two-dimensional frame, each joint may be conceived as consisting of a small cylindrical pin fitting accurately and smoothly into holes drilled through the members which it connects. This supposition is a somewhat ideal one, and is often only roughly approximated to in practice. We shall suppose, in the first instance, that extraneous forces act on the frame at the joints only, i.e. on the pins.

On this assumption, the reactions on any member at its two joints must be equal and opposite. This combination of equal and opposite forces is called the stress in the member; it may be a tension or a thrust. For diagrammatic purposes each member is sufficiently represented by a straight line terminating at the two joints; these lines will be referred to as the bars of the frame.  
  
---  
Fig. 32.  
  
In structural applications a frame must be stiff, or rigid, i.e. it must be incapable of deformation without alteration of length in at least one of its bars. It is said to be just rigid if it ceases to be rigid when any one of its bars is removed. A frame which has more bars than are essential for rigidity may be called over-rigid; such a frame is in general self-stressed, i.e. it is in a state of stress independently of the action of extraneous forces. A plane frame of n joints which is just rigid (as regards deformation in its own plane) has 2n − 3 bars, for if one bar be held fixed the 2(n − 2) co-ordinates of the remaining n − 2 joints must just be determined by the lengths of the remaining bars. The total number of bars is therefore 2(n − 2) + 1. When a plane frame which is just rigid is subject to a given system of equilibrating extraneous forces (in its own plane) acting on the joints, the stresses in the bars are in general uniquely determinate. For the conditions of equilibrium of the forces on each pin furnish 2n equations, viz. two for each point, which are linear in respect of the stresses and the extraneous forces. This system of equations must involve the three conditions of equilibrium of the extraneous forces which are already identically satisfied, by hypothesis; there remain therefore 2n − 3 independent relations to determine the 2n − 3 unknown stresses. A frame of n joints and 2n − 3 bars may of course fail to be rigid owing to some parts being over-stiff whilst others are deformable; in such a case it will be found that the statical equations, apart from the three identical relations imposed by the equilibrium of the extraneous forces, are not all independent but are equivalent to less than 2n − 3 relations. Another exceptional case, known as the critical case, will be noticed later (§ 9).

A plane frame which can be built up from a single bar by successive steps, at each of which a new joint is introduced by two new bars meeting there, is called a simple frame; it is obviously just rigid. The stresses produced by extraneous forces in a simple frame can be found by considering the equilibrium of the various joints in a proper succession; and if the graphical method be employed the various polygons of force can be combined into a single force-diagram. This procedure was introduced by W. J. M. Rankine and J. Clerk Maxwell (1864). It may be noticed that if we take an arbitrary pole in the force-diagram, and draw a corresponding funicular in the skeleton diagram which represents the frame together with the lines of action of the extraneous forces, we obtain two complete reciprocal figures, in Maxwell’s sense. It is accordingly convenient to use Bow’s notation (§ 5), and to distinguish the several compartments of the frame-diagram by letters. See fig. 33, where the successive triangles in the diagram of forces may be constructed in the order XYZ, ZXA, AZB. The class of “simple” frames includes many of the frameworks used in the construction of roofs, lattice girders and suspension bridges; a number of examples will be found in the article Bridges. By examining the senses in which the respective forces act at each joint we can ascertain which members are in tension and which are in thrust; in fig. 33 this is indicated by the directions of the arrowheads.  
  
---  
Fig. 33.  
---  
Fig. 34.  
  
When a frame, though just rigid, is not “simple” in the above sense, the preceding method must be replaced, or supplemented, by one or other of various artifices. In some cases the method of sections is sufficient for the purpose. If an ideal section be drawn across the frame, the extraneous forces on either side must be in equilibrium with the forces in the bars cut across; and if the section can be drawn so as to cut only three bars, the forces in these can be found, since the problem reduces to that of resolving a given force into three components acting in three given lines (§ 4). The “critical case” where the directions of the three bars are concurrent is of course excluded. Another method, always available, will be explained under “Work” (§ 9).

When extraneous forces act on the bars themselves the stress in each bar no longer consists of a simple longitudinal tension or thrust. To find the reactions at the joints we may proceed as follows. Each extraneous force W acting on a bar may be replaced (in an infinite number of ways) by two components P, Q in lines through the centres of the pins at the extremities. In practice the forces W are usually vertical, and the components P, Q are then conveniently taken to be vertical also. We first alter the problem by transferring the forces P, Q to the pins. The stresses in the bars, in the problem as thus modified, may be supposed found by the preceding methods; it remains to infer from the results thus obtained the reactions in the original form of the problem. To find the pressure exerted by a bar AB on the pin A we compound with the force in AB given by the diagram a force equal to P. Conversely, to find the pressure of the pin A on the bar AB we must compound with the force given by the diagram a force equal and opposite to P. This question arises in practice in the theory of “three-jointed” structures; for the purpose in hand such a structure is sufficiently represented by two bars AB, BC. The right-hand figure represents a portion of the force-diagram; in particular ZX> represents the pressure of AB on B in the modified problem where the loads W1 and W2 on the two bars are replaced by loads P1, Q1, and P2, Q2 respectively, acting on the pins. Compounding with this XV>, which represents Q1, we get the actual pressure ZV> exerted by AB on B. The directions and magnitudes of the reactions at A and C are then easily ascertained. On account of its practical importance several other graphical solutions of this problem have been devised.  
  
---  
Fig. 35.  
  
§ 7. Three-dimensional Kinematics of a Rigid Body.—The position of a rigid body is determined when we know the positions of three points A, B, C of it which are not collinear, for the position of any other point P is then determined by the three distances PA, PB, PC. The nine co-ordinates (Cartesian or other) of A, B, C are subject to the three relations which express the invariability of the distances BC, CA, AB, and are therefore equivalent to six independent quantities. Hence a rigid body not constrained in any way is said to have six degrees of freedom. Conversely, any six geometrical relations restrict the body in general to one or other of a series of definite positions, none of which can be departed from without violating the conditions in question. For instance, the position of a theodolite is fixed by the fact that its rounded feet rest in contact with six given plane surfaces. Again, a rigid three-dimensional frame can be rigidly fixed relatively to the earth by means of six links.  
  
---  
Fig. 36. | Fig. 37.  
  
The six independent quantities, or “co-ordinates,” which serve to specify the position of a rigid body in space may of course be chosen in an endless variety of ways. We may, for instance, employ the three Cartesian co-ordinates of a particular point O of the body, and three angular co-ordinates which express the orientation of the body with respect to O. Thus in fig. 36, if OA, OB, OC be three mutually perpendicular lines in the solid, we may denote by θ the angle which OC makes with a fixed direction OZ, by ψ the azimuth of the plane ZOC measured from some fixed plane through OZ, and by φ the inclination of the plane COA to the plane ZOC. In fig. 36 these various lines and planes are represented by their intersections with a unit sphere having O as centre. This very useful, although unsymmetrical, system of angular co-ordinates was introduced by L. Euler. It is exemplified in “Cardan’s suspension,” as used in connexion with a compass-bowl or a gyroscope. Thus in the gyroscope the “flywheel” (represented by the globe in fig. 37) can turn about a diameter OC of a ring which is itself free to turn about a diametral axis OX at right angles to the former; this axis is carried by a second ring which is free to turn about a fixed diameter OZ, which is at right angles to OX.  
  
---  
Fig. 10.  
Fig. 38.  
Fig. 39.  
  
We proceed to sketch the theory of the finite displacements of a rigid body. It was shown by Euler (1776) that any displacement in which one point O of the body is fixed is equivalent to a pure rotation about some axis through O. Imagine two spheres of equal radius with O as their common centre, one fixed in the body and moving with it, the other fixed in space. In any displacement about O as a fixed point, the former sphere slides over the latter, as in a “ball-and-socket” joint. Suppose that as the result of the displacement a point of the moving sphere is brought from A to B, whilst the point which was at B is brought to C (cf. fig. 10). Let J be the pole of the circle ABC (usually a “small circle” of the fixed sphere), and join JA, JB, JC, AB, BC by great-circle arcs. The spherical isosceles triangles AJB, BJC are congruent, and we see that AB can be brought into the position BC by a rotation about the axis OJ through an angle AJB.

It is convenient to distinguish the two senses in which rotation may take place about an axis OA by opposite signs. We shall reckon a rotation as positive when it is related to the direction from O to A as the direction of rotation is related to that of translation in a right-handed screw. Thus a negative rotation about OA may be regarded as a positive rotation about OA′, the prolongation of AO. Now suppose that a body receives first a positive rotation α about OA, and secondly a positive rotation β about OB; and let A, B be the intersections of these axes with a sphere described about O as centre. If we construct the spherical triangles ABC, ABC′ (fig. 38), having in each case the angles at A and B equal to 1⁄2α and 1⁄2β respectively, it is evident that the first rotation will bring a point from C to C′ and that the second will bring it back to C; the result is therefore equivalent to a rotation about OC. We note also that if the given rotations had been effected in the inverse order, the axis of the resultant rotation would have been OC′, so that finite rotations do not obey the “commutative law.” To find the angle of the equivalent rotation, in the actual case, suppose that the second rotation (about OB) brings a point from A to A′. The spherical triangles ABC, A′BC (fig. 39) are “symmetrically equal,” and the angle of the resultant rotation, viz. ACA′, is 2π − 2C. This is equivalent to a negative rotation 2C about OC, whence the theorem that the effect of three successive positive rotations 2A, 2B, 2C about OA, OB, OC, respectively, is to leave the body in its original position, provided the circuit ABC is left-handed as seen from O. This theorem is due to O. Rodrigues (1840). The composition of finite rotations about parallel axes is a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.

In any continuous motion of a solid about a fixed point O, the limiting position of the axis of the rotation by which the body can be brought from any one of its positions to a consecutive one is called the instantaneous axis. This axis traces out a certain cone in the body, and a certain cone in space, and the continuous motion in question may be represented as consisting in a rolling of the former cone on the latter. The proof is similar to that of the corresponding theorem of plane kinematics (§ 3).

It follows from Euler’s theorem that the most general displacement of a rigid body may be effected by a pure translation which brings any one point of it to its final position O, followed by a pure rotation about some axis through O. Those planes in the body which are perpendicular to this axis obviously remain parallel to their original positions. Hence, if σ, σ′ denote the initial and final positions of any figure in one of these planes, the displacement could evidently have been effected by (1) a translation perpendicular to the planes in question, bringing σ into some position σ″ in the plane of σ′, and (2) a rotation about a normal to the planes, bringing σ″ into coincidence with σ (§ 3). In other words, the most general displacement is equivalent to a translation parallel to a certain axis combined with a rotation about that axis; i.e. it may be described as a twist about a certain screw. In particular cases, of course, the translation, or the rotation, may vanish.

The preceding theorem, which is due to Michel Chasles (1830), may be proved in various other interesting ways. Thus if a point of the body be displaced from A to B, whilst the point which was at B is displaced to C, and that which was at C to D, the four points A, B, C, D lie on a helix whose axis is the common perpendicular to the bisectors of the angles ABC, BCD. This is the axis of the required screw; the amount of the translation is measured by the projection of AB or BC or CD on the axis; and the angle of rotation is given by the inclination of the aforesaid bisectors. This construction was given by M. W. Crofton. Again, H. Wiener and W. Burnside have employed the half-turn (i.e. a rotation through two right angles) as the fundamental operation. This has the advantage that it is completely specified by the axis of the rotation, the sense being immaterial. Successive half-turns about parallel axes a, b are equivalent to a translation measured by double the distance between these axes in the direction from a to b. Successive half-turns about intersecting axes a, b are equivalent to a rotation about the common perpendicular to a, b at their intersection, of amount equal to twice the acute angle between them, in the direction from a to b. Successive half-turns about two skew axes a, b are equivalent to a twist about a screw whose axis is the common perpendicular to a, b, the translation being double the shortest distance, and the angle of rotation being twice the acute angle between a, b, in the direction from a to b. It is easily shown that any displacement whatever is equivalent to two half-turns and therefore to a screw.  
  
---  
Fig. 16.  
  
In mechanics we are specially concerned with the theory of infinitesimal displacements. This is included in the preceding, but it is simpler in that the various operations are commutative. An infinitely small rotation about any axis is conveniently represented geometrically by a length AB measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a right-handed screw. The consequent displacement of any point P will then be at right angles to the plane PAB, its amount will be represented by double the area of the triangle PAB, and its sense will depend on the cyclical order of the letters P, A, B. If AB, AC represent infinitesimal rotations about intersecting axes, the consequent displacement of any point O in the plane BAC will be at right angles to this plane, and will be represented by twice the sum of the areas OAB, OAC, taken with proper signs. It follows by analogy with the theory of moments (§ 4) that the resultant rotation will be represented by AD, the vector-sum of AB, AC (see fig. 16). It is easily inferred as a limiting case, or proved directly, that two infinitesimal rotations α, β about parallel axes are equivalent to a rotation α \+ β about a parallel axis in the same plane with the two former, and dividing a common perpendicular AB in a point C so that AC/CB = β/α. If the rotations are equal and opposite, so that α \+ β = 0, the point C is at infinity, and the effect is a translation perpendicular to the plane of the two given axes, of amount α·AB. It thus appears that an infinitesimal rotation is of the nature of a “localized vector,” and is subject in all respects to the same mathematical laws as a force, conceived as acting on a rigid body. Moreover, that an infinitesimal translation is analogous to a couple and follows the same laws. These results are due to Poinsot.

The analytical treatment of small displacements is as follows. We first suppose that one point O of the body is fixed, and take this as the origin of a “right-handed” system of rectangular co-ordinates; i.e. the positive directions of the axes are assumed to be so arranged that a positive rotation of 90° about Ox would bring Oy into the position of Oz, and so on. The displacement will consist of an infinitesimal rotation ε about some axis through O, whose direction-cosines are, say, l, m, n. From the equivalence of a small rotation to a localized vector it follows that the rotation ε will be equivalent to rotations ξ, η, ζ about Ox, Oy, Oz, respectively, provided

ξ = lε,  η = mε,  ζ = nε,

(1)

and we note that

ξ2 \+ η2 \+ ζ2 = ε2.

(2)

Thus in the case of fig. 36 it may be required to connect the infinitesimal rotations ξ, η, ζ about OA, OB, OC with the variations of the angular co-ordinates θ, ψ, φ. The displacement of the point C of the body is made up of δθ tangential to the meridian ZC and sin θ δψ perpendicular to the plane of this meridian. Hence, resolving along the tangents to the arcs BC, CA, respectively, we have

ξ = δθ sin φ − sin θ δψ cos φ,  η = δθ cos φ \+ sin θ δψ sin φ.

(3)  
  
---  
Fig. 40.  
  
Again, consider the point of the solid which was initially at A′ in the figure. This is displaced relatively to A′ through a space δψ perpendicular to the plane of the meridian, whilst A′ itself is displaced through a space cos θ δψ in the same direction. Hence

ζ = δφ \+ cos θ δψ.

(4)

To find the component displacements of a point P of the body, whose co-ordinates are x, y, z, we draw PL normal to the plane yOz, and LH, LK perpendicular to Oy, Oz, respectively. The displacement of P parallel to Ox is the same as that of L, which is made up of ηz and −ζy. In this way we obtain the formulae

δx = ηz − ζy,  δy = ζx − ξz,  δz = ξy − ηx.

(5)

The most general case is derived from this by adding the component displacements λ, μ, ν (say) of the point which was at O; thus

δx = λ \+ ηz − ζy,

δy = μ \+ ζx − ξz,

δz = ν \+ ξy − ηx.  
  
---  
  
(6)

The displacement is thus expressed in terms of the six independent quantities ξ, η, ζ, λ, μ, ν. The points whose displacements are in the direction of the resultant axis of rotation are determined by δx : δy : δz = ξ : η : ζ, or

(λ \+ ηz − ζy)/ξ = (μ \+ ζx − ξz)/η = (ν \+ ξy − ηx)/ζ.

(7)

These are the equations of a straight line, and the displacement is in fact equivalent to a twist about a screw having this line as axis. The translation parallel to this axis is

lδx + mδy + nδz = (λξ \+ μη \+ νζ)/ε.

(8)

The linear magnitude which measures the ratio of translation to rotation in a screw is called the pitch. In the present case the pitch is

(λξ \+ μη \+ νζ) / (ξ2 \+ η2 \+ ζ2).

(9)

Since ξ2 \+ η2 \+ ζ2, or ε2, is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that λξ \+ μη \+ νζ is also an absolute invariant. When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.

If the small displacements of a rigid body be subject to one constraint, e.g. if a point of the body be restricted to lie on a given surface, the mathematical expression of this fact leads to a homogeneous linear equation between the infinitesimals ξ, η, ζ, λ, μ, ν, say

Aξ \+ Bη \+ Cζ \+ Fλ \+ Gμ \+ Hν = 0.

(10)

The quantities ξ, η, ζ, λ, μ, ν are no longer independent, and the body has now only five degrees of freedom. Every additional constraint introduces an additional equation of the type (10) and reduces the number of degrees of freedom by one. In Sir R. S. Ball’s Theory of Screws an analysis is made of the possible displacements of a body which has respectively two, three, four, five degrees of freedom. We will briefly notice the case of two degrees, which involves an interesting generalization of the method (already explained) of compounding rotations about intersecting axes. We assume that the body receives arbitrary twists about two given screws, and it is required to determine the character of the resultant displacement. We examine first the case where the axes of the two screws are at right angles and intersect. We take these as axes of x and y; then if ξ, η be the component rotations about them, we have

λ = hξ,  μ = kη,  ν = 0,

(11)

where h, k, are the pitches of the two given screws. The equations (7) of the axis of the resultant screw then reduce to

x/ξ = y/η,  z(ξ2 \+ η2) = (k − h) ξη.

(12)

Hence, whatever the ratio ξ : η, the axis of the resultant screw lies on the conoidal surface

z (x2 \+ y2) = cxy,

(13)

where c = 1⁄2(k − h). The co-ordinates of any point on (13) may be written

x = r cos θ,  y = r sin θ,  z = c sin 2θ;

(14)

hence if we imagine a curve of sines to be traced on a circular cylinder so that the circumference just includes two complete undulations, a straight line cutting the axis of the cylinder at right angles and meeting this curve will generate the surface. This is called a cylindroid. Again, the pitch of the resultant screw is

p = (λξ \+ μη) / (ξ2 \+ η2) = h cos2 θ \+ k sin2 θ.

(15)  
  
---  
From Sir Robert S. Ball’s Theory of Screws.  
Fig. 41.  
  
The distribution of pitch among the various screws has therefore a simple relation to the pitch-conic

hx2 \+ ky2 = const;

(16)

viz. the pitch of any screw varies inversely as the square of that diameter of the conic which is parallel to its axis. It is to be noticed that the parameter c of the cylindroid is unaltered if the two pitches h, k be increased by equal amounts; the only change is that all the pitches are increased by the same amount. It remains to show that a system of screws of the above type can be constructed so as to contain any two given screws whatever. In the first place, a cylindroid can be constructed so as to have its axis coincident with the common perpendicular to the axes of the two given screws and to satisfy three other conditions, for the position of the centre, the parameter, and the orientation about the axis are still at our disposal. Hence we can adjust these so that the surface shall contain the axes of the two given screws as generators, and that the difference of the corresponding pitches shall have the proper value. It follows that when a body has two degrees of freedom it can twist about any one of a singly infinite system of screws whose axes lie on a certain cylindroid. In particular cases the cylindroid may degenerate into a plane, the pitches being then all equal.

§ 8. Three-dimensional Statics.—A system of parallel forces can be combined two and two until they are replaced by a single resultant equal to their sum, acting in a certain line. As special cases, the system may reduce to a couple, or it may be in equilibrium.

In general, however, a three-dimensional system of forces cannot be replaced by a single resultant force. But it may be reduced to simpler elements in a variety of ways. For example, it may be reduced to two forces in perpendicular skew lines. For consider any plane, and let each force, at its intersection with the plane, be resolved into two components, one (P) normal to the plane, the other (Q) in the plane. The assemblage of parallel forces P can be replaced in general by a single force, and the coplanar system of forces Q by another single force.

If the plane in question be chosen perpendicular to the direction of the vector-sum of the given forces, the vector-sum of the components Q is zero, and these components are therefore equivalent to a couple (§ 4). Hence any three-dimensional system can be reduced to a single force R acting in a certain line, together with a couple G in a plane perpendicular to the line. This theorem was first given by L. Poinsot, and the line of action of R was called by him the central axis of the system. The combination of a force and a couple in a perpendicular plane is termed by Sir R. S. Ball a wrench. Its type, as distinguished from its absolute magnitude, may be specified by a screw whose axis is the line of action of R, and whose pitch is the ratio G/R.  
  
---  
Fig. 42.  
  
The case of two forces may be specially noticed. Let AB be the shortest distance between the lines of action, and let AA′, BB′ (fig. 42) represent the forces. Let α, β be the angles which AA′, BB′ make with the direction of the vector-sum, on opposite sides. Divide AB in O, so that

AA′ · cos α · AO = BB′ · cos β · OB,

(1)

and draw OC parallel to the vector-sum. Resolving AA′, BB′ each into two components parallel and perpendicular to OC, we see that the former components have a single resultant in OC, of amount

R = AA′ cos α \+ BB′ cos β,

(2)

whilst the latter components form a couple of moment

G = AA′ · AB · sin α = BB′ · AB · sin β.

(3)

Conversely it is seen that any wrench can be replaced in an infinite number of ways by two forces, and that the line of action of one of these may be chosen quite arbitrarily. Also, we find from (2) and (3) that

G · R = AA′ · BB′ · AB · sin (α \+ β).

(4)

The right-hand expression is six times the volume of the tetrahedron of which the lines AA′, BB′ representing the forces are opposite edges; and we infer that, in whatever way the wrench be resolved into two forces, the volume of this tetrahedron is invariable.

To define the moment of a force about an axis HK, we project the force orthogonally on a plane perpendicular to HK and take the moment of the projection about the intersection of HK with the plane (see § 4). Some convention as to sign is necessary; we shall reckon the moment to be positive when the tendency of the force is right-handed as regards the direction from H to K. Since two concurrent forces and their resultant obviously project into two concurrent forces and their resultant, we see that the sum of the moments of two concurrent forces about any axis HK is equal to the moment of their resultant. Parallel forces may be included in this statement as a limiting case. Hence, in whatever way one system of forces is by successive steps replaced by another, no change is made in the sum of the moments about any assigned axis. By means of this theorem we can show that the previous reduction of any system to a wrench is unique.

From the analogy of couples to translations which was pointed out in § 7, we may infer that a couple is sufficiently represented by a “free” (or non-localized) vector perpendicular to its plane. The length of the vector must be proportional to the moment of the couple, and its sense must be such that the sum of the moments of the two forces of the couple about it is positive. In particular, we infer that couples of the same moment in parallel planes are equivalent; and that couples in any two planes may be compounded by geometrical addition of the corresponding vectors. Independent statical proofs are of course easily given. Thus, let the plane of the paper be perpendicular to the planes of two couples, and therefore perpendicular to the line of intersection of these planes. By § 4, each couple can be replaced by two forces ±P (fig. 43) perpendicular to the plane of the paper, and so that one force of each couple is in the line of intersection (B); the arms (AB, BC) will then be proportional to the respective moments. The two forces at B will cancel, and we are left with a couple of moment P·AC in the plane AC. If we draw three vectors to represent these three couples, they will be perpendicular and proportional to the respective sides of the triangle ABC; hence the third vector is the geometric sum of the other two. Since, in this proof the magnitude of P is arbitrary, It follows incidentally that couples of the same moment in parallel planes, e.g. planes parallel to AC, are equivalent.  
  
---  
Fig. 43.  
---  
Fig. 44.  
  
Hence a couple of moment G, whose axis has the direction (l, m, n) relative to a right-handed system of rectangular axes, is equivalent to three couples lG, mG, nG in the co-ordinate planes. The analytical reduction of a three-dimensional system can now be conducted as follows. Let (x1, y1, z1) be the co-ordinates of a point P1 on the line of action of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two equal and opposite forces ±X1. The force X1 at P1 with −X1 in KH forms a couple about Oz, of moment −y1X1. Next, introduce along Ox two equal and opposite forces ±X1. The force X1 in KH with −X1 in Ox forms a couple about Oy, of moment z1X1. Hence the force X1 can be transferred from P1 to O, provided we introduce couples of moments z1X1 about Oy and −y1X1, about Oz. Dealing in the same way with the forces Y1, Z1 at P1, we find that all three components of the force at P1 can be transferred to O, provided we introduce three couples L1, M1, N1 about Ox, Oy, Oz respectively, viz.

L1 = y1Z1 − z1Y1,  M1 = z1X1 − x1Z1,  N1 = x1Y1 − y1X1.

(5)

It is seen that L1, M1, N1 are the moments of the original force at P1 about the co-ordinate axes. Summing up for all the forces of the given system, we obtain a force R at O, whose components are

X = Σ(Xr),  Y = Σ(Yr),  Z = Σ(Zr),

(6)

and a couple G whose components are

L = Σ(Lr),  M = Σ(Mr),  N = Σ(Nr),

(7)

where r = 1, 2, 3 ... Since R2 = X2 \+ Y2 \+ Z2, G2 = L2 \+ M2 \+ N2, it is necessary and sufficient for equilibrium that the six quantities X, Y, Z, L, M, N, should all vanish. In words: the sum of the projections of the forces on each of the co-ordinate axes must vanish; and, the sum of the moments of the forces about each of these axes must vanish.

If any other point O′, whose co-ordinates are x, y, z, be chosen in place of O, as the point to which the forces are transferred, we have to write x1 − x, y1 − y, z1 − z for x1, y1, z1, and so on, in the preceding process. The components of the resultant force R are unaltered, but the new components of couple are found to be

L′ = L − yZ + zY,

M′ = M − zX + xZ,

N′ = N − xY + yX.  
  
---  
  
(8)

By properly choosing O′ we can make the plane of the couple perpendicular to the resultant force. The conditions for this are L′ : M′ : N′ = X : Y : Z, or

L − yZ + zY | = | M − zX + xZ | = | N − xY + yX  
---|---|---|---|---  
X | Y | Z  
  
(9)

These are the equations of the central axis. Since the moment of the resultant couple is now

G′ = | X | L′ + | Y | M′ + | Z | N′ = | LX + MY + NZ | ,  
---|---|---|---|---|---|---|---|---  
R | R | R | R  
  
(10)

the pitch of the equivalent wrench is

(LX + MY + NZ) / (X2 \+ Y2 \+ Z2).

It appears that X2 \+ Y2 \+ Z2 and LX + MY + NZ are absolute invariants (cf. § 7). When the latter invariant, but not the former, vanishes, the system reduces to a single force.

The analogy between the mathematical relations of infinitely small displacements on the one hand and those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other. For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily. Again, that wrenches of arbitrary amounts about two given screws compound into a wrench the locus of whose axis is a cylindroid.

The mathematical properties of a twist or of a wrench have been the subject of many remarkable investigations, which are, however, of secondary importance from a physical point of view. In the “Null-System” of A. F. Möbius (1790-1868), a line such that the moment of a given wrench about it is zero is called a null-line. The triply infinite system of null-lines form what is called in line-geometry a “complex.” As regards the configuration of this complex, consider a line whose shortest distance from the central axis is r, and whose inclination to the central axis is θ. The moment of the resultant force R of the wrench about this line is − Rr sin θ, and that of the couple G is G cos θ. Hence the line will be a null-line provided

tan θ = k/r,

(11)

where k is the pitch of the wrench. The null-lines which are at a given distance r from a point O of the central axis will therefore form one system of generators of a hyperboloid of revolution; and by varying r we get a series of such hyperboloids with a common centre and axis. By moving O along the central axis we obtain the whole complex of null-lines. It appears also from (11) that the null-lines whose distance from the central axis is r are tangent lines to a system of helices of slope tan−1 (r/k); and it is to be noticed that these helices are left-handed if the given wrench is right-handed, and vice versa.

Since the given wrench can be replaced by a force acting through any assigned point P, and a couple, the locus of the null-lines through P is a plane, viz. a plane perpendicular to the vector which represents the couple. The complex is therefore of the type called “linear” (in relation to the degree of this locus). The plane in question is called the null-plane of P. If the null-plane of P pass through Q, the null-plane of Q will pass through P, since PQ is a null-line. Again, any plane ω is the locus of a system of null-lines meeting in a point, called the null-point of ω. If a plane revolve about a fixed straight line p in it, its null-point describes another straight line p′, which is called the conjugate line of p. We have seen that the wrench may be replaced by two forces, one of which may act in any arbitrary line p. It is now evident that the second force must act in the conjugate line p′, since every line meeting p, p′ is a null-line. Again, since the shortest distance between any two conjugate lines cuts the central axis at right angles, the orthogonal projections of two conjugate lines on a plane perpendicular to the central axis will be parallel (fig. 42). This property was employed by L. Cremona to prove the existence under certain conditions of “reciprocal figures” in a plane (§ 5). If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former. Projecting orthogonally on a plane perpendicular to the central axis we obtain two reciprocal figures.

In the analogous theory of infinitely small displacements of a solid, a “null-line” is a line such that the lengthwise displacement of any point on it is zero.

Since a wrench is defined by six independent quantities, it can in general be replaced by any system of forces which involves six adjustable elements. For instance, it can in general be replaced by six forces acting in six given lines, e.g. in the six edges of a given tetrahedron. An exception to the general statement occurs when the six lines are such that they are possible lines of action of a system of six forces in equilibrium; they are then said to be in involution. The theory of forces in involution has been studied by A. Cayley, J. J. Sylvester and others. We have seen that a rigid structure may in general be rigidly connected with the earth by six links, and it now appears that any system of forces acting on the structure can in general be balanced by six determinate forces exerted by the links. If, however, the links are in involution, these forces become infinite or indeterminate. There is a corresponding kinematic peculiarity, in that the connexion is now not strictly rigid, an infinitely small relative displacement being possible. See § 9.

When parallel forces of given magnitudes act at given points, the resultant acts through a definite point, or centre of parallel forces, which is independent of the special direction of the forces. If Pr be the force at (xr, yr, zr), acting in the direction (l, m, n), the formulae (6) and (7) reduce to

X = Σ(P)·l,  Y = Σ(P)·m,  Z = Σ(P)·n,

(12)

and

L = Σ(P)·(ny − mz),  M = Σ(P)·(lz − nx),  N = Σ(P)·(mx − ly),

(13)

provided

x = | Σ(Px) | ,  y = | Σ(Py) | ,  z = | Σ(Pz) | .  
---|---|---|---|---|---|---  
Σ(P) | Σ(P) | Σ(P)  
  
(14)

These are the same as if we had a single force Σ(P) acting at the point (x, y, z), which is the same for all directions (l, m, n). We can hence derive the theory of the centre of gravity, as in § 4. An exceptional case occurs when Σ(P) = 0.

If we imagine a rigid body to be acted on at given points by forces of given magnitudes in directions (not all parallel) which are fixed in space, then as the body is turned about the resultant wrench will assume different configurations in the body, and will in certain positions reduce to a single force. The investigation of such questions forms the subject of “Astatics,” which has been cultivated by Möbius, Minding, G. Darboux and others. As it has no physical bearing it is passed over here.  
  
---  
Fig. 45.  
  
§ 9. Work.—The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e. it is equal to F·δs cos θ, where F is the force, δs the displacement, and θ is the angle between the directions of F and δs. In the language of vector analysis (q.v.) it is the “scalar product” of the vector representing the force and the displacement. In the same way, the work done by a force acting on a rigid body in any infinitely small displacement of the body is the scalar product of the force into the displacement of any point on the line of action. This product is the same whatever point on the line of action be taken, since the lengthwise components of the displacements of any two points A, B on a line AB are equal, to the first order of small quantities. To see this, let A′, B′ be the displaced positions of A, B, and let φ be the infinitely small angle between AB and A′B′. Then if α, β be the orthogonal projections of A′, B′ on AB, we have

Aα − Bβ = AB − αβ = AB (1 − cos φ) = 1⁄2AB·φ2,

ultimately. Since this is of the second order, the products F·Aα and F·Bβ are ultimately equal.

|   
---|---  
Fig. 46. | Fig. 47.  
  
The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant. Let AB, AC (fig. 46) represent the forces, AD their resultant, and let AH be the direction of the displacement δs of the point A. The proposition follows at once from the fact that the sum of orthogonal projections of AB>, AC> on AH is equal to the projection of AD>. It is to be noticed that AH need not be in the same plane with AB, AC.

It follows from the preceding statements that any two systems of forces which are statically equivalent, according to the principles of §§ 4, 8, will (to the first order of small quantities) do the same amount of work in any infinitely small displacement of a rigid body to which they may be applied. It is also evident that the total work done in two or more successive infinitely small displacements is equal to the work done in the resultant displacement.

The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed. Let the couple consist of two forces P, P (fig. 47) in the plane of the paper, and let J be the point where this plane is met by the axis of rotation. Draw JBA perpendicular to the lines of action, and let ε be the angle of rotation. The work of the couple is

P·JA·ε − P·JB·ε = P·AB·ε = Gε,

if G be the moment of the couple.

The analytical calculation of the work done by a system of forces in any infinitesimal displacement is as follows. For a two-dimensional system we have, in the notation of §§ 3, 4,

Σ(Xδx + Yδy) | = Σ{X(λ − yε) + Y(μ \+ xε)}  
---|---  
  | = Σ(X)·λ \+ Σ(Y)·μ \+ Σ(xY − yX) ε  
  | = Xλ \+ Yμ \+ Nε.  
  
(1)

Again, for a three-dimensional system, in the notation of §§ 7, 8,

Σ(Xδx + Yδy + Zδz)  
---  
= Σ{(X(λ \+ ηz − ζy) + Y(μ \+ ζx − ξx) + Z(ν \+ ξy − ηx)}  
= Σ(X)·λ \+ Σ(Y)·μ \+ Σ(Z)·ν \+ Σ(yZ − zY)·ξ \+ Σ(zX − xZ)·η \+ Σ(xY − yX)·ζ  
= Xλ \+ Yμ \+ Zν \+ Lξ \+ Mη \+ Nζ.  
  
(2)

This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axes. The first three terms express the work done by the components of a force (X, Y, Z) acting at O, and the remaining three terms express the work of a couple (L, M, N).  
  
---  
Fig. 48.  
  
The work done by a wrench about a given screw, when the body twists about a second given screw, may be calculated directly as follows. In fig. 48 let R, G be the force and couple of the wrench, ε,τ the rotation and translation in the twist. Let the axes of the wrench and the twist be inclined at an angle θ, and let h be the shortest distance between them. The displacement of the point H in the figure, resolved in the direction of R, is τ cos θ − εh sin θ. The work is therefore

R (τ cos θ − εh sin θ) + G cos θ  
= Rε {(p + p′) cos θ − h sin θ},

(3)

if G = pR, τ = p′ε, i.e. p, p′ are the pitches of the two screws. The factor (p + p′) cos θ − h sin θ is called the virtual coefficient of the two screws which define the types of the wrench and twist, respectively.

A screw is determined by its axis and its pitch, and therefore involves five Independent elements. These may be, for instance, the five ratios ξ : η : ζ : λ : μ : ν of the six quantities which specify an infinitesimal twist about the screw. If the twist is a pure rotation, these quantities are subject to the relation

λξ \+ μη \+ νζ = 0.

(4)

In the analytical investigations of line geometry, these six quantities, supposed subject to the relation (4), are used to specify a line, and are called the six “co-ordinates” of the line; they are of course equivalent to only four independent quantities. If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its co-ordinates are accordingly subject to the further relation

Lξ \+ Mη \+ Nζ \+ Xλ \+ Yμ \+ Zν = 0,

(5)

where the coefficients are constant. This is the equation of a “linear complex” (cf. § 8).

Two screws are reciprocal when a wrench about one does no work on a body which twists about the other. The condition for this is

λξ′ \+ μη′ \+ νζ′ \+ λ′ξ \+ μ′η \+ ν′ζ = 0,

(6)

if the screws be defined by the ratios ξ : η : ζ : λ : μ : ν and ξ′ : η′ : ζ′ : λ′ : μ′ : ν′, respectively. The theory of the screw-systems which are reciprocal to one, two, three, four given screws respectively has been investigated by Sir R. S. Ball.

Considering a rigid body in any given position, we may contemplate the whole group of infinitesimal displacements which might be given to it. If the extraneous forces are in equilibrium the total work which they would perform in any such displacement would be zero, since they reduce to a zero force and a zero couple. This is (in part) the celebrated principle of virtual velocities, now often described as the principle of virtual work, enunciated by John Bernoulli (1667-1748). The word “virtual” is used because the displacements in question are not regarded as actually taking place, the body being in fact at rest. The “velocities” referred to are the velocities of the various points of the body in any imagined motion of the body through the position in question; they obviously bear to one another the same ratios as the corresponding infinitesimal displacements. Conversely, we can show that if the virtual work of the extraneous forces be zero for every infinitesimal displacement of the body as rigid, these forces must be in equilibrium. For by giving the body (in imagination) a displacement of translation we learn that the sum of the resolved parts of the forces in any assigned direction is zero, and by giving it a displacement of pure rotation we learn that the sum of the moments about any assigned axis is zero. The same thing follows of course from the analytical expression (2) for the virtual work. If this vanishes for all values of λ, μ, ν, ξ, η, ζ we must have X, Y, Z, L, M, N = 0, which are the conditions of equilibrium.

The principle can of course be extended to any system of particles or rigid bodies, connected together in any way, provided we take into account the internal stresses, or reactions, between the various parts. Each such reaction consists of two equal and opposite forces, both of which may contribute to the equation of virtual work.

The proper significance of the principle of virtual work, and of its converse, will appear more clearly when we come to kinetics (§ 16); for the present it may be regarded merely as a compact and (for many purposes) highly convenient summary of the laws of equilibrium. Its special value lies in this, that by a suitable adjustment of the hypothetical displacements we are often enabled to eliminate unknown reactions. For example, in the case of a particle lying on a smooth curve, or on a smooth surface, if it be displaced along the curve, or on the surface, the virtual work of the normal component of the pressure may be ignored, since it is of the second order. Again, if two bodies are connected by a string or rod, and if the hypothetical displacements be adjusted so that the distance between the points of attachment is unaltered, the corresponding stress may be ignored. This is evident from fig. 45; if AB, A′B′ represent the two positions of a string, and T be the tension, the virtual work of the two forces ±T at A, B is T(Aα − Bβ), which was shown to be of the second order. Again, the normal pressure between two surfaces disappears from the equation, provided the displacements be such that one of these surfaces merely slides relatively to the other. It is evident, in the first place, that in any displacement common to the two surfaces, the work of the two equal and opposite normal pressures will cancel; moreover if, one of the surfaces being fixed, an infinitely small displacement shifts the point of contact from A to B, and if A′ be the new position of that point of the sliding body which was at A, the projection of AA′ on the normal at A is of the second order. It is to be noticed, in this case, that the tangential reaction (if any) between the two surfaces is not eliminated. Again, if the displacements be such that one curved surface rolls without sliding on another, the reaction, whether normal or tangential, at the point of contact may be ignored. For the virtual work of two equal and opposite forces will cancel in any displacement which is common to the two surfaces; whilst, if one surface be fixed, the displacement of that point of the rolling surface which was in contact with the other is of the second order. We are thus able to imagine a great variety of mechanical systems to which the principle of virtual work can be applied without any regard to the internal stresses, provided the hypothetical displacements be such that none of the connexions of the system are violated.

If the system be subject to gravity, the corresponding part of the virtual work can be calculated from the displacement of the centre of gravity. If W1, W2, ... be the weights of a system of particles, whose depths below a fixed horizontal plane of reference are z1, z2, ..., respectively, the virtual work of gravity is

W1δ·z1 \+ W2δz2 \+ ... = δ(W1z1 \+ W2z2 \+ ...) = (W1 \+ W2 \+ ...) δz,

where z is the depth of the centre of gravity (see § 8 (14) and § 11 (6)). This expression is the same as if the whole mass were concentrated at the centre of gravity, and displaced with this point. An important conclusion is that in any displacement of a system of bodies in equilibrium, such that the virtual work of all forces except gravity may be ignored, the depth of the centre of gravity is “stationary.”

The question as to stability of equilibrium belongs essentially to kinetics; but we may state by anticipation that in cases where gravity is the only force which does work, the equilibrium of a body or system of bodies is stable only if the depth of the centre of gravity be a maximum.

Consider, for instance, the case of a bar resting with its ends on two smooth inclines (fig. 18). If the bar be displaced in a vertical plane so that its ends slide on the two inclines, the instantaneous centre is at the point J. The displacement of G is at right angles to JG; this shows that for equilibrium JG must be vertical. Again, the locus of G is an arc of an ellipse whose centre is in the intersection of the planes; since this arc is convex upwards the equilibrium is unstable. A general criterion for the case of a rigid body movable in two dimensions, with one degree of freedom, can be obtained as follows. We have seen (§ 3) that the sequence of possible positions is obtained if we imagine the “body-centrode” to roll on the “space-centrode.” For equilibrium, the altitude of the centre of gravity G must be stationary; hence G must lie in the same vertical line with the point of contact J of the two curves. Further, it is known from the theory of “roulettes” that the locus of G will be concave or convex upwards according as

cos φ | = | 1 | + | 1 | ,  
---|---|---|---|---|---  
h | ρ | ρ′  
  
(8)  
  
---  
Fig. 49.  
  
where ρ, ρ′ are the radii of curvature of the two curves at J, φ is the inclination of the common tangent at J to the horizontal, and h is the height of G above J. The signs of ρ, ρ′ are to be taken positive when the curvatures are as in the standard case shown in fig. 49. Hence for stability the upper sign must obtain in (8). The same criterion may be arrived at in a more intuitive manner as follows. If the body be supposed to roll (say to the right) until the curves touch at J′, and if JJ′ = δs, the angle through which the upper figure rotates is δs/ρ \+ δs/ρ′, and the horizontal displacement of G is equal to the product of this expression into h. If this displacement be less than the horizontal projection of JJ′, viz. δs cosφ, the vertical through the new position of G will fall to the left of J′ and gravity will tend to restore the body to its former position. It is here assumed that the remaining forces acting on the body in its displaced position have zero moment about J′; this is evidently the case, for instance, in the problem of “rocking stones.”

The principle of virtual work is specially convenient in the theory of frames (§ 6), since the reactions at smooth joints and the stresses in inextensible bars may be left out of account. In particular, in the case of a frame which is just rigid, the principle enables us to find the stress in any one bar independently of the rest. If we imagine the bar in question to be removed, equilibrium will still persist if we introduce two equal and opposite forces S, of suitable magnitude, at the joints which it connected. In any infinitely small deformation of the frame as thus modified, the virtual work of the forces S, together with that of the original extraneous forces, must vanish; this determines S.

As a simple example, take the case of a light frame, whose bars form the slides of a rhombus ABCD with the diagonal BD, suspended from A and carrying a weight W at C; and let it be required to find the stress in BD. If we remove the bar BD, and apply two equal and opposite forces S at B and D, the equation is  
  
---  
Fig. 50.  
  
W·δ(2l cosθ) + 2S·δ (l sin θ) = 0,

where l is the length of a side of the rhombus, and θ its inclination to the vertical. Hence

S = W tan θ = W · BD/AC.

(8)

The method is specially appropriate when the frame, although just rigid, is not “simple” in the sense of § 6, and when accordingly the method of reciprocal figures is not immediately available. To avoid the intricate trigonometrical calculations which would often be necessary, graphical devices have been introduced by H. Müller-Breslau and others. For this purpose the infinitesimal displacements of the various joints are replaced by finite lengths proportional to them, and therefore proportional to the velocities of the joints in some imagined motion of the deformable frame through its actual configuration; this is really (it may be remarked) a reversion to the original notion of “virtual velocities.” Let J be the instantaneous centre for any bar CD (fig. 12), and let s1, s2 represent the virtual velocities of C, D. If these lines be turned through a right angle in the same sense, they take up positions such as CC′, DD′, where C′, D′ are on JC, JD, respectively, and C′D′ is parallel to CD. Further, if F1 (fig. 51) be any force acting on the joint C, its virtual work will be equal to the moment of F1 about C′; the equation of virtual work is thus transformed into an equation of moments.  
  
---  
Fig. 12. | Fig. 51.  
---  
Fig. 52.  
  
Consider, for example, a frame whose sides form the six sides of a hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it is required to find the stress in CF due to a given system of extraneous forces in equilibrium, acting on the joints. Imagine the bar CF to be removed, and consider a deformation in which AB is fixed. The instantaneous centre of CD will be at the intersection of AD, BC, and if C′D′ be drawn parallel to CD, the lines CC′, DD′ may be taken to represent the virtual velocities of C, D turned each through a right angle. Moreover, if we draw D′E′ parallel to DE, and E′F′ parallel to EF, the lines CC′, DD′, EE′, FF′ will represent on the same scale the virtual velocities of the points C, D, E, F, respectively, turned each through a right angle. The equation of virtual work is then formed by taking moments about C′, D′, E′, F′ of the extraneous forces which act at C, D, E, F, respectively. Amongst these forces we must include the two equal and opposite forces S which take the place of the stress in the removed bar FC.

The above method lends itself naturally to the investigation of the critical forms of a frame whose general structure is given. We have seen that the stresses produced by an equilibrating system of extraneous forces in a frame which is just rigid, according to the criterion of § 6, are in general uniquely determinate; in particular, when there are no extraneous forces the bars are in general free from stress. It may however happen that owing to some special relation between the lengths of the bars the frame admits of an infinitesimal deformation. The simplest case is that of a frame of three bars, when the three joints A, B, C fall into a straight line; a small displacement of the joint B at right angles to AC would involve changes in the lengths of AB, BC which are only of the second order of small quantities. Another example is shown in fig. 53. The graphical method leads at once to the detection of such cases. Thus in the hexagonal frame of fig. 52, if an infinitesimal deformation is possible without removing the bar CF, the instantaneous centre of CF (when AB is fixed) will be at the intersection of AF and BC, and since CC′, FF′ represent the virtual velocities of the points C, F, turned each through a right angle, C′F′ must be parallel to CF. Conversely, if this condition be satisfied, an infinitesimal deformation is possible. The result may be generalized into the statement that a frame has a critical form whenever a frame of the same structure can be designed with corresponding bars parallel, but without complete geometric similarity. In the case of fig. 52 it may be shown that an equivalent condition is that the six points A, B, C, D, E, F should lie on a conic (M. W. Crofton). This is fulfilled when the opposite sides of the hexagon are parallel, and (as a still more special case) when the hexagon is regular.  
  
---  
Fig. 53.  
  
When a frame has a critical form it may be in a state of stress independently of the action of extraneous forces; moreover, the stresses due to extraneous forces are indeterminate, and may be infinite. For suppose as before that one of the bars is removed. If there are no extraneous forces the equation of virtual work reduces to S·δs = 0, where S is the stress in the removed bar, and δs is the change in the distance between the joints which it connected. In a critical form we have δs = 0, and the equation is satisfied by an arbitrary value of S; a consistent system of stresses in the remaining bars can then be found by preceding rules. Again, when extraneous forces P act on the joints, the equation is

Σ(P·δp) + S·δs = 0,

where δp is the displacement of any joint in the direction of the corresponding force P. If Σ(P·δp) = 0, the stresses are merely indeterminate as before; but if Σ (P·δp) does not vanish, the equation cannot be satisfied by any finite value of S, since δs = 0. This means that, if the material of the frame were absolutely unyielding, no finite stresses in the bars would enable it to withstand the extraneous forces. With actual materials, the frame would yield elastically, until its configuration is no longer “critical.” The stresses in the bars would then be comparatively very great, although finite. The use of frames which approximate to a critical form is of course to be avoided in practice.

A brief reference must suffice to the theory of three dimensional frames. This is important from a technical point of view, since all structures are practically three-dimensional. We may note that a frame of n joints which is just rigid must have 3n − 6 bars; and that the stresses produced in such a frame by a given system of extraneous forces in equilibrium are statically determinate, subject to the exception of “critical forms.”

§ 10. Statics of Inextensible Chains.—The theory of bodies or structures which are deformable in their smallest parts belongs properly to elasticity (q.v.). The case of inextensible strings or chains is, however, so simple that it is generally included in expositions of pure statics.

It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element δs must satisfy the conditions of equilibrium laid down in § 1. It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (§ 4).  
  
---  
Fig. 54.  
  
We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= δs) in the directions of the tangent and normal respectively. If T, T + δT be the tensions at P, Q, and δψ be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + δT) cos ψ − T, or δT, ultimately; whilst for the component along the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius of curvature.

Suppose, for example, that we have a light string stretched over a smooth curve; and let Rδs denote the normal pressure (outwards from the centre of curvature) on δs. The two resolutions give δT = 0, Tδψ = Rδs, or

T = const.,  R = T/ρ.

(1)

The tension is constant, and the pressure per unit length varies as the curvature.

Next suppose that the curve is “rough”; and let Fδs be the tangential force of friction on δs. We have δT ± Fδs = 0, Tδψ = Rδs, where the upper or lower sign is to be taken according to the sense in which F acts. We assume that in limiting equilibrium we have F = μR, everywhere, where μ is the coefficient of friction. If the string be on the point of slipping in the direction in which ψ increases, the lower sign is to be taken; hence δT = Fδs = μTδψ, whence

T = T0 eμψ,

(2)

if T0 be the tension corresponding to ψ = 0. This illustrates the resistance to dragging of a rope coiled round a post; e.g. if we put μ = .3, ψ = 2π, we find for the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared, and so on.

Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane. Let ψ denote the inclination to the horizontal, and wδs the weight of an element δs. The tangential and normal components of wδs are −s sinψ and −wδs cosψ. Hence

δT = wδs sin ψ,  Tδψ = wδs cos ψ \+ Rδs.

(3)

If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have δy = sin ψ δs, whence δT = wδy. If the string be uniform, w is constant, and

T = wy + const. = w (y − y0),

(4)

say; hence the tension varies as the height above some fixed level (y0). The pressure is then given by the formula

R = T | dψ | −w cos ψ.  
---|---|---  
ds  
  
(5)

In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ. The forces on this reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three forces will be concurrent, and their ratios will be given by a triangle of forces. In particular, if we consider a length AP beginning at the lowest point A, then resolving horizontally and vertically we have

T cos ψ = T0,  T sinψ = W,

(6)

where T0 is the tension at A, and W is the weight of PA. The former equation expresses that the horizontal tension is constant.  
  
---  
Fig. 55.  
  
If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0 tan ψ. If we write T0 = wa, so that a is the length of a portion of the chain whose weight would equal the horizontal tension, this becomes

s = a tan ψ.

(7)

This is the “intrinsic” equation of the curve. If the axes of x and y be taken horizontal and vertical (upwards), we derive

x = a log (sec ψ \+ tan ψ),  y = a sec ψ.

(8)

Eliminating ψ we obtain the Cartesian equation

y = a cosh | x  
---|---  
a  
  
(9)

of the common catenary, as it is called (fig. 56). The omission of the additive arbitrary constants of integration in (8) is equivalent to a special choice of the origin O of co-ordinates; viz. O is at a distance a vertically below the lowest point (ψ = 0) of the curve. The horizontal line through O is called the directrix. The relations

s = a sinh | x | ,  y2 = a2 \+ s2,  T = T0 sec ψ = wy,  
---|---|---  
a  
  
(10)  
  
---  
Fig. 56.  
  
which are involved in the preceding formulae are also noteworthy. It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a chain of given length hanging between fixed points must be stationary (§ 9). The length a is called the parameter of the catenary; it determines the scale of the curve, all catenaries being geometrically similar. If weights be suspended from various points of a hanging chain, the intervening portions will form arcs of equal catenaries, since the horizontal tension (wa) is the same for all. Again, if a chain pass over a perfectly smooth peg, the catenaries in which it hangs on the two sides, though usually of different parameters, will have the same directrix, since by (10) y is the same for both at the peg.

As an example of the use of the formulae we may determine the maximum span for a wire of given material. The condition is that the tension must not exceed the weight of a certain length λ of the wire. At the ends we shall have y = λ, or

λ = a cosh | x | ,  
---|---|---  
a  
  
(11)

and the problem is to make x a maximum for variations of a. Differentiating (11) we find that, if dx/da = 0,

x | tanh | x | = 1.  
---|---|---|---  
a | a  
  
(12)

It is easily seen graphically, or from a table of hyperbolic tangents, that the equation u tanh u = 1 has only one positive root (u = 1.200); the span is therefore

2x = 2au = 2λ/sinh u = 1.326 λ,

and the length of wire is

2s = 2λ/u = 1.667 λ.

The tangents at the ends meet on the directrix, and their inclination to the horizontal is 56° 30′.  
  
---  
Fig. 57.  
  
The relation between the sag, the tension, and the span of a wire (e.g. a telegraph wire) stretched nearly straight between two points A, B at the same level is determined most simply from first principles. If T be the tension, W the total weight, k the sag in the middle, and ψ the inclination to the horizontal at A or B, we have 2Tψ = W, AB = 2ρψ, approximately, where ρ is the radius of curvature. Since 2kρ = (1⁄2AB)2, ultimately, we have

k = 1⁄8W · AB/T.

(13)

The same formula applies if A, B be at different levels, provided k be the sag, measured vertically, half way between A and B.

In relation to the theory of suspension bridges the case where the weight of any portion of the chain varies as its horizontal projection is of interest. The vertical through the centre of gravity of the arc AP (see fig. 55) will then bisect its horizontal projection AN; hence if PS be the tangent at P we shall have AS = SN. This property is characteristic of a parabola whose axis is vertical. If we take A as origin and AN as axis of x, the weight of AP may be denoted by wx, where w is the weight per unit length at A. Since PNS is a triangle of forces for the portion AP of the chain, we have wx/T0 = PN/NS, or

y = w · x2/2T0,

(14)

which is the equation of the parabola in question. The result might of course have been inferred from the theory of the parabolic funicular in § 2.

Finally, we may refer to the catenary of uniform strength, where the cross-section of the wire (or cable) is supposed to vary as the tension. Hence w, the weight per foot, varies as T, and we may write T = wλ, where λ is a constant length. Resolving along the normal the forces on an element δs, we find Tδψ = wδs cos ψ, whence

ρ = | ds | = λ sec ψ.  
---|---|---  
dψ  
  
(15)

From this we derive

x = λψ,  y = λ log sec | x | ,  
---|---|---  
λ  
  
(16)

where the directions of x and y are horizontal and vertical, and the origin is taken at the lowest point. The curve (fig. 58) has two vertical asymptotes x = ± 1⁄2πλ; this shows that however the thickness of a cable be adjusted there is a limit πλ to the horizontal span, where λ depends on the tensile strength of the material. For a uniform catenary the limit was found above to be 1.326λ.  
  
---  
Fig. 58.  
  
For investigations relating to the equilibrium of a string in three dimensions we must refer to the textbooks. In the case of a string stretched over a smooth surface, but in other respects free from extraneous force, the tensions at the ends of a small element δs must be balanced by the normal reaction of the surface. It follows that the osculating plane of the curve formed by the string must contain the normal to the surface, i.e. the curve must be a “geodesic,” and that the normal pressure per unit length must vary as the principal curvature of the curve.

§ 11. Theory of Mass-Systems.—This is a purely geometrical subject. We consider a system of points P1, P2 ..., Pn, with which are associated certain coefficients m1, m2, ... mn, respectively. In the application to mechanics these coefficients are the masses of particles situate at the respective points, and are therefore all positive. We shall make this supposition in what follows, but it should be remarked that hardly any difference is made in the theory if some of the coefficients have a different sign from the rest, except in the special case where Σ(m) = 0. This has a certain interest in magnetism.

In a given mass-system there exists one and only one point G such that

Σ(m·GP>) = 0. 

(1)

For, take any point O, and construct the vector

OG> = | Σ(m·OP>) | .  
---|---|---  
Σ(m)  
  
(2)

Then

Σ(m·GP>) = Σ {m(GO> \+ OP>)} = Σ(m)·GO> \+ Σ(m)·OP> = 0.

(3)

Also there cannot be a distinct point G′ such that Σ(m·G′P) = 0, for we should have, by subtraction,

Σ {m(GP> \+ PG>′)} = 0,  or Σ(m)·GG′ = 0;

(4)

i.e. G′ must coincide with G. The point G determined by (1) is called the mass-centre or centre of inertia of the given system. It is easily seen that, in the process of determining the mass-centre, any group of particles may be replaced by a single particle whose mass is equal to that of the group, situate at the mass-centre of the group.

If through P1, P2, ... Pn we draw any system of parallel planes meeting a straight line OX in the points M1, M2 ... Mn, the collinear vectors OM>1, OM>2 ... OM>n may be called the “projections” of OP>1, OP>2, ... OP>n on OX. Let these projections be denoted algebraically by x1, x2, ... xn, the sign being positive or negative according as the direction is that of OX or the reverse. Since the projection of a vector-sum is the sum of the projections of the several vectors, the equation (2) gives

x = | Σ(mx) | ,  
---|---|---  
Σ(m)  
  
(5)

if x be the projection of OG>. Hence if the Cartesian co-ordinates of P1, P2, ... Pn relative to any axes, rectangular or oblique be (x1, y1, z1), (x2, y2, z2), ..., (xn, yn, zn), the mass-centre (x, y, z) is determined by the formulae

x = | Σ(mx) | ,  y = | Σ(my) | ,  z = | Σ(mz) | .  
---|---|---|---|---|---|---  
Σ(m) | Σ(m) | Σ(m)  
  
(6)

If we write x = x \+ ξ, y = y \+ η, z = z \+ ζ, so that ξ, η, ζ denote co-ordinates relative to the mass-centre G, we have from (6)

Σ(mξ) = 0,  Σ(mη) = 0,  Σ(mζ) = 0.

(7)

One or two special cases may be noticed. If three masses α, β, γ be situate at the vertices of a triangle ABC, the mass-centre of β and γ is at a point A′ in BC, such that β·BA′ = γ·A′C. The mass-centre (G) of α, β, γ will then divide AA′ so that α·AG = (β \+ γ) GA′. It is easily proved that

α : β : γ = ΔBGA : ΔGCA : ΔGAB;

also, by giving suitable values (positive or negative) to the ratios α : β : γ we can make G assume any assigned position in the plane ABC. We have here the origin of the “barycentric co-ordinates” of Möbius, now usually known as “areal” co-ordinates. If α \+ β \+ γ = 0, G is at infinity; if α = β = γ, G is at the intersection of the median lines of the triangle; if α : β : γ = a : b : c, G is at the centre of the inscribed circle. Again, if G be the mass-centre of four particles α, β, γ, δ situate at the vertices of a tetrahedron ABCD, we find

α : β : γ : δ = tetn GBCD : tetn GCDA : tetn GDAB : tetn GABC,

and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space. If α \+ β \+ γ \+ δ = O, G is at infinity; if α = β = γ = δ, G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if α : β : γ : δ = ΔBCD : ΔCDA : ΔDAB : ΔABC, G is at the centre of the inscribed sphere.

If we have a continuous distribution of matter, instead of a system of discrete particles, the summations in (6) are to be replaced by integrations. Examples will be found in textbooks of the calculus and of analytical statics. As particular cases: the mass-centre of a uniform thin triangular plate coincides with that of three equal particles at the corners; and that of a uniform solid tetrahedron coincides with that of four equal particles at the vertices. Again, the mass-centre of a uniform solid right circular cone divides the axis in the ratio 3 : 1; that of a uniform solid hemisphere divides the axial radius in the ratio 3 : 5.

It is easily seen from (6) that if the configuration of a system of particles be altered by “homogeneous strain” (see Elasticity) the new position of the mass-centre will be at that point of the strained figure which corresponds to the original mass-centre.

The formula (2) shows that a system of concurrent forces represented by m1·OP>1, m2·OP>2, ... mn·OP>n will have a resultant represented hy Σ(m)·OG>. If we imagine O to recede to infinity in any direction we learn that a system of parallel forces proportional to m1, m2,... mn, acting at P1, P2 ... Pn have a resultant proportional to Σ(m) which acts always through a point G fixed relatively to the given mass-system. This contains the theory of the “centre of gravity” (§§ 4, 9). We may note also that if P1, P2, ... Pn, and P1′, P2′, ... Pn′ represent two configurations of the series of particles, then

Σ(m·PP>′) = Sigma(m)·GG>′,

(8)

where G, G′ are the two positions of the mass-centre. The forces m1·P>1P1′, m2·P>2P2′, ... mn·P>nPn′, considered as localized vectors, do not, however, as a rule reduce to a single resultant.

We proceed to the theory of the plane, axial and polar quadratic moments of the system. The axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions. If h1, h2, ... hn be the perpendicular distances of the particles from any fixed plane, the sum Σ(mh2) is the quadratic moment with respect to the plane. If p1, p2, ... pn be the perpendicular distances from any given axis, the sum Σ(mp2) is the quadratic moment with respect to the axis; it is also called the moment of inertia about the axis. If r1, r2, ... rn be the distances from a fixed point, the sum Σ(mr2) is the quadratic moment with respect to that point (or pole). If we divide any of the above quadratic moments by the total mass Σ(m), the result is called the mean square of the distances of the particles from the respective plane, axis or pole. In the case of an axial moment, the square root of the resulting mean square is called the radius of gyration of the system about the axis in question. If we take rectangular axes through any point O, the quadratic moments with respect to the co-ordinate planes are

Ix = Σ(mx2),  Iy = Σ(my2),  Iz = Σ(mz2);

(9)

those with respect to the co-ordinate axes are

Iyz = Σ {m (y2 \+ z2)},  Izx = Σ {m (z2 \+ x2)},  Ixy = Σ {m (x2 \+ y2)};

(10)

whilst the polar quadratic moment with respect to O is

I0 = Σ {m (x2 \+ y2 \+ z2)}.

(11)

We note that

Iyz = Iy \+ Iz,  Izx = Iz \+ Ix,  Ixy = Ix \+ Iy,

(12)

and

I0 = Ix \+ Iy \+ Iz = 1⁄2 (Iyz \+ Izx \+ Ixy).

(13)

In the case of continuous distributions of matter the summations in (9), (10), (11) are of course to be replaced by integrations. For a uniform thin circular plate, we find, taking the origin at its centre, and the axis of z normal to its plane, I0 = 1⁄2Ma2, where M is the mass and a the radius. Since Ix = Iy, Iz = 0, we deduce Izx = 1⁄2Ma2, Ixy = 1⁄2Ma2; hence the value of the squared radius of gyration is for a diameter 1⁄4a2, and for the axis of symmetry 1⁄2a2. Again, for a uniform solid sphere having its centre at the origin we find I0 = 3⁄5Ma2, Ix = Iy = Iz = 1⁄5Ma2, Iyz = Izx = lxy = 3⁄5Ma2; i.e. the square of the radius of gyration with respect to a diameter is 2⁄5a2. The method of homogeneous strain can be applied to deduce the corresponding results for an ellipsoid of semi-axes a, b, c. If the co-ordinate axes coincide with the principal axes, we find Ix = 1⁄5Ma2, Iy = 1⁄5Mb2, Iz = 1⁄5Mc2, whence Iyz = 1⁄5M (b2 \+ c2), &c.

If φ(x, y, z) be any homogeneous quadratic function of x, y, z, we have

Σ {mφ (x, y, z)} = Σ {mφ (x \+ ξ, y \+ η, z \+ ζ) }  
= Σ {mφ (x, y, z)} + Σ {mφ (ξ, η, ζ)},

(14)

since the terms which are bilinear in respect to x, y, z, and ξ, η, ζ vanish, in virtue of the relations (7). Thus

Ix = Iξ \+ Σ(m)x2,

(15)

Iyz = Iηζ \+ Σ(m) · (y2 \+ z2),

(16)

with similar relations, and

IO = IG \+ Σ(m) · OG2.

(17)

The formula (16) expresses that the squared radius of gyration about any axis (Ox) exceeds the squared radius of gyration about a parallel axis through G by the square of the distance between the two axes. The formula (17) is due to J. L. Lagrange; it may be written

Σ(m · OP2) | = | Σ(m · GP2) | \+ OG2,  
---|---|---|---  
Σ(m) | Σ(m)  
  
(18)

and expresses that the mean square of the distances of the particles from O exceeds the mean square of the distances from G by OG2. The mass-centre is accordingly that point the mean square of whose distances from the several particles is least. If in (18) we make O coincide with P1, P2, ... Pn in succession, we obtain

0  | \+ m2·P1P22 | + ... | \+ mn·P1Pn2 | = Σ(m · GP2) + Σ(m) · GP12,  
---|---|---|---|---  
m1·P2P12 | \+  0 | + ... | \+ mn·P2Pn2 | = Σ(m · GP2) + Σ(m) · GP22,  
.........  
m1·PnP12 | \+ m2·PnP22 | \+  ... | \+  0 | = Σ(m · GP2) + Σ(m) · GPn2.  
  
(19)

If we multiply these equations by m1, m2 ... mn, respectively, and add, we find

ΣΣ (mrms · PrPs2) = Σ (m) · Σ (m · GP2),

(20)

provided the summation ΣΣ on the left hand be understood to include each pair of particles once only. This theorem, also due to Lagrange, enables us to express the mean square of the distances of the particles from the centre of mass in terms of the masses and mutual distances. For instance, considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R of the circumscribing sphere is given by R2 = 3⁄8a2, if a be the length of an edge.

Another type of quadratic moment is supplied by the deviation-moments, or products of inertia of a distribution of matter. Thus the sum Σ(m·yz) is called the “product of inertia” with respect to the planes y = 0, z = 0. This may be expressed In terms of the product of inertia with respect to parallel planes through G by means of the formula (14); viz.:—

Σ (m · yz) = Σ (m · ηζ) + Σ (m) · yz

(21)

The quadratic moments with respect to different planes through a fixed point O are related to one another as follows. The moment with respect to the plane

λx + μy + νz = 0,

(22)

where λ, μ, ν are direction-cosines, is

Σ {m (λx + μy + νz)2} = Σ (mx2)·λ2 \+ Σ (my2)·μ2 \+ Σ (mz2)·ν2 \+ 2Σ (myz)·μν \+ 2Σ (mzx)·νλ \+ 2Σ (mxy)·λμ,

(23)

and therefore varies as the square of the perpendicular drawn from O to a tangent plane of a certain quadric surface, the tangent plane in question being parallel to (22). If the co-ordinate axes coincide with the principal axes of this quadric, we shall have

Σ(myz) = 0,  Σ(mzx) = 0,  Σ(mxy) = 0;

(24)

and if we write

Σ(mx2) = Ma2,  Σ(my2) = Mb2,  Σ(mz2) = Mc2,

(25)

where M = Σ(m), the quadratic moment becomes M(a2λ2 \+ b2μ2 \+ c2ν2), or Mp2, where p is the distance of the origin from that tangent plane of the ellipsoid

x2 | + | y2 | + | z2 | = 1,  
---|---|---|---|---|---  
a2 | b2 | c2  
  
(26)

which is parallel to (22). It appears from (24) that through any assigned point O three rectangular axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes vanishes; these are called the principal axes of inertia at O. The ellipsoid (26) was first employed by J. Binet (1811), and may be called “Binet’s Ellipsoid” for the point O. Evidently the quadratic moment for a variable plane through O will have a “stationary” value when, and only when, the plane coincides with a principal plane of (26). It may further be shown that if Binet’s ellipsoid be referred to any system of conjugate diameters as co-ordinate axes, its equation will be

x′2 | + | y′2 | + | z′2 | = 1,  
---|---|---|---|---|---  
a′2 | b′2 | c′2  
  
(27)

provided

Σ(mx′2) = Ma′2,  Σ(my′2) Mb′2,  Σ(mz′2) = Mc′2;

also that

Σ(my′z′) = 0,  Σ(mz′x′) = 0,  Σ(mx′y′) = 0.

(28)

Let us now take as co-ordinate axes the principal axes of inertia at the mass-centre G. If a, b, c be the semi-axes of the Binet’s ellipsoid of G, the quadratic moment with respect to the plane λx + μy + νz = 0 will be M(a2λ2 \+ b2μ2 \+ c2ν2), and that with respect to a parallel plane

λx + μy + νz = p

(29)

will be M (a2λ2 \+ b2μ2 \+ c2ν2 \+ p2), by (15). This will have a given value Mk2, provided

p2 = (k2 − a2) λ2 \+ (k2 − b2) μ2 \+ (k2 − c2) ν2.

(30)

Hence the planes of constant quadratic moment Mk2 will envelop the quadric

x2 | + | y2 | + | z2 | = 1,  
---|---|---|---|---|---  
k2 − a2 | k2 − b2 | k2 − c2  
  
(31)

and the quadrics corresponding to different values of k2 will be confocal. If we write

k2 = a2 \+ b2 \+ c2 \+ θ,  
b2 \+ c2 = α2,  c2 \+ a2 = β2,  a2 \+ b2 = γ2

(32)

the equation (31) becomes

x2 | + | y2 | + | z2 | = 1;  
---|---|---|---|---|---  
α2 \+ θ | β2 \+ θ | γ2 \+ θ  
  
(33)

for different values of θ this represents a system of quadrics confocal with the ellipsoid

x2 | + | y2 | + | z2 | = 1,  
---|---|---|---|---|---  
α2 | β2 | γ2  
  
(34)

which we shall meet with presently as the “ellipsoid of gyration” at G. Now consider the tangent plane ω at any point P of a confocal, the tangent plane ω′ at an adjacent point N′, and a plane ω″ through P parallel to ω′. The distance between the planes ω′ and ω″ will be of the second order of small quantities, and the quadratic moments with respect to ω′ and ω″ will therefore be equal, to the first order. Since the quadratic moments with respect to ω and ω′ are equal, it follows that ω is a plane of stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes of inertia at P arc the normals to the three confocals of the system (33) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of θ; and if θ1, θ2, θ3 be the roots we find

θ1 \+ θ2 \+ θ3 = r2 − α2 − β2 − γ2,

(35)

where r2 = x2 \+ y2 \+ z2. The squares of the radii of gyration about the principal axes at P may be denoted by k22 \+ k32, k32 \+ k12, k12 \+ k22; hence by (32) and (35) they are r2 −θ1, r2 − θ2, r2 − θ3, respectively.

To find the relations between the moments of inertia about different axes through any assigned point O, we take O as origin. Since the square of the distance of a point (x, y, z) from the axis

x | = | y | = | z  
---|---|---|---|---  
λ | μ | ν  
  
(36)

is x2 \+ y2 \+ z2 − (λx + μy + νz)2, the moment of inertia about this axis is

I = Σ [m { (λ2 \+ μ2 \+ ν2) (x2 \+ y2 \+ z2) − (λx + μy + νz)2} ]  
= Aλ2 \+ Bμ2 \+ Cν2 − 2Fμν − 2Gνλ − 2Hλμ,

(37)

provided

A = Σ {m (y2 \+ z2)},  B = Σ {m (z2 \+ x2)},  C = Σ {m (x2 \+ y2)},  
F = Σ (myz),  G = Σ (mzx),  H = Σ (mxy);

(38)

i.e. A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of inertia with respect to the pairs of co-ordinate planes. If we construct the quadric

Ax2 \+ By2 \+ Cz2 − 2Fyz − 2Gzx − 2Hxy = Mε4

(39)

where ε is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction λ, μ, ν is found by putting x, y, z = λr, μr, νr. Hence, by comparison with (37),

I = Mε4 / r2.

(40)

The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius. When referred to its principal axes, the equation of the quadric takes the form

Ax2 \+ By2 \+ Cz2 = Mε4.

(41)

The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at O, as already defined in connexion with the theory of plane quadratic moments. The new A, B, C are called the principal moments of inertia at O. Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at O. Since, by (12), B + C > A, &c., the sum of the two lesser principal moments must exceed the greatest principal moment. A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g. in the case of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the ellipsoid cannot be less than 1/√2.

If we write A = Mα2, B = Mβ2, C = Mγ2, the formula (37), when referred to the principal axes at O, becomes

I = M (α2λ2 \+ β2μ2 \+ γ2ν2) = Mp2,

(42)

if p denotes the perpendicular drawn from O in the direction (λ, μ, ν) to a tangent plane of the ellipsoid

x2 | + | y2 | + | z2 | = 1  
---|---|---|---|---|---  
α2 | β2 | γ2  
  
(43)

This is called the ellipsoid of gyration at O; it was introduced into the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having O as centre.

If A = B = C, the momental ellipsoid becomes a sphere; all axes through O are then principal axes, and the moment of inertia is the same for each. The mass-system is then said to possess kinetic symmetry about O.

If all the masses lie in a plane (z = 0) we have, in the notation of (25), c2 = 0, and therefore A = Mb2, B = Ma2, C = M(a2 \+ b2), so that the equation of the momental ellipsoid takes the form

b2x2 \+ a2y2 \+ (a2 \+ b2) z2 = ε4.

(44)

The section of this by the plane z = 0 is similar to

x2 | + | y2 |  = 1,  
---|---|---|---  
a2 | b2  
  
(45)

which may be called the momental ellipse at O. It possesses the property that the radius of gyration about any diameter is half the distance between the two tangents which are parallel to that diameter. In the case of a uniform triangular plate it may be shown that the momental ellipse at G is concentric, similar and similarly situated to the ellipse which touches the sides of the triangle at their middle points.  
  
---  
Fig. 59.  
  
The graphical methods of determining the moment of inertia of a plane system of particles with respect to any line in its plane may be briefly noticed. It appears from § 5 (fig. 31) that the linear moment of each particle about the line may be found by means of a funicular polygon. If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line. For if the line in question be the axis of y, the first process gives us the values of mx, and the second the value of Σ(mx·x) or Σ(mx2). The construction of a second funicular may be dispensed with by the employment of a planimeter, as follows. In fig. 59 p is the line with respect to which moments are to be taken, and the masses of the respective particles are indicated by the corresponding segments of a line in the force-diagram, drawn parallel to p. The funicular ZABCD ... corresponding to any pole O is constructed for a system of forces acting parallel to p through the positions of the particles and proportional to the respective masses; and its successive sides are produced to meet p in the points H, K, L, M, ... As explained in § 5, the moment of the first particle is represented on a certain scale by HK, that of the second by KL, and so on. The quadratic moment of the first particle will then be represented by twice the area AHK, that of the second by twice the area BKL, and so on. The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA. Since a quadratic moment is essentially positive, the various areas are to taken positive in all cases. If k be the radius of gyration about p we find

k2 = 2 × area AHEDCBA × ON ÷ αβ,  
  
---  
Fig. 60.  
  
where αβ is the line in the force-diagram which represents the sum of the masses, and ON is the distance of the pole O from this line. If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added. This is illustrated in fig. 60, where the total quadratic moment is represented by the sum of the shaded areas. It is seen that for a given direction of p this moment is least when p passes through the intersection X of the first and last sides of the funicular; i.e. when p goes through the mass-centre of the given system; cf. equation (15).

Part II.—Kinetics

§ 12. Rectilinear Motion.—Let x denote the distance OP of a moving point P at time t from a fixed origin O on the line of motion, this distance being reckoned positive or negative according as it lies to one side or the other of O. At time t + δt let the point be at Q, and let OQ = x + δx. The mean velocity of the point in the interval δt is δx/δt. The limiting value of this when δt is infinitely small, viz. dx/dt, is adopted as the definition of the velocity at the instant t. Again, let u be the velocity at time t, u + δu that at time t + δt. The mean rate of increase of velocity, or the mean acceleration, in the interval δt is then δu/δt. The limiting value of this when δt is infinitely small, viz., du/dt, is adopted as the definition of the acceleration at the instant t. Since u = dx/dt, the acceleration is also denoted by d2x/dt2. It is often convenient to use the “fluxional” notation for differential coefficients with respect to time; thus the velocity may be represented by ẋ and the acceleration by u̇ or ẍ. There is another formula for the acceleration, in which u is regarded as a function of the position; thus du/dt = (du/dx) (dx/dt) = u(du/dx). The relation between x and t in any particular case may be illustrated by means of a curve constructed with t as abscissa and x as ordinate. This is called the curve of positions or space-time curve; its gradient represents the velocity. Such curves are often traced mechanically in acoustical and other experiments. A, curve with t as abscissa and u as ordinate is called the curve of velocities or velocity-time curve. Its gradient represents the acceleration, and the area (∫u dt) included between any two ordinates represents the space described in the interval between the corresponding instants (see fig. 62).

So far nothing has been said about the measurement of time. From the purely kinematic point of view, the t of our formulae may be any continuous independent variable, suggested (it may be) by some physical process. But from the dynamical standpoint it is obvious that equations which represent the facts correctly on one system of time-measurement might become seriously defective on another. It is found that for almost all purposes a system of measurement based ultimately on the earth’s rotation is perfectly adequate. It is only when we come to consider such delicate questions as the influence of tidal friction that other standards become necessary.

The most important conception in kinetics is that of “inertia.” It is a matter of ordinary observation that different bodies acted on by the same force, or what is judged to be the same force, undergo different changes of velocity in equal times. In our ideal representation of natural phenomena this is allowed for by endowing each material particle with a suitable mass or inertia-coefficient m. The product mu of the mass into the velocity is called the momentum or (in Newton’s phrase) the quantity of motion. On the Newtonian system the motion of a particle entirely uninfluenced by other bodies, when referred to a suitable base, would be rectilinear, with constant velocity. If the velocity changes, this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change of momentum which it produces, we have the equation

d | (mu) = X.  
---|---  
dt  
  
(1)

From this point of view the equation is a mere truism, its real importance resting on the fact that by attributing suitable values to the masses m, and by making simple assumptions as to the value of X in each case, we are able to frame adequate representations of whole classes of phenomena as they actually occur. The question remains, of course, as to how far the measurement of force here implied is practically consistent with the gravitational method usually adopted in statics; this will be referred to presently.

The practical unit or standard of mass must, from the nature of the case, be the mass of some particular body, e.g. the imperial pound, or the kilogramme. In the “C.G.S.” system a subdivision of the latter, viz. the gramme, is adopted, and is associated with the centimetre as the unit of length, and the mean solar second as the unit of time. The unit of force implied in (1) is that which produces unit momentum in unit time. On the C.G.S. system it is that force which acting on one gramme for one second produces a velocity of one centimetre per second; this unit is known as the dyne. Units of this kind are called absolute on account of their fundamental and invariable character as contrasted with gravitational units, which (as we shall see presently) vary somewhat with the locality at which the measurements are supposed to be made.

If we integrate the equation (1) with respect to t between the limits t, t′ we obtain

mu′ − mu = ∫t′t X dt.

(2)

The time-integral on the right hand is called the impulse of the force on the interval t′ − t. The statement that the increase of momentum is equal to the impulse is (it maybe remarked) equivalent to Newton’s own formulation of his Second Law. The form (1) is deduced from it by putting t′ − t = δt, and taking δt to be infinitely small. In problems of impact we have to deal with cases of practically instantaneous impulse, where a very great and rapidly varying force produces an appreciable change of momentum in an exceedingly minute interval of time.

In the case of a constant force, the acceleration u̇ or ẍ is, according to (1), constant, and we have

d2x | = α,  
---|---  
dt2  
  
(3)

say, the general solution of which is

x = 1⁄2 αt2 \+ At + B.

(4)

The “arbitrary constants” A, B enable us to represent the circumstances of any particular case; thus if the velocity ẋ and the position x be given for any one value of t, we have two conditions to determine A, B. The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line. We may take it as an experimental result, although the best evidence is indirect, that a particle falling freely under gravity experiences a constant acceleration which at the same place is the same for all bodies. This acceleration is denoted by g; its value at Greenwich is about 981 centimetre-second units, or 32.2 feet per second. It increases somewhat with the latitude, the extreme variation from the equator to the pole being about 1⁄2%. We infer that on our reckoning the force of gravity on a mass m is to be measured by mg, the momentum produced per second when this force acts alone. Since this is proportional to the mass, the relative masses to be attributed to various bodies can be determined practically by means of the balance. We learn also that on account of the variation of g with the locality a gravitational system of force-measurement is inapplicable when more than a moderate degree of accuracy is desired.

We take next the case of a particle attracted towards a fixed point O in the line of motion with a force varying as the distance from that point. If μ be the acceleration at unit distance, the equation of motion becomes

d2x | = −μx,  
---|---  
dt2  
  
(5)

the solution of which may be written in either of the forms

x = A cos σt + B sin σt, x = a cos (σt + ε),

(6)  
  
---  
Fig. 61.  
  
where σ= √μ, and the two constants A, B or a, ε are arbitrary. The particle oscillates between the two positions x = ±a, and the same point is passed through in the same direction with the same velocity at equal intervals of time 2π/σ. The type of motion represented by (6) is of fundamental importance in the theory of vibrations (§ 23); it is called a simple-harmonic or (shortly) a simple vibration. If we imagine a point Q to describe a circle of radius a with the angular velocity σ, its orthogonal projection P on a fixed diameter AA′ will execute a vibration of this character. The angle σt + ε (or AOQ) is called the phase; the arbitrary elements a, ε are called the amplitude and epoch (or initial phase), respectively. In the case of very rapid vibrations it is usual to specify, not the period (2π/σ), but its reciprocal the frequency, i.e. the number of complete vibrations per unit time. Fig. 62 shows the curves of position and velocity; they both have the form of the “curve of sines.” The numbers correspond to an amplitude of 10 centimetres and a period of two seconds.

The vertical oscillations of a weight which hangs from a fixed point by a spiral spring come under this case. If M be the mass, and x the vertical displacement from the position of equilibrium, the equation of motion is of the form

M | d2x | = − Kx,  
---|---|---  
dt2  
  
(7)

provided the inertia of the spring itself be neglected. This becomes identical with (5) if we put μ = K/M; and the period is therefore 2π√(M/K), the same for all amplitudes. The period is increased by an increase of the mass M, and diminished by an increase in the stiffness (K) of the spring. If c be the statical increase of length which is produced by the gravity of the mass M, we have Kc = Mg, and the period is 2π√(c/g).  
  
---  
Fig. 62.  
  
The small oscillations of a simple pendulum in a vertical plane also come under equation (5). According to the principles of § 13, the horizontal motion of the bob is affected only by the horizontal component of the force acting upon it. If the inclination of the string to the vertical does not exceed a few degrees, the vertical displacement of the particle is of the second order, so that the vertical acceleration may be neglected, and the tension of the string may be equated to the gravity mg of the particle. Hence if l be the length of the string, and x the horizontal displacement of the bob from the equilibrium position, the horizontal component of gravity is mgx/l, whence

d2x | = − | gx | ,  
---|---|---|---  
dt2 | l  
  
(8)

The motion is therefore simple-harmonic, of period τ = 2π√(l/g). This indicates an experimental method of determining g with considerable accuracy, using the formula g = 4π2l/τ2.

In the case of a repulsive force varying as the distance from the origin, the equation of motion is of the type

d2x | = μx,  
---|---  
dt2  
  
(9)

the solution of which is

x = Aent \+ Be−nt,

(10)

where n = √μ. Unless the initial conditions be adjusted so as to make A = 0 exactly, x will ultimately increase indefinitely with t. The position x = 0 is one of equilibrium, but it is unstable. This applies to the inverted pendulum, with μ = g/l, but the equation (9) is then only approximate, and the solution therefore only serves to represent the initial stages of a motion in the neighbourhood of the position of unstable equilibrium.

In acoustics we meet with the case where a body is urged towards a fixed point by a force varying as the distance, and is also acted upon by an “extraneous” or “disturbing” force which is a given function of the time. The most important case is where this function is simple-harmonic, so that the equation (5) is replaced by

d2x | \+ μx = ƒ cos (σ1t + α),  
---|---  
dt2  
  
(11)

where σ1 is prescribed. A particular solution is

x = | ƒ | cos (σ1t + α).  
---|---|---  
μ − σ12  
  
(12)

This represents a forced oscillation whose period 2π/σ1, coincides with that of the disturbing force; and the phase agrees with that of the force, or is opposed to it, according as σ12 < or > μ; i.e. according as the imposed period is greater or less than the natural period 2π/√μ. The solution fails when the two periods agree exactly; the formula (12) is then replaced by

x = | ƒt | sin (σ1t + α),  
---|---|---  
2σ1  
  
(13)

which represents a vibration of continually increasing amplitude. Since the equation (12) is in practice generally only an approximation (as in the case of the pendulum), this solution can only be accepted as a representation of the initial stages of the forced oscillation. To obtain the complete solution of (11) we must of course superpose the free vibration (6) with its arbitrary constants in order to obtain a complete representation of the most general motion consequent on arbitrary initial conditions.

A simple mechanical illustration is afforded by the pendulum. If the point of suspension have an imposed simple vibration ξ = a cos σt in a horizontal line, the equation of small motion of the bob is

mẍ = −mg | x − ξ | ,  
---|---|---  
l  
  
or

ẍ + | gx | = g | ξ | .  
---|---|---|---|---  
l | l  
  
(14)  
  
---  
Fig. 63.  
  
This is the same as if the point of suspension were fixed, and a horizontal disturbing force mgξ/l were to act on the bob. The difference of phase of the forced vibration in the two cases is illustrated and explained in the annexed fig. 63, where the pendulum virtually oscillates about C as a fixed point of suspension. This illustration was given by T. Young in connexion with the kinetic theory of the tides, where the same point arises.

We may notice also the case of an attractive force varying inversely as the square of the distance from the origin. If μ be the acceleration at unit distance, we have

u | du | = − | μ  
---|---|---|---  
dx | x2  
  
(15)

whence

u2 = | 2μ | \+ C.  
---|---|---  
x  
  
(16)

In the case of a particle falling directly towards the earth from rest at a very great distance we have C = 0 and, by Newton’s Law of Gravitation, μ/a2 = g, where a is the earth’s radius. The deviation of the earth’s figure from sphericity, and the variation of g with latitude, are here ignored. We find that the velocity with which the particle would arrive at the earth’s surface (x = a) is √(2ga). If we take as rough values a = 21 × 106 feet, g = 32 foot-second units, we get a velocity of 36,500 feet, or about seven miles, per second. If the particles start from rest at a finite distance c, we have in (16), C = − 2μ/c, and therefore

dx | = u = − √ { | 2μ (c − x) | },  
---|---|---|---  
dt | cx  
  
(17)

the minus sign indicating motion towards the origin. If we put x = c cos2 1⁄2φ, we find

t = | c3/2 | (φ \+ sin φ),  
---|---|---  
√(8μ)  
  
(18)

no additive constant being necessary if t be reckoned from the instant of starting, when φ = 0. The time t of reaching the origin (φ = π) is

t1 = | π c3/2 | .  
---|---|---  
√(8μ)  
  
(19)

This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz. 2πc3/2 / √μ (§ 14). We learn that if the orbital motion of a planet, or a satellite, were arrested, the body would fall into the sun, or into its primary, in the fraction 0.1768 of its actual periodic time. Thus the moon would reach the earth in about five days. It may be noticed that if the scales of x and t be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.

In any case of rectilinear motion, if we integrate both sides of the equation

mu | du | = X,  
---|---|---  
dx  
  
(20)

which is equivalent to (1), with respect to x between the limits x0, x1, we obtain

1⁄2 mu12 − 1⁄2 mu02 = ∫x1x0 X dx.

(21)

We recognize the right-hand member as the work done by the force X on the particle as the latter moves from the position x0 to the position x1. If we construct a curve with x as abscissa and X as ordinate, this work is represented, as in J. Watt’s “indicator-diagram,” by the area cut off by the ordinates x = x0, x = x1. The product 1⁄2mu2 is called the kinetic energy of the particle, and the equation (21) is therefore equivalent to the statement that the increment of the kinetic energy is equal to the work done on the particle. If the force X be always the same in the same position, the particle may be regarded as moving in a certain invariable “field of force.” The work which would have to be supplied by other forces, extraneous to the field, in order to bring the particle from rest in some standard position P0 to rest in any assigned position P, will depend only on the position of P; it is called the statical or potential energy of the particle with respect to the field, in the position P. Denoting this by V, we have δV − Xδx = 0, whence

X = − | dV | .  
---|---|---  
dx  
  
(22)

The equation (21) may now be written

1⁄2 mu12 \+ V1 = 1⁄2 mu02 \+ V0,

(23)

which asserts that when no extraneous forces act the sum of the kinetic and potential energies is constant. Thus in the case of a weight hanging by a spiral spring the work required to increase the length by x is V = ∫x0 Kx dx = 1⁄2Kx2, whence 1⁄2Mu2 \+ 1⁄2Kx2 = const., as is easily verified from preceding results. It is easily seen that the effect of extraneous forces will be to increase the sum of the kinetic and potential energies by an amount equal to the work done by them. If this amount be negative the sum in question is diminished by a corresponding amount. It appears then that this sum is a measure of the total capacity for doing work against extraneous resistances which the particle possesses in virtue of its motion and its position; this is in fact the origin of the term “energy.” The product mv2 had been called by G. W. Leibnitz the “vis viva”; the name “energy” was substituted by T. Young; finally the name “actual energy” was appropriated to the expression 1⁄2mv2 by W. J. M. Rankine.

The laws which regulate the resistance of a medium such as air to the motion of bodies through it are only imperfectly known. We may briefly notice the case of resistance varying as the square of the velocity, which is mathematically simple. If the positive direction of x be downwards, the equation of motion of a falling particle will be of the form

du | = g − ku2;  
---|---  
dt  
  
(24)

this shows that the velocity u will send asymptotically to a certain limit V (called the terminal velocity) such that kV2 = g. The solution is

u = V tanh | gt | ,  x = | V2 | log cosh | gt | ,  
---|---|---|---|---|---|---  
V | g | V  
  
(25)

if the particle start from rest in the position x = 0 at the instant t = 0. In the case of a particle projected vertically upwards we have

du | = −g − ku2,  
---|---  
dt  
  
(26)

the positive direction being now upwards. This leads to

tan−1 | u | = tan−1 | u0 | − | gt | ,  x = | V2 | log | V2 \+ u02 | ,  
---|---|---|---|---|---|---|---|---|---|---  
V | V | V | 2g | V2 \+ u2  
  
(27)

where u0 is the velocity of projection. The particle comes to rest when

t = | V | tan−1 | u0 | ,  x = | V2 | log ( 1 + | u02 | ).  
---|---|---|---|---|---|---|---|---  
g | V | 2g | V2  
  
(28)

For small velocities the resistance of the air is more nearly proportional to the first power of the velocity. The effect of forces of this type on small vibratory motions may be investigated as follows. The equation (5) when modified by the introduction of a frictional term becomes

ẍ = −μx − kẋ.

(29)

If k2 < 4μ the solution is

x = a e−t/τ cos (σt + ε),

(30)

where

τ = 2/k,  σ = √(μ − 1⁄4k2),

(31)

and the constants a, ε are arbitrary. This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law e−t/τ. The constant τ is called the modulus of decay of the oscillations; if it is large compared with 2π/σ the effect of friction on the period is of the second order of small quantities and may in general be ignored. We have seen that a true simple-harmonic vibration may be regarded as the orthogonal projection of uniform circular motion; it was pointed out by P. G. Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector. When k2 > 4μ, the solution of (29) is, in real form,

x = a1e−t/τ1 \+ a2e−t/τ2,

(32)

where

1/τ1, 1/τ2 = 1⁄2k ± √(1⁄4k2 − μ).

(33)

The body now passes once (at most) through its equilibrium position, and the vibration is therefore styled aperiodic.

To find the forced oscillation due to a periodic force we have

ẍ \+ kẋ \+ μx = ƒ cos (σ1t + ε).

(34)

The solution is

x = | ƒ | cos (σ1t + ε − ε1),  
---|---|---  
R  
  
(35)

provided

R = { (μ − σ12)2 \+ k2σ12}1/2,  tan ε1 = | kσ1 | .  
---|---|---  
μ − σ12  
  
(36)

Hence the phase of the vibration lags behind that of the force by the amount ε1, which lies between 0 and 1⁄2π or between 1⁄2π and π, according as σ12 ≶ μ. If the friction be comparatively slight the amplitude is greatest when the imposed period coincides with the free period, being then equal to ƒ/kσ1, and therefore very great compared with that due to a slowly varying force of the same average intensity. We have here, in principle, the explanation of the phenomenon of “resonance” in acoustics. The abnormal amplitude is greater, and is restricted to a narrower range of frequency, the smaller the friction. For a complete solution of (34) we must of course superpose the free vibration (30); but owing to the factor e−t/τ the influence of the initial conditions gradually disappears.

For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short. The whole effect is summed up in the value of the instantaneous impulse, which is the time-integral of the force. Thus if an instantaneous impulse ξ changes the velocity of a mass m from u to u′ we have

mu′ − mu = ξ.

(37)

The effect of ordinary finite forces during the infinitely short duration of this impulse is of course ignored.

We may apply this to the theory of impact. If two masses m1, m2 moving in the same straight line impinge, with the result that the velocities are changed from u1, u2, to u1′, u2′, then, since the impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.

m1u1′ \+ m2u2′ = m1u1 \+ m2u2.

(38)

The complete determination of the result of a collision under given circumstances is not a matter of abstract dynamics alone, but requires some auxiliary assumption. If we assume that there is no loss of apparent kinetic energy we have also

m1u1′2 \+ m2u2′2 = m1u12 \+ m2u22.

(39)

Hence, and from (38),

u2′ − u1′ = −(u2 − u1),

(40)

i.e. the relative velocity of the two bodies is reversed in direction, but unaltered in magnitude. This appears to be the case very approximately with steel or glass balls; generally, however, there is some appreciable loss of apparent energy; this is accounted for by vibrations produced in the balls and imperfect elasticity of the materials. The usual empirical assumption is that

u2′ − u1′ = −e (u2 − u1),

(41)

where e is a proper fraction which is constant for the same two bodies. It follows from the formula § 15 (10) for the internal kinetic energy of a system of particles that as a result of the impact this energy is diminished by the amount

1⁄2 (1 − e2) | m1m2 | (u1 − u2)2.  
---|---|---  
m1 \+ m2  
  
(42)

The further theoretical discussion of the subject belongs to Elasticity.

This is perhaps the most suitable place for a few remarks on the theory of “dimensions.” (See also Units, Dimensions of.) In any absolute system of dynamical measurement the fundamental units are those of mass, length and time; we may denote them by the symbols M, L, T, respectively. They may be chosen quite arbitrarily, e.g. on the C.G.S. system they are the gramme, centimetre and second. All other units are derived from these. Thus the unit of velocity is that of a point describing the unit of length in the unit of time; it may be denoted by LT−1, this symbol indicating that the magnitude of the unit in question varies directly as the unit of length and inversely as the unit of time. The unit of acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly denoted by LT−2. The unit of momentum is MLT−1; the unit force generates unit momentum in unit time and is therefore denoted by MLT−2. The unit of work on the same principles is ML2T−2, and it is to be noticed that this is identical with the unit of kinetic energy. Some of these derivative units have special names assigned to them; thus on the C.G.S. system the unit of force is called the dyne, and the unit of work or energy the erg. The number which expresses a physical quantity of any particular kind will of course vary inversely as the magnitude of the corresponding unit. In any general dynamical equation the dimensions of each term in the fundamental units must be the same, for a change of units would otherwise alter the various terms in different ratios. This principle is often useful as a check on the accuracy of an equation.

The theory of dimensions often enables us to forecast, to some extent, the manner in which the magnitudes involved in any particular problem will enter into the result. Thus, assuming that the period of a small oscillation of a given pendulum at a given place is a definite quantity, we see that it must vary as √(l/g). For it can only depend on the mass m of the bob, the length l of the string, and the value of g at the place in question; and the above expression is the only combination of these symbols whose dimensions are those of a time, simply. Again, the time of falling from a distance a into a given centre of force varying inversely as the square of the distance will depend only on a and on the constant μ of equation (15). The dimensions of μ/x2 are those of an acceleration; hence the dimensions of μ are L3T−2. Assuming that the time in question varies as axμy, whose dimensions are Lx+3yT−2y, we must have x + 3y = 0, −2y = 1, so that the time of falling will vary as a3/2/√μ, in agreement with (19).

The argument appears in a more demonstrative form in the theory of “similar” systems, or (more precisely) of the similar motion of similar systems. Thus, considering the equations

d2x | = − | μ | ,  | d2x′ | = − | μ′ | ,  
---|---|---|---|---|---|---|---  
dt2 | x2 | dt′2 | x′2  
  
(43)

which refer to two particles falling independently into two distinct centres of force, it is obvious that it is possible to have x in a constant ratio to x′, and t in a constant ratio to t′, provided that

x | : | x′ | = | μ | : | μ′ | ,  
---|---|---|---|---|---|---|---  
t2 | t′2 | x2 | x′2  
  
(44)

and that there is a suitable correspondence between the initial conditions. The relation (44) is equivalent to

t : t′ = |  x3/2 | : |  x′3/2 | ,  
---|---|---|---|---  
μ1/2 | μ′1/2  
  
(45)

where x, x′ are any two corresponding distances; e.g. they may be the initial distances, both particles being supposed to start from rest. The consideration of dimensions was introduced by J. B. Fourier (1822) in connexion with the conduction of heat.  
  
---  
Fig. 64.  
  
§ 13. General Motion of a Particle.—Let P, Q be the positions of a moving point at times t, t + δt respectively. A vector OU> drawn parallel to PQ, of length proportional to PQ/δt on any convenient scale, will represent the mean velocity in the interval δt, i.e. a point moving with a constant velocity having the magnitude and direction indicated by this vector would experience the same resultant displacement PQ> in the same time. As δt is indefinitely diminished, the vector OU> will tend to a definite limit OV>; this is adopted as the definition of the velocity of the moving point at the instant t. Obviously OV> is parallel to the tangent to the path at P, and its magnitude is ds/dt, where s is the arc. If we project OV> on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called the component velocities parallel to the axes. If x, y, z be the co-ordinates of P it is easily proved that

u = | dx | ,  v = | dy | ,  w = | dz | .  
---|---|---|---|---|---|---  
dt | dt | dt  
  
(1)

The momentum of a particle is the vector obtained by multiplying the velocity by the mass m. The impulse of a force in any infinitely small interval of time δt is the product of the force into δt; it is to be regarded as a vector. The total impulse in any finite interval of time is the integral of the impulses corresponding to the infinitesimal elements δt into which the interval may be subdivided; the summation of which the integral is the limit is of course to be understood in the vectorial sense.

Newton’s Second Law asserts that change of momentum is equal to the impulse; this is a statement as to equality of vectors and so implies identity of direction as well as of magnitude. If X, Y, Z are the components of force, then considering the changes in an infinitely short time δt we have, by projection on the co-ordinate axes, δ(mu) = Xδt, and so on, or

m | du | = X,  m | dv | = Y,  m | dw | = Z.  
---|---|---|---|---|---|---  
dt | dt | dt  
  
(2)

For example, the path of a particle projected anyhow under gravity will obviously be confined to the vertical plane through the initial direction of motion. Taking this as the plane xy, with the axis of x drawn horizontally, and that of y vertically upwards, we have X = 0, Y = −mg; so that

d2x | = 0,  | d2y | = −g.  
---|---|---|---  
dt2 | dt2  
  
(3)

The solution is

x = At + B,  y = −1⁄2gt2 \+ Ct + D.

(4)

If the initial values of x, y, ẋ, ẏ are given, we have four conditions to determine the four arbitrary constants A, B, C, D. Thus if the particle start at time t = 0 from the origin, with the component velocities u0, v0, we have

x = u0t,  y = v0t − 1⁄2gt2.

(5)

Eliminating t we have the equation of the path, viz.

y = | v0 | x − | gx2 | .  
---|---|---|---|---  
u0 | 2u2  
  
(6)

This is a parabola with vertical axis, of latus-rectum 2u02/g. The range on a horizontal plane through O is got by putting y = 0, viz. it is 2u0v0/g. we denote the resultant velocity at any instant by ṡ we have

ṡ2 = ẋ2 \+ ẏ2 = ṡ02 − 2gy.

(7)

Another important example is that of a particle subject to an acceleration which is directed always towards a fixed point O and is proportional to the distance from O. The motion will evidently be in one plane, which we take as the plane z = 0. If μ be the acceleration at unit distance, the component accelerations parallel to axes of x and y through O as origin will be −μx, −μy, whence

d2x | = −μx,  | d2y | = − μy.  
---|---|---|---  
dt2 | dt2  
  
(8)

The solution is

x = A cos nt + B sin nt,  y = C cos nt + D sin nt,

(9)

where n = √μ. If P be the initial position of the particle, we may conveniently take OP as axis of x, and draw Oy parallel to the direction of motion at P. If OP = a, and ṡ0 be the velocity at P, we have, initially, x = a, y = 0, ẋ = 0, ẏ = ṡ0 whence

x = a cos nt,  y = b sin nt,

(10)

if b = ṡ0/n. The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 2π/√μ; moreover, the velocity at any point P is equal to √μ·OD, where OD is the semi-diameter conjugate to OP. This type of motion is called elliptic harmonic. If the co-ordinate axes are the principal axes of the ellipse, the angle nt in (10) is identical with the “excentric angle.” The motion of the bob of a “spherical pendulum,” i.e. a simple pendulum whose oscillations are not confined to one vertical plane, is of this character, provided the extreme inclination of the string to the vertical be small. The acceleration is towards the vertical through the point of suspension, and is equal to gr/l, approximately, if r denote distance from this vertical. Hence the path is approximately an ellipse, and the period is 2π √(l/g).

|   
---|---  
Fig. 65. | Fig. 66.  
  
The above problem is identical with that of the oscillation of a particle in a smooth spherical bowl, in the neighbourhood of the lowest point. If the bowl has any other shape, the axes Ox, Oy may be taken tangential to the lines of curvature at the lowest point O; the equations of small motion then are

d2x | = −g | x | ,  | d2y | = −g | y | ,  
---|---|---|---|---|---|---|---  
dt2 | ρ1 | dt2 | ρ2  
  
(11)

where ρ1, ρ2, are the principal radii of curvature at O. The motion is therefore the resultant of two simple vibrations in perpendicular directions, of periods 2π √(ρ1/g), 2π √(ρ2/g). The circumstances are realized in “Blackburn’s pendulum,” which consists of a weight P hanging from a point C of a string ACB whose ends A, B are fixed. If E be the point in which the line of the string meets AB, we have ρ1 = CP, ρ2 = EP. Many contrivances for actually drawing the resulting curves have been devised.

It is sometimes convenient to resolve the accelerations in directions having a more intrinsic relation to the path. Thus, in a plane path, let P, Q be two consecutive positions, corresponding to the times t, t + δt; and let the normals at P, Q meet in C, making an angle δψ. Let v (= ṡ) be the velocity at P, v + δv that at Q. In the time δt the velocity parallel to the tangent at P changes from v to v + δv, ultimately, and the tangential acceleration at P is therefore dv/dt or s̈. Again, the velocity parallel to the normal at P changes from 0 to vδψ, ultimately, so that the normal acceleration is v dψ/dt. Since

dv | = | dv |   | ds | = v | dv | ,  v | dψ | = v | dψ |   | ds | = | v2 | ,  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
dt | ds | dt | ds | dt | ds | dt | ρ  
  
(12)

where ρ is the radius of curvature of the path at P, the tangential and normal accelerations are also expressed by v dv/ds and v2/ρ, respectively. Take, for example, the case of a particle moving on a smooth curve in a vertical plane, under the action of gravity and the pressure R of the curve. If the axes of x and y be drawn horizontal and vertical (upwards), and if ψ be the inclination of the tangent to the horizontal, we have

mv | dv | = − mg sin ψ = − mg | dy | ,  | mv2 | = − mg cos ψ \+ R.  
---|---|---|---|---|---|---  
ds | ds | ρ  
  
(13)

The former equation gives

v2 = C − 2gy,

(14)

and the latter then determines R.

In the case of the pendulum the tension of the string takes the place of the pressure of the curve. If l be the length of the string, ψ its inclination to the downward vertical, we have δs = lδψ, so that v = ldψ/dt. The tangential resolution then gives

l | d2ψ | = − g sin ψ.  
---|---|---  
dt2  
  
(15)

If we multiply by 2dψ/dt and integrate, we obtain

( | dψ | )2 = | 2g | cos ψ \+ const.,  
---|---|---|---|---  
dt | l  
  
(16)

which is seen to be equivalent to (14). If the pendulum oscillate between the limits ψ = ±α, we have

( | δψ | )2 = | 2g | (cos ψ − cos α) = | 4g | (sin2 1⁄2α − sin2 1⁄2ψ);  
---|---|---|---|---|---|---  
dt | l | l  
  
(17)

and, putting sin 1⁄2ψ = sin 1⁄2α. sin φ, we find for the period (τ) of a complete oscillation

τ = 4 ∫1⁄2π0 | dt | dφ = 4√ | l | · ∫1⁄2π0 | dφ  
---|---|---|---|---|---  
dφ | g | √(1 − sin2 1⁄2α · sin2 φ)  
= 4√ | l | · F1 (sin 1⁄2α),  
---|---|---  
g  
  
(18)

in the notation of elliptic integrals. The function F1 (sin β) was tabulated by A. M. Legendre for values of β ranging from 0° to 90°. The following table gives the period, for various amplitudes α, in terms of that of oscillation in an infinitely small arc [viz. 2π√(l/g)] as unit.

α/π | τ | α/π | τ  
---|---|---|---  
.1 | 1.0062 |  .6 | 1.2817  
.2 | 1.0253 |  .7 | 1.4283  
.3 | 1.0585 |  .8 | 1.6551  
.4 | 1.1087 |  .9 | 2.0724  
.5 | 1.1804 | 1.0 | ∞  
  
The value of τ can also be obtained as an infinite series, by expanding the integrand in (18) by the binomial theorem, and integrating term by term. Thus

τ = 2π √ | l | · { 1 + | 12 | sin2 1⁄2α + | 12 · 32 | sin4 1⁄2α \+ ... }.  
---|---|---|---|---|---|---  
g | 22 | 22 · 42  
  
(19)

If α be small, an approximation (usually sufficient) is

τ = 2π √(l/g) · (1 + 1⁄16α2).

In the extreme case of α = π, the equation (17) is immediately integrable; thus the time from the lowest position is

t = √(l/g) · log tan (1⁄4π \+ 1⁄4ψ).

(20)

This becomes infinite for ψ = π, showing that the pendulum only tends asymptotically to the highest position.

The variation of period with amplitude was at one time a hindrance to the accurate performance of pendulum clocks, since the errors produced are cumulative. It was therefore sought to replace the circular pendulum by some other contrivance free from this defect. The equation of motion of a particle in any smooth path is

d2s | = −g sin ψ,  
---|---  
dt2  
  
(21)

where ψ is the inclination of the tangent to the horizontal. If sin ψ were accurately and not merely approximately proportional to the arc s, say

s = k sin ψ,

(22)  
  
---  
Fig. 67.  
  
the equation (21) would assume the same form as § 12 (5). The motion along the arc would then be accurately simple-harmonic, and the period 2π √(k/g) would be the same for all amplitudes. Now equation (22) is the intrinsic equation of a cycloid; viz. the curve is that traced by a point on the circumference of a circle of radius 1⁄4k which rolls on the under side of a horizontal straight line. Since the evolute of a cycloid is an equal cycloid the object is attained by means of two metal cheeks, having the form of the evolute near the cusp, on which the string wraps itself alternately as the pendulum swings. The device has long been abandoned, the difficulty being met in other ways, but the problem, originally investigated by C. Huygens, is important in the history of mathematics.

The component accelerations of a point describing a tortuous curve, in the directions of the tangent, the principal normal, and the binormal, respectively, are found as follows. If OV>, OV′> be vectors representing the velocities at two consecutive points P, P′ of the path, the plane VOV′ is ultimately parallel to the osculating plane of the path at P; the resultant acceleration is therefore in the osculating plane. Also, the projections of VV′> on OV and on a perpendicular to OV in the plane VOV′ are δv and vδε, where δε is the angle between the directions of the tangents at P, P′. Since δε = δs/ρ, where δs = PP′ = vδt and ρ is the radius of principal curvature at P, the component accelerations along the tangent and principal normal are dv/dt and vdε/dt, respectively, or vdv/ds and v2/ρ. For example, if a particle moves on a smooth surface, under no forces except the reaction of the surface, v is constant, and the principal normal to the path will coincide with the normal to the surface. Hence the path is a “geodesic” on the surface.

If we resolve along the tangent to the path (whether plane or tortuous), the equation of motion of a particle may be written

mv | dv | = T,  
---|---|---  
ds  
  
(23)

where T is the tangential component of the force. Integrating with respect to s we find

1⁄2mv12 − 1⁄2mv02 = ∫s1s0 T ds;

(24)

i.e. the increase of kinetic energy between any two positions is equal to the work done by the forces. The result follows also from the Cartesian equations (2); viz. we have

m (ẋẍ \+ ẏÿ \+ żz̈) = Xẋ \+ Yẏ \+ Zż,

(25)

whence, on integration with respect to t,

1⁄2m (ẋ2 \+ ẏ2 \+ ż2) | = ∫ (Xẋ \+ Yẏ \+ Zż) dt + const.  
---|---  
  | = ∫ (X dx + Y dy + Z dz) + const.  
  
(26)

If the axes be rectangular, this has the same interpretation as (24).

Suppose now that we have a constant field of force; i.e. the force acting on the particle is always the same at the same place. The work which must be done by forces extraneous to the field in order to bring the particle from rest in some standard position A to rest in any other position P will not necessarily be the same for all paths between A and P. If it is different for different paths, then by bringing the particle from A to P by one path, and back again from P to A by another, we might secure a gain of work, and the process could be repeated indefinitely. If the work required is the same for all paths between A and P, and therefore zero for a closed circuit, the field is said to be conservative. In this case the work required to bring the particle from rest at A to rest at P is called the potential energy of the particle in the position P; we denote it by V. If PP′ be a linear element δs drawn in any direction from P, and S be the force due to the field, resolved in the direction PP′, we have δV = −Sδs or

S = − | ∂V | .  
---|---|---  
∂s  
  
(27)

In particular, by taking PP′ parallel to each of the (rectangular) co-ordinate axes in succession, we find

X = − | ∂V | ,  Y = − | ∂V | ,  Z = − | ∂V | .  
---|---|---|---|---|---|---  
∂x | ∂y | ∂z  
  
(28)

The equation (24) or (26) now gives

1⁄2mv12 \+ V1 = 1⁄2mv02 \+ V0;

(29)

i.e. the sum of the kinetic and potential energies is constant when no work is done by extraneous forces. For example, if the field be that due to gravity we have V = ƒmg dy = mgy + const., if the axis of y be drawn vertically upwards; hence

1⁄2mv2 \+ mgy = const.

(30)

This applies to motion on a smooth curve, as well as to the free motion of a projectile; cf. (7), (14). Again, in the case of a force Kr towards O, where r denotes distance from O we have V = ∫ Kr dr = 1⁄2Kr2 \+ const., whence

1⁄2mv2 \+ 1⁄2Kr2 = const.

(31)

It has been seen that the orbit is in this case an ellipse; also that if we put μ = K/m the velocity at any point P is v = √μ. OD, where OD is the semi-diameter conjugate to OP. Hence (31) is consistent with the known property of the ellipse that OP2 \+ OD2 is constant.

The forms assumed by the dynamical equations when the axes of reference are themselves in motion will be considered in § 21. At present we take only the case where the rectangular axes Ox, Oy rotate in their own plane, with angular velocity ω about Oz, which is fixed. In the interval δt the projections of the line joining the origin to any point (x, y, z) on the directions of the co-ordinate axes at time t are changed from x, y, z to (x + δx) cos ω δt − (y + δy) sin ωδt, (x + δx) sin ω δt + (y + δy) cos ω δt, z respectively. Hence the component velocities parallel to the instantaneous positions of the co-ordinate axes at time t are

u = ẋ − ωy,  v = ẏ \+ ωz,  ω = ż.

(32)

In the same way we find that the component accelerations are

u̇ − ωv,  v̇ \+ ωu,  ω̇

(33)

Hence if ω be constant the equations of motion take the forms

m (ẍ − 2ωẏ − ω2ẋ) = X,  m (ÿ \+ 2ωẋ − ω2y) = Y,  mz̈ = Z.

(34)

These become identical with the equations of motion relative to fixed axes provided we introduce a fictitious force mω2r acting outwards from the axis of z, where r = √(x2 \+ y2), and a second fictitious force 2mωv at right angles to the path, where v is the component of the relative velocity parallel to the plane xy. The former force is called by French writers the force centrifuge ordinaire, and the latter the force centrifuge composée, or force de Coriolis. As an application of (34) we may take the case of a symmetrical Blackburn’s pendulum hanging from a horizontal bar which is made to rotate about a vertical axis half-way between the points of attachment of the upper string. The equations of small motion are then of the type

ẍ − 2ωẏ − ω2x = −p2x,  ÿ \+ 2ωẋ − ω2y = −q2y.

(35)

This is satisfied by

ẍ = A cos (σt + ε),  y = B sin (σt + ε),

(36)

provided

(σ2 \+ ω2 − p2) A + 2σωB = 0,  
---  
2σωA + (σ2 \+ ω2 − q2) B = 0.  
  
(37)

Eliminating the ratio A : B we have

(σ2 \+ ω2 − p2) (σ2 \+ ω2 − q2) − 4σ2ω2 = 0.

(38)

It is easily proved that the roots of this quadratic in σ2 are always real, and that they are moreover both positive unless ω2 lies between p2 and q2. The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, ε. Since the equations (35) are linear, these two solutions are to be superposed. If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x = 0, y = 0 is unstable. This occurs only when the period (2π/ω) of revolution of the arm lies between the two periods (2π/p, 2π/q) of oscillation when the arm is fixed.

§ 14. Central Forces. Hodograph.—The motion of a particle subject to a force which passes always through a fixed point O is necessarily in a plane orbit. For its investigation we require two equations; these may be obtained in a variety of forms.

Since the impulse of the force in any element of time δt has zero moment about O, the same will be true of the additional momentum generated. Hence the moment of the momentum (considered as a localized vector) about O will be constant. In symbols, if v be the velocity and p the perpendicular from O to the tangent to the path,

pv = h,

(1)

where h is a constant. If δs be an element of the path, pδs is twice the area enclosed by δs and the radii drawn to its extremities from O. Hence if δA be this area, we have δA = 1⁄2 pδs = 1⁄2 hδt, or

dA | = 1⁄2h  
---|---  
dt  
  
(2)

Hence equal areas are swept over by the radius vector in equal times.

If P be the acceleration towards O, we have

v | dv | = −P | dr |   
---|---|---|---|---  
ds | ds  
  
(3)

since dr/ds is the cosine of the angle between the directions of r and δs. We will suppose that P is a function of r only; then integrating (3) we find

1⁄2v2 = − ∫ P dr + const.,

(4)

which is recognized as the equation of energy. Combining this with (1) we have

h2 | = C − 2 ∫ P dr,  
---|---  
p2  
  
(5)

which completely determines the path except as to its orientation with respect to O.

If the law of attraction be that of the inverse square of the distance, we have P = μ/r2, and

h2 | = C + | 2μ | .  
---|---|---|---  
p2 | τ  
  
(6)

Now in a conic whose focus is at O we have

l | = | 2 | ± | 1 | ,  
---|---|---|---|---|---  
p2 | r | a  
  
(7)

where l is half the latus-rectum, a is half the major axis, and the upper or lower sign is to be taken according as the conic is an ellipse or hyperbola. In the intermediate case of the parabola we have a = ∞ and the last term disappears. The equations (6) and (7) are identified by putting

l = h2/μ,  a = ± μ/C.

(8)

Since

v2 = | h2 | = μ ( | 2 | ± | 1 | ),  
---|---|---|---|---|---|---  
p2 | r | a  
  
(9)

it appears that the orbit is an ellipse, parabola or hyperbola, according as v2 is less than, equal to, or greater than 2μ/r. Now it appears from (6) that 2μ/r is the square of the velocity which would be acquired by a particle falling from rest at infinity to the distance r. Hence the character of the orbit depends on whether the velocity at any point is less than, equal to, or greater than the velocity from infinity, as it is called. In an elliptic orbit the area πab is swept over in the time

r = | πab | = | 2πa3/2 | ,  
---|---|---|---|---  
1⁄2h | √μ  
  
(10)

since h = μ1/2l1/2 = μ1/2ba−1/2 by (8).

The converse problem, to determine the law of force under which a given orbit can be described about a given pole, is solved by differentiating (5) with respect to r; thus

P = | h2 dp | .  
---|---|---  
p3 dr  
  
(11)

In the case of an ellipse described about the centre as pole we have

a2b2 | = a2 \+ b2 − r2;  
---|---  
p2  
  
(12)

hence P = μr, if μ = h2/a2b2. This merely shows that a particular ellipse may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted. But since an ellipse can always be constructed with a given centre so as to touch a given line at a given point, and to have a given value of ab (= h/√μ) we infer that the orbit will be elliptic whatever the initial circumstances. Also the period is 2πab/h = 2π/√μ, as previously found.  
  
---  
Fig. 68.  
  
Again, in the equiangular spiral we have p = r sinα, and therefore P = μ/r3, if μ = h2/sin2 α. But since an equiangular spiral having a given pole is completely determined by a given point and a given tangent, this type of orbit is not a general one for the law of the inverse cube. In order that the spiral may be described it is necessary that the velocity of projection should be adjusted to make h = √μ·sinα. Similarly, in the case of a circle with the pole on the circumference we have p2 = r2/2a, P = μ/r5, if μ = 8h2a2; but this orbit is not a general one for the law of the inverse fifth power.

In astronomical and other investigations relating to central forces it is often convenient to use polar co-ordinates with the centre of force as pole. Let P, Q be the positions of a moving point at times t, t + δt, and write OP = r, OQ = r + δr, ∠POQ = δθ, O being any fixed origin. If u, v be the component velocities at P along and perpendicular to OP (in the direction of θ increasing), we have

u = lim. | δr | = | dr | ,  v = lim. | r δθ | = r | dθ | .  
---|---|---|---|---|---|---|---|---  
δt | dt | δt | dt  
  
(13)

Again, the velocities parallel and perpendicular to OP change in the time δt from u, v to u − v δθ, v + u δθ, ultimately. The component accelerations at P in these directions are therefore

du | − v | dθ | = | d2r | − r ( | dθ | )2,  
---|---|---|---|---|---|---|---  
dt | dt | dt2 | dt  
dv | \+ u | dθ | = | 1 |   | d | ( r2 | dθ | ),  
---|---|---|---|---|---|---|---|---|---  
dt | dt | r | dt | dt  
  
(14)

respectively.

In the case of a central force, with O as pole, the transverse acceleration vanishes, so that

r2 dθ / dt = h,

(15)

where h is constant; this shows (again) that the radius vector sweeps over equal areas in equal times. The radial resolution gives

d2r | − r ( | dθ | )2 = −P,  
---|---|---|---  
dt2 | dt  
  
(16)

where P, as before, denotes the acceleration towards O. If in this we put r = 1/u, and eliminate t by means of (15), we obtain the general differential equation of central orbits, viz.

d2u | \+ u = | P | .  
---|---|---|---  
dθ2 | h2u2  
  
(17)

If, for example, the law be that of the inverse square, we have P = μu2, and the solution is of the form

u = | μ | {1 + e cos (θ − α)},  
---|---|---  
h2  
  
(18)

where e, α are arbitrary constants. This is recognized as the polar equation of a conic referred to the focus, the half latus-rectum being h2/μ.

The law of the inverse cube P = μu3 is interesting by way of contrast. The orbits may be divided into two classes according as h2 ≷ μ, i.e. according as the transverse velocity (hu) is greater or less than the velocity √μ·u appropriate to a circular orbit at the same distance. In the former case the equation (17) takes the form

d2u | \+ m2u = 0,  
---|---  
dθ2  
  
(19)

the solution of which is

au = sin m (θ − α).

(20)

The orbit has therefore two asymptotes, inclined at an angle π/m. In the latter case the differential equation is of the form

d2u | = m2u,  
---|---  
dθ2  
  
(21)

so that

u = Aemθ \+ Be−mθ

(22)

If A, B have the same sign, this is equivalent to

au = cosh mθ,

(23)

if the origin of θ be suitably adjusted; hence r has a maximum value α, and the particle ultimately approaches the pole asymptotically by an infinite number of convolutions. If A, B have opposite signs the form is

au = sinh mθ,

(24)

this has an asymptote parallel to θ = 0, but the path near the origin has the same general form as in the case of (23). If A or B vanish we have an equiangular spiral, and the velocity at infinity is zero. In the critical case of h2 = μ, we have d2u/dθ2 = 0, and

u = Aθ \+ B;

(25)

the orbit is therefore a “reciprocal spiral,” except in the special case of A = 0, when it is a circle. It will be seen that unless the conditions be exactly adjusted for a circular orbit the particle will either recede to infinity or approach the pole asymptotically. This problem was investigated by R. Cotes (1682-1716), and the various curves obtained arc known as Coles’s spirals.

A point on a central orbit where the radial velocity (dr/dt) vanishes is called an apse, and the corresponding radius is called an apse-line. If the force is always the same at the same distance any apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be reversed. It follows that the angle between successive apse-lines is constant; it is called the apsidal angle of the orbit.

If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),

h2 | = 2 ∫∞r P dr;  
---|---  
p2  
  
(26)

this determines the form of the critical orbit, as it is called. If P = μ/rn, its polar equation is

rm cos mθ = am,

(27)

where m = 1⁄2(3 − n), except in the case n = 3, when the orbit is an equiangular spiral. The case n = 2 gives the parabola as before.

If we eliminate dθ/dt between (15) and (16) we obtain

d2r | − | h2 | = −P = −ƒ(r),  
---|---|---|---  
dt2 | r3  
  
say. We may apply this to the investigation of the stability of a circular orbit. Assuming that r = a + x, where x is small, we have, approximately,

d2x | − | h2 | ( 1 − | 3x | ) = −ƒ(a) − xƒ′(a).  
---|---|---|---|---|---  
dt2 | a3 | a  
  
Hence if h and a be connected by the relation h2 = a3ƒ(a) proper to a circular orbit, we have

d2x | \+ { ƒ′(a) + | 3 | ƒ(a) } x = 0  
---|---|---|---  
dt2 | a  
  
(28)

If the coefficient of x be positive the variations of x are simple-harmonic, and x can remain permanently small; the circular orbit is then said to be stable. The condition for this may be written

d | { a3ƒ(a) } > 0,  
---|---  
da  
  
(29)

i.e. the intensity of the force in the region for which r = a, nearly, must diminish with increasing distance less rapidly than according to the law of the inverse cube. Again, the half-period of x is π / √{ƒ′(a) + 3−1ƒ(a)}, and since the angular velocity in the orbit is h/a2, approximately, the apsidal angle is, ultimately,

π √{ | ƒ(a) | },  
---|---|---  
aƒ′(a) + 3ƒ(a)  
  
(30)

or, in the case of ƒ(a) = μ/rn, π/√(3 − n). This is in agreement with the known results for n = 2, n = −1.

We have seen that under the law of the inverse square all finite orbits are elliptical. The question presents itself whether there then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves. If this is the case, the apsidal angle must evidently be commensurable with π, and since it cannot vary discontinuously the apsidal angle in a nearly circular orbit must be constant. Equating the expression (30) to π/m, we find that ƒ(a) = C/an, where n = 3 − m2. The force must therefore vary as a power of the distance, and n must be less than 3. Moreover, the case n = 2 is the only one in which the critical orbit (27) can be regarded as the limiting form of a closed curve. Hence the only law of force which satisfies the conditions is that of the inverse square.

At the beginning of § 13 the velocity of a moving point P was represented by a vector OV> drawn from a fixed origin O. The locus of the point V is called the hodograph (q.v.); and it appears that the velocity of the point V along the hodograph represents in magnitude and in direction the acceleration in the original orbit. Thus in the case of a plane orbit, if v be the velocity of P, ψ the inclination of the direction of motion to some fixed direction, the polar co-ordinates of V may be taken to be v, ψ; hence the velocities of V along and perpendicular to OV will be dv/dt and v dψ/dt. These expressions therefore give the tangential and normal accelerations of P; cf. § 13 (12).  
  
---  
Fig. 69.  
  
In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity. In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit. In the case of a central orbit described under the law of the inverse square we have v = h/SY = h. SZ/b2, where S is the centre of force, SY is the perpendicular to the tangent at P, and Z is the point where YS meets the auxiliary circle again. Hence the hodograph is similar and similarly situated to the locus of Z (the auxiliary circle) turned about S through a right angle. This applies to an elliptic or hyperbolic orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case. The annexed fig. 70 exhibits the various cases, with the hodograph in its proper orientation. The pole O of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola. In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the “reciprocal polar” of the orbit with respect to the centre of force. Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a conic with the pole as focus. In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hodograph is a circle through the pole, described with constant velocity.

§ 15. Kinetics of a System of Discrete Particles.—The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (§ 8). Thus taking any point O as base, we have first a linear momentum whose components referred to rectangular axes through O are

Σ(mẋ),  Σ(mẏ),  Σ(mż);

(1)

its representative vector is the same whatever point O be chosen. Secondly, we have an angular momentum whose components are

Σ {m (yż − zẏ) },  Σ {m (zẋ − xż) },  Σ {m (xẏ − yẋ) },

(2)

these being the sums of the moments of the momenta of the several particles about the respective axes. This is subject to the same relations as a couple in statics; it may be represented by a vector which will, however, in general vary with the position of O.

The linear momentum is the same as if the whole mass were concentrated at the centre of mass G, and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we imagine the two configurations of the system there referred to to be those corresponding to the instants t, t + δt. Thus

Σ ( m· | PP> | ) = Σ(m)· | GG′> | .  
---|---|---|---|---  
δt | δt  
  
(3)

Analytically we have

Σ(mẋ) = | d | Σ(mx) = Σ(m)· | dx | ,  
---|---|---|---|---  
dt | dt  
  
(4)

with two similar formulae.  
  
---  
Fig. 70.  
  
Again, if the instantaneous position of G be taken as base, the angular momentum of the absolute motion is the same as the angular momentum of the motion relative to G. For the velocity of a particle m at P may be replaced by two components one of which (v) is identical in magnitude and direction with the velocity of G, whilst the other (v) is the velocity relative to G. The aggregate of the components mv of momentum is equivalent to a single localized vector Σ(m)·v in a line through G, and has therefore zero moment about any axis through G; hence in taking moments about such an axis we need only regard the velocities relative to G. In symbols, we have

Σ { m(yż − zẏ) } = Σ(m)· ( y | dz | − z | dy | ) \+ Σ { m (ηζ − ζ̇η̇) }.  
---|---|---|---|---  
dt | dt  
  
(5)  
  
---  
Fig. 71.  
  
since Σ(mξ) = 0, Σ(mξ̇) = 0, and so on, the notation being as in § 11. This expresses that the moment of momentum about any fixed axis (e.g. Ox) is equal to the moment of momentum of the motion relative to G about a parallel axis through G, together with the moment of momentum of the whole mass supposed concentrated at G and moving with this point. If in (5) we make O coincide with the instantaneous position of G, we have x, z, z = 0, and the theorem follows.

Finally, the rates of change of the components of the angular momentum of the motion relative to G referred to G as a moving base, are equal to the rates of change of the corresponding components of angular momentum relative to a fixed base coincident with the instantaneous position of G. For let G′ be a consecutive position of G. At the instant t + δt the momenta of the system are equivalent to a linear momentum represented by a localized vector Σ(m)·(v \+ δv) in a line through G′ tangential to the path of G′, together with a certain angular momentum. Now the moment of this localized vector with respect to any axis through G is zero, to the first order of δt, since the perpendicular distance of G from the tangent line at G′ is of the order (δt)2. Analytically we have from (5),

d | Σ { m (yż − zẏ) } = Σ(m)· ( y | dz2 | − z | d2y | ) + | d | Σ { m(ηζ − ζη̇) }  
---|---|---|---|---|---|---|---  
dt | dt2 | dt2 | dt  
  
(6)

If we put x, y, z = 0, the theorem is proved as regards axes parallel to Ox.

Next consider the kinetic energy of the system. If from a fixed point O we draw vectors OV1>, OV2> to represent the velocities of the several particles m1, m2 ..., and if we construct the vector

OK> = | Σ ( m·OV> )  
---|---  
Σ(m)  
  
(7)

this will represent the velocity of the mass-centre, by (3). We find, exactly as in the proof of Lagrange’s First Theorem (§ 11), that

1⁄2Σ (m·OV2) = 1⁄2Σ (m)·OK2 \+ 1⁄2Σ (m·KV2);

(8)

i.e. the total kinetic energy is equal to the kinetic energy of the whole mass supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G. The latter may be called the internal kinetic energy of the system. Analytically we have

1⁄2Σ { m (ẋ2 \+ ẏ2 \+ ż2) } = 1⁄2Σ(m)· { ( | dx | )2 \+ ( | dy | )2 \+ ( | dz | )2 } \+ 1⁄2Σ{ m(ζ2 \+ η̇2 \+ ζ̇2) }.  
---|---|---|---|---|---|---  
dt | dt | dt  
  
(9)

There is also an analogue to Lagrange’s Second Theorem, viz.

1⁄2Σ (m·KV2) = 1⁄2 | ΣΣ (mpmq · VpVq2) | ,  
---|---|---  
Σm  
  
(10)

which expresses the internal kinetic energy in terms of the relative velocities of the several pairs of particles. This formula is due to Möbius.

The preceding theorems are purely kinematical. We have now to consider the effect of the forces acting on the particles. These may be divided into two categories; we have first, the extraneous forces exerted on the various particles from without, and, secondly, the mutual or internal forces between the various pairs of particles. It is assumed that these latter are subject to the law of equality of action and reaction. If the equations of motion of each particle be formed separately, each such internal force will appear twice over, with opposite signs for its components, viz. as affecting the motion of each of the two particles between which it acts. The full working out is in general difficult, the comparatively simple problem of “three bodies,” for instance, in gravitational astronomy being still unsolved, but some general theorems can be formulated.

The first of these may be called the Principle of Linear Momentum. If there are no extraneous forces, the resultant linear momentum is constant in every respect. For consider any two particles at P and Q, acting on one another with equal and opposite forces in the line PQ. In the time δt a certain impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite impulse is given to the second in the direction from Q to P. Since these impulses produce equal and opposite momenta in the two particles, the resultant linear momentum of the system is unaltered. If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric addition of the total impulse of the extraneous forces. It follows, by the preceding kinematic theory, that the mass-centre G of the system will move exactly as if the whole mass were concentrated there and were acted on by the extraneous forces applied parallel to their original directions. For example, the mass-centre of a system free from extraneous force will describe a straight line with constant velocity. Again, the mass-centre of a chain of particles connected by strings, projected anyhow under gravity, will describe a parabola.

The second general result is the Principle of Angular Momentum. If there are no extraneous forces, the moment of momentum about any fixed axis is constant. For in time δt the mutual action between two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have equal and opposite moments about the fixed axis. If extraneous forces act, the total angular momentum about any fixed axis is in time δt increased by the total extraneous impulse about that axis. The kinematical relations above explained now lead to the conclusion that in calculating the effect of extraneous forces in an infinitely short time δt we may take moments about an axis passing through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the same whether in this process we employ the true velocities of the particles or merely their velocities relative to G. If there are no extraneous forces, or if the extraneous forces have zero moment about any axis through G, the vector which represents the resultant angular momentum relative to G is constant in every respect. A plane through G perpendicular to this vector has a fixed direction in space, and is called the invariable plane; it may sometimes be conveniently used as a plane of reference.

For example, if we have two particles connected by a string, the invariable plane passes through the string, and if ω be the angular velocity in this plane, the angular momentum relative to G is

m1ω1r1·r1 \+ m2ωr2·r2 = (m1r12 \+ m2r22) ω,

where r1, r2 are the distances of m1, m2 from their mass-centre G. Hence if the extraneous forces (e.g. gravity) have zero moment about G, ω will be constant. Again, the tension R of the string is given by

R = m1ω2r1 = | m1m2 | ω2a,  
---|---|---  
m1 \+ m2  
  
where a = r1 \+ r2. Also by (10) the internal kinetic energy is

1⁄2 |  m1m2 | ω2a2  
---|---|---  
m1 \+ m2  
  
The increase of the kinetic energy of the system in any interval of time will of course be equal to the total work done by all the forces acting on the particles. In many questions relating to systems of discrete particles the internal force Rpq (which we will reckon positive when attractive) between any two particles mp, mq is a function only of the distance rpq between them. In this case the work done by the internal forces will be represented by

−Σ ∫ Rpg drpq,

when the summation includes every pair of particles, and each integral is to be taken between the proper limits. If we write

V = Σ ∫ Rpq drpq,

(11)

when rpq ranges from its value in some standard configuration A of the system to its value in any other configuration P, it is plain that V represents the work which would have to be done in order to bring the system from rest in the configuration A to rest in the configuration P. Hence V is a definite function of the configuration P; it is called the internal potential energy. If T denote the kinetic energy, we may say then that the sum T + V is in any interval of time increased by an amount equal to the work done by the extraneous forces. In particular, if there are no extraneous forces T + V is constant. Again, if some of the extraneous forces are due to a conservative field of force, the work which they do may be reckoned as a diminution of the potential energy relative to the field as in § 13.

§ 16. Kinetics of a Rigid Body. Fundamental Principles.—When we pass from the consideration of discrete particles to that of continuous distributions of matter, we require some physical postulate over and above what is contained in the Laws of Motion, in their original formulation. This additional postulate may be introduced under various forms. One plan is to assume that any body whatever may be treated as if it were composed of material particles, i.e. mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction. In the case of a rigid body we must suppose that those forces adjust themselves so as to preserve the mutual distances of the various particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as in § 15.

An alternative procedure is to adopt the principle first formally enunciated by J. Le R. d’Alembert and since known by his name. If x, y, z be the rectangular co-ordinates of a mass-element m, the expressions mẍ, mÿ, mz̈ must be equal to the components of the total force on m, these forces being partly extraneous and partly forces exerted on m by other mass-elements of the system. Hence (mẍ, mÿ, mz̈) is called the actual or effective force on m. According to d’Alembert’s formulation, the extraneous forces together with the effective forces reversed fulfil the statical conditions of equilibrium. In other words, the whole assemblage of effective forces is statically equivalent to the extraneous forces. This leads, by the principles of § 8, to the equations

Σ(mẍ) = X,  Σ(mÿ) = Y,  Σ(mz̈) = Z,  
Σ {m (yz̈ − zÿ) } = L,  Σ {m (zẍ − xz̈) } = M,  Σ{m (xÿ − yẍ) } = N,

(1)

where (X, Y, Z) and (L, M, N) are the force—and couple—constituents of the system of extraneous forces, referred to O as base, and the summations extend over all the mass-elements of the system. These equations may be written

d | Σ(mẋ) = X,  | d | Σ(mẏ) = Y,  | d | Σ(mż) = Z,  
---|---|---|---|---|---  
dt | dt | dt  
d | Σ {m (yż − zẏ) } = L,  | d | Σ {m (zẋ − xż) } = M,  | d | Σ {m (xẏ − yẋ) } = N,  
---|---|---|---|---|---  
dt | dt | dt  
  
(2)

and so express that the rate of change of the linear momentum in any fixed direction (e.g. that of Ox) is equal to the total extraneous force in that direction, and that the rate of change of the angular momentum about any fixed axis is equal to the moment of the extraneous forces about that axis. If we integrate with respect to t between fixed limits, we obtain the principles of linear and angular momentum in the form previously given. Hence, whichever form of postulate we adopt, we are led to the principles of linear and angular momentum, which form in fact the basis of all our subsequent work. It is to be noticed that the preceding statements are not intended to be restricted to rigid bodies; they are assumed to hold for all material systems whatever. The peculiar status of rigid bodies is that the principles in question are in most cases sufficient for the complete determination of the motion, the dynamical equations (1 or 2) being equal in number to the degrees of freedom (six) of a rigid solid, whereas in cases where the freedom is greater we have to invoke the aid of other supplementary physical hypotheses (cf. Elasticity; Hydromechanics).

The increase of the kinetic energy of a rigid body in any interval of time is equal to the work done by the extraneous forces acting on the body. This is an immediate consequence of the fundamental postulate, in either of the forms above stated, since the internal forces do on the whole no work. The statement may be extended to a system of rigid bodies, provided the mutual reactions consist of the stresses in inextensible links, or the pressures between smooth surfaces, or the reactions at rolling contacts (§ 9).

§ 17. Two-dimensional Problems.—In the case of rotation about a fixed axis, the principles take a very simple form. The position of the body is specified by a single co-ordinate, viz. the angle θ through which some plane passing through the axis and fixed in the body has turned from a standard position in space. Then dθ/dt, = ω say, is the angular velocity of the body. The angular momentum of a particle m at a distance r from the axis is mωr·r, and the total angular momentum is Σ(mr2)·ω, or Iω, if I denote the moment of inertia (§ 11) about the axis. Hence if N be the moment of the extraneous forces about the axis, we have

d | (Iω) = N.  
---|---  
dt  
  
(1)

This may be compared with the equation of rectilinear motion of a particle, viz. d/dt·(Mu) = X; it shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as regards translation. If N = 0, ω is constant.  
  
---  
Fig. 72. | Fig. 73.  
  
As a first example, suppose we have a flywheel free to rotate about a horizontal axis, and that a weight m hangs by a vertical string from the circumferences of an axle of radius b (fig. 72). Neglecting frictional resistance we have, if R be the tension of the string,

Iω̇ = Rb, mu̇ = mg − R,

whence

bω̇ = | mb2 | g.  
---|---|---  
1 + mb2  
  
(2)

This gives the acceleration of m as modified by the inertia of the wheel.

A “compound pendulum” is a body of any form which is free to rotate about a fixed horizontal axis, the only extraneous force (other than the pressures of the axis) being that of gravity. If M be the total mass, k the radius of gyration (§ 11) about the axis, we have

d | ( Mk2 | dθ | ) = −Mgh sin θ,  
---|---|---|---  
dt | dt  
  
(3)

where θ is the angle which the plane containing the axis and the centre of gravity G makes with the vertical, and h is the distance of G from the axis. This coincides with the equation of motion of a simple pendulum [§ 13 (15)] of length l, provided l = k2/h. The plane of the diagram (fig. 73) is supposed to be a plane through G perpendicular to the axis, which it meets in O. If we produce OG to P, making OP = l, the point P is called the centre of oscillation; the bob of a simple pendulum of length OP suspended from O will keep step with the motion of P, if properly started. If κ be the radius of gyration about a parallel axis through G, we have k2 = κ2 \+ h2 by § 11 (16), and therefore l = h + κ2/h, whence

GO · GP = κ2.

(4)

This shows that if the body were swung from a parallel axis through P the new centre of oscillation would be at O. For different parallel axes, the period of a small oscillation varies as √l, or √(GO + OP); this is least, subject to the condition (4), when GO = GP = κ. The reciprocal relation between the centres of suspension and oscillation is the basis of Kater’s method of determining g experimentally. A pendulum is constructed with two parallel knife-edges as nearly as possible in the same plane with G, the position of one of them being adjustable. If it could be arranged that the period of a small oscillation should be exactly the same about either edge, the two knife-edges would in general occupy the positions of conjugate centres of suspension and oscillation; and the distances between them would be the length l of the equivalent simple pendulum. For if h1 \+ κ2/h1 = h2 \+ κ2/h2, then unless h1 = h2, we must have κ2 = h1h2, l = h1 \+ h2. Exact equality of the two observed periods (τ1, τ2, say) cannot of course be secured in practice, and a modification is necessary. If we write l1 = h1 \+ κ2/h1, l2 = h2 \+ κ2/h2, we find, on elimination of κ,

1⁄2 | l1 \+ l2 | \+ 1⁄2 | l1 − l2 | = 1,  
---|---|---|---|---  
h1 \+ h2 | h1 − h2  
  
whence

4π2 | = | 1⁄2 (τ12 \+ τ22) | + | 1⁄2 (τ12 − τ22) | .  
---|---|---|---|---|---  
g | h1 \+ h2 | h1 − h2  
  
(5)

The distance h1 \+ h2, which occurs in the first term on the right hand can be measured directly. For the second term we require the values of h1, h2 separately, but if τ1, τ2 are nearly equal whilst h1, h2 are distinctly unequal this term will be relatively small, so that an approximate knowledge of h1, h2 is sufficient.

As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position. The equation of motion is of the type

I θ̈ = −Kθ,

(6)

and the period is therefore τ = 2π√(I/K). If by the attachment of another body of known moment of inertia I′, the period is altered from τ to τ′, we have τ′ = 2π√{ (I + I′)/K }. We are thus enabled to determine both I and K, viz.

I / I′ = τ2 / (τ′2 − τ2),  K = 4π2τ2I / (τ′2 − τ2).

(7)

The couple may be due to the earth’s magnetism, or to the torsion of a suspending wire, or to a “bifilar” suspension. In the latter case, the body hangs by two vertical threads of equal length l in a plane through G. The motion being assumed to be small, the tensions of the two strings may be taken to have their statical values Mgb/(a + b), Mga/(a + b), where a, b are the distances of G from the two threads. When the body is twisted through an angle θ the threads make angles aθ/l, bθ/l with the vertical, and the moment of the tensions about the vertical through G is accordingly −Kθ, where K = M gab/l.

For the determination of the motion it has only been necessary to use one of the dynamical equations. The remaining equations serve to determine the reactions of the rotating body on its bearings. Suppose, for example, that there are no extraneous forces. Take rectangular axes, of which Oz coincides with the axis of rotation. The angular velocity being constant, the effective force on a particle m at a distance r from Oz is mω2r towards this axis, and its components are accordingly −ω2mx, −ω2my, O. Since the reactions on the bearings must be statically equivalent to the whole system of effective forces, they will reduce to a force (X Y Z) at O and a couple (L M N) given by

X = −ω2Σ(mx) = −ω2Σ(m)x,  Y = −ω2Σ(my) = −ω2Σ(m)y,  Z = 0,  
L = ω2Σ(myz),  M = −ω2Σ(mzx),  N = 0, 

(8)

where x, y refer to the mass-centre G. The reactions do not therefore reduce to a single force at O unless Σ(myz) = 0, Σ(msx) = 0, i.e. unless the axis of rotation be a principal axis of inertia (§ 11) at O. In order that the force may vanish we must also have x, y = 0, i.e. the mass-centre must lie in the axis of rotation. These considerations are important in the “balancing” of machinery. We note further that if a body be free to turn about a fixed point O, there are three mutually perpendicular lines through this point about which it can rotate steadily, without further constraint. The theory of principal or “permanent” axes was first investigated from this point of view by J. A. Segner (1755). The origin of the name “deviation moment” sometimes applied to a product of inertia is also now apparent.  
  
---  
Fig. 74.  
  
Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle θ through which the body has turned from some standard position. The components of linear momentum are then Mẋ, Mẏ, and the angular momentum relative to G as base is Iθ̇, where M is the mass and I the moment of inertia about G. If the extraneous forces be reduced to a force (X, Y) at G and a couple N, we have

Mẍ = X,  Mÿ = Y,  Iθ̈ = N.

(9)

If the extraneous forces have zero moment about G the angular velocity θ̇ is constant. Thus a circular disk projected under gravity in a vertical plane spins with constant angular velocity, whilst its centre describes a parabola.

We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal on a plane of inclination α. If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we have

Mẍ = Mg sin α − F,  0 = Mg cos α − R,  Mκ2θ̈ = Fa,

(10)

where κ is the radius of gyration about the axis of symmetry, a is the constant distance of G from the plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig. 74. We have also the kinematical relation ẋ = aθ̇. Hence

ẍ = | a2 | g sin α, R = Mg cos α,  F = | κ2 | Mg sin α.  
---|---|---|---|---  
κ2 \+ a2 | κ2 \+ a2  
  
(11)

The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a2/(κ2 \+ a2). For a homogeneous sphere this ratio is 5⁄7, for a uniform circular cylinder or disk 2⁄3, for a circular hoop or a thin cylindrical shell 1⁄2.

The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions. It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9). We have

M (ẋẍ \+ ẏÿ) + lθ̇θ̈ \+ Xẋ \+ Yẏ \+ Nθ̇,

(12)

whence, integrating with respect to t,

1⁄2M (ẋ2 \+ ẏ2) + 1⁄2Iθ̇2 = ∫ (X dx + Y dy + N dθ) + const.

(13)

The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G (§ 15); and the right-hand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.  
  
---  
Fig. 75.  
  
The formula (13) may be easily verified in the case of the compound pendulum, or of the solid rolling down an incline. As another example, suppose we have a circular cylinder whose mass-centre is at an excentric point, rolling on a horizontal plane. This includes the case of a compound pendulum in which the knife-edge is replaced by a cylindrical pin. If α be the radius of the cylinder, h the distance of G from its axis (O), κ the radius of gyration about a longitudinal axis through G, and θ the inclination of OG to the vertical, the kinetic energy is 1⁄2Mκ2θ̇2 \+ 1⁄2M·CG2·thetȧ2, by § 3, since the body is turning about the line of contact (C) as instantaneous axis, and the potential energy is −Mgh cos θ. The equation of energy is therefore

1⁄2M (κ2 \+ α2 \+ h2 − 2 ah cos θ) θ̇2 − Mgh cos θ − const.

(14)

Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete determination of the motion. If q be any variable co-ordinate defining the position or (in the case of a system of bodies) the configuration, the velocity of each particle at any instant will be proportional to q̇, and the total kinetic energy may be expressed in the form 1⁄2Aq̇2, where A is in general a function of q [cf. equation (14)]. This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the co-ordinate q.  
  
---  
Fig. 76.  
  
Thus in the case of a railway truck travelling with velocity u the kinetic energy is 1⁄2 (M + mκ2/α2)u2, where M is the total mass, α the radius and κ the radius of gyration of each wheel, and m is the sum of the masses of the wheels; the reduced inertia is therefore M + mκ2/α2. Again, take the system composed of the flywheel, connecting rod, and piston of a steam-engine. We have here a limiting case of three-bar motion (§ 3), and the instantaneous centre J of the connecting-rod PQ will have the position shown in the figure. The velocities of P and Q will be in the ratio of JP to JQ, or OR to OQ; the velocity of the piston is therefore yθ̇, where y = OR. Hence if, for simplicity, we neglect the inertia of the connecting-rod, the kinetic energy will be 1⁄2 (I + My2)thetȧ2, where I is the moment of inertia of the flywheel, and M is the mass of the piston. The effect of the mass of the piston is therefore to increase the apparent moment of inertia of the flywheel by the variable amount My2. If, on the other hand, we take OP (= x) as our variable, the kinetic energy is 1⁄2 (M + I/y2)ẋ2. We may also say, therefore, that the effect of the flywheel is to increase the apparent mass of the piston by the amount I/y2; this becomes infinite at the “dead-points” where the crank is in line with the connecting-rod.

If the system be “conservative,” we have

1⁄2Aq2 \+ V = const.,

(15)

where V is the potential energy. If we differentiate this with respect to t, and divide out by q̇, we obtain

Aq̈ \+ 1⁄2 | dA | q̇2 + | dV | = 0  
---|---|---|---|---  
dq | dq  
  
(16)

as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this must be satisfied by q̇ = O; this requires that dV/dq = 0, i.e. the potential energy must be “stationary.” To examine the effect of a small disturbance from equilibrium we put V = ƒ(q), and write q = q0 \+ η, where q0 is a root of ƒ′ (q0) = 0 and η is small. Neglecting terms of the second order in η we have dV/dq = ƒ′(q) = ƒ″(q0)·η, and the equation (16) reduces to

Aη̈ \+ ƒ″ (q0)η = 0,

(17)

where A may be supposed to be constant and to have the value corresponding to q = q0. Hence if ƒ″ (q0) > 0, i.e. if V is a minimum in the configuration of equilibrium, the variation of η is simple-harmonic, and the period is 2π √{A/ƒ″(q0) }. This depends only on the constitution of the system, whereas the amplitude and epoch will vary with the initial circumstances. If ƒ″ (q0) < 0, the solution of (17) will involve real exponentials, and η will in general increase until the neglect of the terms of the second order is no longer justified. The configuration q = q0, is then unstable.

As an example of the method, we may take the problem to which equation (14) relates. If we differentiate, and divide by θ, and retain only the terms of the first order in θ, we obtain

{x2 \+ (h − α)2} θ̈ \+ ghθ = 0,

(18)

as the equation of small oscillations about the position θ = 0. The length of the equivalent simple pendulum is {κ2 \+ (h − α)2}/h.

The equations which express the change of motion (in two dimensions) due to an instantaneous impulse are of the forms

M (u′ − u) = ξ,  M (ν′ − ν) = η,  I (ω′ − ω) = ν.

(19)  
  
---  
Fig. 77.  
  
Here u′, ν′ are the values of the component velocities of G just before, and u, ν their values just after, the impulse, whilst ω′, ω denote the corresponding angular velocities. Further, ξ, η are the time-integrals of the forces parallel to the co-ordinate axes, and ν is the time-integral of their moment about G. Suppose, for example, that a rigid lamina at rest, but free to move, is struck by an instantaneous impulse F in a given line. Evidently G will begin to move parallel to the line of F; let its initial velocity be u′, and let ω′ be the initial angular velocity. Then Mu′ = F, Iω′ = F·GP, where GP is the perpendicular from G to the line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be

u′ − ω′·GC = (F/M) · (I − GC·CP/κ2),

where κ2 is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided GC·GP = κ2. If this condition be satisfied there would be no impulsive reaction at C even if this point were fixed. The point P is therefore called the centre of percussion for the axis at C. It will be noted that the relation between C and P is the same as that which connects the centres of suspension and oscillation in the compound pendulum.

§ 18. Equations of Motion in Three Dimensions.—It was proved in § 7 that a body moving about a fixed point O can be brought from its position at time t to its position at time t + δt by an infinitesimal rotation ε about some axis through O; and the limiting position of this axis, when δt is infinitely small, was called the “instantaneous axis.” The limiting value of the ratio ε/δt is called the angular velocity of the body; we denote it by ω. If ξ, η, ζ are the components of ε about rectangular co-ordinate axes through O, the limiting values of ξ/δt, η/δt, ζ/δt are called the component angular velocities; we denote them by p, q, r. If l, m, n be the direction-cosines of the instantaneous axis we have

p = lω,  q = mω,  r = nω,

(1)

p2 \+ q2 \+ r2 = ω2.

(2)

If we draw a vector OJ to represent the angular velocity, then J traces out a certain curve in the body, called the polhode, and a certain curve in space, called the herpolhode. The cones generated by the instantaneous axis in the body and in space are called the polhode and herpolhode cones, respectively; in the actual motion the former cone rolls on the latter (§ 7).  
  
---  
Fig. 78.  
  
The special case where both cones are right circular and ω is constant is important in astronomy and also in mechanism (theory of bevel wheels). The “precession of the equinoxes” is due to the fact that the earth performs a motion of this kind about its centre, and the whole class of such motions has therefore been termed precessional. In fig. 78, which shows the various cases, OZ is the axis of the fixed and OC that of the rolling cone, and J is the point of contact of the polhode and herpolhode, which are of course both circles. If αbe the semi-angle of the rolling cone, β the constant inclination of OC to OZ, and ψ̇ the angular velocity with which the plane ZOC revolves about OZ, then, considering the velocity of a point in OC at unit distance from O, we have

ω sin α = ±ψ̇ sin β,

(3)

where the lower sign belongs to the third case. The earth’s precessional motion is of this latter type, the angles being α = .0087″, β = 23° 28′.

If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T is given by

2T = Σ {m (ω·PN)2 } = ω2·Σ (m·PN2) = Iω2,

(4)

where I is the moment of inertia about the instantaneous axis. With the same notation for moments and products of inertia as in § 11 (38), we have

I = Al2 \+ Bm2 \+ Cn2 − 2Fmn − 2Gnl − 2Hlm,

and therefore by (1),

2T = Ap2 \+ Bq2 \+ Cr2 − 2Fqr − 2Grp − 2Hpq.

(5)

Again, if x, y, z be the co-ordinates of P, the component velocities of m are

qz − ry,  rx − pz,  py − qx,

(6)

by § 7 (5); hence, if λ, μ, ν be now used to denote the component angular momenta about the co-ordinate axes, we have λ = Σ {m (py − qx)y − m(rx − pz) z }, with two similar formulae, or

λ =  Ap −Hq − Gr = | ∂T | ,  
---|---|---  
∂p  
μ = −Hp + Bq − Fr = | ∂T | ,  
---|---|---  
∂q  
ν = −Gp − Fq + Cr = | ∂T | .  
---|---|---  
∂r  
  
(7)

If the co-ordinate axes be taken to coincide with the principal axes of inertia at O, at the instant under consideration, we have the simpler formulae

2T = Ap2 \+ Bq2 \+ Cr2,

(8)

λ = Ap, μ = Bq, ν = Cr.

(9)

It is to be carefully noticed that the axis of resultant angular momentum about O does not in general coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means of the momental ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § 11 (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or λ, μ, ν. The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis. Again, if Γ be the resultant angular momentum, so that

λ2 \+ μ2 \+ ν2 = Γ2,

(10)

the length of the perpendicular OH on the tangent plane at J is

OH = | Ap | · | p | ρ + | Bq | · | q | ρ + | Cr | · | r | ρ = | 2T | · | ρ | ,  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
Γ | ω | Γ | ω | Γ | ω | Γ | ω  
  
(11)

where ρ = OJ. This relation will be of use to us presently (§ 19).

The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point O of it which is taken as base, and the component angular velocities p, q, r. The component velocities of any point whose co-ordinates relative to O are x, y, z are then

u + qz − ry,  v + rx − pz,  w + py − qx

(12)

by § 7 (6). It is usually convenient to take as our base-point the mass-centre of the body. In this case the kinetic energy is given by

2T = M0 (u2 \+ v2 \+ w2) + Ap2 \+ Bq2 \+ Cr2 − 2Fqr − 2Grp − 2Hpg,

(13)

where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-centre; cf. § 15 (9).

The components ξ, η, ζ of linear momentum are

ξ = M0u = | ∂T | ,  η = M0v = | ∂T | ,  ζ = M0w = | ∂T |   
---|---|---|---|---|---|---  
∂u | ∂v | ∂w  
  
(14)

whilst those of the relative angular momentum are given by (7). The preceding formulae are sufficient for the treatment of instantaneous impulses. Thus if an impulse (ξ, η, ζ, λ, μ, ν) change the motion from (u, v, w, p, q, r) to (u′, v′, w′, p′, q′, r′) we have

M0 (u′ − u) = ξ, | M0 (v′ − v) = η, | M0(w′ − w) = ζ,  
---|---|---  
A (p′ − p) = λ, | B (q′ − q) = μ, | C (r′ − r) = ν,  
  
(15)

where, for simplicity, the co-ordinate axes are supposed to coincide with the principal axes at the mass-centre. Hence the change of kinetic energy is

T′ − T = ξ · 1⁄2 (u + u′) + η · 1⁄2 (v + v′) + ζ · 1⁄2 (w + w′),  
\+ λ · 1⁄2 (p + p′) + μ · 1⁄2 (q + q′) + ν · 1⁄2 (r + r′).

(16)

The factors of ξ, η, ζ, λ, μ, ν on the right-hand side are proportional to the constituents of a possible infinitesimal displacement of the solid, and the whole expression is proportional (on the same scale) to the work done by the given system of impulsive forces in such a displacement. As in § 9 this must be equal to the total work done in such a displacement by the several forces, whatever they are, which make up the impulse. We are thus led to the following statement: the change of kinetic energy due to any system of impulsive forces is equal to the sum of the products of the several forces into the semi-sum of the initial and final velocities of their respective points of application, resolved in the directions of the forces. Thus in the problem of fig. 77 the kinetic energy generated is 1⁄2M (κ2 \+ Cq2)ω′2, if C be the instantaneous centre; this is seen to be equal to 1⁄2F·ω′·CP, where ω′·CP represents the initial velocity of P.

The equations of continuous motion of a solid are obtained by substituting the values of ξ, η, ζ, λ, μ, ν from (14) and (7) in the general equations

dξ | = X,  | dη | = Y,  | dζ | = Z,  
---|---|---|---|---|---  
dt | dt | dt  
dλ | = L,  | dμ | = M,  | dν | = N,  
---|---|---|---|---|---  
dt | dt | dt  
  
(17)  
  
---  
Fig. 79.  
  
where (X, Y, Z, L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction. The resulting equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in consequence of the changing orientation of the body with respect to the co-ordinate axes.

An exception occurs, however, in the case of a solid which is kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform sphere. The equations then take the forms

M0u̇ = X, | M0v̇ = Y, | M0ẇ = Z,  
---|---|---  
Cṗ = L, | Cq̇ = M, | Cṙ = N,  
  
(18)

where C is the constant moment of inertia about any axis through the mass-centre. Take, for example, the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to the plane, so that the equation of the latter is z = −a. We will suppose that the extraneous forces consist of a known force (X, Y, Z) at the centre, and of the reactions (F1, F2, R) at the point of contact. Hence

M0u̇ = X + F1,  M0v̇ = Y + F2,  0 = Z + R,  
Cṗ = F2a,  Cq̇ = −F1a,  Cṙ = 0.

(19)

The last equation shows that the angular velocity about the normal to the plane is constant. Again, since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the geometrical relations

u + qa = 0,  v + pa = 0,  w = 0,

(20)

by (12). Eliminating p, q, we get

(M0 \+ Ca−2) u̇ = X,  (M0 \+ Ca−2) v̇ = Y.

(21)

The acceleration of the centre is therefore the same as if the plane were smooth and the mass of the sphere were increased by C/α2. Thus the centre of a sphere rolling under gravity on a plane of inclination a describes a parabola with an acceleration

g sin α/(1 + C/Ma2)

parallel to the lines of greatest slope.

Take next the case of a sphere rolling on a fixed spherical surface. Let a be the radius of the rolling sphere, c that of the spherical surface which is the locus of its centre, and let x, y, z be the co-ordinates of this centre relative to axes through O, the centre of the fixed sphere. If the only extraneous forces are the reactions (P, Q, R) at the point of contact, we have

M0ẍ = P,  M0ÿ = Q,  M0z̈ = R,

Cṗ = − | a | (yR − zQ),  Cq̇ = − | a | (zP − xR),  Cṙ = − | a | (xQ − yP),  
---|---|---|---|---|---|---  
c | c | c  
  
(22)

the standard case being that where the rolling sphere is outside the fixed surface. The opposite case is obtained by reversing the sign of a. We have also the geometrical relations

ẋ = (a/c) (qz − ry),  ẏ = (a/c) (rx − pz),  ż = (a/c) (py − gx),

(23)

If we eliminate P, Q, R from (22), the resulting equations are integrable with respect to t; thus

p = − | M0a | (yż − zẏ) + α,  q = − | M0a | (zẋ − xż) + β,  r = − | M0a | (xẏ − yẋ) + γ,  
---|---|---|---|---|---|---  
Cc | Cc | Cc  
  
(24)

where α, β, γ are arbitrary constants. Substituting in (23) we find

( 1 + | M0a2 | ) ẋ = | a | (βz − γy),  ( 1 + | M0a2 | ) ẏ = | a | (γx − αz),  ( 1 + | M0a2 | ) ż = | a | (αy − βx).  
---|---|---|---|---|---|---|---|---|---|---|---|---  
C | c | C | c | C | c  
  
(25)

Hence αẋ \+ βẏ \+ γż = 0, or

αx + βy + γz = const.;

(26)

which shows that the centre of the rolling sphere describes a circle. If the axis of z be taken normal to the plane of this circle we have α = 0, β = 0, and

( 1 + | M0a2 | ) ẋ = −γ | a | y,  ( 1 + | M0a2 | ) ẏ = γ | a | x.  
---|---|---|---|---|---|---|---|---  
C | c | C | c  
  
(27)

The solution of these equations is of the type

x = b cos (στ \+ ε),  y = b sin (σt + ε),

(28)

where b, ε are arbitrary, and

σ = | γa/c | .  
---|---|---  
1 + M0a2/C  
  
(29)

The circle is described with the constant angular velocity σ.

When the gravity of the rolling sphere is to be taken into account the preceding method is not in general convenient, unless the whole motion of G is small. As an example of this latter type, suppose that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical diameter with the angular velocity n; it will appear that under a certain condition the motion of G consequent on a slight disturbance will be oscillatory. If Oz be drawn vertically upwards, then in the beginning of the disturbed motion the quantities x, y, p, q, P, Q will all be small. Hence, omitting terms of the second order, we find

M0ẍ = P,  M0ẏ = Q,  R = M0g,  
Cṗ = −(M0ga/c) y + aQ,  Cq̇ = (M0ga/c) x − aP,  Cṙ = 0.

(30)

The last equation shows that the component r of the angular velocity retains (to the first order) the constant value n. The geometrical relations reduce to

ẋ = aq − (na/c) y,  ẏ = −ap + (na/c) x.

(31)

Eliminating p, g, P, Q, we obtain the equations

(C + M0a2) ẍ \+ (Cna/c) y − (M0ga2/c) x = 0,  
(C + M0a2) ÿ − (Cna/c) x − (M0ga2/c) y = 0,

(32)

which are both contained in

{ (C + M0a2) | d2 | − i | Cna |   | d | − | M0ga2 | } (x + iy) = 0.  
---|---|---|---|---|---|---|---|---  
dt2 | c | dt | c  
  
(33)

This has two solutions of the type x + iy = αei(σt + ε), where α, ε are arbitrary, and σ is a root of the quadratic

(C + M0a2) σ2 − (Cna/c) σ \+ M0ga2/c = 0.

(34)

If

n2 > (4Mgc/C) (1 + M0a2/C),

(35)

both roots are real, and have the same sign as n. The motion of G then consists of two superposed circular vibrations of the type

x = α cos (σt + ε),  y = α sin (σt + ε),

(36)

in each of which the direction of revolution is the same as that of the initial spin of the sphere. It follows therefore that the original position is stable provided the spin n exceed the limit defined by (35). The case of a sphere spinning about a vertical axis at the lowest point of a spherical bowl is obtained by reversing the signs of α and c. It appears that this position is always stable.

It is to be remarked, however, that in the first form of the problem the stability above investigated is practically of a limited or temporary kind. The slightest frictional forces—such as the resistance of the air—even if they act in lines through the centre of the rolling sphere, and so do not directly affect its angular momentum, will cause the centre gradually to descend in an ever-widening spiral path.

§ 19. Free Motion of a Solid.—Before proceeding to further problems of motion under extraneous forces it is convenient to investigate the free motion of a solid relative to its mass-centre O, in the most general case. This is the same as the motion about a fixed point under the action of extraneous forces which have zero moment about that point. The question was first discussed by Euler (1750); the geometrical representation to be given is due to Poinsot (1851).

The kinetic energy T of the motion relative to O will be constant. Now T = 1⁄2Iω2, where ω is the angular velocity and I is the moment of inertia about the instantaneous axis. If ρ be the radius-vector OJ of the momental ellipsoid

Ax2 \+ By2 \+ Cz2 = Mε4

(1)

drawn in the direction of the instantaneous axis, we have I = Mε4/ρ2 (§ 11); hence ω varies as ρ. The locus of J may therefore be taken as the “polhode” (§ 18). Again, the vector which represents the angular momentum with respect to O will be constant in every respect. We have seen (§ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that

OH = | 2T | · | ρ | ,  
---|---|---|---|---  
Γ | ω  
  
(2)

where Γ is the resultant angular momentum about O. Since ω varies as ρ, it follows that OH is constant, and the tangent plane at J is therefore fixed in space. The motion of the body relative to O is therefore completely represented if we imagine the momental ellipsoid at O to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact. The fixed plane is parallel to the invariable plane at O, and the line OH is called the invariable line. The trace of the point of contact J on the fixed plane is the “herpolhode.”

If p, q, r be the component angular velocities about the principal axes at O, we have

(A2p2 \+ B2q2 \+ C2r2) / Γ2 = (Ap2 \+ Bq2 \+ Cr2) / 2T,

(3)

each side being in fact equal to unity. At a point on the polhode cone x : y : z = p : q : r, and the equation of this cone is therefore

A2 ( 1 − | Γ2 | ) x2 \+ B2 ( 1 − | Γ2 | ) y2 \+ C2 ( 1 − | Γ2 | ) z2 = 0.  
---|---|---|---|---|---|---  
2AT | 2BT | 2CT  
  
(4)

Since 2AT − Γ2 = B (A − B)q2 \+ C(A − C)r2, it appears that if A > B > C the coefficient of x2 in (4) is positive, that of z2 is negative, whilst that of y2 is positive or negative according as 2BT ≷ Γ2. Hence the polhode cone surrounds the axis of greatest or least moment according as 2BT ≷ Γ2. In the critical case of 2BT = Γ2 it breaks up into two planes through the axis of mean moment (Oy). The herpolhode curve in the fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in general a re-entrant curve. It has been shown by De Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B + C > A, the curve has no points of inflexion. The invariable line OH describes another cone in the body, called the invariable cone. At any point of this we have x : y : z = Ap · Bq : Cr, and the equation is therefore

( 1 − | Γ2 | ) x2 \+ ( 1 − | Γ2 | ) y2 \+ ( 1 − | Γ2 | ) z2 = 0.  
---|---|---|---|---|---|---  
2AT | 2BT | 2CT  
  
(5)  
  
---  
Fig. 80.  
  
The signs of the coefficients follow the same rule as in the case of (4). The possible forms of the invariable cone are indicated in fig. 80 by means of the intersections with a concentric spherical surface. In the critical case of 2BT = Γ2 the cone degenerates into two planes. It appears that if the body be sightly disturbed from a state of rotation about the principal axis of greatest or least moment, the invariable cone will closely surround this axis, which will therefore never deviate far from the invariable line. If, on the other hand, the body be slightly disturbed from a state of rotation about the mean axis a wide deviation will take place. Hence a rotation about the axis of greatest or least moment is reckoned as stable, a rotation about the mean axis as unstable. The question is greatly simplified when two of the principal moments are equal, say A = B. The polhode and herpolhode cones are then right circular, and the motion is “precessional” according to the definition of § 18. If α be the inclination of the instantaneous axis to the axis of symmetry, β the inclination of the latter axis to the invariable line, we have

Γ cos β = C ω cos α,  Γ sin β = A ω sin α,

(6)

whence

tan β | A | tan α.  
---|---|---  
C  
  
(7)  
  
---  
Fig. 81.  
  
Hence β ≷ α, and the circumstances are therefore those of the first or second case in fig. 78, according as A ≷ C. If ψ be the rate at which the plane HOJ revolves about OH, we have

ψ = | sin α | ω = | C cos α | ω,  
---|---|---|---|---  
sin β | A cos β  
  
(8)

by § 18 (3). Also if χ̇ be the rate at which J describes the polhode, we have ψ̇ sin (β − α) = χ̇ sin β, whence

χ̇ = | sin (α − β) | ω.  
---|---|---  
sin α  
  
(9)

If the instantaneous axis only deviate slightly from the axis of symmetry the angles α, β are small, and χ̇ = (A − C) A·ω; the instantaneous axis therefore completes its revolution in the body in the period

2π | = | A − C | ω.  
---|---|---|---  
χ̇ | A  
  
(10)

In the case of the earth it is inferred from the independent phenomenon of luni-solar precession that (C − A)/A = .00313. Hence if the earth’s axis of rotation deviates slightly from the axis of figure, it should describe a cone about the latter in 320 sidereal days. This would cause a periodic variation in the latitude of any place on the earth’s surface, as determined by astronomical methods. There appears to be evidence of a slight periodic variation of latitude, but the period would seem to be about fourteen months. The discrepancy is attributed to a defect of rigidity in the earth. The phenomenon is known as the Eulerian nutation, since it is supposed to come under the free rotations first discussed by Euler.

§ 20. Motion of a Solid of Revolution.—In the case of a solid of revolution, or (more generally) whenever there is kinetic symmetry about an axis through the mass-centre, or through a fixed point O, a number of interesting problems can be treated almost directly from first principles. It frequently happens that the extraneous forces have zero moment about the axis of symmetry, as e.g. in the case of the flywheel of a gyroscope if we neglect the friction at the bearings. The angular velocity (r) about this axis is then constant. For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered. And a continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.  
  
---  
Fig. 82.  
  
Suppose, for example, that a flywheel is rotating with angular velocity n about its axis, which is (say) horizontal, and that this axis is made to rotate with the angular velocity ψ̇ in the horizontal plane. The components of angular momentum about the axis of the flywheel and about the vertical will be Cn and A ψ̇ respectively, where A is the moment of inertia about any axis through the mass-centre (or through the fixed point O) perpendicular to that of symmetry. If OK> be the vector representing the former component at time t, the vector which represents it at time t + δt will be OK′>, equal to OK> in magnitude and making with it an angle δψ. Hence KK′> (= Cn δψ) will represent the change in this component due to the extraneous forces. Hence, so far as this component is concerned, the extraneous forces must supply a couple of moment Cnψ̇ in a vertical plane through the axis of the flywheel. If this couple be absent, the axis will be tilted out of the horizontal plane in such a sense that the direction of the spin n approximates to that of the azimuthal rotation ψ̇. The remaining constituent of the extraneous forces is a couple Aψ̈ about the vertical; this vanishes if ψ̇ is constant. If the axis of the flywheel make an angle θ with the vertical, it is seen in like manner that the required couple in the vertical plane through the axis is Cn sin θ ψ̇. This matter can be strikingly illustrated with an ordinary gyroscope, e.g. by making the larger movable ring in fig. 37 rotate about its vertical diameter.  
  
---  
Fig. 83.  
  
If the direction of the axis of kinetic symmetry be specified by means of the angular co-ordinates θ, ψ of § 7, then considering the component velocities of the point C in fig. 83, which are θ̇ and sin θψ̇ along and perpendicular to the meridian ZC, we see that the component angular velocities about the lines OA′, OB′ are −sin θ ψ̇ and θ̇ respectively. Hence if the principal moments of inertia at O be A, A, C, and if n be the constant angular velocity about the axis OC, the kinetic energy is given by

2T = A (θ̇2 \+ sin2 θψ̇2) + Cn2.

(1)

Again, the components of angular momentum about OC, OA′ are Cn, −A sin θ ψ̇, and therefore the angular momentum (μ, say) about OZ is

μ = A sin2 θψ̇ \+ Cn cos θ.

(2)

We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is supposed to be free to turn about a fixed point O on its axis of symmetry, its mass-centre G being in this axis at a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the direction OG. If θ is constant the points C, A′ will in time δt come to positions C″, A″ such that CC″ = sin θ δψ, A′A″ = cos θ δψ, and the angular momentum about OB′ will become Cn sin θ δψ − A sin θ ψ̇ · cos θ δψ. Equating this to Mgh sin θ δt, and dividing out by sin θ, we obtain

A cos θ ψ̇2 − Cnψ̇ \+ Mgh = 0,

(3)

as the condition in question. For given values of n and θ we have two possible values of ψ̇ provided n exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with √(4AMgh cos θ), one value of ψ̇ is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos θ approximately. The absence of g from the latter expression indicates that the circumstances of the rapid precession are very nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.  
  
---  
Fig. 84.  
  
Again, take the case of a circular disk rolling in steady motion on a horizontal plane. The centre O of the disk is supposed to describe a horizontal circle of radius c with the constant angular velocity ψ̇, whilst its plane preserves a constant inclination θ to the horizontal. The components of the reaction of the horizontal lane will be Mcψ̇2 at right angles to the tangent line at the point of contact and Mg vertically upwards, and the moment of these about the horizontal diameter of the disk, which corresponds to OB′ in fig. 83, is Mcψ̇2. α sin θ − Mgα cos θ, where α is the radius of the disk. Equating this to the rate of increase of the angular momentum about OB′, investigated as above, we find

( C + Ma2 \+ A | a | cos θ ) ψ̇2 = Mg | a2 | cot θ,  
---|---|---|---|---  
c | c  
  
(4)

where use has been made of the obvious relation nα = cψ̇. If c and θ be given this formula determines the value of ψ̇ for which the motion will be steady.

In the case of the top, the equation of energy and the condition of constant angular momentum (μ) about the vertical OZ are sufficient to determine the motion of the axis. Thus, we have

1⁄2A (θ̇2 \+ sin2 θψ̇2) + 1⁄2Cn2 \+ Mgh cos θ = const.,

(5)

A sin2 θψ̇ \+ ν cos θ = μ,

(6)

where ν is written for Cn. From these ψ̇ may be eliminated, and on differentiating the resulting equation with respect to t we obtain

Aθ̈ − | (μ − ν cos θ) (μ cos θ − ν) | − Mgh sin θ = 0.  
---|---|---  
A sin3 θ  
  
(7)

If we put θ̈ = 0 we get the condition of steady precessional motion in a form equivalent to (3). To find the small oscillation about a state of steady precession in which the axis makes a constant angle α with the vertical, we write θ = α \+ χ, and neglect terms of the second order in χ. The result is of the form

χ̈ \+ σ2χ = 0,

(8)

where

σ2 = { (μ − ν cos α)2 \+ 2 (μ − ν cos α) (μ cos α − ν) cos α +  
(μ cos α − ν)2 } / A2 sin4 α.

(9)

When ν is large we have, for the “slow” precession σ = ν/A, and for the “rapid” precession σ = A/ν cos α = ψ̇, approximately. Further, on examining the small variation in ψ̇, it appears that in a slightly disturbed slow precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady precession, so that the resultant path has a trochoidal character. This is a type of motion commonly observed in a top spun in the ordinary way, although the successive undulations of the trochoid may be too small to be easily observed. In a slightly disturbed rapid precession the superposed vibration is elliptic-harmonic, with a period equal to that of the precession itself. The ratio of the axes of the ellipse is sec α, the longer axis being in the plane of θ. The result is that the axis of the top describes a circular cone about a fixed line making a small angle with the vertical. This is, in fact, the “invariable line” of the free Eulerian rotation with which (as already remarked) we are here virtually concerned. For the more general discussion of the motion of a top see Gyroscope.

§ 21. Moving Axes of Reference.—For the more general treatment of the kinetics of a rigid body it is usually convenient to adopt a system of moving axes. In order that the moments and products of inertia with respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.

We will assume for the present that the origin O is fixed. The moving axes Ox, Oy, Oz form a rigid frame of reference whose motion at time t may be specified by the three component angular velocities p, q, r. The components of angular momentum about Ox, Oy, Oz will be denoted as usual by λ, μ, ν. Now consider a system of fixed axes Ox′, Oy′, Oz′ chosen so as to coincide at the instant t with the moving system Ox, Oy, Oz. At the instant t + δt, Ox, Oy, Oz will no longer coincide with Ox′, Oy′, Oz′; in particular they will make with Ox′ angles whose cosines are, to the first order, 1, −rδt, qδt, respectively. Hence the altered angular momentum about Ox′ will be λ \+ δλ \+ (μ \+ δμ) (−rδt) + (ν \+ δν) qδt. If L, M, N be the moments of the extraneous forces about Ox, Oy, Oz this must be equal to λ \+ Lδt. Hence, and by symmetry, we obtain

dλ | − rν \+ qν = L,  
---|---  
dt  
dμ | − pν \+ rλ = M,  
---|---  
dt  
dν | − qλ \+ pν = N.  
---|---  
dt  
  
(1)

These equations are applicable to any dynamical system whatever. If we now apply them to the case of a rigid body moving about a fixed point O, and make Ox, Oy, Oz coincide with the principal axes of inertia at O, we have λ, μ, ν = Ap, Bq, Cr, whence

A | dp | − (B − C) qr = L,  
---|---|---  
dt  
B | dq | − (C − A) rp = M,  
---|---|---  
C | dr | − (A − B) pq = N.  
---|---|---  
dt  
  
(2)

If we multiply these by p, q, r and add, we get

d | · 1⁄2 (Ap2 \+ Bq2 \+ Cr2) = Lp + Mq + Nr,  
---|---  
dt  
  
(3)

which is (virtually) the equation of energy.

As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity ω about a fixed axis (l, m, n). Since p, q, r are then constant, the requisite constraining couple is

L = (C − B) mnω2,  M = (A − C) nlω2,  N = (B − A) lmω2.

(4)

If we reverse the signs, we get the “centrifugal couple” exerted by the solid on its bearings. This couple vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).

If in (2) we put, L, M, N = O we get the case of free rotation; thus

A | dp | (B − C) qr,  
---|---|---  
dt  
B | dq | (C − A) rp,  
---|---|---  
dt  
C | dr | (A − B) pq.  
---|---|---  
dt  
  
(5)

These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap2 \+ Bq2 \+ Cr2 and A2p2 \+ B2q2 \+ C2r2 are both constant. The former is, in fact, equal to 2T, and the latter to Γ2, where T is the kinetic energy and Γ the resultant angular momentum.

To complete the solution of (2) a third integral is required; this involves in general the use of elliptic functions. The problem has been the subject of numerous memoirs; we will here notice only the form of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we write

u = ∫φ0 | dφ | ,  Δφ = √(1 − k2 sin2 φ),  
---|---|---  
Δφ  
  
we have, in the notation of elliptic functions, φ = am u. If we assume

p = p0 cos am (σt + ε),  q = q0sin am (σt + ε),  r = r0Δ am (σt + ε),

(7)

we find

ṗ = − | σp0 | qr,  q̇ = | σq0 | rp,  ṙ = | k2σr0 | pq.  
---|---|---|---|---|---|---  
q0r0 | r0p0 | p0q0  
  
(8)

Hence (5) will be satisfied, provided

−σp0 | = | B − C | ,  | σq0 | = | C − A | ,  | −k2σr0 | = | A − B | .  
---|---|---|---|---|---|---|---|---|---|---|---  
q0r0 | A | r0p0 | B | p0q0 | C  
  
(9)

These equations, together with the arbitrary initial values of p, q, r, determine the six constants which we have denoted by p0, q0, r0, k2, σ, ε. We will suppose that A > B > C. From the form of the polhode curves referred to in § 19 it appears that the angular velocity q about the axis of mean moment must vanish periodically. If we adopt one of these epochs as the origin of t, we have ε = 0, and p0, r0 will become identical with the initial values of p, r. The conditions (9) then lead to

q02 = | A (A − C) | p02,  σ2 = | (A − C) (B − C) | r02,  k2 = | A (A − B) | · | p02 | .  
---|---|---|---|---|---|---|---|---  
B (B − C) | AB | C (B − C) | r02  
  
(10)

For a real solution we must have k2 < 1, which is equivalent to 2BT > Γ2. If the initial conditions are such as to make 2BT < Γ2, we must interchange the forms of p and r in (7). In the present case the instantaneous axis returns to its initial position in the body whenever φ increases by 2π, i.e. whenever t increases by 4K/σ, when K is the “complete” elliptic integral of the first kind with respect to the modulus k.

The elliptic functions degenerate into simpler forms when k2 = 0 or k2 = 1. The former case arises when two of the principal moments are equal; this has been sufficiently dealt with in § 19. If k2 = 1, we must have 2BT = Γ2. We have seen that the alternative 2BT ≷ Γ2 determines whether the polhode cone surrounds the principal axis of least or greatest moment. The case of 2BT = Γ2, exactly, is therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with the axis of mean moment or approaches it asymptotically.

When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are

dξ | − rη \+ qζ = X,  | dη | − pζ \+ rχ = Y,  | dζ | − qχ \+ pη = Z,  
---|---|---|---|---|---  
dt | dt | dt  
  
(11)

dλ | − rμ \+ qν − wη \+ vζ = L,  | dμ | − pν \+ rλ\- uζ \+ wξ = M,  
---|---|---|---  
dt | dt  
dν | − qλ \+ pμ − vξ \+ uη = N.  
---|---  
dt  
  
(12)

To prove these, we may take fixed axes O′x′, O′y′, O′z′ coincident with the moving axes at time t, and compare the linear and angular momenta ξ \+ δξ, η \+ δη, ζ \+ δζ, λ \+ δλ, μ \+ δμ, ν \+ δν relative to the new position of the axes, Ox, Oy, Oz at time t + δt with the original momenta ξ, η, ζ, λ, μ, ν relative to O′x′, O′y′, O′z′ at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the moving origin coincide always with the mass-centre, we have ξ, η, ζ = M0u, M0v, M0w, where M0 is the total mass, and the equations simplify.

When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of motion about a fixed point O; the angular co-ordinates θ, φ, ψ of Euler may then be used for the purpose. Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their instantaneous positions are

p = θ̇ sin φ − sin θ cos φψ̇,  q = θ̇ cos φ \+ sin θ sin φψ̇,  
r = φ̇ \+ cos θψ̇,

(13)

by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding moments of inertia, the kinetic energy is given by

2T = A (θ̇ sin φ − sin θ cos φψ̇)2 \+ B (θ̇ cos φ \+ sin θ sin θψ)2  
\+ C (φ̇ \+ cos θψ̇)2.

(14)

If A = B this reduces to

2T = A (θ̇2 \+ sin2 θ ψ̇2) + C (φ̇ \+ cos θ ψ̇)2; 

(15)

cf. § 20 (1).

§ 22. Equations of Motion in Generalized Co-ordinates.—Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized co-ordinates of the system. These co-ordinates may be chosen in an endless variety of ways, but their number is determinate, and expresses the number of degrees of freedom of the system. We denote these co-ordinates by q1, q2, ... qn. It is implied in the above description of the system that the Cartesian co-ordinates x, y, z of any particle of the system are known functions of the q’s, varying in form (of course) from particle to particle. Hence the kinetic energy T is given by

2T = Σ {m (ẋ2 \+ ẏ2 \+ ż2) }  
---  
= a11q̇12 \+ a22q̇22 \+ ... + 2a12q̇1q̇2 \+ ...,  
  
(1)

where

arr = Σ [ m { ( | ∂x | )2 \+ ( | ∂y | )2 \+ ( | ∂z | )2 } ],  
---|---|---|---|---|---|---  
∂qr | ∂qr | ∂qr  
ars = Σ { m ( | ∂x |   | ∂x | + | ∂y |   | ∂y | + | ∂z |   | ∂z | ) } = asr.  
---|---|---|---|---|---|---|---|---|---|---|---|---  
∂qr | ∂qs | ∂qr | ∂qs | ∂qr | ∂qs  
  
(2)

Thus T is expressed as a homogeneous quadratic function of the quantities q̇1, q̇2, ... q̇n, which are called the generalized components of velocity. The coefficients arr, ars are called the coefficients of inertia; they are not in general constants, being functions of the q’s and so variable with the configuration. Again, If (X, Y, Z) be the force on m, the work done in an infinitesimal change of configuration is

Σ (Xδx + Yδy + Zδz) = Q1δq1 \+ Q2δq2 \+ ... + Qnδqn,

(3)

where

Qr = Σ ( X | ∂x | \+ Y | ∂y | \+ Z | ∂z | ).  
---|---|---|---|---|---|---  
∂qr | ∂qr | ∂qr  
  
(4)

The quantities Qr are called the generalized components of force.

The equations of motion of m being

mẍ = X,  mÿ = Y,  mz̈ = Z,

(5)

we have

Σ { m ( ẍ | ∂x | \+ ÿ | ∂y | \+ z̈ | ∂z | ) } = Qr.  
---|---|---|---|---|---|---  
∂qr | ∂qr | ∂qr  
  
(6)

Now

(7)

ẋ = | ∂x | q̇1 + | ∂x | q̇2 \+ ... + | ∂x | q̇n,  
---|---|---|---|---|---|---  
∂q1 | ∂q2 | ∂qn  
  
whence

(8)

∂ẋ | = | ∂x | .  
---|---|---|---  
∂q̇r | ∂qr  
  
Also

(9)

d | ( | ∂x | ) = | ∂2x | q̇1 + | ∂2x | q̇2 \+ ... + | ∂2x | q̇r = | ∂ẋ | .  
---|---|---|---|---|---|---|---|---|---|---|---  
dt | ∂qr | ∂q1∂qr | ∂q2∂qr | ∂qn∂qr | ∂qr  
  
Hence

(10)

ẍ | ∂x | = | d | ( ẋ | ∂x | ) − ẋ | d | ( | ∂x | ) = | d | ( ẋ | ∂ẋ | ) − ẋ | ∂ẋ | .  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
∂qr | dt | ∂qr | dt | ∂qr | dt | ∂q̇r | ∂qr  
  
By these and the similar transformations relating to y and z the equation (6) takes the form

d | ( | ∂T | ) − | ∂T | = Qr.  
---|---|---|---|---|---  
dt | ∂q̇r | ∂qr  
  
(11)

If we put r = 1, 2, ... n in succession, we get the n independent equations of motion of the system. These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated. The above proof was given by Sir W. R. Hamilton (1835). Lagrange’s own proof will be found under Dynamics, § Analytical. In a conservative system free from extraneous force we have

Σ (X δx + Y δy + Z δz) = −δV,

(12)

where V is the potential energy. Hence

Qr = − | ∂V | ,  
---|---|---  
∂qr  
  
(13)

and

d | ( | ∂T | ) − | ∂T | = − | ∂V | .  
---|---|---|---|---|---|---|---  
dt | ∂q̇r | ∂qr | ∂qr  
  
(14)

If we imagine any given state of motion (q̇1, q̇2 ... q̇n) through the configuration (q1, q2, ... qn) to be generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11) with respect to t over the infinitely short duration of the impulse

∂T | = Qr′,  
---|---  
∂q̇r  
  
(15)

where Qr′ is the time integral of Qr and so represents a generalized component of impulse. By an obvious analogy, the expressions ∂T/∂q̇r may be called the generalized components of momentum; they are usually denoted by pr thus

pr = ∂T / ∂q̇r = a1rq̇1 \+ a2rq̇2 \+ ... + anrq̇n.

(16)

Since T is a homogeneous quadratic function of the velocities q̇1, q̇2, ... q̇n, we have

2T = | ∂T | q̇1 + | ∂T | q̇2 \+ ... + | ∂T | q̇n = p1q̇2 \+ p2q̇2 \+ ... + pnq̇n.  
---|---|---|---|---|---|---  
∂q̇1 | ∂q̇2 | ∂q̇n  
  
(17)

Hence

2 | dT | = ṗ1q̇1 \+ ṗ2q̇2 \+ ... + ṗnq̇n \+ ṗ1q̈1 \+ ṗ2q̈2 \+ ... + ṗnq̈n  
---|---|---  
dt  
= ( | ∂T | \+ Q1 ) q̇1 \+ ( | ∂T | \+ Q2 ) q̇2 \+ ... + ( | ∂T | \+ Qn ) q̇n + | ∂T | q̈1 + | ∂T | q̈2 \+ ... + | ∂T | q̈n  
---|---|---|---|---|---|---|---|---|---|---|---|---  
∂q̇1 | ∂q̇2 | ∂q̇n | ∂q̇1 | ∂q̇2 | ∂q̇n  
= | dT | \+ Q1q̇1 \+ Q2q̇2 \+ ... + Qnq̇n,  
---|---|---  
dt  
  
(18)

or

dT | = Q1q̇1 \+ Q2q̇2 \+ ... + Qnq̇n.  
---|---  
dt  
  
(19)

This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being done by the forces. In the case of a conservative system free from extraneous force it becomes the equation of energy

d | (T + V) = 0, or T + V = const.,  
---|---  
dt  
  
(20)

in virtue of (13).

As a first application of Lagrange’s formula (11) we may form the equations of motion of a particle in spherical polar co-ordinates. Let r be the distance of a point P from a fixed origin O, θ the angle which OP makes with a fixed direction OZ, ψ the azimuth of the plane ZOP relative to some fixed plane through OZ. The displacements of P due to small variations of these co-ordinates are ∂r along OP, r δθ perpendicular to OP in the plane ZOP, and r sin θ δψ perpendicular to this plane. The component velocities in these directions are therefore ṙ, rθ̇, r sin θψ̇, and if m be the mass of a moving particle at P we have

2T = m (ṙ2 \+ r2θ;̇2 \+ r2 sin2 θψ;̇2).

(21)

Hence the formula (11) gives

m (r̈ − rθ̇2 − r sin2 θψ̇2) | = R,  
---|---  
d/dt (mr2θ̇) − mr2 · sin θ cos θψ̇2 | = Θ,  
d/dt (mr2 sin2 θψ̇) | = Ψ.  
  
(22)

The quantities R, Θ, Ψ are the coefficients in the expression R δr + Θ δθ \+ Ψ δψ for the work done in an infinitely small displacement; viz. R is the radial component of force, Θ is the moment about a line through O perpendicular to the plane ZOP, and Ψ is the moment about OZ. In the case of the spherical pendulum we have r = l, Θ = − mgl sin θ, Ψ = 0, if OZ be drawn vertically downwards, and therefore

θ̈ − sin θ cos θψ̇2 | = − (g/l) sin θ,  
---|---  
sin2 θψ̇ | = h,  
  
(23)

where h is a constant. The latter equation expresses that the angular momentum ml2 sin2 θψ̇ about the vertical OZ is constant. By elimination of ψ̇ we obtain

θ̈ − h2 cos2 θ / sin3 θ = − | g | sin θ.  
---|---|---  
l  
  
(24)

If the particle describes a horizontal circle of angular radius α with constant angular velocity Ω, we have ω̇ = 0, h = Ω2 sin α, and therefore

Ω2 = | g | cos α,  
---|---|---  
l  
  
(25)

as is otherwise evident from the elementary theory of uniform circular motion. To investigate the small oscillations about this state of steady motion we write θ = α \+ χ in (24) and neglect terms of the second order in χ. We find, after some reductions,

χ̈ \+ (1 + 3 cos2 α) Ω2χ = 0;

(26)

this shows that the variation of χ is simple-harmonic, with the period

2π / √(1 + 3 cos2 α)·Ω

As regards the most general motion of a spherical pendulum, it is obvious that a particle moving under gravity on a smooth sphere cannot pass through the highest or lowest point unless it describes a vertical circle. In all other cases there must be an upper and a lower limit to the altitude. Again, a vertical plane passing through O and a point where the motion is horizontal is evidently a plane of symmetry as regards the path. Hence the path will be confined between two horizontal circles which it touches alternately, and the direction of motion is never horizontal except at these circles. In the case of disturbed steady motion, just considered, these circles are nearly coincident. When both are near the lowest point the horizontal projection of the path is approximately an ellipse, as shown in § 13; a closer investigation shows that the ellipse is to be regarded as revolving about its centre with the angular velocity 2⁄3 abΩ/l2, where a, b are the semi-axes.

To apply the equations (11) to the case of the top we start with the expression (15) of § 21 for the kinetic energy, the simplified form (1) of § 20 being for the present purpose inadmissible, since it is essential that the generalized co-ordinates employed should be competent to specify the position of every particle. If λ, μ, ν be the components of momentum, we have

λ = ∂T / ∂θ̇ | = Aθ̇,  
---|---  
μ = ∂T / ∂ψ̇ | = A sin2 θψ̇ \+ C (φ̇ \+ cos θψ̇) cos θ,  
ν = ∂T / ∂φ̇ | = C (θ̇ \+ cos θψ̇).  
  
(27)

The meaning of these quantities is easily recognized; thus λ is the angular momentum about a horizontal axis normal to the plane of θ, μ is the angular momentum about the vertical OZ, and ν is the angular momentum about the axis of symmetry. If M be the total mass, the potential energy is V = Mgh cos θ, if OZ be drawn vertically upwards. Hence the equations (11) become

Aθ̇ − A sin θ cos θψ̇2 \+ C (φ̇ \+ cos θψ̇) ψ̇ sin θ = Mgh sin θ,  
---  
d/dt · { A sin2 θψ̇ \+ C(φ̇ \+ cos θψ̇) cos θ } = 0,  
d/dt · { C (φ̇ \+ cos θψ̇) } = 0,  
  
(28)

of which the last two express the constancy of the momenta μ, ν. Hence

Aθ̈ − A sin θ cos θψ̇2 \+ ν sin θψ̇ = Mgh sin θ,  
A sin2 θψ̇ \+ ν cosθ = μ.

(29)

If we eliminate ψ̇ we obtain the equation (7) of § 20. The theory of disturbed precessional motion there outlined does not give a convenient view of the oscillations of the axis about the vertical position. If θ be small the equations (29) may be written

θ̈ − θω̇2 = − | ν2 − 4AMgh | θ,  
---|---|---  
4A2  
  
θ2ω̇ = const.,

(30)

where

ω = ψ − | ν | t.  
---|---|---  
2A  
  
(31)

Since θ, ω are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative to an initial line which revolves with constant angular velocity ν/2A, we see by comparison with § 14 (15) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation ν/2A, provided ν2 > 4AMgh. This gives (in essentials) the theory of the “gyroscopic pendulum.”

§ 23. Stability of Equilibrium. Theory of Vibrations.—If, in a conservative system, the configuration (q1, q2, ... qn) be one of equilibrium, the equations (14) of § 22 must be satisfied by q̇1, q̇2 ... q̇n = 0, whence

∂V / ∂qr = 0.

(1)

A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates. If, further, V be a minimum, the equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet (1846). In the motion consequent on any slight disturbance the total energy T + V is constant, and since T is essentially positive it follows that V can never exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance. This implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance. No such simple proof is available to show without qualification that the above condition is necessary. If, however, we recognize the existence of dissipative forces called into play by any motion whatever of the system, the conclusion can be drawn as follows. However slight these forces may be, the total energy T + V must continually diminish so long as the velocities q̇1, q̇2, ... q̇n differ from zero. Hence if the system be started from rest in a configuration for which V is less than in the equilibrium configuration considered, this quantity must still further decrease (since T cannot be negative), and it is evident that either the system will finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879).

In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient so to choose the generalized cc-ordinates q1, q2, ... qn that they shall vanish in the configuration in question. The potential energy is then given with sufficient approximation by an expression of the form

2V = c11q12 \+ c22q22 \+ ... + 2c12q1q2 \+ ...,

(2)

a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V is stationary. The coefficients crr, crs are called coefficients of stability. We may further treat the coefficients of inertia arr, ars of § 22 (1) as constants. The Lagrangian equations of motion are then of the type

a1rq̈1 \+ a2rq̈2 \+ ... + anrq̈n \+ c1rq1 \+ c2rq2 \+ ... + cnrqn = Qr,

(3)

where Qr now stands for a component of extraneous force. In a free oscillation we have Q1, Q2, ... Qn = 0, and if we assume

qr = Ar eiσt,

(4)

we obtain n equations of the type

(c1r − σ2a1r) A1 \+ (c2r − σ2a2r) A2 \+ ... + (cnr − σ2anr) An = 0.

(5)

Eliminating the n − 1 ratios A1 : A2 : ... : An we obtain the determinantal equation

Δ (σ2) = 0,

(6)

where

Δ(σ2) = | c11 − σ2a11, | c21 − σ2a21, | ..., | Cn1 − σ2anl  
---|---|---|---|---  
  | c12 − σ2a12, | c22 − σ2a22, | ..., | Cn2 − σ2an2  
  | . | . | ... | .  
  | . | . | ... | .  
  | . | . | ... | .  
  | c1n − σ2a1n, | c2n − σ2a2n, | ..., | Cnn − σ2ann  
  
(7)

The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability. It may be shown algebraically that under these conditions the n roots of the above equation in σ2 are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities A1, A2, ... An, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors of any one row in the determinate Δ(σ2). By combining the solutions corresponding to a pair of equal and opposite values of σ we obtain a solution in real form:

qr = Car cos (σt + ε),

(8)  
  
---  
Fig. 85.  
  
where a1, a2 ... ar are a determinate series of quantities having to one another the above-mentioned ratios, whilst the constants C, ε are arbitrary. This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the q’s) executes a simple vibration of period 2π/σ. The amplitudes of oscillation of the various particles have definite ratios to one another, and the phases are in agreement, the absolute amplitude (depending on C) and the phase-constant (ε) being alone arbitrary. A vibration of this character is called a normal mode of vibration of the system; the number n of such modes is equal to that of the degrees of freedom possessed by the system. These statements require some modification when two or more of the roots of the equation (6) are equal. In the case of a multiple root the minors of Δ(σ2) all vanish, and the basis for the determination of the quantities ar disappears. Two or more normal modes then become to some extent indeterminate, and elliptic vibrations of the individual particles are possible. An example is furnished by the spherical pendulum (§ 13).

As an example of the method of determination of the normal modes we may take the “double pendulum.” A mass M hangs from a fixed point by a string of length a, and a second mass m hangs from M by a string of length b. For simplicity we will suppose that the motion is confined to one vertical plane. If θ, φ be the inclinations of the two strings to the vertical, we have, approximately,

2T = Ma2θ̇2 \+ m (aθ̇ \+ bψ̇)2  
---  
2V = Mgaθ2 \+ mg (aθ2 \+ bψ2).  
  
(9)

The equations (3) take the forms

aθ ̈ \+ μbφ̈ \+ gθ = 0,  
---  
aθ ̈ \+ bφ̈ \+ gφ = 0.  
  
(10)

where μ = m/(M + m). Hence

(σ2 − g/a) aθ \+ μσ2bφ = 0,  
---  
σ2aθ \+ (σ2 − g/b) bφ = 0.  
  
(11)

The frequency equation is therefore

(σ2 − g/a) (σ2 − g/b) − μσ4 = 0.

(12)

The roots of this quadratic in σ2 are easily seen to be real and positive. If M be large compared with m, μ is small, and the roots are g/a and g/b, approximately. In the normal mode corresponding to the former root, M swings almost like the bob of a simple pendulum of length a, being comparatively uninfluenced by the presence of m, whilst m executes a “forced” vibration (§ 12) of the corresponding period. In the second mode, M is nearly at rest [as appears from the second of equations (11)], whilst m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two values of σ2 can never be exactly equal, but they are approximately equal if a, b are nearly equal and μ is very small. A curious phenomenon is then to be observed; the motion of each particle, being made up (in general) of two superposed simple vibrations of nearly equal period, is seen to fluctuate greatly in extent, and if the amplitudes be equal we have periods of approximate rest, as in the case of “beats” in acoustics. The vibration then appears to be transferred alternately from m to M at regular intervals. If, on the other hand, M is small compared with m, μ is nearly equal to unity, and the roots of (12) are σ2 = g/(a + b) and σ2 = mg/M·(a + b)/ab, approximately. The former root makes θ = φ, nearly; in the corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the second mode aθ \+ bφ = 0, nearly, so that m is approximately at rest. The oscillation of M then resembles that of a particle at a distance a from one end of a string of length a + b fixed at the ends and subject to a tension mg.

The motion of the system consequent on arbitrary initial conditions may be obtained by superposition of the n normal modes with suitable amplitudes and phases. We have then

qr = αrθ \+ αr′θ′ \+ αr″θ″ \+ ...,

(13)

where

θ = C cos (σt + ε),  θ′ = C′ cos (σ′t + ε),  θ″ = C″ cos (σ″t + ε), ...

(14)

provided σ2, σ′2, σ″2, ... are the n roots of (6). The coefficients of θ, θ′, θ″, ... in (13) satisfy the conjugate or orthogonal relations

a11α1α1′ \+ a22α2α2′ \+ ... + a12 (α1α2′ \+ α2α1′) + ... = 0,

(15)

c11α1α1′ \+ c22α2α2′ \+ ... + c12 (α1α2′ \+ α2α1′) + ... = 0,

(16)

provided the symbols αr, αr′ correspond to two distinct roots σ2, σ′2 of (6). To prove these relations, we replace the symbols A1, A2, ... An in (5) by α1, α2, ... αn respectively, multiply the resulting equations by a′1, a′2, ... a′n, in order, and add. The result, owing to its symmetry, must still hold if we interchange accented and unaccented Greek letters, and by comparison we deduce (15) and (16), provided σ2 and σ′2 are unequal. The actual determination of C, C′, C″, ... and ε, ε′, ε″, ... in terms of the initial conditions is as follows. If we write

C cos ε = H,  −C sin ε = K,

(17)

we must have

αrH + αr′H′ \+ αr″H″ \+ ... | = [qr]0,  
---|---  
σαrH + σ′αr′H′ \+ σ″αr″H″ \+ ... | = [q̇r]0,  
  
(18)

where the zero suffix indicates initial values. These equations can be at once solved for H, H′, H″, ... and K, K′, K″, ... by means of the orthogonal relations (15).

By a suitable choice of the generalized co-ordinates it is possible to reduce T and V simultaneously to sums of squares. The transformation is in fact effected by the assumption (13), in virtue of the relations (15) (16), and we may write

2T = aθ̇2 \+ a′θ̇′2 \+ a″θ̇″2 \+ ...,  
2V = cθ2 \+ c′θ′2 \+ c″θ″2 \+ ....

(19)

The new co-ordinates θ, θ′, θ″ ... are called the normal co-ordinates of the system; in a normal mode of vibration one of these varies alone. The physical characteristics of a normal mode are that an impulse of a particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a particular normal type maintains a displacement of that type only. The normal modes are further distinguished by an important “stationary” property, as regards the frequency. If we imagine the system reduced by frictionless constraints to one degree of freedom, so that the co-ordinates θ, θ′, θ″, ... have prescribed ratios to one another, we have, from (19),

σ2 = |  cθ2 \+ c′θ′2 = c″θ″2 \+ ... | ,  
---|---|---  
aθ2 \+ a′θ′2 \+ a″θ″2 \+ ...  
  
(20)

This shows that the value of σ2 for the constrained mode is intermediate to the greatest and least of the values c/a, c′/a′, c″/a″, ... proper to the several normal modes. Also that if the constrained mode differs little from a normal mode of free vibration (e.g. if θ′, θ″, ... are small compared with θ), the change in the frequency is of the second order. This property can often be utilized to estimate the frequency of the gravest normal mode of a system, by means of an assumed approximate type, when the exact determination would be difficult. It also appears that an estimate thus obtained is necessarily too high.

From another point of view it is easily recognized that the equations (5) are exactly those to which we are led in the ordinary process of finding the stationary values of the function

V (q1, q2, ... qn) | ,  
---|---  
T (q1, q2, ... qn)  
  
where the denominator stands for the same homogeneous quadratic function of the q’s that T is for the q̇’s. It is easy to construct in this connexion a proof that the n values of σ2 are all real and positive.

The case of three degrees of freedom is instructive on account of the geometrical analogies. With a view to these we may write

2T = aẋ2 \+ bẏ2 \+ cż2 \+ 2fẏż \+ 2gżẋ \+ 2hẋẏ,  
---  
2V = Ax2 \+ By2 \+ Cz2 \+ 2Fyz + 2Gzx + 2Hxy.  
  
(21)

It is obvious that the ratio

V (x, y, z)  
---  
T (x, y, z)  
  
(22)

must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive. Denoting this value by σ12, we have

Ax1 \+ Hy1 \+ Gz1 = σ12 (ax1 \+ hy1 \+ ∂gz1),  
---  
Hx1 \+ By1 \+ Fz1 = σ12 (hx1 \+ by1 \+ fz1),  
Gx1 \+ Fy1 \+ Cz1 = σ12 (gx1 \+ fy1 \+ cz1),  
  
(23)

provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z. Again, the expression (22) will also have a least value when the ratios x : y : z are subject to the condition

x1 | ∂V  | \+ y1 | ∂V  | \+ z1 | ∂V  | = 0;  
---|---|---|---|---|---|---  
∂x | ∂y | ∂z  
  
(24)

and if this be denoted by σ22 we have a second system of equations similar to (23). The remaining value σ22 is the value of (22) when x : y : z arc chosen so as to satisfy (24) and

x2 | ∂V  | \+ y2 | ∂V  | \+ z2 | ∂V  | = 0;  
---|---|---|---|---|---|---  
∂x | ∂y | ∂z  
  
(25)

The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y, z) = const., V(x, y, z) = const. If in (21) we imagine that x, y, z denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of σ are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.

We proceed to the forced vibrations of the system. The typical case is where the extraneous forces are of the simple-harmonic type cos (σt + ε); the most general law of variation with time can be derived from this by superposition, in virtue of Fourier’s theorem. Analytically, it is convenient to put Qr, equal to eiσt multiplied by a complex coefficient; owing to the linearity of the equations the factor eiσt will run through them all, and need not always be exhibited. For a system of one degree of freedom we have

aq̈ \+ cq = Q,

(26)

and therefore on the present supposition as to the nature of Q

q = | Q | .  
---|---|---  
c − σ2a  
  
(27)

This solution has been discussed to some extent in § 12, in connexion with the forced oscillations of a pendulum. We may note further that when σ is small the displacement q has the “equilibrium value” Q/c, the same as would be produced by a steady force equal to the instantaneous value of the actual force, the inertia of the system being inoperative. On the other hand, when σ2 is great q tends to the value −Q/σ2a, the same as if the potential energy were ignored. When there are n degrees of freedom we have from 

(3)

(c1r − σ2a2r) q1 \+ (c22r − σ2a2r) q2 \+ ... + (cnr − σ2anr) qn = Qr,

(28)

and therefore

Δ(σ2) · qr = a1rQ1 \+ a2rQ2 \+ ... + anrQn,

(29)

where a1r, a2r, ... anr are the minors of the rth row of the determinant (7). Every particle of the system executes in general a simple vibration of the imposed period 2π/σ, and all the particles pass simultaneously through their equilibrium positions. The amplitude becomes very great when σ2 approximates to a root of (6), i.e. when the imposed period nearly coincides with one of the free periods. Since ars = asr, the coefficient of Qs in the expression for qr is identical with that of Qr in the expression for qs. Various important “reciprocal theorems” formulated by H. Helmholtz and Lord Rayleigh are founded on this relation. Free vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the complete solution of the dynamical equations.

In practice the vibrations of a system are more or less affected by dissipative forces. In order to obtain at all events a qualitative representation of these it is usual to introduce into the equations frictional terms proportional to the velocities. Thus in the case of one degree of freedom we have, in place of (26),

aq̈ \+ bq̇ \+ cq = Q,

(30)

where a, b, c are positive. The solution of this has been sufficiently discussed in § 12. In the case of multiple freedom, the equations of small motion when modified by the introduction of terms proportional to the velocities are of the type

d |   | ∂T | \+ B1rq̇1 \+ B2rq̇2 \+ ... + Bnrq̇n + | ∂V | = Qr.  
---|---|---|---|---|---  
dt | ∂q̇r | ∂qr  
  
(31)

If we put

brs = bsr = 1⁄2 (Brs \+ Bsr),  βrs = −βsr = 1⁄2 (Brs − Bsr),

(32)

this may be written

d |   | ∂T | + | ∂F | \+ β1rq̇1 \+ β2rq̇2 \+ ... + βnrq̇r + | ∂V | = Qr,  
---|---|---|---|---|---|---|---  
dt | ∂q̇r | ∂q̇r | ∂qr  
  
(33)

provided

2F = b11q̇12 \+ b22q̇22 \+ ... + 2b12q̇1q̇2 \+ ...

(34)

The terms due to F in (33) are such as would arise from frictional resistances proportional to the absolute velocities of the particles, or to mutual forces of resistance proportional to the relative velocities; they are therefore classed as frictional or dissipative forces. The terms affected with the coefficients βrs on the other hand are such as occur in “cyclic” systems with latent motion (Dynamics, § Analytical); they are called the gyrostatic terms. If we multiply (33) by q̇r and sum with respect to r from 1 to n, we obtain, in virtue of the relations βrs = −βsr, βrr = 0,

d | (T + V) = 2F + Q1q̇1 \+ Q2q̇2 \+ ... + Qnq̇n.  
---|---  
dt  
  
(35)

This shows that mechanical energy is lost at the rate 2F per unit time. The function F is therefore called by Lord Rayleigh the dissipation function.

If we omit the gyrostatic terms, and write qr = Creλt, we find, for a free vibration,

(a1rλ2 \+ b1rλ \+ c1r) C1 \+ (a2rλ2 \+ b2rλ \+ c2r) C2 \+ ...  
\+ (anrλ2 \+ bnrλ \+ cnr) Cn = 0.

(36)

This leads to a determinantal equation in λ whose 2n roots are either real and negative, or complex with negative real parts, on the present hypothesis that the functions T, V, F are all essentially positive. If we combine the solutions corresponding to a pair of conjugate complex roots, we obtain, in real form,

qr = Cαr e−t/τ cos (σt + ε − εr),

(37)

where σ, τ, αr, εr are determined by the constitution of the system, whilst C, ε are arbitrary, and independent of r. The n formulae of this type represent a normal mode of free vibration: the individual particles revolve as a rule in elliptic orbits which gradually contract according to the law indicated by the exponential factor. If the friction be relatively small, all the normal modes are of this character, and unless two or more values of σ are nearly equal the elliptic orbits are very elongated. The effect of friction on the period is moreover of the second order.

In a forced vibration of eiσt the variation of each co-ordinate is simple-harmonic, with the prescribed period, but there is a retardation of phase as compared with the force. If the friction be small the amplitude becomes relatively very great if the imposed period approximate to a free period. The validity of the “reciprocal theorems” of Helmholtz and Lord Rayleigh, already referred to, is not affected by frictional forces of the kind here considered.

The most important applications of the theory of vibrations are to the case of continuous systems such as strings, bars, membranes, plates, columns of air, where the number of degrees of freedom is infinite. The series of equations of the type (3) is then replaced by a single linear partial differential equation, or by a set of two or three such equations, according to the number of dependent variables. These variables represent the whole assemblage of generalized co-ordinates qr; they are continuous functions of the independent variables x, y, z whose range of variation corresponds to that of the index r, and of t. For example, in a one-dimensional system such as a string or a bar, we have one dependent variable, and two independent variables x and t. To determine the free oscillations we assume a time factor eiσt; the equations then become linear differential equations between the dependent variables of the problem and the independent variables x, or x, y, or x, y, z as the case may be. If the range of the independent variable or variables is unlimited, the value of σ is at our disposal, and the solution gives us the laws of wave-propagation (see Wave). If, on the other hand, the body is finite, certain terminal conditions have to be satisfied. These limit the admissible values of σ, which are in general determined by a transcendental equation corresponding to the determinantal equation (6).

Numerous examples of this procedure, and of the corresponding treatment of forced oscillations, present themselves in theoretical acoustics. It must suffice here to consider the small oscillations of a chain hanging vertically from a fixed extremity. If x be measured upwards from the lower end, the horizontal component of the tension P at any point will be Pδy/δx, approximately, if y denote the lateral displacement. Hence, forming the equation of motion of a mass-element, ρδx, we have

ρ δx · ÿ = δ (P · ∂y/∂x).

(38)

Neglecting the vertical acceleration we have P = gρx, whence

∂2y | = g | ∂ | ( x | ∂y | ).  
---|---|---|---|---|---  
∂t2 | ∂x | ∂x  
  
(39)

Assuming that y varies as eiσt we have

∂ | ( x | ∂y | ) \+ ky = 0.  
---|---|---|---  
∂x | ∂x  
  
(40)

provided k = σ2/g. The solution of (40) which is finite for x = 0 is readily obtained in the form of a series, thus

y = C ( 1 − | kx | + | k2x2 | − ... ) = CJ0(z),  
---|---|---|---|---  
12 | 1222  
  
(41)

in the notation of Bessel’s functions, if z2 = 4kx. Since y must vanish at the upper end (x = l), the admissible values of σ are determined by

σ2 = gz2/4l,  J0(z) = 0.

(42)

The function J0(z) has been tabulated; its lower roots are given by

z/π= .7655, 1.7571, 2.7546,...,

approximately, where the numbers tend to the form s − 1⁄4. The frequency of the gravest mode is to that of a uniform bar in the ratio .9815 That this ratio should be less than unity agrees with the theory of “constrained types” already given. In the higher normal modes there are nodes or points of rest (y = 0); thus in the second mode there is a node at a distance .190l from the lower end.

Authorities.—For indications as to the earlier history of the subject see W. W. R. Ball, Short Account of the History of Mathematics; M. Cantor, Geschichte der Mathematik (Leipzig, 1880 ... ); J. Cox, Mechanics (Cambridge, 1904); E. Mach, Die Mechanik in ihrer Entwickelung (4th ed., Leipzig, 1901; Eng. trans.). Of the classical treatises which have had a notable influence on the development of the subject, and which may still be consulted with advantage, we may note particularly, Sir I. Newton, Philosophiae naturalis Principia Mathematica (1st ed., London, 1687); J. L. Lagrange, Mécanique analytique (2nd ed., Paris, 1811-1815); P. S. Laplace, Mécanique céleste (Paris, 1799-1825); A. F. Möbius, Lehrbuch der Statik (Leipzig, 1837), and Mechanik des Himmels; L. Poinsot, Éléments de statique (Paris, 1804), and Théorie nouvelle de la rotation des corps (Paris, 1834).

Of the more recent general treatises we may mention Sir W. Thomson (Lord Kelvin) and P. G. Tait, Natural Philosophy (2nd ed., Cambridge, 1879-1883); E. J. Routh, Analytical Statics (2nd ed., Cambridge, 1896), Dynamics of a Particle (Cambridge, 1898), Rigid Dynamics (6th ed., Cambridge 1905); G. Minchin, Statics (4th ed., Oxford, 1888); A. E. H. Love, Theoretical Mechanics (2nd ed., Cambridge, 1909); A. G. Webster, Dynamics of Particles, &c. (1904); E. T. Whittaker, Analytical Dynamics (Cambridge, 1904); L. Arnal, Traitê de mécanique (1888-1898); P. Appell, Mécanique rationelle (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., 1896); G. Kirchhoff, Vorlesungen über Mechanik (Leipzig, 1896); H. Helmholtz, Vorlesungen über theoretische Physik, vol. i. (Leipzig, 1898); J. Somoff, Theoretische Mechanik (Leipzig, 1878-1879).

The literature of graphical statics and its technical applications is very extensive. We may mention K. Culmann, Graphische Statik (2nd ed., Zürich, 1895); A. Föppl, Technische Mechanik, vol. ii. (Leipzig, 1900); L. Henneberg, Statik des starren Systems (Darmstadt, 1886); M. Lévy, La statique graphique (2nd ed., Paris, 1886-1888); H. Müller-Breslau, Graphische Statik (3rd ed., Berlin, 1901). Sir R. S. Ball’s highly original investigations in kinematics and dynamics were published in collected form under the title Theory of Screws (Cambridge, 1900).

Detailed accounts of the developments of the various branches of the subject from the beginning of the 19th century to the present time, with full bibliographical references, are given in the fourth volume (edited by Professor F. Klein) of the Encyclopädie der mathematischen Wissenschaften (Leipzig). There is a French translation of this work. (See also Dynamics.)

(H. Lb.)

II.—Applied Mechanics1

§ 1. The practical application of mechanics may be divided into two classes, according as the assemblages of material objects to which they relate are intended to remain fixed or to move relatively to each other—the former class being comprehended under the term “Theory of Structures” and the latter under the term “Theory of Machines.”

PART I.—OUTLINE OF THE THEORY OF STRUCTURES

§ 2. Support of Structures.—Every structure, as a whole, is maintained in equilibrium by the joint action of its own weight, of the external load or pressure applied to it from without and tending to displace it, and of the resistance of the material which supports it. A structure is supported either by resting on the solid crust of the earth, as buildings do, or by floating in a fluid, as ships do in water and balloons in air. The principles of the support of a floating structure form an important part of Hydromechanics (q.v.). The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure that the only principle which seems to require special mention here is one which comprehends in one statement the power both of liquids and of loose earth to support structures. This was first demonstrated in a paper “On the Stability of Loose Earth,” read to the Royal Society on the 19th of June 1856 (Phil. Trans. 1856), as follows:—

Let E represent the weight of the portion of a horizontal stratum of earth which is displaced by the foundation of a structure, S the utmost weight of that structure consistently with the power of the earth to resist displacement, φ the angle of repose of the earth; then

S | = ( | 1 + sin φ | )2.  
---|---|---|---  
E | 1 − sin φ  
  
To apply this to liquids φ must be made zero, and then S/E = 1, as is well known. For a proof of this expression see Rankine’s Applied Mechanics, 17th ed., p. 219.

§ 3. Composition of a Structure, and Connexion of its Pieces.—A structure is composed of pieces,—such as the stones of a building in masonry, the beams of a timber framework, the bars, plates and bolts of an iron bridge. Those pieces are connected at their joints or surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist mortar or without mortar), by pressure and adhesion (as in masonry with cement or with hardened mortar, and timber with glue), or by the resistance of fastenings of different kinds, whether made by means of the form of the joint (as dovetails, notches, mortices and tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, hoops, straps and sockets.)

§ 4. Stability, Stiffness and Strength.—A structure may be damaged or destroyed in three ways:—first, by displacement of its pieces from their proper positions relatively to each other or to the earth; secondly by disfigurement of one or more of those pieces, owing to their being unable to preserve their proper shapes under the pressures to which they are subjected; thirdly, by breaking of one or more of those pieces. The power of resisting displacement constitutes stability, the power of each piece to resist disfigurement is its stiffness; and its power to resist breaking, its strength.

§ 5. Conditions of Stability.—The principles of the stability of a structure can be to a certain extent investigated independently of the stiffness and strength, by assuming, in the first instance, that each piece has strength sufficient to be safe against being broken, and stiffness sufficient to prevent its being disfigured to an extent inconsistent with the purposes of the structure, by the greatest forces which are to be applied to it. The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by the resistances or pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position and direction of the resistances required at each joint in order to produce equilibrium; and the conditions of stability are, first, that the position, and, secondly, that the direction, of the resistance required at each joint shall, under all the variations to which the load is subject, be such as the joint is capable of exerting—conditions which are fulfilled by suitably adjusting the figures and positions of the joints, and the ratios of the gross loads of the pieces. As for the magnitude of the resistance, it is limited by conditions, not of stability, but of strength and stiffness.

§ 6. Principle of Least Resistance.—Where more than one system of resistances are alike capable of balancing the same system of loads applied to a given structure, the smallest of those alternative systems, as was demonstrated by the Rev. Henry Moseley in his Mechanics of Engineering and Architecture, is that which will actually be exerted—because the resistances to displacement are the effect of a strained state of the pieces, which strained state is the effect of the load, and when the load is applied the strained state and the resistances produced by it increase until the resistances acquire just those magnitudes which are sufficient to balance the load, after which they increase no further.

This principle of least resistance renders determinate many problems in the statics of structures which were formerly considered indeterminate.

§ 7. Relations between Polygons of Loads and of Resistances.—In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two,—all the three distances being measured along one direction.  
  
---  
Fig. 86.  
  
Considering, in the first place, the case in which the load and the two resistances by which each piece is balanced meet in one point, which may be called the centre of load, there will be as many such points of intersection, or centres of load, as there are pieces in the structure; and the directions and positions of the resistances or mutual pressures exerted between the pieces will be represented by the sides of a polygon joining those points, as in fig. 86 where P1, P2, P3, P4 represent the centres of load in a structure of four pieces, and the sides of the polygon of resistances P1 P2 P3 P4 represent respectively the directions and positions of the resistances exerted at the joints. Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then will those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangle; and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.  
  
---  
Fig. 87.  
  
According to a well-known principle of statics, because the loads or external pressures P1L1, &c., balance each other, they must be proportional to the sides of a closed polygon drawn respectively parallel to their directions. In fig. 87 construct such a polygon of loads by drawing the lines L1, &c., parallel and proportional to, and joined end to end in the order of, the gross loads on the pieces of the structure. Then from the proportionality and parallelism of the load and the two resistances applied to each piece of the structure to the three sides of a triangle, there results the following theorem (originally due to Rankine):—

If from the angles of the polygon of loads there be drawn lines (R1, R2, &c.), each of which is parallel to the resistance (as P1P2, &c.) exerted at the joint between the pieces to which the two loads represented by the contiguous sides of the polygon of loads (such as L1, L2, &c.) are applied; then will all those lines meet in one point (O), and their lengths, measured from that point to the angles of the polygon, will represent the magnitudes of the resistances to which they are respectively parallel.

When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases to be closed, two of the sides becoming parallel to each other and to the load in question, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point O.2

§ 8. How the Earth’s Resistance is to be treated.... When the pressure exerted by a structure on the earth (to which the earth’s resistance is equal and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure. But in other cases the earth is to be treated as one of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it.

§ 9. Partial Polygons of Resistance.—In a structure in which there are pieces supported at more than two joints, let a polygon be constructed of lines connecting the centres of load of any continuous series of pieces. This may be called a partial polygon of resistances. In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of § 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.

§ 10. Line of Pressures—Centres and Line of Resistance.—The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. The centre of resistance at any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. The line of resistance is a line traversing all the centres of resistance of a series of joints,—its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be subdivided by the introduction of intermediate joints ad infinitum, and finding the continuous line, curved or straight, in which the intermediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley.  
  
---  
Fig. 88.  
  
§ 11.* The principles of the two preceding sections may be illustrated by the consideration of a particular case of a buttress of blocks forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd represent plane joints. Let the centre of pressure C at the first joint aa be known, and also the pressure P acting at C in direction and magnitude. Find R1 the resultant of this pressure, the weight of the block aabb acting through its centre of gravity, and any other external force which may be acting on the block, and produce its line of action to cut the joint bb in C1. C1 is then the centre of pressure for the joint bb, and R1 is the total force acting there. Repeating this process for each block in succession there will be found the centres of pressure C2, C3, &c., and also the resultant pressures R2, R3, &c., acting at these respective centres. The centres of pressure at the joints are also called centres of resistance, and the curve passing through these points is called a line of resistance. Let all the resultants acting at the several centres of resistance be produced until they cut one another in a series of points so as to form an unclosed polygon. This polygon is the partial polygon of resistance. A curve tangential to all the sides of the polygon is the line of pressures.

§ 12. Stability of Position, and Stability of Friction.—The resistances at the several joints having been determined by the principles set forth in §§ 6, 7, 8, 9 and 10, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative displacement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place either by turning or by sliding. Safety against displacement by turning is called stability of position; safety against displacement by sliding, stability of friction.

§ 13. Condition of Stability of Position.—If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load. In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials. That least distance is called by Moseley the modulus of stability. The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice:—

Retaining walls, as designed by British engineers | 1 : 8  
---|---  
Retaining walls, as designed by French engineers | 1 : 5  
Rectangular piers of bridges and other buildings, and arch-stones | 1 : 3  
Rectangular foundations, firm ground | 1 : 3  
Rectangular foundations, very soft ground | 1 : 2  
Rectangular foundations, intermediate kinds of ground | 1 : 3 to 1 : 2  
Thin, hollow towers (such as furnace chimneys exposed to high winds), square | 1 : 6  
Thin, hollow towers, circular | 1 : 4  
Frames of timber or metal, under their ordinary or average distribution of load | 1 : 3  
Frames of timber or metal, under the greatest irregularities of load | 1 : 3  
  
In the case of the towers, the depth of the joint is to be understood to mean the diameter of the tower.  
  
---  
Fig. 89.  
  
§ 14. Condition of Stability of Friction.—If the resistance to be exerted at a joint is always perpendicular to the surfaces which abut at and form that joint, there is no tendency of the pieces to be displaced by sliding. If the resistance be oblique, let JK (fig. 89) be the joint, C its centre of resistance, CR a line representing the resistance, CN a perpendicular to the joint at the centre of resistance. The angle NCR is the obliquity of the resistance. From R draw RP parallel and RQ perpendicular to the joint; then, by the principles of statics, the component of the resistance normal to the joint is—

CP = CR · cos PCR;

and the component tangential to the joint is—

CQ = CR · sin PCR = CP · tan PCR.

If the joint be provided either with projections and recesses, such as mortises and tenons, or with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question of the utmost amount of the tangential resistance CQ which it is capable of exerting depends on the strength of such projections, recesses, or fastenings; and belongs to the subject of strength, and not to that of stability. In other cases the safety of the joint against displacement by sliding depends on its power of exerting friction, and that power depends on the law, known by experiment, that the friction between two surfaces bears a constant ratio, depending on the nature of the surfaces, to the force by which they are pressed together. In order that the surfaces which abut at the joint JK may be pressed together, the resistance required by the conditions of equilibrium CR, must be a thrust and not a pull; and in that case the force by which the surfaces are pressed together is equal and opposite to the normal component CP of the resistance. The condition of stability of friction is that the tangential component CQ of the resistance required shall not exceed the friction due to the normal component; that is, that

CQ ≯ ƒ · CP,

where ƒ denotes the coefficient of friction for the surfaces in question. The angle whose tangent is the coefficient of friction is called the angle of repose, and is expressed symbolically by—

φ = tan −1 ƒ.

Now CQ = CP · tan PCR;

consequently the condition of stability of friction is fulfilled if the angle PCR is not greater than φ; that is to say, if the obliquity of the resistance required at the joint does not exceed the angle of repose; and this condition ought to be fulfilled under all possible variations of the load.

It is chiefly in masonry and earthwork that stability of friction is relied on.

§ 15. Stability of Friction in Earth.—The grains of a mass of loose earth are to be regarded as so many separate pieces abutting against each other at joints in all possible positions, and depending for their stability on friction. To determine whether a mass of earth is stable at a given point, conceive that point to be traversed by planes in all possible positions, and determine which position gives the greatest obliquity to the total pressure exerted between the portions of the mass which abut against each other at the plane. The condition of stability is that this obliquity shall not exceed the angle of repose of the earth. The consequences of this principle are developed in a paper, “On the Stability of Loose Earth,” already cited in § 2.

§ 16. Parallel Projections of Figures.—If any figure be referred to a system of co-ordinates, rectangular or oblique, and if a second figure be constructed by means of a second system of co-ordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the co-ordinates of the first figure that for each point in the first figure there shall be a corresponding point in the second figure, the lengths of whose co-ordinates shall bear respectively to the three corresponding co-ordinates of the corresponding point in the first figure three ratios which are the same for every pair of corresponding points in the two figures, these corresponding figures are called parallel projections of each other. The properties of parallel projections of most importance to the subject of the present article are the following:—

(1) A parallel projection of a straight line is a straight line.

(2) A parallel projection of a plane is a plane.

(3) A parallel projection of a straight line or a plane surface divided in a given ratio is a straight line or a plane surface divided in the same ratio.

(4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is a pair of equal and parallel straight lines, or plane surfaces; whence it follows

(5) That a parallel projection of a parallelogram is a parallelogram, and

(6) That a parallel projection of a parallelepiped is a parallelepiped.

(7) A parallel projection of a pair of solids having a given ratio is a pair of solids having the same ratio.

Though not essential for the purposes of the present article, the following consequence will serve to illustrate the principle of parallel projections:—

(8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve or a surface of the same order.

For example, all ellipsoids referred to co-ordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular co-ordinates.

§ 17. Parallel Projections of Systems of Forces.—If a balanced system of forces be represented by a system of lines, then will every parallel projection of that system of lines represent a balanced system of forces.

For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio.

The resultant of a parallel projection of any system of forces is the projection of their resultant; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid.

§ 18. Principle of the Transformation of Structures.—Here we have the following theorem: If a structure of a given figure have stability of position under a system of forces represented by a given system of lines, then will any structure whose figure is a parallel projection of that of the first structure have stability of position under a system of forces represented by the corresponding projection of the first system of lines.

For in the second structure the weights, external pressures, and resistances will balance each other as in the first structure; the weights of the pieces and all other parallel systems of forces will have the same ratios as in the first structure; and the several centres of resistance will divide the depths of the joints in the same proportions as in the first structure.

If the first structure have stability of friction, the second structure will have stability of friction also, so long as the effect of the projection is not to increase the obliquity of the resistance at any joint beyond the angle of repose.

The lines representing the forces in the second figure show their relative directions and magnitudes. To find their absolute directions and magnitudes, a vertical line is to be drawn in the first figure, of such a length as to represent the weight of a particular portion of the structure. Then will the projection of that line in the projected figure indicate the vertical direction, and represent the weight of the part of the second structure corresponding to the before-mentioned portion of the first structure.

The foregoing “principle of the transformation of structures” was first announced, though in a somewhat less comprehensive form, to the Royal Society on the 6th of March 1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of equilibrium and stability of structures of complex and unsymmetrical figures from those of structures of simple and symmetrical figures. By its aid, for example, the whole of the properties of elliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a certain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.

§ 19. Conditions of Stiffness and Strength.—After the arrangement of the pieces of a structure and the size and figure of their joints or surfaces of contact have been determined so as to fulfil the conditions of stability,—conditions which depend mainly on the position and direction of the resultant or total load on each piece, and the relative magnitude of the loads on the different pieces—the dimensions of each piece singly have to be adjusted so as to fulfil the conditions of stiffness and strength—conditions which depend not only on the absolute magnitude of the load on each piece, and of the resistances by which it is balanced, but also on the mode of distribution of the load over the piece, and of the resistances over the joints.

The effect of the pressures applied to a piece, consisting of the load and the supporting resistances, is to force the piece into a state of strain or disfigurement, which increases until the elasticity, or resistance to strain, of the material causes it to exert a stress, or effort to recover its figure, equal and opposite to the system of applied pressures. The condition of stiffness is that the strain or disfigurement shall not be greater than is consistent with the purposes of the structure; and the condition of strength is that the stress shall be within the limits of that which the material can bear with safety against breaking. The ratio in which the utmost stress before breaking exceeds the safe working stress is called the factor of safety, and is determined empirically. It varies from three to twelve for various materials and structures. (See Strength of Materials.)

PART II. THEORY OF MACHINES

§ 20. Parts of a Machine: Frame and Mechanism.—The parts of a machine may be distinguished into two principal divisions,—the frame, or fixed parts, and the mechanism, or moving parts. The frame is a structure which supports the pieces of the mechanism, and to a certain extent determines the nature of their motions.

The form and arrangement of the pieces of the frame depend upon the arrangement and the motions of the mechanism; the dimensions of the pieces of the frame required in order to give it stability and strength are determined from the pressures applied to it by means of the mechanism. It appears therefore that in general the mechanism is to be designed first and the frame afterwards, and that the designing of the frame is regulated by the principles of the stability of structures and of the strength and stiffness of materials,—care being taken to adapt the frame to the most severe load which can be thrown upon it at any period of the action of the mechanism.

Each independent piece of the mechanism also is a structure, and its dimensions are to be adapted, according to the principles of the strength and stiffness of materials, to the most severe load to which it can be subjected during the action of the machine.

§ 21. Definition and Division of the Theory of Machines.—From what has been said in the last section it appears that the department of the art of designing machines which has reference to the stability of the frame and to the stiffness and strength of the frame and mechanism is a branch of the art of construction. It is therefore to be separated from the theory of machines, properly speaking, which has reference to the action of machines considered as moving. In the action of a machine the following three things take place:—

Firstly, Some natural source of energy communicates motion and force to a piece or pieces of the mechanism, called the receiver of power or prime mover.

Secondly, The motion and force are transmitted from the prime mover through the train of mechanism to the working piece or pieces, and during that transmission the motion and force are modified in amount and direction, so as to be rendered suitable for the purpose to which they are to be applied.

Thirdly, The working piece or pieces by their motion, or by their motion and force combined, produce some useful effect.

Such are the phenomena of the action of a machine, arranged in the order of causation. But in studying or treating of the theory of machines, the order of simplicity is the best; and in this order the first branch of the subject is the modification of motion and force by the train of mechanism; the next is the effect or purpose of the machine; and the last, or most complex, is the action of the prime mover.

The modification of motion and the modification of force take place together, and are connected by certain laws; but in the study of the theory of machines, as well as in that of pure mechanics, much advantage has been gained in point of clearness and simplicity by first considering alone the principles of the modification of motion, which are founded upon what is now known as Kinematics, and afterwards considering the principles of the combined modification of motion and force, which are founded both on geometry and on the laws of dynamics. The separation of kinematics from dynamics is due mainly to G. Monge, Ampère and R. Willis.

The theory of machines in the present article will be considered under the following heads:—

I. Pure Mechanism, or Applied Kinematics; being the theory of machines considered simply as modifying motion.

II. Applied Dynamics; being the theory of machines considered as modifying both motion and force.

Chap. I. On Pure Mechanism

§ 22. Division of the Subject.—Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Kinematics, may be thus divided:—

Division 1.—Motion of a point.

Division 2.—Motion of the surface of a fluid.

Division 3.—Motion of a rigid solid.

Division 4.—Motions of a pair of connected pieces, or of an “elementary combination” in mechanism.

Division 5.—Motions of trains of pieces of mechanism.

Division 6.—Motions of sets of more than two connected pieces, or of “aggregate combinations.”

A point is the boundary of a line, which is the boundary of a surface, which is the boundary of a volume. Points, lines and surfaces have no independent existence, and consequently those divisions of this chapter which relate to their motions are only preliminary to the subsequent divisions, which relate to the motions of bodies.

Division 1. Motion of a Point.

§ 23. Comparative Motion.—The comparative motion of two points is the relation which exists between their motions, without having regard to their absolute amounts. It consists of two elements,—the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the directional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.

It is obvious that the motions of a pair of points may be varied in any manner, whether by direct or by lateral deviation, and yet that their comparative motion may remain constant, in consequence of the deviations taking place in the same proportions, in the same directions and at the same instants for both points.

Robert Willis (1800-1875) has the merit of having been the first to simplify considerably the theory of pure mechanism, by pointing out that that branch of mechanics relates wholly to comparative motions.

The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the velocity ratio, and the “versor” the directional relation.

Graphical methods of analysis founded on this way of representing velocity and acceleration were developed by R. H. Smith in a paper communicated to the Royal Society of Edinburgh in 1885, and illustrations of the method will be found below.

Division 2. Motion of the Surface of a Fluid Mass.

§ 24. General Principle.—A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (called pistons or plungers) of the solid envelope or vessel in which the fluid is contained; and, when such transmission is the sole action, or the only appreciable action of the fluid mass, its volume is either absolutely constant, by reason of its temperature and pressure being maintained constant, or not sensibly varied.

Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward. Then the variation of the cubic contents of the vessel in a unit of time by reason of the motion of one piston is va. The condition that the volume of the fluid mass shall remain unchanged requires that there shall be more than one piston, and that the velocities and areas of the pistons shall be connected by the equation—

Σ · va = 0.

(1)

§ 25. Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are a1 and a2, and their velocities v1 and v2, their comparative motion is expressed by the equation—

v2/v1 = −a1/a2;

(2)

that is to say, their velocities are opposite as to inwardness and outwardness and inversely proportional to their areas.

§ 26. Applications: Hydraulic Press: Pneumatic Power-Transmitter.—In the hydraulic press the vessel consists of two cylinders, viz. the pump-barrel and the press-barrel, each having its piston, and of a passage connecting them having a valve opening towards the press-barrel. The action of the enclosed water in transmitting motion takes place during the inward stroke of the pump-plunger, when the above-mentioned valve is open; and at that time the press-plunger moves outwards with a velocity which is less than the inward velocity of the pump-plunger, in the same ratio that the area of the pump-plunger is less than the area of the press-plunger. (See Hydraulics.)

In the pneumatic power-transmitter the motion of one piston is transmitted to another at a distance by means of a mass of air contained in two cylinders and an intervening tube. When the pressure and temperature of the air can be maintained constant, this machine fulfils equation (2), like the hydraulic press. The amount and effect of the variations of pressure and temperature undergone by the air depend on the principles of the mechanical action of heat, or Thermodynamics (q.v.), and are foreign to the subject of pure mechanism.

Division 3. Motion of a Rigid Solid.

§ 27. Motions Classed.—In problems of mechanism, each solid piece of the machine is supposed to be so stiff and strong as not to undergo any sensible change of figure or dimensions by the forces applied to it—a supposition which is realized in practice if the machine is skilfully designed.

This being the case, the various possible motions of a rigid solid body may all be classed under the following heads: (1) Shifting or Translation; (2) Turning or Rotation; (3) Motions compounded of Shifting and Turning.

The most common forms for the paths of the points of a piece of mechanism, whose motion is simple shifting, are the straight line and the circle.

Shifting in a straight line is regulated either by straight fixed guides, in contact with which the moving piece slides, or by combinations of link-work, called parallel motions, which will be described in the sequel. Shifting in a straight line is usually reciprocating; that is to say, the piece, after shifting through a certain distance, returns to its original position by reversing its motion.

Circular shifting is regulated by attaching two or more points of the shifting piece to ends of equal and parallel rotating cranks, or by combinations of wheel-work to be afterwards described. As an example of circular shifting may be cited the motion of the coupling rod, by which the parallel and equal cranks upon two or more axles of a locomotive engine are connected and made to rotate simultaneously. The coupling rod remains always parallel to itself, and all its points describe equal and similar circles relatively to the frame of the engine, and move in parallel directions with equal velocities at the same instant.

§ 28. Rotation about a Fixed Axis: Lever, Wheel and Axle.—The fixed axis of a turning body is a line fixed relatively to the body and relatively to the fixed space in which the body turns. In mechanism it is usually the central line either of a rotating shaft or axle having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead centre round which a rotating bush turns; but it may sometimes be entirely beyond the limits of the turning body. For example, if a sliding piece moves in circular fixed guides, that piece rotates about an ideal fixed axis traversing the centre of those guides.

Let the angular velocity of the rotation be denoted by α = dθ/dt, then the linear velocity of any point A at the distance r from the axis is αr; and the path of that point is a circle of the radius r described about the axis.

This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.

In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of different radii described about their common fixed axis of turning, their velocity-ratio being that of their radii.  
  
---  
Fig. 90.  
  
§ 29. Velocity Ratio of Components of Motion.—As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal. Hence it follows that, if A in fig. 90 be a point in a rigid body CD, rotating round the fixed axis F, the component of the velocity of A in any direction AP parallel to the plane of rotation is equal to the total velocity of the point m, found by letting fall Fm perpendicular to AP; that is to say, is equal to

α · Fm.

Hence also the ratio of the components of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fm and Fn.

§ 30. Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose axis of figure is B and angular velocity γ, roll on a fixed cylinder ααα, whose axis of figure is A, either outside (as in fig. 91), when the rolling will be towards the same hand as the rotation, or inside (as in fig. 92), when the rolling will be towards the opposite hand; and at a given instant let T be the line of contact of the two cylindrical surfaces, which is at their common intersection with the plane AB traversing the two axes of figure.

The line T on the surface bbb has for the instant no velocity in a direction perpendicular to AB; because for the instant it touches, without sliding, the line T on the fixed surface aaa.

The line T on the surface bbb has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.

The line of contact T, therefore, on the surface of the cylinder bbb, is for the instant at rest, and is the “instantaneous axis” about which the cylinder bbb turns, together with any body rigidly attached to that cylinder.  
  
---  
Fig. 91. | Fig. 92.  
  
To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P will be perpendicular to that plane, and towards the right or left hand according to the direction of the rotation of bbb; and the velocity of P will be

vP = γ·PT,

(3)

PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind called epitrochoids. If P is in the circumference of bbb, that path becomes an epicycloid.

The velocity of any point in the axis of figure B is

vB = γ·TB;

(4)

and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.

Let α denote the angular velocity with which the plane of axes AB rotates about the fixed axis A. Then it is evident that

vB = α·AB,

(5)

and consequently that

α = γ·TB/AB.

(6)

For internal rolling, as in fig. 92, AB is to be treated as negative, which will give a negative value to α, indicating that in this case the rotation of AB round A is contrary to that of the cylinder bbb.

The angular velocity of the rolling cylinder, relatively to the plane of axes AB, is obviously given by the equation—

β = γ − α  
---  
whence β = γ · TA/AB  
  
(7)

care being taken to attend to the sign of α, so that when that is negative the arithmetical values of γ and α are to be added in order to give that of β.

The whole of the foregoing reasonings are applicable, not merely when aaa and bbb are actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration.  
  
---  
Fig. 93.  
  
§ 31. Instantaneous Axis of a Cone rolling on a Cone.—Let Oaa (fig. 93) be a fixed cone, OA its axis, Obb a cone rolling on it, OB the axis of the rolling cone, OT the line of contact of the two cones at the instant under consideration. By reasoning similar to that of § 30, it appears that OT is the instantaneous axis of rotation of the rolling cone.

Let γ denote the total angular velocity of the rotation of the cone B about the instantaneous axis, β its angular velocity about the axis OB relatively to the plane AOB, and α the angular velocity with which the plane AOB turns round the axis OA. It is required to find the ratios of those angular velocities.

Solution.—In OT take any point E, from which draw EC parallel to OA, and ED parallel to OB, so as to construct the parallelogram OCED. Then

OD : OC : OE :: α : β : γ.

(8)

Or because of the proportionality of the sides of triangles to the sines of the opposite angles,

sin TOB : sin TOA : sin AOB :: α : β : γ,

(8 A)

that is to say, the angular velocity about each axis is proportional to the sine of the angle between the other two.

Demonstration.—From C draw CF perpendicular to OA, and CG perpendicular to OE

Then CF = 2 × | area EC | ,  
---|---|---  
CE  
and CG = 2 × | area ECO | ;  
---|---|---  
OE  
  
∴ CG : CF :: CE = OD : OE.

Let vc denote the linear velocity of the point C. Then

vc = α · CF = γ · CG  
∴ γ : α :: CF : CG :: OE : OD,

which is one part of the solution above stated. From E draw EH perpendicular to OB, and EK to OA. Then it can be shown as before that

EK : EH :: OC : OD.

Let vE be the linear velocity of the point E fixed in the plane of axes AOB. Then

vK = α · EK.

Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone. That is to say,

β · EH = vE = α · EK;

therefore

α : β :: EH : EK :: OD : OC,

which is the remainder of the solution.

The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the instantaneous axis. Then the motion of P is perpendicular to the plane OPQ, and its velocity is

vP = γ · PQ.

(9)

The whole of the foregoing reasonings are applicable, not merely when A and B are actual regular cones, but also when they are the osculating regular cones of a pair of irregular conical surfaces, having a common apex at O.

§ 32. Screw-like or Helical Motion.—Since any displacement in a plane can be represented in general by a rotation, it follows that the only combination of translation and rotation, in which a complex movement which is not a mere rotation is produced, occurs when there is a translation perpendicular to the plane and parallel to the axis of rotation.  
  
---  
Fig. 94.  
  
Such a complex motion is called screw-like or helical motion; for each point in the body describes a helix or screw round the axis of rotation, fixed or instantaneous as the case may be. To cause a body to move in this manner it is usually made of a helical or screw-like figure, and moves in a guide of a corresponding figure. Helical motion and screws adapted to it are said to be right- or left-handed according to the appearance presented by the rotation to an observer looking towards the direction of the translation. Thus the screw G in fig. 94 is right-handed.

The translation of a body in helical motion is called its advance. Let vx denote the velocity of advance at a given instant, which of course is common to all the particles of the body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the circumference of a circle of the radius unity. Then

T = 2π/α

(10)

is the time of one turn at the rate α; and

p = vxT = 2πvx/α

(11)

is the pitch or advance per turn—a length which expresses the comparative motion of the translation and the rotation.

The pitch of a screw is the distance, measured parallel to its axis, between two successive turns of the same thread or helical projection.

Let r denote the perpendicular distance of a point in a body moving helically from the axis. Then

vr = αr 

(12)

is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity is

v = √ {vx2 \+ vr2}.

(13)

The ratio of the two components of that velocity is

vx/vr = p/2πr = tan θ.

(14)

where θ denotes the angle made by the helical path of the point with a plane perpendicular to the axis.

Division 4. Elementary Combinations in Mechanism

§ 33. Definitions.—An elementary combination in mechanism consists of two pieces whose kinds of motion are determined by their connexion with the frame, and their comparative motion by their connexion with each other—that connexion being effected either by direct contact of the pieces, or by a connecting piece, which is not connected with the frame, and whose motion depends entirely on the motions of the pieces which it connects.

The piece whose motion is the cause is called the driver; the piece whose motion is the effect, the follower.

The connexion of each of those two pieces with the frame is in general such as to determine the path of every point in it. In the investigation, therefore, of the comparative motion of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connexion of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of the directional relation, so far only as it expresses the connexion between forward and backward movements of the driver and follower. When a continuous motion of the driver produces a continuous motion of the follower, forward or backward, and a reciprocating motion a motion reciprocating at the same instant, the directional relation is said to be constant. When a continuous motion produces a reciprocating motion, or vice versa, or when a reciprocating motion produces a motion not reciprocating at the same instant, the directional relation is said to be variable.

The line of action or of connexion of the driver and follower is a line traversing a pair of points in the driver and follower respectively, which are so connected that the component of their velocity relatively to each other, resolved along the line of connexion, is null. There may be several or an indefinite number of lines of connexion, or there may be but one; and a line of connexion may connect either the same pair of points or a succession of different pairs.

§ 34. General Principle.—From the definition of a line of connexion it follows that the components of the velocities of a pair of connected points along their line of connexion are equal. And from this, and from the property of a rigid body, already stated in § 29, it follows, that the components along a line of connexion of all the points traversed by that line, whether in the driver or in the follower, are equal; and consequently, that the velocities of any pair of points traversed by a line of connexion are to each other inversely as the cosines, or directly as the secants, of the angles made by the paths of those points with the line of connexion.

The general principle stated above in different forms serves to solve every problem in which—the mode of connexion of a pair of pieces being given—it is required to find their comparative motion at a given instant, or vice versa.  
  
---  
Fig. 95.  
  
§ 35. Application to a Pair of Shifting Pieces.—In fig. 95, let P1P2 be the line of connexion of a pair of pieces, each of which has a motion of translation or shifting. Through any point T in that line draw TV1, TV2, respectively parallel to the simultaneous direction of motion of the pieces; through any other point A in the line of connexion draw a plane perpendicular to that line, cutting TV1, TV2 in V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1 : TV2. Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V1V2 represents their velocity relatively to each other.

§ 36. Application to a Pair of Turning Pieces.—Let α1, α2 be the angular velocities of a pair of turning pieces; θ1, θ2 the angles which their line of connexion makes with their respective planes of rotation; r1, r2 the common perpendiculars let fall from the line of connexion upon the respective axes of rotation of the pieces. Then the equal components, along the line of connexion, of the velocities of the points where those perpendiculars meet that line are—

α1r1 cos θ1 = α2r2 cos θ2;

consequently, the comparative motion of the pieces is given by the equation

α2 | = | r1 cos θ1 | .  
---|---|---|---  
α1 | r2 cos θ2  
  
(15)

§ 37. Application to a Shifting Piece and a Turning Piece.—Let a shifting piece be connected with a turning piece, and at a given instant let α1 be the angular velocity of the turning piece, r1 the common perpendicular of its axis of rotation and the line of connexion, θ1 the angle made by the line of connexion with the plane of rotation, θ2 the angle made by the line of connexion with the direction of motion of the shifting piece, v2 the linear velocity of that piece. Then

α1r1 cos θ1 = v2 cos θ2;

(16)

which equation expresses the comparative motion of the two pieces.

§ 38. Classification of Elementary Combinations in Mechanism.—The first systematic classification of elementary combinations in mechanism was that founded by Monge, and fully developed by Lanz and Bétancourt, which has been generally received, and has been adopted in most treatises on applied mechanics. But that classification is founded on the absolute instead of the comparative motions of the pieces, and is, for that reason, defective, as Willis pointed out in his admirable treatise On the Principles of Mechanism.

Willis’s classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connexion of the driver and follower. He divides the elementary combinations in mechanism into three classes, of which the characters are as follows:—

Class A: Directional relation constant; velocity ratio constant.

Class B: Directional relation constant; velocity ratio varying.

Class C: Directional relation changing periodically; velocity ratio constant or varying.

Each of those classes is subdivided by Willis into five divisions, of which the characters are as follows:—

Division | A: | Connexion | by | rolling contact.  
---|---|---|---|---  
” | B: | ” | ” | sliding contact.  
” | C: | ” | ” | wrapping connectors.  
” | D: | ” | ” | link-work.  
” | E: | ” | ” | reduplication.  
  
In the Reuleaux system of analysis of mechanisms the principle of comparative motion is generalized, and mechanisms apparently very diverse in character are shown to be founded on the same sequence of elementary combinations forming a kinematic chain. A short description of this system is given in § 80, but in the present article the principle of Willis’s classification is followed mainly. The arrangement is, however, modified by taking the mode of connexion as the basis of the primary classification, and by removing the subject of connexion by reduplication to the section of aggregate combinations. This modified arrangement is adopted as being better suited than the original arrangement to the limits of an article in an encyclopaedia; but it is not disputed that the original arrangement may be the best for a separate treatise.

§ 39. Rolling Contact: Smooth Wheels and Racks.—In order that two pieces may move in rolling contact, it is necessary that each pair of points in the two pieces which touch each other should at the instant of contact be moving in the same direction with the same velocity. In the case of two shifting pieces this would involve equal and parallel velocities for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces would move like one; hence, in the case of rolling contact, either one or both of the pieces must rotate.

The direction of motion of a point in a turning piece being perpendicular to a plane passing through its axis, the condition that each pair of points in contact with each other must move in the same direction leads to the following consequences:—

I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane.

II. That, when one piece rotates, and the other shifts, the axis of the rotating piece, and all the points of contact, lie in a plane perpendicular to the direction of motion of the shifting piece.

The condition that the velocity of each pair of points of contact must be equal leads to the following consequences:—

III. That the angular velocities of a pair of turning pieces in rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.

IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.

The line of contact is that line in which the points of contact are all situated. Respecting this line, the above Principles III. and IV. lead to the following conclusions:—

V. That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that the rolling surfaces are either plane or cylindrical (the term “cylindrical” including all surfaces generated by the motion of a straight line parallel to itself).

VI. That for a pair of turning pieces with intersecting axes the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term “conical” including all surfaces generated by the motion of a straight line which traverses a fixed point).

Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in rolling contact with turning pieces may be called smooth or toothless racks.

VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion.

§ 40. Cylindrical Wheels and Smooth Racks.—In designing cylindrical wheels and smooth racks, and determining their comparative motion, it is sufficient to consider a section of the pair of pieces made by a plane perpendicular to the axis or axes.

The points where axes intersect the plane of section are called centres; the point where the line of contact intersects it, the point of contact, or pitch-point; and the wheels are described as circular, elliptical, &c., according to the forms of their sections made by that plane.

When the point of contact of two wheels lies between their centres, they are said to be in outside gearing; when beyond their centres, in inside gearing, because the rolling surface of the larger wheel must in this case be turned inward or towards its centre.

From Principle III. of § 39 it appears that the angular velocity-ratio of a pair of wheels is the inverse ratio of the distances of the point of contact from the centres respectively.  
  
---  
Fig. 96.  
  
For outside gearing that ratio is negative, because the wheels turn contrary ways; for inside gearing it is positive, because they turn the same way.

If the velocity ratio is to be constant, as in Willis’s Class A, the wheels must be circular; and this is the most common form for wheels.

If the velocity ratio is to be variable, as in Willis’s Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.

The following is the geometrical relation which must exist between such a pair of curves:—

Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2 any pair of points of contact; U1, U2 any other pair of points of contact. Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled:—

Sum of radii, C1U1 \+ C2U2 = C1T1 \+ C2T2 = constant;  
arc, T2U2 = T1U1.

(17)

A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radii-vectores shall be equal and contrary; or, denoting by r1, r2 the radii-vectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact—

dr2/ds = −dr1/ds;

(18)

which is the differential equation of a pair of rolling curves whose poles are at a constant distance apart.

For full details as to rolling curves, see Willis’s work, already mentioned, and Clerk Maxwell’s paper on Rolling Curves, Trans. Roy. Soc. Edin., 1849.

A rack, to work with a circular wheel, must be straight. To work with a wheel of any other figure, its section must be a rolling curve, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant. Let r1 be the radius-vector of a point of contact on the wheel, x2 the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then

dx2/ds = −dr1/ds

(19)

is the differential equation of the pair of rolling curves.

To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity ratio varying from (1 + eccentricity)/(1 − eccentricity) to (1 − eccentricity)/(1 + eccentricity); an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyperbolas, and the velocity ratio varying between (eccentricity + 1)/(eccentricity − 1) and unity; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix.  
  
---  
Fig. 97.  
  
§ 41. Conical or Bevel and Disk Wheels.—From Principles III. and VI. of § 39 it appears that the angular velocities of a pair of wheels whose axes meet in a point are to each other inversely as the sines of the angles which the axes of the wheels make with the line of contact. Hence we have the following construction (figs. 97 and 98).—Let O be the apex or point of intersection of the two axes OC1, OC2. The angular velocity ratio being given, it is required to find the line of contact. On OC1, OC2 take lengths OA1, OA2, respectively proportional to the angular velocities of the pieces on whose axes they are taken. Complete the parallelogram OA1EA2; the diagonal OET will be the line of contact required.

When the velocity ratio is variable, the line of contact will shift its position in the plane C1OC2, and the wheels will be cones, with eccentric or irregular bases. In every case which occurs in practice, however, the velocity ratio is constant; the line of contact is constant in position, and the rolling surfaces of the wheels are regular circular cones (when they are called bevel wheels); or one of a pair of wheels may have a flat disk for its rolling surface, as W2 in fig. 98, in which case it is a disk wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or disks, as shown by W1, W2 in figs. 97 and 98.  
  
---  
Fig. 98.  
Fig. 99.  
Fig. 100.  
  
§ 42. Sliding Contact (lateral): Skew-Bevel Wheels.—An hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids E, F, equal or unequal, be placed in the closest possible contact, as in fig. 99, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes AG, BH in opposite directions. The axes will not be parallel, nor will they intersect each other.

The motion of two such hyperboloids, turning in contact with each other, has hitherto been classed amongst cases of rolling contact; but that classification is not strictly correct, for, although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, as the axes are parallel neither to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of contact which are unequal, and their difference constitutes a lateral sliding.

The directions and positions of the axes being given, and the required angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated:—

In fig. 100, let B1C1, B2C2 be the two axes; B1B2 their common perpendicular. Through any point O in this common perpendicular draw OA1 parallel to B1C1 and OA2 parallel to B2C2; make those lines proportional to the angular velocities about the axes to which they are respectively parallel; complete the parallelogram OA1EA2, and draw the diagonal OE; divide B1B2 in D into two parts, inversely proportional to the angular velocities about the axes which they respectively adjoin; through D parallel to OE draw DT. This will be the line of contact.

A pair of thin frusta of a pair of hyperboloids are used in practice to communicate motion between a pair of axes neither parallel nor intersecting, and are called skew-bevel wheels.

In skew-bevel wheels the properties of a line of connexion are not possessed by every line traversing the line of contact, but only by every line traversing the line of contact at right angles.

If the velocity ratio to be communicated were variable, the point D would alter its position, and the line DT its direction, at different periods of the motion, and the wheels would be hyperboloids of an eccentric or irregular cross-section; but forms of this kind are not used in practice.

§ 43. Sliding Contact (circular): Grooved Wheels.—As the adhesion or friction between a pair of smooth wheels is seldom sufficient to prevent their slipping on each other, contrivances are used to increase their mutual hold. One of those consists in forming the rim of each wheel into a series of alternate ridges and grooves parallel to the plane of rotation; it is applicable to cylindrical and bevel wheels, but not to skew-bevel wheels. The comparative motion of a pair of wheels so ridged and grooved is the same as that of a pair of smooth wheels in rolling contact, whose cylindrical or conical surfaces lie midway between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces are called the pitch surfaces of the wheels.

The relative motion of the faces of contact of the ridges and grooves is a rotatory sliding or grinding motion, about the line of contact of the pitch-surfaces as an instantaneous axis.

Grooved wheels have hitherto been but little used.

§ 44. Sliding Contact (direct): Teeth of Wheels, their Number and Pitch.—The ordinary method of connecting a pair of wheels, or a wheel and a rack, and the only method which ensures the exact maintenance of a given numerical velocity ratio, is by means of a series of alternate ridges and hollows parallel or nearly parallel to the successive lines of contact of the ideal smooth wheels whose velocity ratio would be the same with that of the toothed wheels. The ridges are called teeth; the hollows, spaces. The teeth of the driver push those of the follower before them, and in so doing sliding takes place between them in a direction across their lines of contact.

The pitch-surfaces of a pair of toothed wheels are the ideal smooth surfaces which would have the same comparative motion by rolling contact that the actual wheels have by the sliding contact of their teeth. The pitch-circles of a pair of circular toothed wheels are sections of their pitch-surfaces, made for spur-wheels (that is, for wheels whose axes are parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described about the common apex. For a pair of skew-bevel wheels the pitch-circles are a pair of contiguous rectangular sections of the pitch-surfaces. The pitch-point is the point of contact of the pitch-circles.

The pitch-surface of a wheel lies intermediate between the points of the teeth and the bottoms of the hollows between them. That part of the acting surface of a tooth which projects beyond the pitch-surface is called the face; that part which lies within the pitch-surface, the flank.

Teeth, when not otherwise specified, are understood to be made in one piece with the wheel, the material being generally cast-iron, brass or bronze. Separate teeth, fixed into mortises in the rim of the wheel, are called cogs. A pinion is a small toothed wheel; a trundle is a pinion with cylindrical staves for teeth.

The radius of the pitch-circle of a wheel is called the geometrical radius; a circle touching the ends of the teeth is called the addendum circle, and its radius the real radius; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called the addendum.

The distance, measured along the pitch-circle, from the face of one tooth to the face of the next, is called the pitch. The pitch and the number of teeth in wheels are regulated by the following principles:—

I. In wheels which rotate continuously for one revolution or more, it is obviously necessary that the pitch should be an aliquot part of the circumference.

In wheels which reciprocate without performing a complete revolution this condition is not necessary. Such wheels are called sectors.

II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each.

III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii and inversely as the angular velocities.

IV. Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.

From this principle arise problems of a kind which will be referred to in treating of Trains of Mechanism.

V. Let n, N be the respective numbers of teeth in a pair of wheels, N being the greater. Let t, T be a pair of teeth in the smaller and larger wheel respectively, which at a particular instant work together. It is required to find, first, how many pairs of teeth must pass the line of contact of the pitch-surfaces before t and T work together again (let this number be called a); and, secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called b); thirdly, with how many different teeth of the smaller wheel the tooth T will work at different times (let this be called c).

Case 1. If n is a divisor of N,

a = N; b = N/n; c = 1.

(20)

Case 2. If the greatest common divisor of N and n be d, a number less than n, so that n = md, N = Md; then

a = mN = Mn = Mmd; b = M; c = m.

(21)

Case 3. If N and n be prime to each other,

a = nN; b = N; c = n.

(22)

It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They therefore study that the numbers of teeth in each pair of wheels which work together shall either be prime to each other, or shall have their greatest common divisor as small as is consistent with a velocity ratio suited for the purposes of the machine.

§ 45. Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point where they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack.

In fig. 101, let C1, C2 be the centres of a pair of spur-wheels; B1IB1′, B2IB2′ portions of their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower.  
  
---  
Fig. 101.  
  
Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of the acting surfaces of a pair of teeth in the driver and follower respectively, touching each other at T; the line of connexion of those teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2 be perpendiculars let fall from the centres of the wheels on the line of contact. Then, by § 36, the angular velocity-ratio is

α2/α1 = C1P1/C2P2.

(23)

The following principles regulate the forms of the teeth and their relative motions:—

I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point.

For, let P1P2 cut the line of centres at I; then, by similar triangles,

α1 : α2 :: C2P2 : C1P1 :: IC2 :: IC1;

(24)

which is also the angular velocity ratio due to the rolling contact of the circles B1IB1′, B2IB2′.

This principle determines the forms of all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE′, parallel to the direction of motion of the rack, and perpendicular to C1IC2, be substituted for a pitch-circle.

II. The component of the velocity of the point of contact of the teeth T along the line of connexion is

α1 · C1P1 = α2 · C2P2.

(25)

III. The relative velocity perpendicular to P1P2 of the teeth at their point of contact—that is, their velocity of sliding on each other—is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C1C2 to rotate backwards round C1 with the angular velocity α1, and the wheel 2 to rotate round C2 as before, with the angular velocity α2 relatively to the line of centres C1C2, so as to have the same motion as if its pitch-circle rolled on the pitch-circle of the first wheel. Thus the relative motion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity α1 \+ α2. Hence the velocity of sliding is that due to this rotation about I, with the radius IT; that is to say, its value is

(α1 \+ α2) · IT;

(26)

so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with the pitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact.

IV. The path of contact is the line traversing the various positions of the point T. If the line of connexion preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved.

It is divided by the pitch-point I into two parts—the arc or line of approach described by T in approaching the line of centres, and the arc or line of recess described by T after having passed the line of centres.

During the approach, the flank D1B1 of the driving tooth drives the face D2B2 of the following tooth, and the teeth are sliding towards each other. During the recess (in which the position of the teeth is exemplified in the figure by curves marked with accented letters), the face B1′A1′ of the driving tooth drives the flank B2′A2′ of the following tooth, and the teeth are sliding from each other.

The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact.

V. The obliquity of the action of the teeth is the angle EIT = IC1, P1 = IC2P2.

In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15°, nor the maximum value 30°.

It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formulae is, that in equation (26) the difference of the angular velocities should be substituted for their sum.

§ 46. Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is obtained in the following manner (see fig. 102). Let C1, C2 be the centres of two wheels, B1IB1′, B2IB2′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line P1IP2 shall represent at once the constant line of connexion of teeth and the path of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and with those lines as radii describe about the centres of the wheels the circles D1D1′, D2D2′, called base-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that

C1P1 = IC1 · cos obliquity  
C2P2 = IC2 · cos obliquity.

(27)

(The obliquity which is found to answer best in practice is about 141⁄2°; its cosine is about 31/22, and its sine about 1⁄4. These values though not absolutely exact, are near enough to the truth for practical purposes.)  
  
---  
Fig. 102.  
  
Suppose the base-circles to be a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is P1IP2. As the line of connexion of those pulleys is the same as that of the proposed teeth, they will rotate with the required velocity ratio. Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of contact P1IP2, that point will trace on a plane rotating along with the wheel 1 part of the involute of the base-circle D1D1′, and on a plane rotating along with the wheel 2 part of the involute of the base-circle D2D2′; and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Principle I. of § 45.

Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.

All involute teeth of the same pitch work smoothly together.

To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along P1IP2, is less than the pitch in the ratio of cos obliquity : I; and consequently that, if distances equal to the pitch be marked off either way from I towards P1 and P2 respectively, as the extremities of the path of contact, and if, according to Principle IV. of § 45, the addendum-circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice it is usual to make the path of contact somewhat longer, viz. about 2.4 times the pitch; and with this length of path, and the obliquity already mentioned of 141⁄2°, the addendum is about 3.1 of the pitch.

The teeth of a rack, to work correctly with wheels having involute teeth, should have plane surfaces perpendicular to the line of connexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action.

§ 47. Teeth for a given Path of Contact: Sang’s Method.—In the preceding section the form of the teeth is found by assuming a figure for the path of contact, viz. the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II. of § 45, and by equation (25). This method of finding the forms of the teeth of wheels forms the subject of an elaborate and most interesting treatise by Edward Sang.

All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set.  
  
---  
Fig. 103.  
  
§ 48. Teeth traced by Rolling Curves.—If any curve R (fig. 103) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from § 30, that the instantaneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-circle for the moment, and that consequently the line AT, traced by a tracing-point T, fixed to the rolling curve upon the plane of the wheel, will be everywhere perpendicular to the straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If the same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the face of a tooth suitable to work with the flank AT.

In like manner, if either the same or any other rolling curve R′ be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T′ shall start from A, it will trace the face AT′ of a tooth suitable to work with a flank traced by rolling the same curve R′ with the same tracing-point T′ inside any other pitch-circle.

The figure of the path of contact is that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E′I′E′, as the case may be) at a fixed point I (or I′).

If the same rolling curve and tracing-point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form a set. The teeth of a rack, of the same set, are traced by rolling the rolling curve on both sides of a straight line.

The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch-surfaces are in rolling contact.  
  
---  
Fig. 104.  
  
§ 49. Epicycloidal Teeth.—The most convenient rolling curve is the circle. The path of contact which it traces is identical with itself; and the flanks of the teeth are internal and their faces external epicycloids for wheels, and both flanks and faces are cycloids for a rack.

For a pitch-circle of twice the radius of the rolling or describing circle (as it is called) the internal epicycloid is a straight line, being, in fact, a diameter of the pitch-circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. For a smaller pitch-circle the flanks would be convex and in-curved or under-cut, which would be inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the radius of the describing circle, so that the flanks may be either straight or concave.

In fig. 104 let BB′ be part of the pitch-circle of a wheel with epicycloidal teeth; CIC′ the line of centres; I the pitch-point; EIE′ a straight tangent to the pitch-circle at that point; R the internal and R′ the equal external describing circles, so placed as to touch the pitch-circle and each other at I. Let DID′ be the path of contact, consisting of the arc of approach DI and the arc of recess ID′. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch.

The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID = E′ID; and the mean obliquity is one-half of that angle.

It appears from experience that the mean obliquity should not exceed 15°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID′ should each be one-sixth of a circumference; therefore the circumference of the describing circle should be six times the pitch.

It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth.

§ 50. Nearly Epicycloidal Teeth: Willis’s Method.—To facilitate the drawing of epicycloidal teeth in practice, Willis showed how to approximate to their figure by means of two circular arcs—one concave, for the flank, and the other convex, for the face—and each having for its radius the mean radius of curvature of the epicycloidal arc. Willis’s formulae are founded on the following properties of epicycloids:—

Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by the normal TI to the epicycloid at a given point T, with a tangent to the circle at I—that is, the obliquity of the action at T.

Then the radius of curvature of the epicycloid at T is—

For an internal epicycloid, ρ = 4r sin θ | R − r  
---|---  
R − 2r  
For an external epicycloid, ρ′ = 4r sin θ | R + r  
---|---  
R + 2r  
  
(28)

Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c = 2r sin θ; and therefore

For the flank, ρ − c = 2r sin θ | R  
---|---  
R − 2r  
For the face, ρ′ − c = 2r sin θ | R  
---|---  
R + 2r  
  
(29)

For the proportions approved of by Willis, sin θ = 1⁄4 nearly; r = p (the pitch) nearly; c = 1⁄2p nearly; and, if N be the number of teeth in the wheel, r/R = 6/N nearly; therefore, approximately,

ρ − c = | p | · | N  
---|---|---|---  
2 | N − 12  
ρ − c = | p | · | N  
---|---|---|---  
2 | N + 12  
  
(30)  
  
---  
Fig. 105.  
  
Hence the following construction (fig. 105). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab = ac − 1⁄2p. Draw radii bd, ce; draw fb, cg, making angles of 751⁄2° with those radii. Make bf = p′ − c, cg = p − c. From f, with the radius fa, draw the circular arc ah; from g, with the radius ga, draw the circular arc ak. Then ah is the face and ak the flank of the tooth required.

To facilitate the application of this rule, Willis published tables of ρ − c and ρ′ − c, and invented an instrument called the “odontograph.”

§ 51. Trundles and Pin-Wheels.—If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels.

§ 52. Backs of Teeth and Spaces.—Toothed wheels being in general intended to rotate either way, the backs of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch-circle, is made about 1⁄6th part wider than the thickness of the tooth on the pitch-circle—that is to say,

Thickness of tooth | = 5⁄11 pitch;  
---|---  
Width of space | = 6⁄11 pitch.  
  
The difference of 1⁄11 of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about one-tenth of the pitch.

§ 53. Stepped and Helical Teeth.—R. J. Hooke invented the making of the fronts of teeth in a series of steps with a view to increase the smoothness of action. A wheel thus formed resembles in shape a series of equal and similar toothed disks placed side by side, with the teeth of each a little behind those of the preceding disk. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel-work of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the aftermost end; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next pair has commenced.

Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and expensive workmanship than common teeth; and helical teeth are, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery.

§ 54. Teeth of Bevel-Wheels.—The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex.  
  
---  
Fig. 106.  
  
The operations of describing the exact figures of the teeth of bevel-wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, substituting poles for centres, and great circles for straight lines.

In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used:—

Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I, OB2I their pitch cones; OC1, OC2 their axes; OI their line of contact. Perpendicular to OI draw A1IA2, cutting the axes in A1, A2; make the outer rims of the patterns and of the wheels portions of the cones A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about O for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs ID1, ID2, with the radii A1I, A2I; those arcs will be the developments of arcs of the pitch-circles B1I, B2I, when the conical surfaces A1B1I, A2B2I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces.

§ 55. Teeth of Skew-Bevel Wheels.—The crests of the teeth of a skew-bevel wheel are parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel whose pitch surface is a cone touching the hyperboloidal surface at the given circle.

§ 56. Cams.—A cam is a single tooth, either rotating continuously or oscillating, and driving a sliding or turning piece either constantly or at intervals. All the principles which have been stated in § 45 as being applicable to teeth are applicable to cams; but in designing cams it is not usual to determine or take into consideration the form of the ideal pitch-surface, which would give the same comparative motion by rolling contact that the cam gives by sliding contact.

§ 57. Screws.—The figure of a screw is that of a convex or concave cylinder, with one or more helical projections, called threads, winding round it. Convex and concave screws are distinguished technically by the respective names of male and female; a short concave screw is called a nut; and when a screw is spoken of without qualification a convex screw is usually understood.

The relation between the advance and the rotation, which compose the motion of a screw working in contact with a fixed screw or helical guide, has already been demonstrated in § 32; and the same relation exists between the magnitudes of the rotation of a screw about a fixed axis and the advance of a shifting nut in which it rotates. The advance of the nut takes place in the opposite direction to that of the advance of the screw in the case in which the nut is fixed. The pitch or axial pitch of a screw has the meaning assigned to it in that section, viz. the distance, measured parallel to the axis, between the corresponding points in two successive turns of the same thread. If, therefore, the screw has several equidistant threads, the true pitch is equal to the divided axial pitch, as measured between two adjacent threads, multiplied by the number of threads.

If a helix be described round the screw, crossing each turn of the thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal helix, may be called the normal pitch; and when the screw has more than one thread the normal pitch from thread to thread may be called the normal divided pitch.

The distance from thread to thread, measured on a circle described about the axis of the screw, called the pitch-circle, may be called the circumferential pitch; for a screw of one thread it is one circumference; for a screw of n threads, (one circumference)/n.

Let r denote the radius of the pitch circle;

n the number of threads;

θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;

Pa | the axial | pitch  
---|---|---  
Pa/n = pa | divided pitch;  
   
Pn | the normal | pitch  
Pn/n = pn | divided pitch;  
   
Pc the circumferential pitch;  
  
then

pc = pa cot θ = pn cos θ = | 2πr | ,  
---|---|---  
n  
pa = pn sec θ = pc tan θ = | 2πr tan θ | ,  
---|---|---  
n  
pn = pc sin θ = pa cos θ = | 2πr sin θ | .  
---|---|---  
n  
  
(31)

If a screw rotates, the number of threads which pass a fixed point in one revolution is the number of threads in the screw.

A pair of convex screws, each rotating about its axis, are used as an elementary combination to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws their threads must be parallel; and their line of connexion is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles:—

I. If the screws are both right-handed or both left-handed, the angle between the directions of their axes is the sum of their obliquities; if one is right-handed and the other left-handed, that angle is the difference of their obliquities.

II. The normal pitch for a screw of one thread, and the normal divided pitch for a screw of more than one thread, must be the same in each screw.

III. The angular velocities of the screws are inversely as their numbers of threads.

Hooke’s wheels with oblique or helical teeth are in fact screws of many threads, and of large diameters as compared with their lengths.

The ordinary position of a pair of endless screws is with their axes at right angles to each other. When one is of considerably greater diameter than the other, the larger is commonly called in practice a wheel, the name screw being applied to the smaller only; but they are nevertheless both screws in fact.

To make the teeth of a pair of endless screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to form a cutting tool; the larger screw, or “wheel,” is cast approximately of the required figure; the larger screw and the steel screw are fitted up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure.  
  
---  
Fig. 107.  
  
§ 58. Coupling of Parallel Axes—Oldham’s Coupling.—A coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction with the same mean angular velocity. If the axes of the shafts are in the same straight line, the coupling consists in so connecting their contiguous ends that they shall rotate as one piece; but if the axes are not in the same straight line combinations of mechanism are required. A coupling for parallel shafts which acts by sliding contact was invented by Oldham, and is represented in fig. 107. C1, C2 are the axes of the two parallel shafts; D1, D2 two disks facing each other, fixed on the ends of the two shafts respectively; E1E1 a bar sliding in a diametral groove in the face of D1; E2E2 a bar sliding in a diametral groove in the face of D2: those bars are fixed together at A, so as to form a rigid cross. The angular velocities of the two disks and of the cross are all equal at every instant; the middle point of the cross, at A, revolves in the dotted circle described upon the line of centres C1C2 as a diameter twice for each turn of the disks and cross; the instantaneous axis of rotation of the cross at any instant is at I, the point in the circle C1C2 diametrically opposite to A.

Oldham’s coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practibility or permanency of their exact continuity.

§ 59. Wrapping Connectors—Belts, Cords and Chains.—Flat belts of leather or of gutta percha, round cords of catgut, hemp or other material, and metal chains are used as wrapping connectors to transmit rotatory motion between pairs of pulleys and drums.

Belts (the most frequently used of all wrapping connectors) require nearly cylindrical pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on the pulley, unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one pulley on to another of equal size alongside of it, by pressing against that part of the belt which is moving towards the pulley.

Cords require either cylindrical drums with ledges or grooved pulleys.

Chains require pulleys or drums, grooved, notched and toothed, so as to fit the links of the chain.

Wrapping connectors for communicating continuous motion are endless.

Wrapping connectors for communicating reciprocating motion have usually their ends made fast to the pulleys or drums which they connect, and which in this case may be sectors.  
  
---  
Fig. 108.  
  
The line of connexion of two pieces connected by a wrapping connector is the centre line of the belt, cord or chain; and the comparative motions of the pieces are determined by the principles of § 36 if both pieces turn, and of § 37 if one turns and the other shifts, in which latter case the motion must be reciprocating.

The pitch-line of a pulley or drum is a curve to which the line of connexion is always a tangent—that is to say, it is a curve parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the wrapping connector.

Pulleys and drums for communicating a constant velocity ratio are circular. The effective radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius added to half the thickness of the connector. The angular velocities of a pair of connected circular pulleys or drums are inversely as the effective radii.

A crossed belt, as in fig. 108, A, reverses the direction of the rotation communicated; an uncrossed belt, as in fig. 108, B, preserves that direction.

The length L of an endless belt connecting a pair of pulleys whose effective radii are r1, r2, with parallel axes whose distance apart is c, is given by the following formulae, in each of which the first term, containing the radical, expresses the length of the straight parts of the belt, and the remainder of the formula the length of the curved parts.

For a crossed belt:—

L = 2 √ {c2 − (r1 \+ r2)2} + (r1 \+ r2) ( π − 2 sin−1 | r1 \+ r2 | );  
---|---|---  
c  
  
(32 A)

and for an uncrossed belt:—

L = 2 √ {c2 − (r1 − r2)2 } + π (r1 \+ r2 \+ 2 (r1 − r2) sin−1 | r1 − r2 | ;  
---|---|---  
c  
  
(32 B)

in which r1 is the greater radius, and r2 the less.

When the axes of a pair of pulleys are not parallel, the pulleys should be so placed that the part of the belt which is approaching each pulley shall be in the plane of the pulley.

§ 60. Speed-Cones.—A pair of speed-cones (fig. 109) is a contrivance for varying and adjusting the velocity ratio communicated between a pair of parallel shafts by means of a belt. The speed-cones are either continuous cones or conoids, as A, B, whose velocity ratio can be varied gradually while they are in motion by shifting the belt, or sets of pulleys whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by shifting the belt from one pair of pulleys to another.  
  
---  
Fig. 109.  
  
In order that the belt may fit accurately in every possible position on a pair of speed-cones, the quantity L must be constant, in equations (32 A) or (32 B), according as the belt is crossed or uncrossed.

For a crossed belt, as in A and C, fig. 109, L depends solely on c and on r1 \+ r2. Now c is constant because the axes are parallel; therefore the sum of the radii of the pitch-circles connected in every position of the belt is to be constant. That condition is fulfilled by a pair of continuous cones generated by the revolution of two straight lines inclined opposite ways to their respective axes at equal angles.

For an uncrossed belt, the quantity L in equation (32 B) is to be made constant. The exact fulfilment of this condition requires the solution of a transcendental equation; but it may be fulfilled with accuracy sufficient for practical purposes by using, instead of (32 B) the following approximate equation:—

L nearly = 2c + π (r1 \+ r2) + (r1 − r2)2 / c.

(33)

The following is the most convenient practical rule for the application of this equation:—

Let the speed-cones be equal and similar conoids, as in B, fig. 109, but with their large and small ends turned opposite ways. Let r1 be the radius of the large end of each, r2 that of the small end, r0 that of the middle; and let v be the sagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length. Then

v = r0 − (r1 \+ r2) / 2 = (r1 − r2)2 / 2πc.

(34)

2π = 6.2832; but 6 may be used in most practical cases without sensible error.

The radii at the middle and end being thus determined, make the generating curve an arc either of a circle or of a parabola.

§ 61. Linkwork in General.—The pieces which are connected by linkwork, if they rotate or oscillate, are usually called cranks, beams and levers. The link by which they are connected is a rigid rod or bar, which may be straight or of any other figure; the straight figure being the most favourable to strength, is always used when there is no special reason to the contrary. The link is known by various names in various circumstances, such as coupling-rod, connecting-rod, crank-rod, eccentric-rod, &c. It is attached to the pieces which it connects by two pins, about which it is free to turn. The effect of the link is to maintain the distance between the axes of those pins invariable; hence the common perpendicular of the axes of the pins is the line of connexion, and its extremities may be called the connected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the arm or crank-arm.

The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connexion in which case the angular velocity ratio at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the line of connexion upon the respective axes of rotation.

If at any instant the direction of one of the crank-arms coincides with the line of connexion, the common perpendicular of the line of connexion and the axis of that crank-arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank-arm in question at such an instant is called a dead-point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point at the same instant, so that the line of connexion is in the plane of the two axes of rotation, in which case the velocity ratio is indeterminate. Examples of dead-points, and of the means of preventing the inconvenience which they tend to occasion, will appear in the sequel.

§ 62. Coupling of Parallel Axes.—Two or more parallel shafts (such as those of a locomotive engine, with two or more pairs of driving wheels) are made to rotate with constantly equal angular velocities by having equal cranks, which are maintained parallel by a coupling-rod of such a length that the line of connexion is equal to the distance between the axes. The cranks pass their dead-points simultaneously. To obviate the unsteadiness of motion which this tends to cause, the shafts are provided with a second set of cranks at right angles to the first, connected by means of a similar coupling-rod, so that one set of cranks pass their dead points at the instant when the other set are farthest from theirs.

§ 63. Comparative Motion of Connected Points.—As the link is a rigid body, it is obvious that its action in communicating motion may be determined by finding the comparative motion of the connected points, and this is often the most convenient method of proceeding.

If a connected point belongs to a turning piece, the direction of its motion at a given instant is perpendicular to the plane containing the axis and crank-arm of the piece. If a connected point belongs to a shifting piece, the direction of its motion at any instant is given, and a plane can be drawn perpendicular to that direction.

The line of intersection of the planes perpendicular to the paths of the two connected points at a given instant is the instantaneous axis of the link at that instant; and the velocities of the connected points are directly as their distances from that axis.  
  
---  
Fig. 110.  
  
In drawing on a plane surface, the two planes perpendicular to the paths of the connected points are represented by two lines (being their sections by a plane normal to them), and the instantaneous axis by a point (fig. 110); and, should the length of the two lines render it impracticable to produce them until they actually intersect, the velocity ratio of the connected points may be found by the principle that it is equal to the ratio of the segments which a line parallel to the line of connexion cuts off from any two lines drawn from a given point, perpendicular respectively to the paths of the connected points.

To illustrate this by one example. Let C1 be the axis, and T1 the connected point of the beam of a steam-engine; T1T2 the connecting or crank-rod; T2 the other connected point, and the centre of the crank-pin; C2 the axis of the crank and its shaft. Let v1 denote the velocity of T1 at any given instant; v2 that of T2. To find the ratio of these velocities, produce C1T1, C2T2 till they intersect in K; K is the instantaneous axis of the connecting rod, and the velocity ratio is

v1 : v2 :: KT1 : KT2.

(35)

Should K be inconveniently far off, draw any triangle with its sides respectively parallel to C1T1, C2T2 and T1T2; the ratio of the two sides first mentioned will be the velocity ratio required. For example, draw C2A parallel to C1T1, cutting T1T2 in A; then

v1 : v2 :: C2A : C2T2.

(36)

§ 64. Eccentric.—An eccentric circular disk fixed on a shaft, and used to give a reciprocating motion to a rod, is in effect a crank-pin of sufficiently large diameter to surround the shaft, and so to avoid the weakening of the shaft which would arise from bending it so as to form an ordinary crank. The centre of the eccentric is its connected point; and its eccentricity, or the distance from that centre to the axis of the shaft, is its crank-arm.

An eccentric may be made capable of having its eccentricity altered by means of an adjusting screw, so as to vary the extent of the reciprocating motion which it communicates.

§ 65. Reciprocating Pieces—Stroke—Dead-Points.—The distance between the extremities of the path of the connected point in a reciprocating piece (such as the piston of a steam-engine) is called the stroke or length of stroke of that piece. When it is connected with a continuously turning piece (such as the crank of a steam-engine) the ends of the stroke of the reciprocating piece correspond to the dead-points of the path of the connected point of the turning piece, where the line of connexion is continuous with or coincides with the crank-arm.

Let S be the length of stroke of the reciprocating piece, L the length of the line of connexion, and R the crank-arm of the continuously turning piece. Then, if the two ends of the stroke be in one straight line with the axis of the crank,

S = 2R;

(37)

and if these ends be not in one straight line with that axis, then S, L − R, and L + R, are the three sides of a triangle, having the angle opposite S at that axis; so that, if θ be the supplement of the arc between the dead-points,

S2 = 2 (L2 \+ R2) − 2 (L2 − R2) cos θ,

cos θ = | 2L2 \+ 2R2 − S2 | .  
---|---|---  
2 (L2 − R2)  
  
(38)  
  
---  
Fig. 111.  
  
§ 66. Coupling of Intersecting Axes—Hooke’s Universal Joint.—Intersecting axes are coupled by a contrivance of Hooke’s, known as the “universal joint,” which belongs to the class of linkwork (see fig. 111). Let O be the point of intersection of the axes OC1, OC2, and θ their angle of inclination to each other. The pair of shafts C1, C2 terminate in a pair of forks F1, F2 in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at O. This cross is the link; the connected points are the centres of the bearings F1, F2. At each instant each of those points moves at right angles to the central plane of its shaft and fork, therefore the line of intersection of the central planes of the two forks at any instant is the instantaneous axis of the cross, and the velocity ratio of the points F1, F2 (which, as the forks are equal, is also the angular velocity ratio of the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity ratio is that of equality, for each successive quarter-turn is made by both shafts in the same time; but its actual value fluctuates between the limits:—

α2 | = | 1 | when F1 is the plane of OC1C2  
---|---|---|---  
α1 | cos θ  
and | α2 | = cos θ when F2 is in that plane.  
---|---|---  
α1  
  
(39)

Its value at intermediate instants is given by the following equations: let φ1, φ2 be the angles respectively made by the central planes of the forks and shafts with the plane OC1C2 at a given instant; then

cos θ = tan φ1 tan φ2,

α2 | = − | dφ2 | = | tan φ1 \+ cot φ1 | .  
---|---|---|---|---|---  
α1 | dφ1 | tan φ2 \+ cot φ2  
  
(40)

§ 67. Intermittent Linkwork—Click and Ratchet.—A click acting upon a ratchet-wheel or rack, which it pushes or pulls through a certain arc at each forward stroke and leaves at rest at each backward stroke, is an example of intermittent linkwork. During the forward stroke the action of the click is governed by the principles of linkwork; during the backward stroke that action ceases. A catch or pall, turning on a fixed axis, prevents the ratchet-wheel or rack from reversing its motion.

Division 5.—Trains of Mechanism.

§ 68. General Principles..—A train of mechanism consists of a series of pieces each of which is follower to that which drives it and driver to that which follows it.

The comparative motion of the first driver and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the directional relations of the several elementary combinations of which the train consists.

§ 69. Trains of Wheelwork.—Let A1, A2, A3, &c., Am−1, Am denote a series of axes, and α1, α2, α3, &c., αm−1, αm their angular velocities. Let the axis A1 carry a wheel of N1 teeth, driving a wheel of n2 teeth on the axis A2, which carries also a wheel of N2 teeth, driving a wheel of n3 teeth on the axis A3, and so on; the numbers of teeth in drivers being denoted by N′s, and in followers by n’s, and the axes to which the wheels are fixed being denoted by numbers. Then the resulting velocity ratio is denoted by

αm | = | α2 | · | α3 | · &c. ... | αm | = | N1 · N2 ... &c. ... Nm−1 | ;  
---|---|---|---|---|---|---|---|---|---  
α1 | α1 | α2 | αm−1 | n2 · n3 ... &c. ... nm  
  
(41)

that is to say, the velocity ratio of the last and first axes is the ratio of the product of the numbers of teeth in the drivers to the product of the numbers of teeth in the followers.

Supposing all the wheels to be in outside gearing, then, as each elementary combination reverses the direction of rotation, and as the number of elementary combinations m − 1 is one less than the number of axes m, it is evident that if m is odd the direction of rotation is preserved, and if even reversed.

It is often a question of importance to determine the number of teeth in a train of wheels best suited for giving a determinate velocity ratio to two axes. It was shown by Young that, to do this with the least total number of teeth, the velocity ratio of each elementary combination should approximate as nearly as possible to 3.59. This would in many cases give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6 ought to be the limit of the velocity ratio of an elementary combination in wheel-work. The smallest number of teeth in a pinion for epicycloidal teeth ought to be twelve (see § 49)—but it is better, for smoothness of motion, not to go below fifteen; and for involute teeth the smallest number is about twenty-four.

Let B/C be the velocity ratio required, reduced to its least terms, and let B be greater than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of teeth (which may be called t) and its double 2t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are, if possible, to be resolved into factors, and those factors (or if they are too small, multiples of them) used for the number of teeth. Should B or C, or both, be at once inconveniently large and prime, then, instead of the exact ratio B/C some ratio approximating to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.

Should B/C be greater than 6, the best number of elementary combinations m − 1 will lie between

log B − log C | and | log B − log C | .  
---|---|---|---  
log 6 | log 3  
  
Then, if possible, B and C themselves are to be resolved each into m − 1 factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.

So far as the resultant velocity ratio is concerned, the order of the drivers N and of the followers n is immaterial: but to secure equable wear of the teeth, as explained in § 44, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and n shall be either 1, or as small as possible.

§ 70. Double Hooke’s Coupling.—It has been shown in § 66 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cos θ and 1/cos θ. Hence one or both of the shafts must have a vibratory and unsteady motion, injurious to the mechanism and framework. To obviate this evil a short intermediate shaft is introduced, making equal angles with the first and last shaft, coupled with each of them by a Hooke’s joint, and having its own two forks in the same plane. Let α1, α2, α3 be the angular velocities of the first, intermediate, and last shaft in this train of two Hooke’s couplings. Then, from the principles of § 60 it is evident that at each instant α2/α1 = α2/α3, and consequently that α3 = α1; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant.

§ 71. Converging and Diverging Trains of Mechanism.—Two or more trains of mechanism may converge into one—as when the two pistons of a pair of steam-engines, each through its own connecting-rod, act upon one crank-shaft. One train of mechanism may diverge into two or more—as when a single shaft, driven by a prime mover, carries several pulleys, each of which drives a different machine. The principles of comparative motion in such converging and diverging trains are the same as in simple trains.

Division 6.—Aggregate Combinations.

§ 72. General Principles.—Willis designated as “aggregate combinations” those assemblages of pieces of mechanism in which the motion of one follower is the resultant of component motions impressed on it by more than one driver. Two classes of aggregate combinations may be distinguished which, though not different in their actual nature, differ in the data which they present to the designer, and in the method of solution to be followed in questions respecting them.

Class I. comprises those cases in which a piece A is not carried directly by the frame C, but by another piece B, relatively to which the motion of A is given—the motion of the piece B relatively to the frame C being also given. Then the motion of A relatively to the frame C is the resultant of the motion of A relatively to B and of B relatively to C; and that resultant is to be found by the principles already explained in Division 3 of this Chapter §§ 27-32.

Class II. comprises those cases in which the motions of three points in one follower are determined by their connexions with two or with three different drivers.

This classification is founded on the kinds of problems arising from the combinations. Willis adopts another classification founded on the objects of the combinations, which objects he divides into two classes, viz. (1) to produce aggregate velocity, or a velocity which is the resultant of two or more components in the same path, and (2) to produce an aggregate path—that is, to make a given point in a rigid body move in an assigned path by communicating certain motions to other points in that body.

It is seldom that one of these effects is produced without at the same time producing the other; but the classification of Willis depends upon which of those two effects, even supposing them to occur together, is the practical object of the mechanism.  
  
---  
Fig. 112.  
  
§ 73. Differential Windlass.—The axis C (fig. 112) carries a larger barrel AE and a smaller barrel DB, rotating as one piece with the angular velocity α1 in the direction AE. The pulley or sheave FG has a weight W hung to its centre. A cord has one end made fast to and wrapped round the barrel AE; it passes from A under the sheave FG, and has the other end wrapped round and made fast to the barrel BD. Required the relation between the velocity of translation v2 of W and the angular velocity α1 of the differential barrel.

In this case v2 is an aggregate velocity, produced by the joint action of the two drivers AE and BD, transmitted by wrapping connectors to FG, and combined by that sheave so as to act on the follower W, whose motion is the same with that of the centre of FG.

The velocity of the point F is α1·AC, upward motion being considered positive. The velocity of the point G is −α1·CB, downward motion being negative. Hence the instantaneous axis of the sheave FG is in the diameter FG, at the distance

FG | · | AC − BC  
---|---|---  
2 | AC + BC  
  
from the centre towards G; the angular velocity of the sheave is

α2 = α1 · | AC + BC | ;  
---|---|---  
FG  
  
and, consequently, the velocity of its centre is

v2 = α2 · | FG | · | AC − BC | = | α1 (AC − BC) | ,  
---|---|---|---|---|---|---  
2 | AC + BC | 2  
  
(42)

or the mean between the velocities of the two vertical parts of the cord.

If the cord be fixed to the framework at the point B, instead of being wound on a barrel, the velocity of W is half that of AF.

A case containing several sheaves is called a block. A fall-block is attached to a fixed point; a running-block is movable to and from a fall-block, with which it is connected by two or more plies of a rope. The whole combination constitutes a tackle or purchase. (See Pulleys for practical applications of these principles.)

§ 74. Differential Screw.—On the same axis let there be two screws of the respective pitches p1 and p2, made in one piece, and rotating with the angular velocity α. Let this piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut A. The velocity of advance of B relatively to C is (according to § 32) αp1, and of A relatively to B (according to § 57) −αp2; hence the velocity of A relatively to C is

α (p1 − p2),

(46)

being the same with the velocity of advance of a screw of the pitch p1 − p2. This combination, called Hunter’s or the differential screw, combines the strength of a large thread with the slowness of motion due to a small one.

§ 75. Epicyclic Trains.—The term epicyclic train is used by Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels or assist in driving them. The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in §§ 30, 31.

§ 76. Link Motion.—A slide valve operated by a link motion receives an aggregate motion from the mechanism driving it. (See Steam-engine for a description of this and other types of mechanism of this class.)  
  
---  
Fig. 113.  
  
§ 77. Parallel Motions.—A parallel motion is a combination of turning pieces in mechanism designed to guide the motion of a reciprocating piece either exactly or approximately in a straight line, so as to avoid the friction which arises from the use of straight guides for that purpose.

Fig. 113 represents an exact parallel motion, first proposed, it is believed, by Scott Russell. The arm CD turns on the axis C, and is jointed at D to the middle of the bar ADB, whose length is double of that of CD, and one of whose ends B is jointed to a slider, sliding in straight guides along the line CB. Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis of the bar ADB; and the direction of motion of A is at every instant perpendicular to EA—that is, along the straight line ACa. While the stroke of A is ACa, extending to equal distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by the sliding of B through a comparatively short stroke, and by rotatory motions at the joints C, D, B.  
  
---  
Fig. 114. |  Fig. 115.  
  
§ 78.* An example of an approximate straight-line motion composed of three bars fixed to a frame is shown in fig. 114. It is due to P. L. Tchebichev of St Petersburg. The links AB and CD are equal in length and are centred respectively at A and C. The ends D and B are joined by a link DB. If the respective lengths are made in the proportions AC : CD : DB = 1 : 1.3 : 0.4 the middle point P of DB will describe an approximately straight line parallel to AC within limits of length about equal to AC. C. N. Peaucellier, a French engineer officer, was the first, in 1864, to invent a linkwork with which an exact straight line could be drawn. The linkwork is shown in fig. 115, from which it will be seen that it consists of a rhombus of four equal bars ABCD, jointed at opposite corners with two equal bars BE and DE. The seventh link AF is equal in length to halt the distance EA when the mechanism is in its central position. The points E and F are fixed. It can be proved that the point C always moves in a straight line at right angles to the line EF. The more general property of the mechanism corresponding to proportions between the lengths FA and EF other than that of equality is that the curve described by the point C is the inverse of the curve described by A. There are other arrangements of bars giving straight-line motions, and these arrangements together with the general properties of mechanisms of this kind are discussed in How to Draw a Straight Line by A. B. Kempe (London, 1877).  
  
---  
Fig. 116.  
Fig. 117.  
  
§ 79.* The Pantograph.—If a parallelogram of links (fig. 116), be fixed at any one point a in any one of the links produced in either direction, and if any straight line be drawn from this point to cut the links in the points b and c, then the points a, b, c will be in a straight line for all positions of the mechanism, and if the point b be guided in any curve whatever, the point c will trace a similar curve to a scale enlarged in the ratio ab : ac. This property of the parallelogram is utilized in the construction of the pantograph, an instrument used for obtaining a copy of a map or drawing on a different scale. Professor J. J. Sylvester discovered that this property of the parallelogram is not confined to points lying in one line with the fixed point. Thus if b (fig. 117) be any point on the link CD, and if a point c be taken on the link DE such that the triangles CbD and DcE are similar and similarly situated with regard to their respective links, then the ratio of the distances ab and ac is constant, and the angle bac is constant for all positions of the mechanism; so that, if b is guided in any curve, the point c will describe a similar curve turned through an angle bac, the scales of the curves being in the ratio ab to ac. Sylvester called an instrument based on this property a plagiograph or a skew pantograph.

The combination of the parallelogram with a straight-line motion, for guiding one of the points in a straight line, is illustrated in Watt’s parallel motion for steam-engines. (See Steam-engine.)

§ 80.* The Reuleaux System of Analysis.—If two pieces, A and B, (fig. 118) are jointed together by a pin, the pin being fixed, say, to A, the only relative motion possible between the pieces is one of turning about the axis of the pin. Whatever motion the pair of pieces may have as a whole each separate piece shares in common, and this common motion in no way affects the relative motion of A and B. The motion of one piece is said to be completely constrained relatively to the other piece. Again, the pieces A and B (fig. 119) are paired together as a slide, and the only relative motion possible between them now is that of sliding, and therefore the motion of one relatively to the other is completely constrained. The pieces may be paired together as a screw and nut, in which case the relative motion is compounded of turning with sliding.  
  
---  
Fig. 118. | Fig. 119.  
  
These combinations of pieces are known individually as kinematic pairs of elements, or briefly kinematic pairs. The three pairs mentioned above have each the peculiarity that contact between the two pieces forming the pair is distributed over a surface. Kinematic pairs which have surface contact are classified as lower pairs. Kinematic pairs in which contact takes place along a line only are classified as higher pairs. A pair of spur wheels in gear is an example of a higher pair, because the wheels have contact between their teeth along lines only.

A kinematic link of the simplest form is made by joining up the halves of two kinematic pairs by means of a rigid link. Thus if A1B1 represent a turning pair, and A2B2 a second turning pair, the rigid link formed by joining B1 to B2 is a kinematic link. Four links of this kind are shown in fig. 120 joined up to form a closed kinematic chain.  
  
---  
Fig. 120.  
  
In order that a kinematic chain may be made the basis of a mechanism, every point in any link of it must be completely constrained with regard to every other link. Thus in fig. 120 the motion of a point a in the link A1A2 is completely constrained with regard to the link B1B4 by the turning pair A1B1, and it can be proved that the motion of a relatively to the non-adjacent link A3A4 is completely constrained, and therefore the four-bar chain, as it is called, can be and is used as the basis of many mechanisms. Another way of considering the question of constraint is to imagine any one link of the chain fixed; then, however the chain be moved, the path of a point, as a, will always remain the same. In a five-bar chain, if a is a point in a link non-adjacent to a fixed link, its path is indeterminate. Still another way of stating the matter is to say that, if any one link in the chain be fixed, any point in the chain must have only one degree of freedom. In a five-bar chain a point, as a, in a link non-adjacent to the fixed link has two degrees of freedom and the chain cannot therefore be used for a mechanism. These principles may be applied to examine any possible combination of links forming a kinematic chain in order to test its suitability for use as a mechanism. Compound chains are formed by the superposition of two or more simple chains, and in these more complex chains links will be found carrying three, or even more, halves of kinematic pairs. The Joy valve gear mechanism is a good example of a compound kinematic chain.  
  
---  
Fig. 121.  
  
A chain built up of three turning pairs and one sliding pair, and known as the slider crank chain, is shown in fig. 121. It will be seen that the piece A1 can only slide relatively to the piece B1, and these two pieces therefore form the sliding pair. The piece A1 carries the pin B4, which is one half of the turning pair A4 B4. The piece A1 together with the pin B4 therefore form a kinematic link A1B4. The other links of the chain are, B1A2, B2B3, A3A4. In order to convert a chain into a mechanism it is necessary to fix one link in it. Any one of the links may be fixed. It follows therefore that there are as many possible mechanisms as there are links in the chain. For example, there is a well-known mechanism corresponding to the fixing of three of the four links of the slider crank chain (fig. 121). If the link d is fixed the chain at once becomes the mechanism of the ordinary steam engine; if the link e is fixed the mechanism obtained is that of the oscillating cylinder steam engine; if the link c is fixed the mechanism becomes either the Whitworth quick-return motion or the slot-bar motion, depending upon the proportion between the lengths of the links c and e. These different mechanisms are called inversions of the slider crank chain. What was the fixed framework of the mechanism in one case becomes a moving link in an inversion.

The Reuleaux system, therefore, consists essentially of the analysis of every mechanism into a kinematic chain, and since each link of the chain may be the fixed frame of a mechanism quite diverse mechanisms are found to be merely inversions of the same kinematic chain. Franz Reuleaux’s Kinematics of Machinery, translated by Sir A. B. W. Kennedy (London, 1876), is the book in which the system is set forth in all its completeness. In Mechanics of Machinery, by Sir A. B. W. Kennedy (London, 1886), the system was used for the first time in an English textbook, and now it has found its way into most modern textbooks relating to the subject of mechanism.

§ 81.* Centrodes, Instantaneous Centres, Velocity Image, Velocity Diagram.—Problems concerning the relative motion of the several parts of a kinematic chain may be considered in two ways, in addition to the way hitherto used in this article and based on the principle of § 34. The first is by the method of instantaneous centres, already exemplified in § 63, and rolling centroids, developed by Reuleaux in connexion with his method of analysis. The second is by means of Professor R. H. Smith’s method already referred to in § 23.

Method 1.—By reference to § 30 it will be seen that the motion of a cylinder rolling on a fixed cylinder is one of rotation about an instantaneous axis T, and that the velocity both as regards direction and magnitude is the same as if the rolling piece B were for the instant turning about a fixed axis coincident with the instantaneous axis. If the rolling cylinder B and its path A now be assumed to receive a common plane motion, what was before the velocity of the point P becomes the velocity of P relatively to the cylinder A, since the motion of B relatively to A still takes place about the instantaneous axis T. If B stops rolling, then the two cylinders continue to move as though they were parts of a rigid body. Notice that the shape of either rolling curve (fig. 91 or 92) may be found by considering each fixed in turn and then tracing out the locus of the instantaneous axis. These rolling cylinders are sometimes called axodes, and a section of an axode in a plane parallel to the plane of motion is called a centrode. The axode is hence the locus of the instantaneous axis, whilst the centrode is the locus of the instantaneous centre in any plane parallel to the plane of motion. There is no restriction on the shape of these rolling axodes; they may have any shape consistent with rolling (that is, no slipping is permitted), and the relative velocity of a point P is still found by considering it with regard to the instantaneous centre.

Reuleaux has shown that the relative motion of any pair of non-adjacent links of a kinematic chain is determined by the rolling together of two ideal cylindrical surfaces (cylindrical being used here in the general sense), each of which may be assumed to be formed by the extension of the material of the link to which it corresponds. These surfaces have contact at the instantaneous axis, which is now called the instantaneous axis of the two links concerned. To find the form of these surfaces corresponding to a particular pair of non-adjacent links, consider each link of the pair fixed in turn, then the locus of the instantaneous axis is the axode corresponding to the fixed link, or, considering a plane of motion only, the locus of the instantaneous centre is the centrode corresponding to the fixed link.

To find the instantaneous centre for a particular link corresponding to any given configuration of the kinematic chain, it is only necessary to know the direction of motion of any two points in the link, since lines through these points respectively at right angles to their directions of motion intersect in the instantaneous centre.  
  
---  
Fig. 122.  
  
To illustrate this principle, consider the four-bar chain shown in fig. 122 made up of the four links, a, b, c, d. Let a be the fixed link, and consider the link c. Its extremities are moving respectively in directions at right angles to the links b and d; hence produce the links b and d to meet in the point Oac. This point is the instantaneous centre of the motion of the link c relatively to the fixed link a, a fact indicated by the suffix ac placed after the letter O. The process being repeated for different values of the angle θ the curve through the several points Oac is the centroid which may be imagined as formed by an extension of the material of the link a. To find the corresponding centroid for the link c, fix c and repeat the process. Again, imagine d fixed, then the instantaneous centre Obd of b with regard to d is found by producing the links c and a to intersect in Obd, and the shapes of the centroids belonging respectively to the links b and d can be found as before. The axis about which a pair of adjacent links turn is a permanent axis, and is of course the axis of the pin which forms the point. Adding the centres corresponding to these several axes to the figure, it will be seen that there are six centres in connexion with the four-bar chain of which four are permanent and two are instantaneous or virtual centres; and, further, that whatever be the configuration of the chain these centres group themselves into three sets of three, each set lying on a straight line. This peculiarity is not an accident or a special property of the four-bar chain, but is an illustration of a general law regarding the subject discovered by Aronhold and Sir A. B. W. Kennedy independently, which may be thus stated: If any three bodies, a, b, c, have plane motion their three virtual centres, Oab, Obc, Oac, are three points on one straight line. A proof of this will be found in The Mechanics of Machinery quoted above. Having obtained the set of instantaneous centres for a chain, suppose a is the fixed link of the chain and c any other link; then Oac is the instantaneous centre of the two links and may be considered for the instant as the trace of an axis fixed to an extension of the link a about which c is turning, and thus problems of instantaneous velocity concerning the link c are solved as though the link c were merely rotating for the instant about a fixed axis coincident with the instantaneous axis.  
  
---  
Fig. 123. | Fig. 124.  
  
Method 2.—The second method is based upon the vector representation of velocity, and may be illustrated by applying it to the four-bar chain. Let AD (fig. 123) be the fixed link. Consider the link BC, and let it be required to find the velocity of the point B having given the velocity of the point C. The principle upon which the solution is based is that the only motion which B can have relatively to an axis through C fixed to the link CD is one of turning about C. Choose any pole O (fig. 124). From this pole set out Oc to represent the velocity of the point C. The direction of this must be at right angles to the line CD, because this is the only direction possible to the point C. If the link BC moves without turning, Oc will also represent the velocity of the point B; but, if the link is turning, B can only move about the axis C, and its direction of motion is therefore at right angles to the line CB. Hence set out the possible direction of B′s motion in the velocity diagram, namely cb1, at right angles to CB. But the point B must also move at right angles to AB in the case under consideration. Hence draw a line through O in the velocity diagram at right angles to AB to cut cb1 in b. Then Ob is the velocity of the point b in magnitude and direction, and cb is the tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the velocities, the instantaneous velocity ratio of the points C and B is given by the ratio Oc/Ob.

A most important property of the diagram (figs. 123 and 124) is the following: If points X and x are taken dividing the link BC and the tangential velocity cb, so that cx:xb = CX:XB, then Ox represents the velocity of the point X in magnitude and direction. The line cb has been called the velocity image of the rod, since it may be looked upon as a scale drawing of the rod turned through 90° from the actual rod. Or, put in another way, if the link CB is drawn to scale on the new length cb in the velocity diagram (fig. 124), then a vector drawn from O to any point on the new drawing of the rod will represent the velocity of that point of the actual rod in magnitude and direction. It will be understood that there is a new velocity diagram for every new configuration of the mechanism, and that in each new diagram the image of the rod will be different in scale. Following the method indicated above for a kinematic chain in general, there will be obtained a velocity diagram similar to that of fig. 124 for each configuration of the mechanism, a diagram in which the velocity of the several points in the chain utilized for drawing the diagram will appear to the same scale, all radiating from the pole O. The lines joining the ends of these several velocities are the several tangential velocities, each being the velocity image of a link in the chain. These several images are not to the same scale, so that although the images may be considered to form collectively an image of the chain itself, the several members of this chain-image are to different scales in any one velocity diagram, and thus the chain-image is distorted from the actual proportions of the mechanism which it represents.  
  
---  
Fig. 125.  
  
§ 82.* Acceleration Diagram. Acceleration Image.—Although it is possible to obtain the acceleration of points in a kinematic chain with one link fixed by methods which utilize the instantaneous centres of the chain, the vector method more readily lends itself to this purpose. It should be understood that the instantaneous centre considered in the preceding paragraphs is available only for estimating relative velocities; it cannot be used in a similar manner for questions regarding acceleration. That is to say, although the instantaneous centre is a centre of no velocity for the instant, it is not a centre of no acceleration, and in fact the centre of no acceleration is in general a quite different point. The general principle on which the method of drawing an acceleration diagram depends is that if a link CB (fig. 125) have plane motion and the acceleration of any point C be given in magnitude and direction, the acceleration of any other point B is the vector sum of the acceleration of C, the radial acceleration of B about C and the tangential acceleration of B about C. Let A be any origin, and let Ac represent the acceleration of the point C, ct the radial acceleration of B about C which must be in a direction parallel to BC, and tb the tangential acceleration of B about C, which must of course be at right angles to ct; then the vector sum of these three magnitudes is Ab, and this vector represents the acceleration of the point B. The directions of the radial and tangential accelerations of the point B are always known when the position of the link is assigned, since these are to be drawn respectively parallel to and at right angles to the link itself. The magnitude of the radial acceleration is given by the expression v2/BC, v being the velocity of the point B about the point C. This velocity can always be found from the velocity diagram of the chain of which the link forms a part. If dw/dt is the angular acceleration of the link, dw/dt × CB is the tangential acceleration of the point B about the point C. Generally this tangential acceleration is unknown in magnitude, and it becomes part of the problem to find it. An important property of the diagram is that if points X and x are taken dividing the link CB and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents the acceleration of the point X in magnitude and direction; cb is called the acceleration image of the rod. In applying this principle to the drawing of an acceleration diagram for a mechanism, the velocity diagram of the mechanism must be first drawn in order to afford the means of calculating the several radial accelerations of the links. Then assuming that the acceleration of one point of a particular link of the mechanism is known together with the corresponding configuration of the mechanism, the two vectors Ac and ct can be drawn. The direction of tb, the third vector in the diagram, is also known, so that the problem is reduced to the condition that b is somewhere on the line tb. Then other conditions consequent upon the fact that the link forms part of a kinematic chain operate to enable b to be fixed. These methods are set forth and exemplified in Graphics, by R. H. Smith (London, 1889). Examples, completely worked out, of velocity and acceleration diagrams for the slider crank chain, the four-bar chain, and the mechanism of the Joy valve gear will be found in ch. ix. of Valves and Valve Gear Mechanism, by W. E. Dalby (London, 1906).

Chapter II. On Applied Dynamics.

§ 83. Laws of Motion.—The action of a machine in transmitting force and motion simultaneously, or performing work, is governed, in common with the phenomena of moving bodies in general, by two “laws of motion.”

Division 1. Balanced Forces in Machines of Uniform Velocity.

§ 84. Application of Force to Mechanism.—Forces are applied in units of weight; and the unit most commonly employed in Britain is the pound avoirdupois. The action of a force applied to a body is always in reality distributed over some definite space, either a volume of three dimensions or a surface of two. An example of a force distributed throughout a volume is the weight of the body itself, which acts on every particle, however small. The pressure exerted between two bodies at their surface of contact, or between the two parts of one body on either side of an ideal surface of separation, is an example of a force distributed over a surface. The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of; but, in questions respecting the action of a force upon a rigid body considered as a whole, the resultant of the distributed force, determined according to the principles of statics, and considered as acting in a single line and applied at a single point, may, for the occasion, be substituted for the force as really distributed. Thus, the weight of each separate piece in a machine is treated as acting wholly at its centre of gravity, and each pressure applied to it as acting at a point called the centre of pressure of the surface to which the pressure is really applied.

§ 85. Forces applied to Mechanism Classed.—If θ be the obliquity of a force F applied to a piece of a machine—that is, the angle made by the direction of the force with the direction of motion of its point of application—then by the principles of statics, F may be resolved into two rectangular components, viz.:—

Along the direction of motion, P = F cos θ  
Across the direction of motion, Q = F sin θ

(49)

If the component along the direction of motion acts with the motion, it is called an effort; if against the motion, a resistance. The component across the direction of motion is a lateral pressure; the unbalanced lateral pressure on any piece, or part of a piece, is deflecting force. A lateral pressure may increase resistance by causing friction; the friction so caused acts against the motion, and is a resistance, but the lateral pressure causing it is not a resistance. Resistances are distinguished into useful and prejudicial, according as they arise from the useful effect produced by the machine or from other causes.

§ 86. Work.—Work consists in moving against resistance. The work is said to be performed, and the resistance overcome. Work is measured by the product of the resistance into the distance through which its point of application is moved. The unit of work commonly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a foot-pound.

Work is distinguished into useful work and prejudicial or lost work, according as it is performed in producing the useful effect of the machine, or in overcoming prejudicial resistance.

§ 87. Energy: Potential Energy.—Energy means capacity for performing work. The energy of an effort, or potential energy, is measured by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work.

When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved.

§ 88. Variable Effort and Resistance.—If an effort has different magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length Δs of the corresponding portion of the path of the point of application; the sum

Σ · PΔs

(50)

is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum approaches continually as the divisions of the path are made smaller and more numerous, and is expressed by

∫ P ds.

(51)

Similar processes are applicable to the finding of the work performed in overcoming a varying resistance.

The work done by a machine can be actually measured by means of a dynamometer (q.v.).

§ 89. Principle of the Equality of Energy and Work.—From the first law of motion it follows that in a machine whose pieces move with uniform velocities the efforts and resistances must balance each other. Now from the laws of statics it is known that, in order that a system of forces applied to a system of connected points may be in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero,—products being considered positive or negative according as the direction of the forces and the possible motions of their points of application are the same or opposite.

In other words, the sum of the negative products is equal to the sum of the positive products. This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time the energy exerted is equal to the work performed.

The symbolical expression of this law is as follows: let efforts be applied to one or any number of points of a machine; let any one of these efforts be represented by P, and the distance traversed by its point of application in a given interval of time by ds; let resistances be overcome at one or any number of points of the same machine; let any one of these resistances be denoted by R, and the distance traversed by its point of application in the given interval of time by ds′; then

Σ · P ds = Σ · R ds′.

(52)

The lengths ds, ds′ are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.

§ 90. Static Equilibrium of Mechanisms.—The principle stated in the preceding section, namely, that the energy exerted is equal to the work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds′ were made, it becomes P ds/dt = R ds′/dt, that is, Pv = Rv′, which shows that the force ratio is the inverse of the velocity ratio. It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points. The relation between the effort and the resistance may be found by means of this principle for all kinds of mechanisms, when the friction produced by the components of the forces across the direction of motion of the two points is neglected. Consider the following example:—  
  
---  
Fig. 126.  
  
A four-bar chain having the configuration shown in fig. 126 supports a load P at the point x. What load is required at the point y to maintain the configuration shown, both loads being supposed to act vertically? Find the instantaneous centre Obd, and resolve each load in the respective directions of motion of the points x and y; thus there are obtained the components P cos θ and R cos φ. Let the mechanism have a small motion; then, for the instant, the link b is turning about its instantaneous centre Obd, and, if ω is its instantaneous angular velocity, the velocity of the point x is ωr, and the velocity of the point y is ωs. Hence, by the principle just stated, P cos θ × ωr = R cos φ × ωs. But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to Pp = Rq, which shows that the ratio of the two forces may be found by taking moments about the instantaneous centre of the link on which they act.

The forces P and R may, however, act on different links. The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force ƒs, acting on the link s, to find the force ƒt acting in a given direction on the link t, which will keep the mechanism in static equilibrium. The graphic solution of this problem may be effected thus:—

(1) Find the three virtual centres Ors, Ort, Ost, which must be three points in a line.

(2) Resolve ƒs into two components, one of which, namely, ƒq, passes through Ors and may be neglected, and the other ƒp passes through Ost.

(3) Find the point M, where ƒp joins the given direction of ƒt, and resolve ƒp into two components, of which one is in the direction MOrt, and may be neglected because it passes through Ort, and the other is in the given direction of ƒt and is therefore the force required.  
  
---  
Fig. 127.  
  
This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is given in ch. 8 of his Mechanics of Machinery. Another general solution of the problem is given in the Proc. Lond. Math. Soc. (1878-1879), by the same author. An example of the method of solution stated above, and taken from the Mechanics of Machinery, is illustrated by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r fixed. Let it be required to find the vertical force which must act at the pitch radius of the last wheel t to balance exactly a force ƒs acting vertically downwards on the arm at the point indicated in the figure. The two links concerned are the last wheel t and the arm s, the wheel r being the fixed link of the mechanism. The virtual centres Ors, Ost are at the respective axes of the wheels r and t, and the centre Ort divides the line through these two points externally in the ratio of the train of wheels. The figure sufficiently indicates the various steps of the solution.

The relation between the effort and the resistance in a machine to include the effect of friction at the joints has been investigated in a paper by Professor Fleeming Jenkin, “On the application of graphic methods to the determination of the efficiency of machinery” (Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented by a frame of links the stresses in which are identical with the pressures at the joints of the mechanism. This self-strained frame is called the dynamic frame of the machine. The driving and resisting efforts are represented by elastic links in the dynamic frame, and when the frame with its elastic links is drawn the stresses in the several members of it may be determined by means of reciprocal figures. Incidentally the method gives the pressures at every joint of the mechanism.

§ 91. Efficiency.—The efficiency of a machine is the ratio of the useful work to the total work—that is, to the energy exerted—and is represented by

Σ · Ruds′ | = | Σ · Ru ds′ | = | Σ · Ru ds′ | = | U | .  
---|---|---|---|---|---|---|---  
Σ · R ds′ | Σ · Ru ds′ \+ Σ · Rp ds′ | Σ · P ds | E  
  
(53)

Ru being taken to represent useful and Rp prejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine.

§ 92. Power and Effect.—The power of a machine is the energy exerted, and the effect the useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.

The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.

§ 93. Modulus of a Machine.—In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances arc generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of the lost work, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine. The general form of the modulus may be expressed thus—

E = U + φ (U, A) + ψ (A),

(54)

where A denotes some quantity or set of quantities depending on the form, arrangement, weight and other properties of the mechanism. Moseley, however, has pointed out that in most cases this equation takes the much more simple form of

E = (1 + A) U + B,

(55)

where A and B are constants, depending on the form, arrangement and weight of the mechanism. The efficiency corresponding to the last equation is

U | = | 1 | .  
---|---|---|---  
E | 1 + A + B/U  
  
(56)

§ 94. Trains of Mechanism.—In applying the preceding principles to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train—each section being treated as a machine, driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances. It is evident that the efficiency of the whole train is the product of the efficiencies of its sections.

§ 95. Rotating Pieces: Couples of Forces.—It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of the moment of the couples of forces acting on it, and of the angular velocity. The ordinary British unit of moment is a foot-pound; but it is to be remembered that this is a foot-pound of a different sort from the unit of energy and work.

If a force be applied to a turning piece in a line not passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the first-mentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force.

A couple is said to be right or left handed with reference to the observer, according to the direction in which it tends to turn the body, and is a driving couple or a resisting couple according as its tendency is with or against that of the actual rotation.

Let dt be an interval of time, α the angular velocity of the piece; then αdt is the angle through which it turns in the interval dt, and ds = v dt = rα dt is the distance through which the point of application of the force moves. Let P represent an effort, so that Pr is a driving couple, then

P ds = Pv dt = Prα dt = Mα dt

(57)

is the energy exerted by the couple M in the interval dt; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece.

§ 96. Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece.—In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, the equivalent force or couple applied to some other point or piece; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.

These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.

§ 97. Balanced Lateral Pressure of Guides and Bearings.—The most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel.

§ 98. Friction. Unguents.—The most important kind of resistance in machines is the friction or rubbing resistance of surfaces which slide over each other. The direction of the resistance of friction is opposite to that in which the sliding takes place. Its magnitude is the product of the normal pressure or force which presses the rubbing surfaces together in a direction perpendicular to themselves into a specific constant already mentioned in § 14, as the coefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, ƒ the coefficient of friction, and φ the angle of repose; then

ƒ = tan φ  
R = ƒN = N tan φ = T sin φ

(58)

Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie, James Wood, D. Rankine and others. The most complete and elaborate experiments are those of Morin, published in his Notions fondamentales de mécanique, and republished in Britain in the works of Moseley and Gordon.

The experiments of Beauchamp Tower (“Report of Friction Experiments,” Proc. Inst. Mech. Eng., 1883) showed that when oil is supplied to a journal by means of an oil bath the coefficient of friction varies nearly inversely as the load on the bearing, thus making the product of the load on the bearing and the coefficient of friction a constant. Mr Tower’s experiments were carried out at nearly constant temperature. The more recent experiments of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the coefficient of friction, the load on the bearing, and the temperature is approximately constant. For further information on this point and on Osborne Reynolds’s theory of lubrication see Bearings and Lubrication.

§ 99. Work of Friction. Moment of Friction.—The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces. If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.

When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called the moment of friction. The total moment of friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of friction of their several portions.

To express this symbolically, let du represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and ƒ the coefficient of friction. Then the moment of friction of du is ƒpr du;

the total moment of friction is ƒ ∫ pr·du;

and the work performed in a unit cf time in overcoming friction, when the angular velocity is α, is αƒ ∫ pr·du.

(59)

It is evident that the moment of friction, and the work lost by being performed in overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a limit is put to the diminution of the radii of journals and pivots by the conditions of durability and of proper lubrication, and also by conditions of strength and stiffness.

§ 100. Total Pressure between Journal and Bearing.—A single piece rotating with a uniform velocity has four mutually balanced forces applied to it: (l) the effort exerted on it by the piece which drives it; (2) the resistance of the piece which follows it—which may be considered for the purposes of the present question as useful resistance; (3) its weight; and (4) the reaction of its own cylindrical bearings. There are given the following data:—

The direction of the effort.

The direction of the useful resistance.

The weight of the piece and the direction in which it acts.

The magnitude of the useful resistance.

The radius of the bearing r.

The angle of repose φ, corresponding to the friction of the journal on the bearing.

And there are required the following:—

The direction of the reaction of the bearing.

The magnitude of that reaction.

The magnitude of the effort.

Let the useful resistance and the weight of the piece be compounded by the principles of statics into one force, and let this be called the given force.  
  
---  
Fig. 128.  
  
The directions of the effort and of the given force are either parallel or meet in a point. If they are parallel, the direction of the reaction of the bearing is also parallel to them; if they meet in a point, the direction of the reaction traverses the same point.

Also, let AAA, fig. 128, be a section of the bearing, and C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle φ with the radius of that circle; that is to say, it must be a line such as PT touching the smaller circle BB, whose radius is r · sin φ. The side on which it touches that circle is determined by the fact that the obliquity of the reaction is such as to oppose the rotation.

Thus is determined the direction of the reaction of the bearing; and the magnitude of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in § 7.

The work lost in overcoming the friction of the bearing is the same as that which would be performed in overcoming at the circumference of the small circle BB a resistance equal to the whole pressure between the journal and bearing.

In order to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the “given force”) ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.

An investigation of the forces acting on a bearing and journal lubricated by an oil bath will be found in a paper by Osborne Reynolds in the Phil. Trans. pt. i. (1886). (See also Bearings.)

§ 101. Friction of Pivots and Collars.—When a shaft is acted upon by a force tending to shift it lengthways, that force must be balanced by the reaction of a bearing against a pivot at the end of the shaft; or, if that be impossible, against one or more collars, or rings projecting from the body of the shaft. The bearing of the pivot is called a step or footstep. Pivots require great hardness, and are usually made of steel. The flat pivot is a cylinder of steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the pressure is

p = N / πr2;

(60)

and, introducing this value into equation 59, the moment of friction of the flat pivot is found to be

2⁄3ƒNr

(61)

or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure.

The friction of a conical pivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.

The moment of friction of a collar is given by the formula—

2⁄3 ƒN | r3 − r′3 | ,  
---|---|---  
r2 − r′2  
  
(62)

where r is the external and r′ the internal radius.  
  
---  
Fig. 129.  
  
In the cup and ball pivot the end of the shaft and the step present two recesses facing each other, into which art fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable from the extreme smallness of the radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.

It appears that the rapidity with which a rubbing surface wears away is proportional to the friction and to the velocity jointly, or nearly so. Hence the pivots already mentioned wear unequally at different points, and tend to alter their figures. Schiele has invented a pivot which preserves its original figure by wearing equally at all points in a direction parallel to its axis. The following are the principles on which this equality of wear depends:—

The rapidity of wear of a surface measured in an oblique direction is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round OX will generate the form of pivot required. Now let PT be a tangent to the curve at P, cutting OX in T; PT = PY × secant obliquity, and this is to be a constant quantity; hence the curve is that known as the tractory of the straight line OX, in which PT = OR = constant. This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T. On that pin turns an arm, carrying at a point P a tracing-point, pencil or pen. Should the pen have a nib of two jaws, like those of an ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid along the axis from O towards X, P will be drawn after it from R towards C along the tractory. This curve, being an asymptote to its axis, is capable of being indefinitely prolonged towards X; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to stick in its bearing. The moment of friction of “Schiele’s anti-friction pivot,” as it is called, is equal to that of a cylindrical journal of the radius OR = PT the constant tangent, under the same pressure.

Records of experiments on the friction of a pivot bearing will be found in the Proc. Inst. Mech. Eng. (1891), and on the friction of a collar bearing ib. May 1888.

§ 102. Friction of Teeth.—Let N be the normal pressure exerted between a pair of teeth of a pair of wheels; s the total distance through which they slide upon each other; n the number of pairs of teeth which pass the plane of axis in a unit of time; then

nƒNs

(63)

is the work lost in unity of time by the friction of the teeth. The sliding s is composed of two parts, which take place during the approach and recess respectively. Let those be denoted by s1 and s2, so that s = s1 \+ s2. In § 45 the velocity of sliding at any instant has been given, viz. u = c (α1 \+ α2), where u is that velocity, c the distance T1 at any instant from the point of contact of the teeth to the pitch-point, and α1, α2 the respective angular velocities of the wheels.

Let v be the common velocity of the two pitch-circles, r1, r2, their radii; then the above equation becomes

u = cv ( | 1 | + | 1 | ).  
---|---|---|---|---  
r1 | r2  
  
To apply this to involute teeth, let c1 be the length of the approach, c2 that of the recess, u1, the mean volocity of sliding during the approach, u2 that during the recess; then

u1 = | c1v | ( | 1 | + | 1 | );  u2 = | c2v | ( | 1 | + | 1 | )  
---|---|---|---|---|---|---|---|---|---|---|---|---  
2 | r1 | r2 | 2 | r1 | r2  
  
also, let θ be the obliquity of the action; then the times occupied by the approach and recess are respectively

c1 | ,  | c2 | ;  
---|---|---|---  
v cos θ | v cos θ  
  
giving, finally, for the length of sliding between each pair of teeth,

s = s1 \+ s2 = | c12 \+ c22 | ( | 1 | + | 1 | )  
---|---|---|---|---|---|---  
2 cos θ | r1 | r2  
  
(64)

which, substituted in equation (63), gives the work lost in a unit of time by the friction of involute teeth. This result, which is exact for involute teeth, is approximately true for teeth of any figure.

For inside gearing, if r1 be the less radius and r2 the greater, 1/r1 − 1/r2 is to be substituted for 1/r1 \+ 1/r2.

§ 103. Friction of Cords and Belts.—A flexible band, such as a cord, rope, belt or strap, may be used either to exert an effort or a resistance upon a pulley round which it wraps. In either case the tangential force, whether effort or resistance, exerted between the band and the pulley is their mutual friction, caused by and proportional to the normal pressure between them.

Let T1 be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; T2 the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; θ the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an indefinitely small element of that arc to the radius; F = T1 − T2 the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc dθ; ƒ the coefficient of friction between the materials of the band and pulley.

Then, according to a well-known principle in statics, the normal pressure at the elementary arc dθ is T dθ, T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF = ƒT dθ. Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or dT = dF = ƒT dθ; which equation, being integrated throughout the entire arc of contact, gives the following formulae:—

hyp log. | T1 | = ƒθ  
---|---|---  
T2  
T1 | = eƒθ  
---|---  
T2  
  
F = T1 − T2 = T1 (1 − e − ƒθ) = T2 (eƒθ − 1)

(65)

When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of them drives the other by means of the belt, it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged. Its value is given by this formula—

T1 \+ T2 | = | eƒθ \+ 1  
---|---|---  
2 | 2 (eƒθ − 1)  
  
(66)

which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.

The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to stop or moderate the velocity of machines by being tightened round a pulley. The strap is usually of iron, and the pulley of hard wood.

Let α denote the arc of contact expressed in turns and fractions of a turn; then

θ = 6.2832a  
eƒθ = number whose common logarithm is 2.7288ƒa

(67)

See also Dynamometer for illustrations of the use of what are essentially friction-straps of different forms for the measurement of the brake horse-power of an engine or motor.

§ 104. Stiffness of Ropes.—Ropes offer a resistance to being bent, and, when bent, to being straightened again, which arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.

The work lost in pulling a given length of rope over a pulley is found by multiplying the length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend it into a curve fitting the pulley, and then to straighten it again.

The following empirical formulae for the stiffness of hempen ropes have been deduced by Morin from the experiments of Coulomb:—

Let F be the stiffness in pounds avoirdupois; d the diameter of the rope in inches, n = 48d2 for white ropes and 35d2 for tarred ropes; r the effective radius of the pulley in inches; T the tension in pounds. Then

For white ropes, F = | n | (0.0012 + 0.001026n + 0.0012T).  
---|---|---  
r  
For tarred ropes, F = | n | (0.006 + 0.001392n + 0.00168T).  
---|---|---  
r  
  
(68)

§ 105. Friction-Couplings.—Friction is useful as a means of communicating motion where sudden changes either of force or velocity take place, because, being limited in amount, it may be so adjusted as to limit the forces which strain the pieces of the mechanism within the bounds of safety. Amongst contrivances for effecting this object are friction-cones. A rotating shaft carries upon a cylindrical portion of its figure a wheel or pulley turning loosely on it, and consequently capable of remaining at rest when the shaft is in motion. This pulley has fixed to one side, and concentric with it, a short frustum of a hollow cone. At a small distance from the pulley the shaft carries a short frustum of a solid cone accurately turned to fit the hollow cone. This frustum is made always to turn along with the shaft by being fitted on a square portion of it, or by means of a rib and groove, or otherwise, but is capable of a slight longitudinal motion, so as to be pressed into, or withdrawn from, the hollow cone by means of a lever. When the cones are pressed together or engaged, their friction causes the pulley to rotate along with the shaft; when they are disengaged, the pulley is free to stand still. The angle made by the sides of the cones with the axis should not be less than the angle of repose. In the friction-clutch, a pulley loose on a shaft has a hoop or gland made to embrace it more or less tightly by means of a screw; this hoop has short projecting arms or ears. A fork or clutch rotates along with the shaft, and is capable of being moved longitudinally by a handle. When the clutch is moved towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to communicate its rotation to the pulley by friction. There are many other contrivances of the same class, but the two just mentioned may serve for examples.

§ 106. Heat of Friction: Unguents.—The work lost in friction is employed in producing heat. This fact is very obvious, and has been known from a remote period; but the exact determination of the proportion of the work lost to the heat produced, and the experimental proof that that proportion is the same under all circumstances and with all materials, solid, liquid and gaseous, are comparatively recent achievements of J. P. Joule. The quantity of work which produces a British unit of heat (or so much heat as elevates the temperature of one pound of pure water, at or near ordinary atmospheric temperatures, by 1° F.) is 772 foot-pounds. This constant, now designated as “Joule’s equivalent,” is the principal experimental datum of the science of thermodynamics.

A more recent determination (Phil. Trans., 1897), by Osborne Reynolds and W. M. Moorby, gives 778 as the mean value of Joule’s equivalent through the range of 32° to 212° F. See also the papers of Rowland in the Proc. Amer. Acad. (1879), and Griffiths, Phil. Trans. (1893).

The heat produced by friction, when moderate in amount, is useful in softening and liquefying thick unguents; but when excessive it is prejudicial, by decomposing the unguents, and sometimes even by softening the metal of the bearings, and raising their temperature so high as to set fire to neighbouring combustible matters.

Excessive heating is prevented by a constant and copious supply of a good unguent. The elevation of temperature produced by the friction of a journal is sometimes used as an experimental test of the quality of unguents. For modern methods of forced lubrication see Bearings.

§ 107. Rolling Resistance.—By the rolling of two surfaces over each other without sliding a resistance is caused which is called sometimes “rolling friction,” but more correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot:—

Oak upon oak | 0.006 (Coulomb).  
---|---  
Lignum vitae on oak | 0.004  ”  
Cast iron on cast iron | 0.002 (Tredgold).  
  
§ 108. Reciprocating Forces: Stored and Restored Energy.—When a force acts on a machine alternately as an effort and as a resistance, it may be called a reciprocating force. Of this kind is the weight of any piece in the mechanism whose centre of gravity alternately rises and falls; for during the rise of the centre of gravity that weight acts as a resistance, and energy is employed in lifting it to an amount expressed by the product of the weight into the vertical height of its rise; and during the fall of the centre of gravity the weight acts as an effort, and exerts in assisting to perform the work of the machine an amount of energy exactly equal to that which had previously been employed in lifting it. Thus that amount of energy is not lost, but has its operation deferred; and it is said to be stored when the weight is lifted, and restored when it falls.

In a machine of which each piece is to move with a uniform velocity, if the effort and the resistance be constant, the weight of each piece must be balanced on its axis, so that it may produce lateral pressure only, and not act as a reciprocating force. But if the effort and the resistance be alternately in excess, the uniformity of speed may still be preserved by so adjusting some moving weight in the mechanism that when the effort is in excess it may be lifted, and so balance and employ the excess of effort, and that when the resistance is in excess it may fall, and so balance and overcome the excess of resistance—thus storing the periodical excess of energy and restoring that energy to perform the periodical excess of work.

Other forces besides gravity may be used as reciprocating forces for storing and restoring energy—for example, the elasticity of a spring or of a mass of air.

In most of the delusive machines commonly called “perpetual motions,” of which so many are patented in each year, and which are expected by their inventors to perform work without receiving energy, the fundamental fallacy consists in an expectation that some reciprocating force shall restore more energy than it has been the means of storing.

Division 2. Deflecting Forces.

§ 109. Deflecting Force for Translation in a Curved Path.—In machinery, deflecting force is supplied by the tenacity of some piece, such as a crank, which guides the deflected body in its curved path, and is unbalanced, being employed in producing deflexion, and not in balancing another force.

§ 110. Centrifugal Force of a Rotating Body.—The centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body.

The particles of a rotating body exert centrifugal forces on each other, which strain the body, and tend to tear it asunder, but these forces balance each other, and do not affect the resultant centrifugal force exerted on the axis of rotation.3

If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing.

Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.

§ 111. Centrifugal Couples of a Rotating Body.—Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body to shift the axis of rotation, they may also tend to turn it out of its original direction. The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis.

It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings.

The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.  
  
---  
(From Balancing of Engines, by permission of Edward Arnold.)  
Fig. 130.  
  
§ 112.* Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis.—The method here briefly explained is taken from a paper by W. E. Dalby, “The Balancing of Engines with special reference to Marine Work,” Trans. Inst. Nav. Arch. (1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held in a bearing at one point, O. The force required to constrain the weight to move in a circle, that is the deviating force, produces an equal and opposite reaction on the shaft, whose amount F is equal to the centrifugal force Wa2r/g ℔, where r is the radius of the mass centre of the weight, and a is its angular velocity in radians per second. Transferring this force to the point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point O

(1) ΣF = 0  
(2) ΣFa = 0

(a)

The plane through O to which the shaft is perpendicular is called the reference plane, because all the transferred forces act in that plane at the point O. The plane through the radius of the weight containing the axis OX is called the axial plane because it contains the forces forming the couple due to the transference of F to the reference plane. Substituting the values of F in (a) the two conditions become

(1) (W1r1 \+ W2r2 \+ W3r3 \+ ...) | α2 | = 0  
---|---|---  
g  
(2) (W1a1r1 \+ W2a2r2 \+ ... ) | α2 | = 0  
---|---|---  
g  
  
(b)

In order that these conditions may obtain, the quantities in the brackets must be zero, since the factor α2/g is not zero. Hence finally the conditions which must be satisfied by the system of weights in order that the axis of rotation may be a permanent axis is

(1) (W1r1 \+ W2r2 \+ W3r3) = 0  
---  
(2) (W1a1r1 \+ W2a2r2 \+ W3a3r3) = 0  
  
(c)

It must be remembered that these are all directed quantities, and that their respective sums are to be taken by drawing vector polygons. In drawing these polygons the magnitude of the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft outwards towards the weight W, parallel to the radius r. For the vector representing a couple of the type War, if the masses are all on the same side of the reference plane, the direction of drawing is from the axis outwards; if the masses are some on one side of the reference plane and some on the other side, the direction of drawing is from the axis outwards towards the weight for all masses on the one side, and from the mass inwards towards the axis for all weights on the other side, drawing always parallel to the direction defined by the radius r. The magnitude of the vector is the product War. The conditions (c) may thus be expressed: first, that the sum of the vectors Wr must form a closed polygon, and, second, that the sum of the vectors War must form a closed polygon. The general problem in practice is, given a system of weights attached to a shaft, to find the respective weights and positions of two balance weights or counterpoises which must be added to the system in order to make the shaft a permanent axis, the planes in which the balance weights are to revolve also being given. To solve this the reference plane must be chosen so that it coincides with the plane of revolution of one of the as yet unknown balance weights. The balance weight in this plane has therefore no couple corresponding to it. Hence by drawing a couple polygon for the given weights the vector which is required to close the polygon is at once found and from it the magnitude and position of the balance weight which must be added to the system to balance the couples follow at once. Then, transferring the product Wr corresponding with this balance weight to the reference plane, proceed to draw the force polygon. The vector required to close it will determine the second balance weight, the work may be checked by taking the reference plane to coincide with the plane of revolution of the second balance weight and then re-determining them, or by taking a reference plane anywhere and including the two balance weights trying if condition (c) is satisfied.

When a weight is reciprocated, the equal and opposite force required for its acceleration at any instant appears as an unbalanced force on the frame of the machine to which the weight belongs. In the particular case, where the motion is of the kind known as “simple harmonic” the disturbing force on the frame due to the reciprocation of the weight is equal to the component of the centrifugal force in the line of stroke due to a weight equal to the reciprocated weight supposed concentrated at the crank pin. Using this principle the method of finding the balance weights to be added to a given system of reciprocating weights in order to produce a system of forces on the frame continuously in equilibrium is exactly the same as that just explained for a system of revolving weights, because for the purpose of finding the balance weights each reciprocating weight may be supposed attached to the crank pin which operates it, thus forming an equivalent revolving system. The balance weights found as part of the equivalent revolving system when reciprocated by their respective crank pins form the balance weights for the given reciprocating system. These conditions may be exactly realized by a system of weights reciprocated by slotted bars, the crank shaft driving the slotted bars rotating uniformly. In practice reciprocation is usually effected through a connecting rod, as in the case of steam engines. In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin. The moving parts of the engine are then divided into two complete and independent systems, namely, one system of revolving weights consisting of crank pins, crank arms, &c., attached to and revolving with the crank shaft, and a second system of reciprocating weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line of stroke with simple harmonic motion. The balance weights are to be separately calculated for each system, the one set being added to the crank shaft as revolving weights, and the second set being included with the reciprocating weights and operated by a properly placed crank on the crank shaft. Balance weights added in this way to a set of reciprocating weights are sometimes called bob-weights. In the case of locomotives the balance weights required to balance the pistons are added as revolving weights to the crank shaft system, and in fact are generally combined with the weights required to balance the revolving system so as to form one weight, the counterpoise referred to in the preceding section, which is seen between the spokes of the wheels of a locomotive. Although this method balances the pistons in the horizontal plane, and thus allows the pull of the engine on the train to be exerted without the variation due to the reciprocation of the pistons, yet the force balanced horizontally is introduced vertically and appears as a variation of pressure on the rail. In practice about two-thirds of the reciprocating weight is balanced in order to keep this variation of rail pressure within safe limits. The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the connecting rod l increases. A more accurate though still approximate expression for the force on the frame due to the acceleration of the piston whose weight is W is given by

W | ω2r { cos θ + | r | cos 2θ }  
---|---|---|---  
g | l  
  
The conditions regulating the balancing of a system of weights reciprocating under the action of accelerating forces given by the above expression are investigated in a paper by Otto Schlick, “On Balancing of Steam Engines,” Trans, Inst. Nav. Arch. (1900), and in a paper by W. E. Dalby, “On the Balancing of the Reciprocating Parts of Engines, including the Effect of the Connecting Rod” (ibid., 1901). A still more accurate expression than the above is obtained by expansion in a Fourier series, regarding which and its bearing on balancing engines see a paper by J. H. Macalpine, “A Solution of the Vibration Problem” (ibid., 1901). The whole subject is dealt with in a treatise, The Balancing of Engines, by W. E. Dalby (London, 1906). Most of the original papers on this subject of engine balancing are to be found in the Transactions of the Institution of Naval Architects.

§ 113.* Centrifugal Whirling of Shafts.—When a system of revolving masses is balanced so that the conditions of the preceding section are fulfilled, the centre of gravity of the system lies on the axis of revolution. If there is the slightest displacement of the centre of gravity of the system from the axis of revolution a force acts on the shaft tending to deflect it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to revolve stably, this force must be balanced at any instant by the elastic resistance of the shaft to deflexion. To take a simple case, suppose a shaft, supported on two bearings to carry a disk of weight W at its centre, and let the centre of gravity of the disk be at a distance e from the axis of rotation, this small distance being due to imperfections of material or faulty construction. Neglecting the mass of the shaft itself, when the shaft rotates with an angular velocity a, the centrifugal force Wa2e/g will act upon the shaft and cause its axis to deflect from the axis of rotation a distance, y say. The elastic resistance evoked by this deflexion is proportional to the deflexion, so that if c is a constant depending upon the form, material and method of support of the shaft, the following equality must hold if the shaft is to rotate stably at the stated speed—

W | (y + e) a2 = cy,  
---|---  
g  
  
from which y = Wa2e / (gc − Wa2).

This expression shows that as a increases y increases until when Wa2 = gc, y becomes infinitely large. The corresponding value of a, namely √(gc/W), is called the critical velocity of the shaft, and is the speed at which the shaft ceases to rotate stably and at which centrifugal whirling begins. The general problem is to find the value of a corresponding to all kinds of loadings on shafts supported in any manner. The question was investigated by Rankine in an article in the Engineer (April 9, 1869). Professor A. G. Greenhill treated the problem of the centrifugal whirling of an unloaded shaft with different supporting conditions in a paper “On the Strength of Shafting exposed both to torsion and to end thrust,” Proc. Inst. Mech. Eng. (1883). Professor S. Dunkerley (“On the Whirling and Vibration of Shafts,” Phil. Trans., 1894) investigated the question for the cases of loaded and unloaded shafts, and, owing to the complication arising from the application of the general theory to the cases of loaded shafts, devised empirical formulae for the critical speeds of shafts loaded with heavy pulleys, based generally upon the following assumption, which is stated for the case of a shaft carrying one pulley: If N1, N2 be the separate speeds of whirl of the shaft and pulley on the assumption that the effect of one is neglected when that of the other is under consideration, then the resulting speed of whirl due to both causes combined may be taken to be of the form N1N2 √(N21 \+ N12) where N means revolutions per minute. This form is extended to include the cases of several pulleys on the same shaft. The interesting and important part of the investigation is that a number of experiments were made on small shafts arranged in different ways and loaded in different ways, and the speed at which whirling actually occurred was compared with the speed calculated from formulae of the general type indicated above. The agreement between the observed and calculated values of the critical speeds was in most cases quite remarkable. In a paper by Dr C. Chree, “The Whirling and Transverse Vibrations of Rotating Shafts,” Proc. Phys. Soc. Lon., vol. 19 (1904); also Phil. Mag., vol. 7 (1904), the question is investigated from a new mathematical point of view, and expressions for the whirling of loaded shafts are obtained without the necessity of any assumption of the kind stated above. An elementary presentation of the problem from a practical point of view will be found in Steam Turbines, by Dr A. Stodola (London, 1905).  
  
---  
Fig. 131.  
  
§ 114. Revolving Pendulum. Governors.—In fig. 131 AO represents an upright axis or spindle; B a weight called a bob, suspended by rod OB from a horizontal axis at O, carried by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle rotates, the bob, being made to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting at O in the direction OB, and then it revolves steadily in a circle. This combination is called a revolving, centrifugal, or conical pendulum. Revolving pendulums are usually constructed with pairs of rods and bobs, as OB, Ob, hung at opposite sides of the spindle, that the centrifugal forces exerted at the point O may balance each other.

In finding the position in which the bob will revolve with a given angular velocity, a, for most practical cases connected with machinery the mass of the rod may be considered as insensible compared with that of the bob. Let the bob be a sphere, and from the centre of that sphere draw BH = y perpendicular to OA. Let OH = z; let W be the weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W : F :: z : y; that is to say, y/z = F/W = yα2/g; consequently

z = g/α2

(69)

Or, if n = α 2π = α/6.2832 be the number of turns or fractions of a turn in a second,

z = | g | = | 0.8165 ft. | = | 9.79771 in.  
---|---|---|---|---|---  
4π2n2 | n2 | n2  
  
(70)

z is called the altitude of the pendulum.  
  
---  
Fig. 132.  
  
If the rod of a revolving pendulum be jointed, as in fig. 132, not to a point in the vertical axis, but to the end of a projecting arm C, the position in which the bob will revolve will be the same as if the rod were jointed to the point O, where its prolongation cuts the vertical axis.

A revolving pendulum is an essential part of most of the contrivances called governors, for regulating the speed of prime movers, for further particulars of which see Steam Engine.

Division 3. Working of Machines of Varying Velocity.

§ 115. General Principles.—In order that the velocity of every piece of a machine may be uniform, it is necessary that the forces acting on each piece should be always exactly balanced. Also, in order that the forces acting on each piece of a machine may be always exactly balanced, it is necessary that the velocity of that piece should be uniform.

An excess of the effort exerted on any piece, above that which is necessary to balance the resistance, is accompanied with acceleration; a deficiency of the effort, with retardation.

When a machine is being started from a state of rest, and brought by degrees up to its proper speed, the effort must be in excess; when it is being retarded for the purpose of stopping it, the resistance must be in excess.

An excess of effort above resistance involves an excess of energy exerted above work performed; that excess of energy is employed in producing acceleration.

An excess of resistance above effort involves an excess of work performed above energy expended; that excess of work is performed by means of the retardation of the machinery.

When a machine undergoes alternate acceleration and retardation, so that at certain instants of time, occurring at the end of intervals called periods or cycles, it returns to its original speed, then in each of those periods or cycles the alternate excesses of energy and of work neutralize each other; and at the end of each cycle the principle of the equality of energy and work stated in § 87, with all its consequences, is verified exactly as in the case of machines of uniform speed.

At intermediate instants, however, other principles have also to be taken into account, which are deduced from the second law of motion, as applied to direct deviation, or acceleration and retardation.

§ 116. Energy of Acceleration and Work of Retardation for a Shifting Body.—Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time Δt let its velocity be increased at a uniform rate of acceleration from v1 to v2. The rate of acceleration will be

dv/dt = const. = (v2 − v1) Δt;

and to produce this acceleration a uniform effort will be required, expressed by

P = w (v2 − v1) gΔt

(71)

(The product wv/g of the mass of a body by its velocity is called its momentum; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)

To find the energy which has to be exerted to produce the acceleration from v1 to v2, it is to be observed that the distance through which the effort P acts during the acceleration is

Δs = (v2 \+ v1) Δt/2;

consequently, the energy of acceleration is

PΔs = w (v2 − v1) (v2 \+ v1) / 2g = w (v22 − v12) 2g,

(72)

being proportional to the increase in the square of the velocity, and independent of the time.

In order to produce a retardation from the greater velocity v2 to the less velocity v1, it is necessary to apply to the body a resistance connected with the retardation and the time by an equation identical in every respect with equation (71), except by the substitution of a resistance for an effort; and in overcoming that resistance the body performs work to an amount determined by equation (72), putting Rds for Pas.

§ 117. Energy Stored and Restored by Deviations of Velocity.—Thus a body alternately accelerated and retarded, so as to be brought back to its original speed, performs work during its retardation exactly equal in amount to the energy exerted upon it during its acceleration; so that that energy may be considered as stored during the acceleration, and restored during the retardation, in a manner analogous to the operation of a reciprocating force (§ 108).

Let there be given the mean velocity V = 1⁄2 (v2 \+ v1) of a body whose weight is w, and let it be required to determine the fluctuation of velocity v2 − v1, and the extreme velocities v1, v2, which that body must have, in order alternately to store and restore an amount of energy E. By equation (72) we have

E = w (v22 − v12) / 2g

which, being divided by V = 1⁄2(v2 \+ v1), gives

E/V = w (v2 − v1) / g;

and consequently

v2 − v1 = gE / Vw

(73)

The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the motion of the body, is

(v2 − v1) V = gE / V2w.

(74)

§ 118. Actual Energy of a Shifting Body.—The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by overcoming resistance while being retarded from the same velocity of translation v to a state of rest, is

wv2 / 2g.

(75)

This is called the actual energy of the motion of the body, and is half the quantity which in some treatises is called vis viva.

The energy stored or restored, as the case may be, by the deviations of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished.

§ 119. Principle of the Conservation of Energy in Machines.—The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in § 89 for machines of uniform speed.

In any given interval during the working of a machine, the energy exerted added to the energy restored is equal to the energy stored added to the work performed.

§ 120. Actual Energy of Circular Translation—Moment of Inertia.—Let a small body of the weight w undergo translation in a circular path of the radius ρ, with the angular velocity of deflexion α, so that the common linear velocity of all its particles is v = αρ. Then the actual energy of that body is

wv2 / 2g = wa2ρ2 / 2g.

(76)

By comparing this with the expression for the centrifugal force (wa2ρ/g), it appears that the actual energy of a revolving body is equal to the potential energy Fρ/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.

The product wρ2/g, by which the half-square of the angular velocity is multiplied, is called the moment of inertia of the revolving body.

§ 121. Flywheels.—A flywheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its angular velocity.

The principles according to which variations of angular velocity store and restore energy are the same as those of § 117, only substituting moment of inertia for mass, and angular for linear velocity.

Let W be the weight of a flywheel, R its radius of gyration, a2 its maximum, a1 its minimum, and A = 1⁄2 (α2 \+ α1) its mean angular velocity. Let

I/S = (α2 − α2) / A

denote the unsteadiness of the motion of the flywheel; the denominator S of this fraction is called the steadiness. Let e denote the quantity by which the energy exerted in each cycle of the working of the machine alternately exceeds and falls short of the work performed, and which has consequently to be alternately stored by acceleration and restored by retardation of the flywheel. The value of this periodical excess is—

e = R2W (α22 − α12), 2g,

(77)

from which, dividing both sides by A2, we obtain the following equations:—

e / A2 = R2W / gS  
R2WA2 / 2g = Se / 2.

(78)

The latter of these equations may be thus expressed in words: The actual energy due to the rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical excess of energy multiplied by the steadiness.

In ordinary machinery S = about 32; in machinery for fine purposes S = from 50 to 60; and when great steadiness is required S = from 100 to 150.

The periodical excess e may arise either from variations in the effort exerted by the prime mover, or from variations in the resistance of the work, or from both these causes combined. When but one flywheel is used, it should be placed in as direct connexion as possible with that part of the mechanism where the greatest amount of the periodical excess originates; but when it originates at two or more points, it is best to have a flywheel in connexion with each of these points. For example, in a machine-work, the steam-engine, which is the prime mover of the various tools, has a flywheel on the crank-shaft to store and restore the periodical excess of energy arising from the variations in the effort exerted by the connecting-rod upon the crank; and each of the slotting machines, punching machines, riveting machines, and other tools has a flywheel of its own to store and restore energy, so as to enable the very different resistances opposed to those tools at different times to be overcome without too great unsteadiness of motion. For tools performing useful work at intervals, and having only their own friction to overcome during the intermediate intervals, e should be assumed equal to the whole work performed at each separate operation.

§ 122. Brakes.—A brake is an apparatus for stopping and diminishing the velocity of a machine by friction, such as the friction-strap already referred to in § 103. To find the distance s through which a brake, exerting the friction F, must rub in order to stop a machine having the total actual energy E at the moment when the brake begins to act, reduce, by the principles of § 96, the various efforts and other resistances of the machine which act at the same time with the friction of the brake to the rubbing surface of the brake, and let R be their resultant—positive if resistance, negative if effort preponderates. Then

s = E / (F + R).

(79)

§ 123. Energy distributed between two Bodies: Projection and Propulsion.—Hitherto the effort by which a machine is moved has been treated as a force exerted between a movable body and a fixed body, so that the whole energy exerted by it is employed upon the movable body, and none upon the fixed body. This conception is sensibly realized in practice when one of the two bodies between which the effort acts is either so heavy as compared with the other, or has so great a resistance opposed to its motion, that it may, without sensible error, be treated as fixed. But there are cases in which the motions of both bodies are appreciable, and must be taken into account—such as the projection of projectiles, where the velocity of the recoil or backward motion of the gun bears an appreciable proportion to the forward motion of the projectile; and such as the propulsion of vessels, where the velocity of the water thrown backward by the paddle, screw or other propeller bears a very considerable proportion to the velocity of the water moved forwards and sideways by the ship. In cases of this kind the energy exerted by the effort is distributed between the two bodies between which the effort is exerted in shares proportional to the velocities of the two bodies during the action of the effort; and those velocities are to each other directly as the portions of the effort unbalanced by resistance on the respective bodies, and inversely as the weights of the bodies.

To express this symbolically, let W1, W2 be the weights of the bodies; P the effort exerted between them; S the distance through which it acts; R1, R2 the resistances opposed to the effort overcome by W1, W2 respectively; E1, E2 the shares of the whole energy E exerted upon W1, W2 respectively. Then

  |  E | : | E1 | : | E2 |    
---|---|---|---|---|---|---  
:: | W2 (P − R1) + W1 (P − R2) | : | P − R1 | : | P − R2 | .  
W1W2 | W1 | W2  
  
(80)

If R1 = R2, which is the case when the resistance, as well as the effort, arises from the mutual actions of the two bodies, the above becomes,

E : E1 : E2  
:: W1 \+ W2 : W2 : W1,

(81)

that is to say, the energy is exerted on the bodies in shares inversely proportional to their weights; and they receive accelerations inversely proportional to their weights, according to the principle of dynamics, already quoted in a note to § 110, that the mutual actions of a system of bodies do not affect the motion of their common centre of gravity.

For example, if the weight of a gun be 160 times that of its ball 160⁄161 of the energy exerted by the powder in exploding will be employed in propelling the ball, and 1⁄161 in producing the recoil of the gun, provided the gun up to the instant of the ball’s quitting the muzzle meets with no resistance to its recoil except the friction of the ball.

§ 124. Centre of Percussion.—It is obviously desirable that the deviations or changes of motion of oscillating pieces in machinery should, as far as possible, be effected by forces applied at their centres of percussion.

If the deviation be a translation—that is, an equal change of motion of all the particles of the body—the centre of percussion is obviously the centre of gravity itself; and, according to the second law of motion, if dv be the deviation of velocity to be produced in the interval dt, and W the weight of the body, then

P = | W | · | dv  
---|---|---|---  
g | dt  
  
(82)

is the unbalanced effort required.

If the deviation be a rotation about an axis traversing the centre of gravity, there is no centre of percussion; for such a deviation can only be produced by a couple of forces, and not by any single force. Let dα be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body about an axis through its centre of gravity; then 1⁄2Id(α2) = Iα dα is the variation of the body’s actual energy. Let M be the moment of the unbalanced couple required to produce the deviation; then by equation 57, § 104, the energy exerted by this couple in the interval dt is Mα dt, which, being equated to the variation of energy, gives

M = I | dα | = | R2W | · | dα | .  
---|---|---|---|---|---|---  
dt | g | dt  
  
(83)

R is called the radius of gyration of the body with regard to an axis through its centre of gravity.  
  
---  
Fig. 133.  
  
Now (fig. 133) let the required deviation be a rotation of the body BB about an axis O, not traversing the centre of gravity G, dα being, as before, the deviation of angular velocity to be produced in the interval dt. A rotation with the angular velocity α about an axis O may be considered as compounded of a rotation with the same angular velocity about an axis drawn through G parallel to O and a translation with the velocity α. OG, OG being the perpendicular distance between the two axes. Hence the required deviation may be regarded as compounded of a deviation of translation dv = OG · dα, to produce which there would be required, according to equation (82), a force applied at G perpendicular to the plane OG—

P = | W | · OG · | dα  
---|---|---|---  
g | dt  
  
(84)

and a deviation dα of rotation about an axis drawn through G parallel to O, to produce which there would be required a couple of the moment M given by equation (83). According to the principles of statics, the resultant of the force P, applied at G perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value is

GC = M / P = R2 / OG.

(85)

Thus is determined the position of the centre of percussion C, corresponding to the axis of rotation O. It is obvious from this equation that, for an axis of rotation parallel to O traversing C, the centre of percussion is at the point where the perpendicular OG meets O.

§ 125.* To find the moment of inertia of a body about an axis through its centre of gravity experimentally.—Suspend the body from any conveniently selected axis O (fig. 48) and hang near it a small plumb bob. Adjust the length of the plumb-line until it and the body oscillate together in unison. The length of the plumb-line, measured from its point of suspension to the centre of the bob, is for all practical purposes equal to the length OC, C being therefore the centre of percussion corresponding to the selected axis O. From equation (85)

R2 = CG × OG = (OC − OG) OG.

The position of G can be found experimentally; hence OG is known, and the quantity R2 can be calculated, from which and the ascertained weight W of the body the moment of inertia about an axis through G, namely, W/g × R2, can be computed.  
  
---  
Fig. 134.  
  
§ 126.* To find the force competent to produce the instantaneous acceleration of any link of a mechanism.—In many practical problems it is necessary to know the magnitude and position of the forces acting to produce the accelerations of the several links of a mechanism. For a given link, this force is the resultant of all the accelerating forces distributed through the substance of the material of the link required to produce the requisite acceleration of each particle, and the determination of this force depends upon the principles of the two preceding sections. The investigation of the distribution of the forces through the material and the stress consequently produced belongs to the subject of the Strength of Materials (q.v.). Let BK (fig. 134) be any link moving in any manner in a plane, and let G be its centre of gravity. Then its motion may be analysed into (1) a translation of its centre of gravity; and (2) a rotation about an axis through its centre of gravity perpendicular to its plane of motion. Let α be the acceleration of the centre of gravity and let A be the angular acceleration about the axis through the centre of gravity; then the force required to produce the translation of the centre of gravity is F = Wα/g, and the couple required to produce the angular acceleration about the centre of gravity is M = IA/g, W and I being respectively the weight and the moment of inertia of the link about the axis through the centre of gravity. The couple M may be produced by shifting the force F parallel to itself through a distance x. such that Fx = M. When the link forms part of a mechanism the respective accelerations of two points in the link can be determined by means of the velocity and acceleration diagrams described in § 82, it being understood that the motion of one link in the mechanism is prescribed, for instance, in the steam-engine’s mechanism that the crank shall revolve uniformly. Let the acceleration of the two points B and K therefore be supposed known. The problem is now to find the acceleration α and A. Take any pole O (fig. 49), and set out Ob equal to the acceleration of B and Ok equal to the acceleration of K. Join bk and take the point g so that KG: GB = kg : gb. Og is then the acceleration of the centre of gravity and the force F can therefore be immediately calculated. To find the angular acceleration A, draw kt, bt respectively parallel to and at right angles to the link KB. Then tb represents the angular acceleration of the point B relatively to the point K and hence tb/KB is the value of A, the angular acceleration of the link. Its moment of inertia about G can be found experimentally by the method explained in § 125, and then the value of the couple M can be computed. The value of x is found immediately from the quotient M/F. Hence the magnitude F and the position of F relatively to the centre of gravity of the link, necessary to give rise to the couple M, are known, and this force is therefore the resultant force required.  
  
---  
Fig. 135.  
  
§ 127.* Alternative construction for finding the position of F relatively to the centre of gravity of the link.—Let B and K be any two points in the link which for greater generality are taken in fig. 135, so that the centre of gravity G is not in the line joining them. First find the value of R experimentally. Then produce the given directions of acceleration of B and K to meet in O; draw a circle through the three points B, K and O; produce the line joining O and G to cut the circle in Y; and take a point Z on the line OY so that YG × GZ = R2. Then Z is a point in the line of action of the force F. This useful theorem is due to G. T. Bennett, of Emmanuel College, Cambridge. A proof of it and three corollaries are given in appendix 4 of the second edition of Dalby’s Balancing of Engines (London, 1906). It is to be noticed that only the directions of the accelerations of two points are required to find the point Z.

For an example of the application of the principles of the two preceding sections to a practical problem see Valve and Valve Gear Mechanisms, by W. E. Dalby (London, 1906), where the inertia stresses brought upon the several links of a Joy valve gear, belonging to an express passenger engine of the Lancashire & Yorkshire railway, are investigated for an engine-speed of 68 m. an hour.  
  
---  
Fig. 136.  
  
§ 128.* The Connecting Rod Problem.—A particular problem of practical importance is the determination of the force producing the motion of the connecting rod of a steam-engine mechanism of the usual type. The methods of the two preceding sections may be used when the acceleration of two points in the rod are known. In this problem it is usually assumed that the crank pin K (fig. 136) moves with uniform velocity, so that if α is its angular velocity and r its radius, the acceleration is α2r in a direction along the crank arm from the crank pin to the centre of the shaft. Thus the acceleration of one point K is known completely. The acceleration of a second point, usually taken at the centre of the crosshead pin, can be found by the principles of § 82, but several special geometrical constructions have been devised for this purpose, notably the construction of Klein,4 discovered also independently by Kirsch.5 But probably the most convenient is the construction due to G. T. Bennett6 which is as follows: Let OK be the crank and KB the connecting rod. On the connecting rod take a point L such that KL × KB = KO2. Then, the crank standing at any angle with the line of stroke, draw LP at right angles to the connecting rod, PN at right angles to the line of stroke OB and NA at right angles to the connecting rod; then AO is the acceleration of the point B to the scale on which KO represents the acceleration of the point K. The proof of this construction is given in The Balancing of Engines.

The finding of F may be continued thus: join AK, then AK is the acceleration image of the rod, OKA being the acceleration diagram. Through G, the centre of gravity of the rod, draw Gg parallel to the line of stroke, thus dividing the image at g in the proportion that the connecting rod is divided by G. Hence Og represents the acceleration of the centre of gravity and, the weight of the connecting rod being ascertained, F can be immediately calculated. To find a point in its line of action, take a point Q on the rod such that KG × GQ = R2, R having been determined experimentally by the method of § 125; join G with O and through Q draw a line parallel to BO to cut GO in Z. Z is a point in the line of action of the resultant force F; hence through Z draw a line parallel to Og. The force F acts in this line, and thus the problem is completely solved. The above construction for Z is a corollary of the general theorem given in § 127.

§ 129. Impact. Impact or collision is a pressure of short duration exerted between two bodies.

The effects of impact are sometimes an alteration of the distribution of actual energy between the two bodies, and always a loss of a portion of that energy, depending on the imperfection of the elasticity of the bodies, in permanently altering their figures, and producing heat. The determination of the distribution of the actual energy after collision and of the loss of energy is effected by means of the following principles:—

I. The motion of the common centre of gravity of the two bodies is unchanged by the collision.

II. The loss of energy consists of a certain proportion of that part of the actual energy of the bodies which is due to their motion relatively to their common centre of gravity.

Unless there is some special reason for using impact in machines, it ought to be avoided, on account not only of the waste of energy which it causes, but from the damage which it occasions to the frame and mechanism.

(W. J. M. R.; W. E. D.)

* * *

1 In view of the great authority of the author, the late Professor Macquorn Rankine, it has been thought desirable to retain the greater part of this article as it appeared in the 9th edition of the Encyclopaedia Britannica. Considerable additions, however, have been introduced in order to indicate subsequent developments of the subject; the new sections are numbered continuously with the old, but are distinguished by an asterisk. Also, two short chapters which concluded the original article have been omitted—ch. iii., “On Purposes and Effects of Machines,” which was really a classification of machines, because the classification of Franz Reuleaux is now usually followed, and ch. iv., “Applied Energetics, or Theory of Prime Movers,” because its subject matter is now treated in various special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil Engine, and fully developed in Rankine’s The Steam Engine and Other Prime Movers (London, 1902). (Ed. E.B.)

2 Since the relation discussed in § 7 was enunciated by Rankine, an enormous development has taken place in the subject of Graphic Statics, the first comprehensive textbook on the subject being Die Graphische Statik by K. Culmann, published at Zürich in 1866. Many of the graphical methods therein given have now passed into the textbooks usually studied by engineers. One of the most beautiful graphical constructions regularly used by engineers and known as “the method of reciprocal figures” is that for finding the loads supported by the several members of a braced structure, having given a system of external loads. The method was discovered by Clerk Maxwell, and the complete theory is discussed and exemplified in a paper “On Reciprocal Figures, Frames and Diagrams of Forces,” Trans. Roy. Soc. Ed., vol. xxvi. (1870). Professor M. W. Crofton read a paper on “Stress-Diagrams in Warren and Lattice Girders” at the meeting of the Mathematical Society (April 13, 1871), and Professor O. Henrici illustrated the subject by a simple and ingenious notation. The application of the method of reciprocal figures was facilitated by a system of notation published in Economics of Construction in relation to framed Structures, by Robert H. Bow (London, 1873). A notable work on the general subject is that of Luigi Cremona, translated from the Italian by Professor T. H. Beare (Oxford, 1890), and a discussion of the subject of reciprocal figures from the special point of view of the engineering student is given in Vectors and Rotors by Henrici and Turner (London, 1903). See also above under “Theoretical Mechanics,” Part 1. § 5.

3 This is a particular case of a more general principle, that the motion of the centre of gravity of a body is not affected by the mutual actions of its parts.

4 J. F. Klein, “New Constructions of the Force of Inertia of Connecting Rods and Couplers and Constructions of the Pressures on their Pins,” Journ. Franklin Inst., vol. 132 (Sept. and Oct., 1891).

5 Prof. Kirsch, “Über die graphische Bestimmung der Kolbenbeschleunigung,” Zeitsch. Verein deutsche Ingen. (1890), p. 1320.

6 Dalby, The Balancing of Engines (London, 1906), app. 1.