Vibration of plates

Short description: Type of mechanical vibration

For the construction machine, see plate compactor.

Vibration mode of a clamped square plate

The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This permits a two-dimensional plate theory to give an excellent approximation to the actual three-dimensional motion of a plate-like object.[1]

There are several theories that have been developed to describe the motion of plates. The most commonly used are the Kirchhoff-Love theory[2] and the Uflyand-Mindlin.[3][4] The latter theory is discussed in detail by Elishakoff.[5] Solutions to the governing equations predicted by these theories can give us insight into the behavior of plate-like objects both under free and forced conditions. This includes the propagation of waves and the study of standing waves and vibration modes in plates. The topic of plate vibrations is treated in books by Leissa,[6][7] Gontkevich,[8] Rao,[9] Soedel,[10] Yu,[11] Gorman[12][13] and Rao.[14]

## Contents

* 1 Kirchhoff-Love plates
* 2 Isotropic Kirchhoff–Love plates
* 3 Free vibrations of isotropic plates
* 3.1 Circular plates
* 3.2 Rectangular plates
* 4 References
* 5 See also

## Kirchhoff-Love plates

The governing equations for the dynamics of a Kirchhoff-Love plate are

Nαβ,β=J1u¨αMαβ,αβ+q(x,t)=J1w¨−J3w¨,αα

where uα are the in-plane displacements of the mid-surface of the plate, w is the transverse (out-of-plane) displacement of the mid-surface of the plate, q is an applied transverse load pointing to x3 (upwards), and the resultant forces and moments are defined as

Nαβ:=∫−hhσαβdx3andMαβ:=∫−hhx3σαβdx3.

Note that the thickness of the plate is 2h and that the resultants are defined as weighted averages of the in-plane stresses σαβ. The derivatives in the governing equations are defined as

u˙i:=∂ui∂t;u¨i:=∂2ui∂t2;ui,α:=∂ui∂xα;ui,αβ:=∂2ui∂xα∂xβ

where the Latin indices go from 1 to 3 while the Greek indices go from 1 to 2. Summation over repeated indices is implied. The x3 coordinates is out-of-plane while the coordinates x1 and x2 are in plane. For a uniformly thick plate of thickness 2h and homogeneous mass density ρ

J1:=∫−hhρdx3=2ρhandJ3:=∫−hhx32ρdx3=23ρh3.

## Isotropic Kirchhoff–Love plates

For an isotropic and homogeneous plate, the stress-strain relations are

[σ11σ22σ12]=E1−ν2[1ν0ν10001−ν][ε11ε22ε12].

where εαβ are the in-plane strains and ν is the Poisson's ratio of the material. The strain-displacement relations for Kirchhoff-Love plates are

εαβ=12(uα,β+uβ,α)−x3w,αβ.

Therefore, the resultant moments corresponding to these stresses are

[M11M22M12]=−2h3E3(1−ν2)[1ν0ν10001−ν][w,11w,22w,12]

If we ignore the in-plane displacements uαβ, the governing equations reduce to

D∇2∇2w=q(x,t)−2ρhw¨
where D is the bending stiffness of the plate. For a uniform plate of thickness 2h,

D:=2h3E3(1−ν2).

The above equation can also be written in an alternative notation:

μΔΔw−q^+ρwtt=0.

In solid mechanics, a plate is often modeled as a two-dimensional elastic body whose potential energy depends on how it is bent from a planar configuration, rather than how it is stretched (which is the instead the case for a membrane such as a drumhead). In such situations, a vibrating plate can be modeled in a manner analogous to a vibrating drum. However, the resulting partial differential equation for the vertical displacement w of a plate from its equilibrium position is fourth order, involving the square of the Laplacian of w, rather than second order, and its qualitative behavior is fundamentally different from that of the circular membrane drum.

## Free vibrations of isotropic plates

For free vibrations, the external force q is zero, and the governing equation of an isotropic plate reduces to

D∇2∇2w=−2ρhw¨

μΔΔw+ρwtt=0.

This relation can be derived in an alternative manner by considering the curvature of the plate.[15] The potential energy density of a plate depends how the plate is deformed, and so on the mean curvature and Gaussian curvature of the plate. For small deformations, the mean curvature is expressed in terms of w, the vertical displacement of the plate from kinetic equilibrium, as Δw, the Laplacian of w, and the Gaussian curvature is the Monge–Ampère operator wxxwyy−w2xy. The total potential energy of a plate Ω therefore has the form

U=∫Ω[(Δw)2+(1−μ)(wxxwyy−wxy2)]dxdy

apart from an overall inessential normalization constant. Here μ is a constant depending on the properties of the material.

The kinetic energy is given by an integral of the form

T=ρ2∫Ωwt2dxdy.

Hamilton's principle asserts that w is a stationary point with respect to variations of the total energy T+U. The resulting partial differential equation is

ρwtt+μΔΔw=0.

### Circular plates

For freely vibrating circular plates, w=w(r,t), and the Laplacian in cylindrical coordinates has the form

∇2w≡1r∂∂r(r∂w∂r).

Therefore, the governing equation for free vibrations of a circular plate of thickness 2h is

1r∂∂r[r∂∂r{1r∂∂r(r∂w∂r)}]=−2ρhD∂2w∂t2.

Expanded out,

∂4w∂r4+2r∂3w∂r3−1r2∂2w∂r2+1r3∂w∂r=−2ρhD∂2w∂t2.

To solve this equation we use the idea of separation of variables and assume a solution of the form

w(r,t)=W(r)F(t).

Plugging this assumed solution into the governing equation gives us

1βW[d4Wdr4+2rd3Wdr3−1r2d2Wdr2+1r3dWdr]=−1Fd2Fdt2=ω2

where ω2 is a constant and β:=2ρh/D. The solution of the right hand equation is

F(t)=Re[Aeiωt+Be−iωt].

The left hand side equation can be written as

d4Wdr4+2rd3Wdr3−1r2d2Wdr2+1r3dWdr=λ4W

where λ4:=βω2. The general solution of this eigenvalue problem that is appropriate for plates has the form

W(r)=C1J0(λr)+C2I0(λr)

where J0 is the order 0 Bessel function of the first kind and I0 is the order 0 modified Bessel function of the first kind. The constants C1 and C2 are determined from the boundary conditions. For a plate of radius a with a clamped circumference, the boundary conditions are

W(r)=0anddWdr=0atr=a.

From these boundary conditions we find that

J0(λa)I1(λa)+I0(λa)J1(λa)=0.

We can solve this equation for λn (and there are an infinite number of roots) and from that find the modal frequencies ωn=λn2/β. We can also express the displacement in the form

w(r,t)=∑n=1∞Cn[J0(λnr)−J0(λna)I0(λna)I0(λnr)][Aneiωnt+Bne−iωnt].

For a given frequency ωn the first term inside the sum in the above equation gives the mode shape. We can find the value of Cn using the appropriate boundary condition at r=0 and the coefficients An and Bn from the initial conditions by taking advantage of the orthogonality of Fourier components.

* mode n = 1

* mode n = 2

### Rectangular plates

A vibration mode of a rectangular plate.

Consider a rectangular plate which has dimensions a×b in the (x1,x2)-plane and thickness 2h in the x3-direction. We seek to find the free vibration modes of the plate.

Assume a displacement field of the form

w(x1,x2,t)=W(x1,x2)F(t).

Then,

∇2∇2w=w,1111+2w,1212+w,2222=[∂4W∂x14+2∂4W∂x12∂x22+∂4W∂x24]F(t)

and

w¨=W(x1,x2)d2Fdt2.

Plugging these into the governing equation gives

D2ρhW[∂4W∂x14+2∂4W∂x12∂x22+∂4W∂x24]=−1Fd2Fdt2=ω2

where ω2 is a constant because the left hand side is independent of t while the right hand side is independent of x1,x2. From the right hand side, we then have

F(t)=Aeiωt+Be−iωt.

From the left hand side,

∂4W∂x14+2∂4W∂x12∂x22+∂4W∂x24=2ρhω2DW=:λ4W

where

λ2=ω2ρhD.

Since the above equation is a biharmonic eigenvalue problem, we look for Fourier expansion solutions of the form

Wmn(x1,x2)=sin⁡mπx1asin⁡nπx2b.

We can check and see that this solution satisfies the boundary conditions for a freely vibrating rectangular plate with simply supported edges:

w(x1,x2,t)=0atx1=0,aandx2=0,bM11=D(∂2w∂x12+ν∂2w∂x22)=0atx1=0,aM22=D(∂2w∂x22+ν∂2w∂x12)=0atx2=0,b.

Plugging the solution into the biharmonic equation gives us

λ2=π2(m2a2+n2b2).

Comparison with the previous expression for λ2 indicates that we can have an infinite number of solutions with

ωmn=(m2a2+n2b2)Dπ42ρh.

Therefore the general solution for the plate equation is

w(x1,x2,t)=∑m=1∞∑n=1∞sin⁡mπx1asin⁡nπx2b(Amneiωmnt+Bmne−iωmnt).

To find the values of Amn and Bmn we use initial conditions and the orthogonality of Fourier components. For example, if

w(x1,x2,0)=φ(x1,x2)onx1∈[0,a]and∂w∂t(x1,x2,0)=ψ(x1,x2)onx2∈[0,b]

we get,

Amn=4ab∫0a∫0bφ(x1,x2)sin⁡mπx1asin⁡nπx2bdx1dx2Bmn=4abωmn∫0a∫0bψ(x1,x2)sin⁡mπx1asin⁡nπx2bdx1dx2.

## References

1. ↑ Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.
2. ↑ A. E. H. Love, On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–549.
3. ↑ Uflyand, Ya. S.,1948, Wave Propagation by Transverse Vibrations of Beams and Plates, PMM: Journal of Applied Mathematics and Mechanics, Vol. 12, pp. 287-300 (in Russian)
4. ↑ Mindlin, R.D. 1951, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, ASME Journal of Applied Mechanics, Vol. 18 pp. 31–38
5. ↑ Elishakoff, I., 2020, Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories, World Scientific, Singapore, ISBN 978-981-3236-51-6
6. ↑ Leissa, A.W.,1969, Vibration of Plates, NASA SP-160, Washington, D.C.: U.S. Government Printing Office
7. ↑ Leissa, A.W. and Qatu, M.S.,2011, Vibration of Continuous Systems, New York: Mc Graw-Hill
8. ↑ Gontkevich, V. S., 1964, Natural Vibrations of Plates and Shells, Kiev: “Naukova Dumka” Publishers, 1964 (in Russian); (English Translation: Lockheed Missiles & Space Co., Sunnyvale, CA)
9. ↑ Rao, S.S., Vibration of Continuous Systems, New York: Wiley
10. ↑ Soedel, W.,1993, Vibrations of Shells and Plates, New York: Marcel Dekker Inc., (second edition)
11. ↑ Yu, Y.Y.,1996, Vibrations of Elastic Plates, New York: Springer
12. ↑ Gorman, D.,1982, Free Vibration Analysis of Rectangular Plates, Amsterdam: Elsevier
13. ↑ Gorman, D.J., 1999, Vibration Analysis of Plates by Superposition Method, Singapore: World Scientific
14. ↑ Rao, J.S., 1999, Dynamics of Plates, New Delhi: Narosa Publishing House
15. ↑ Courant, Richard; Hilbert, David (1953), Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y.

## See also

* Bending
* Bending of plates
* Chladni figures
* Infinitesimal strain theory
* Kirchhoff–Love plate theory
* Linear elasticity
* Mindlin–Reissner plate theory
* Plate theory
* Stress (mechanics)
* Stress resultants
* Structural acoustics

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