Forced oscillations

Oscillations in some material system under the effect of an external time-variable force. In a linear dissipative system under the action of an external force which varies in a harmonic manner, the frequency of the forced oscillation is that of the external force. The amplitude of the forced oscillation is determined by the parameters of the external force (amplitude, frequency) and by the resistance coefficients of the material medium in which the oscillation is taking place. If the frequency of the external force is close to an eigen oscillation of the system, the amplitude of the forced oscillation may have a rather large value, and will be the larger the lower the resistance of the medium. If the resistance of the medium is zero, while the frequency of the external force is equal to one of the eigen frequencies of oscillation of the system, the amplitude of the forced oscillation increases without limit and varies linearly with time. This effect is known as resonance.

If an external force is comprised in the material system, there result both forced and eigen oscillations. In a dissipative system the eigen oscillations become damped, and forced oscillations alone remain in the system (a stabilizing state). The conditions of transition to a stabilizing oscillation are known as transient conditions. The duration of the transient state is the shorter, the larger the resistance of the medium. If the external force is a periodic function of the time with period $ T = 2 \pi / p $, which may be represented by a Fourier series, in the linear system forced oscillations are generated which are the sum of harmonics with frequencies $ np $, where $ n = 1, 2 , . . . $. The amplitude of these harmonics decreases with increasing $ n $, but not uniformly. In practical work, a finite number of harmonics is usually taken.

If, for some value of $ n $, the frequency $ np $ is close to one of the eigen frequencies of the system, the amplitude of the harmonic of the forced oscillation may be considerably great and, in the absence of resistance of the medium, will increase without limit. For instance, for a dissipative system with one degree of freedom, the equation of motion of which is of the form

$$ \dot{x} dot + 2h {\dot{x} } + k ^ {2} x = \ \sum _ {n = 1 } ^ \infty H _ {n} \sin ( npt + \delta _ {n} ) , $$

where $ h $, $ k ^ {2} $, $ H _ {n} $, $ p $, $ \delta _ {n} $ are constant coefficients, the forced oscillation will obey the law

$$ x = \sum _ {n = 1 } ^ \infty \frac{H _ {n} }{\sqrt {( k ^ {2} - n ^ {2} p ^ {2} ) ^ {2} + 4h ^ {2} n ^ {2} p ^ {2} } } \sin ( npt + \delta _ {n} - \gamma _ {n} ), $$

where $ \gamma _ {n} = \mathop{\rm arctan} [ 2hnp/( k ^ {2} - n ^ {2} p ^ {2} )] $. If $ h = 0 $( medium without resistance) and $ sp = k $, one has

$$ x = \sum _ {n = 1 } ^ \infty {} ^ {(} s) \frac{H _ {n} }{k ^ {2} - n ^ {2} p ^ {2} } \sin ( npt + \delta _ {n} ) - \frac{H _ {s} t }{2sp } \cos ( spt + \delta _ {s} ), $$

where the superscript $ ( s) $ at the summation sign means that the term corresponding to $ n = s $ is not to be summed.

In the presence of an external aperiodic disturbance, the forced oscillation generated in the system will also be aperiodic. When an external harmonic force is acting on a non-linear dissipative system, the frequency of the forced oscillation may be that of the external force, but may also be an integral multiple of it (subharmonic oscillations).

In an auto-oscillating system (cf. Auto-oscillation) acted upon by an external harmonic force, quasi-periodic states become established; such conditions are distinguished by the presence of an aperiodic oscillation having a frequency close to that of the auto-oscillation, and a forced oscillation having the frequency of the external disturbance. If the frequency of the external perturbation is close to that of an auto-oscillation, the system executes oscillations at the frequency of the external force alone. This effect is known as forced synchronization (capture).

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