# damping

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## Damping

A term broadly used to denote either the dissipation of energy in, and the consequent decay of, oscillations of all types or the extent of the dissipation and decay. The energy losses arise from frictional (or analogous) forces which are unavoidable in any system or from the radiation of energy to space or to other systems. For sufficiently small oscillations, the analogous forces are proportional to the velocity of the vibrating member and oppositely directed thereto; the ratio of force to velocity is *-R*, the mechanical resistance. For the role of damping in the case of forced oscillations, where it is decisive for the frequency response, *See* Forced oscillation, **Resonance (acoustics and mechanics)**, Harmonic motion, Oscillation, Vibration

An undamped system of mass *m* and stiffness *s* oscillates at an angular frequency &ogr;_{0} = (s/m)^{1/2}. The effect of a mechanical resistance *R* is twofold: It produces a change in the frequency of oscillation, and it causes the oscillations to decay with time. If *u* is one of the oscillating quantities (displacement, velocity, acceleration) of amplitude *A*, then Eq. (1) holds in the damped case, whereas in the undamped case Eq. (2)

The damped angular frequency ω_{d} in Eq. (1) is always less than ω_{0}. According to Eq. (1), the amplitude of the oscillation decays exponentially; the time required for the amplitude to decrease to the fraction 1/*e* of its initial value is equal to 1/α.

A common measure of the damping is the logarithmic decrement δ, defined as the natural logarithm of the ratio of two successive maxima of the decaying sinusoid. If *T* is the period of the oscillation, then Eq. (3) holds. Then 1/δ is the number

*e*in the same way that 1/α is the time required.

The *Q* of a system is a measure of damping usually defined from energy considerations. The *Q* is π times the ratio of peak energy stored to energy dissipated per cycle and is equal to π/δ.

If α in Eq. (1) exceeds ω_{0}, then the system is not oscillatory and is said to be overdamped. If the mass is displaced, it returns to its equilibrium position without overshoot, and the return is slower as the ratio α/ω_{0} increases. If α = ω_{0} (that is, *Q* = 1/2), the oscillator is critically damped. In this case, the motion is again nonoscillatory, but the return to equilibrium is faster than for any overdamped case.