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In engineering, the damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system is "trying" to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero. The damping ratio is a measure of describing how rapidly the oscillations decay from one bounce to the next.
The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include control engineering, mechanical engineering, structural engineering and electrical engineering. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of an electric motor, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior.
The damping ratio is a parameter, usually denoted by ζ (zeta), that characterizes the frequency response of a second order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator.
The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient:
where the system's equation of motion is
and the corresponding critical damping coefficient is
The damping ratio is dimensionless, being the ratio of two coefficients of identical units.
This equation can be solved with the approach.
where C and s are both complex constants. That approach assumes a solution that is oscillatory and/or decaying exponentially. Using it in the ODE gives a condition on the frequency of the damped oscillations,
The factors Q, damping ratio ζ, and exponential decay rate α are related such that
When a second-order system has (that is, when the system is underdamped), it has two complex conjugate poles that each have a real part of ; that is, the decay rate parameter represents the rate of exponential decay of the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. For example, a high quality tuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer.
The damping ratio is also related to the logarithmic decrement for underdamped vibrations via the relation
This relation is only meaningful for underdamped systems because the logarithmic decrement is defined as the natural log of the ratio of any two successive amplitudes, and only underdamped systems exhibit oscillation.