Damping ratio: Definition from Answers.com

In engineering, the damping ratio is a 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, will, if pulled and released, bounce up and down. On each bounce, the system is trying to return to its equilibrium position, but overshoots it. Frictional losses damp the system and 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 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.

1 Oscillation modes
2 Damping ratio
3 Derivation of the damping ratio
4 Q factor
5 See also
6 References

Oscillation modes

Were the spring-mass system completely lossless, the mass would oscillate indefinitely, each bounce of equal height to the last. This hypothetical case is called undamped.
If the system contained high losses, for example if the spring-mass experiment were conducted in a viscous fluid, the mass could slowly return to its rest position without ever overshooting. This case is called overdamped.
Commonly, the mass tends to overshoot its starting position, and then return, overshooting again. With each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. This case is called underdamped.
Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. This case is called critical damping.

Damping ratio

The effect of varying damping ratio on a second-order system.

The damping ratio provides a mathematical means of expressing the level of damping in a system and which one of the cases above is in effect. It is defined as the ratio of the damping constant to the critical damping constant:

$\zeta = \frac{c}{c_c}.$

The damping ratio is unitless, because it is the result of dividing the units of the damping constant (N·s/m) by the critical damping constant (N·s/m); the units cancel out.

The damping ratio is a parameter, usually denoted by ζ (zeta)^[1], 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.

For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, the damping ratio is

$\zeta = \frac{c}{2 \sqrt{km}}.$

The damping ratio is also related to the logarithmic decrement $δ$ for underdamped vibrations via the relation

$\zeta = \frac{\delta}{\sqrt{(2\pi)^2+\delta^2}} \qquad \text{where} \qquad \delta \triangleq \ln\frac{x_1}{x_2}.$

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.

Derivation of the damping ratio

The ordinary differential equation governing a damped harmonic oscillator is

$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0.$

Using the natural frequency of the simple harmonic oscillator $\omega_0=\sqrt{k/m}$ and the definition of the damping ratio above, we can rewrite this as:

$\frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x = 0.$

This equation can be solved with the ansatz

$x(t)=C e^{-\omega t},\,$

where C and ω are both complex constants. That ansatz 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,

$\omega = \omega_0 (\zeta \pm \sqrt{\zeta^2-1})$ .

Overdamped:If ω is a real number, then the solution is simply a decaying exponential with no oscillation. This case occurs for $ζ > 1$ , and is referred to as overdamped.

Underdamped:If ω is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like $\exp(i \omega_0 \sqrt{1-\zeta^2})$ . This case occurs for $ζ < 1$ , and is referred to as underdamped. (The case where $\zeta \to 0$ corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like $exp(i ω 0)$ , as expected.)

Critically damped:The case where $ζ = 1$ is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism).

Q factor

Main article: Q factor

The factors Q, damping ratio ζ, and attenuation α are related such that^[2]

$\zeta = \frac{1}{2 Q} = { \alpha \over \omega_0 }.$

So the quality factor can be expressed as

$Q = \frac{1}{2 \zeta} = { \omega_0 \over 2 \alpha },$

and the exponential attenuation rate can be expressed as

$\alpha = \zeta \omega_0 = { \omega_0 \over 2 Q }.$

When a second-order system has $ζ < 1$ (i.e, when the system is underdamped), it has two complex conjugate poles that each have a real part of $α$ . That is, the attenuation parameter $α$ represents the rate of exponential decay of the oscillations (e.g., after an impulse) of the system. A lower damping ratio implies a lower attenuation, and so very underdamped systems oscillate for long times. For example, high quality bells have an approximately pure sinusoidal tone for a long time after being struck by a hammer.

References

^ Alciatore, David G. (2007). Introduction to Mechatronics and Measurement Systems (3rd ed.). McGraw Hill. ISBN 978-0-07-296305-2.
^ William McC. Siebert. Circuits, Signals, and Systems. MIT Press.

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Damping ratio

Contents

Oscillation modes

Damping ratio

Derivation of the damping ratio

Q factor

See also

References