Damping ratio

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

In control theory, the damping ratio is a measure of the damping of the system, and therefore how long the system will take to reach its final value and the amount of overshoot. If the damping ratio is low, the system can oscillate for longer (underdamping), if it is large, it may not oscillate at all (overdamping). 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), 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 for underdamped vibrations only, via the following relation

$\zeta = \frac{\delta}{\sqrt{(2\pi)^2+\delta^2}}.$

This relation only works for underdamped vibrations because the logarithmic decrement is the natural log of the ratio of any two successive amplitudes. For the three cases, overdamped, critically damped & Underdamped, only underdamped has more than 1 amplitude, meaning it's the only one to oscillate.

[edit] 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).