Damping ratio

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The damping ratio is a parameter, usually denoted by ζ, 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

$\zeta = \frac{c}{2 \sqrt{k*m}}.$

[edit] Mathematical Logic for Definition of 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 $e x p (i ω 0)$ , 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).