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.
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