Damping

The damping ratio is defined as the ratio of the damping constant to the critical damping constant

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

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

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

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 then 1 amplitude, meaning its the only one to oscilate.

Mathematical logic for definition of damping ratio

The ordinary differential equation governing a damped harmonic oscillator is

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

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:

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

This equation can be solved with the ansatz

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

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 $\zeta >1$, 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 $\zeta <1$, 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 $\exp(i\omega _{0})$, as expected.)

• Critically damped:The case where $\zeta =1$ 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).

See also

• Damping