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Damping

From Academic Kids

Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce oscillations.

In applied mathematics, damping is mathematically modelled as a force with magnitude proportional to that of the velocity of the object but opposite in direction to it.

In playing stringed instruments such as Guitar, Ukulele and Steel Guitar, Damping is the quieting or abrupt silencing of the strings after they have been sounded, by pressing with the edge of the palm, or other parts of the hand such as the fingers on one or more strings near the bridge of the instrument.

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Springdampermass.png
Image:Springdampermass.png

An ideal Mass-Spring-Damper system with mass <math>m (kg)<math>, spring constant <math>k (\frac{N}{m})<math> and damper constant <math>R (\frac{Ns}{m})<math> can be described with the following formulae:

<math> \begin{matrix} F_{spring} & = & - k x \\ F_{damper} \ & = & - R \dot{x} = - R \frac{dx}{dt} \\ \Sigma\ F \ & = & m \ddot{x} = m \frac{d^2x}{dt^t} \end{matrix} <math>

Where <math>x<math> is the displacement of the centre of the mass. These equations combine to:

<math> m \ddot{x} + R \dot{x} + k x = 0 <math>

This is a second order differential equation in <math>t<math>. It can be solved by assuming <math>x = e^{\gamma t}<math> with <math>\gamma<math> complex. Then:

<math> m \gamma^2 + R \gamma + k = 0 <math>

Which can be solved to:

<math> \gamma = \frac{-R \pm \sqrt{R^2 - 4 m k}}{2m} <math>

When <math>R^2 - 4 m k = 0<math>, <math>\gamma<math> is real and the system has critical damping.


The solution can be generally written as:

<math> x (t) = A e^{-\frac{R}{2m} t} cos(\sqrt{\frac{k}{m} - \frac{R}{2m}} t + \phi\ ) <math>

Where <math>A<math> and <math>\phi<math> are determined by the initial position and velocity of the mass.

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