Damped oscillations

Simple harmonic motion occurs in ideal systems, where there is no friction. In real systems, non-conservative forces, such as friction, retard the motion and the mechanical energy diminishes in time. For example, this happens in a mass-spring system in a viscous liquid.

The retarding forces (such as friction) can be often written in the form

\[\mathbf{F}_{\mathrm{R}} = -\gamma \mathbf{v} = -\gamma \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t},\]

where \(\gamma\) is called the damping coefficient.

The net force in one dimension, including the harmonic force \(-k x\) and the retarding force:

\[\sum{F_x} = - k x -\gamma \frac{\mathrm{d}x}{\mathrm{d}t}.\]

So the basic equation of damped oscillations is
\[m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x - \gamma \frac{\mathrm{d}x}{\mathrm{d}t}.\]

This can be rewritten in another form:

\[\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \frac{\gamma}{m} \frac{\mathrm{d}x}{\mathrm{d}t}+ \frac{k}{m} x = 0.\]

The solution of this equation can be of two types, depending on the damping coefficient \(\gamma,\) in relation to the so called critical damping coefficient:

\[\gamma_{\mathrm{c}} = 2 m \omega _0 = 2 \sqrt{k m},\]

where \(\omega_0\) is the angular frequency in the absence of a retarding force (simple harmonic oscillator):
\[\omega_0 := \sqrt{\frac{k}{m}}.\]