Dumped oscillations

When the losses are small, the solution to this equation is a sinusoid slowly decreasing with time.
Play with the simulation, change the value of the damping factor (viscosity) and look the results.
The frequency of the oscillations is, for the undamped system is:

and so, the period, T = 2p/w 0 , increases with m and decreases with k. This frequency is the one we have called "natural frequency" of the system. The oscillation frequency slightly changes from the value reported when the system is lossy.
The effect of the friction, represented by a non zero value of g, is to produce a gradual loss of energy of the vibrating system, usually in the form of heat. If you have a look to the system when enough time has passed, the oscillations will be damped and the mass will eventually be at rest. To keep the system vibrating an external force should be applied to it in order to overcome the loss of energy due to friction. Then we can find a resonant behavior.