Damped Oscillations

In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.

Although friction and other non-conservative forces can be made negligibly small, complete undamped motion is rare. In fact, these non-conservative forces are often required to dampen oscillations, such as in-car shock absorbers. For a system that has a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion, but the amplitude gradually decreases. This occurs because the non-conservative damping force removes energy from the system, usually in the form of thermal energy. In general, energy removal by non-conservative forces is described as

where W_nc is work done by a non-conservative force (here, the damping force). For a damped harmonic oscillator, W_nc is negative because it removes mechanical energy (KE + PE) from the system.

This text is adapted from Openstax, College Physics, Section 16.7: Damped Harmonic Motion and Openstax, University Physics Volume 1, Section 15.4: Damped Oscillations.