Damping and Resonance

A $\text{free oscillation}$ happens when a system is displaced and left to oscillate on its own, with no external driving force — it oscillates at its own $\text{natural frequency}$, $f_0$, set entirely by its own physical properties (mass, stiffness, length, etc.). A $\text{forced oscillation}$ happens when an external periodic force continuously drives the system at some chosen $\text{driving frequency}$, which does not have to match the natural frequency at all.

$\text{Damping}$ is the loss of energy from an oscillating system, usually to resistive forces like friction or air resistance, causing the amplitude to decrease over time. $\text{Light damping}$ causes the amplitude to decay gradually while the system continues oscillating for many cycles. $\text{Critical damping}$ is the precise amount of damping that returns the system to equilibrium in the shortest possible time, without overshooting or oscillating at all — this is exactly what a well-designed car suspension system aims for. $\text{Heavy (over-)damping}$ returns the system to equilibrium without any oscillation, but more slowly than the critically damped case.

When a system is forced to oscillate, its response amplitude depends strongly on how close the driving frequency is to its natural frequency. As the driving frequency approaches the natural frequency, the amplitude of oscillation grows dramatically — this dramatic build-up is $\text{resonance}$, and it occurs (for light damping) essentially exactly at the natural frequency.

Increasing the amount of damping in a resonant system reduces the maximum amplitude reached at resonance, broadens the range of driving frequencies over which a significant response occurs, and shifts the frequency of peak response slightly below the natural frequency. In the limit of very heavy damping, the resonance peak can disappear almost entirely.

Resonance is deliberately used in musical instruments (a guitar body resonating with a plucked string to amplify sound) and in microwave ovens (tuned to resonate with water molecules to heat food). It can also be dangerous and destructive: soldiers are traditionally ordered to break step crossing a bridge, since marching in step at a frequency close to the bridge’s natural frequency could drive it into large, potentially damaging oscillations. London’s Millennium Bridge had to be closed and retrofitted with dampers shortly after opening, because pedestrians’ footsteps synchronised with (and were amplified by) the bridge’s natural sideways sway.

Engineers combat unwanted resonance in two main ways: designing a structure so its natural frequency is far away from any frequency it’s likely to be driven at, or deliberately adding damping (such as dampers or shock absorbers) to limit the amplitude if resonance does occur.