Resonance and Damping

Free oscillation — an object oscillating with no external driving force and no resistive force, at its natural frequency (f₀), determined by its own properties such as mass and stiffness.

Forced oscillation — an object made to oscillate by a periodic external force, at the driving frequency.

Resonance — when the driving frequency equals the natural frequency, energy is transferred to the oscillator most efficiently and the amplitude rises sharply.

Why the transfer is efficient at resonance — the driving force stays in step with the oscillator's motion, pushing in the direction it is already moving throughout each cycle. Off resonance the two drift out of step, so the driver spends part of each cycle opposing the motion and the amplitude stays small.

Example: pushing a child on a swing. Push once per swing, in time with the motion, and the amplitude builds quickly. Push at random moments and some pushes help while others fight, so nothing much happens. The frequency matters more than the force.

Damping — any resistive force that removes energy from an oscillator, reducing its amplitude over time.

Light damping — amplitude decays slowly over many oscillations, in an exponential envelope. The period is almost unchanged.

Heavy damping — amplitude falls quickly, over few oscillations, and the period lengthens noticeably.

Critical damping — the system returns to equilibrium in the shortest possible time without oscillating at all. It does not oscillate once, which is a common misconception.

Overdamping — returns to equilibrium without oscillating, but more slowly than critical damping.

Damping and the resonance curve — plotting amplitude against driving frequency gives a peak at the natural frequency. Increasing the damping:

lowers the peak amplitude,

broadens the peak, and

shifts the maximum to a slightly lower frequency.

Applications — car suspension is close to critically damped, so the car settles quickly after a bump rather than bouncing. Buildings in earthquake zones use dampers to reduce resonant response to ground motion. A loudspeaker cone is damped so that it reproduces sound faithfully rather than ringing at its own natural frequency.

Resonance put to use — it is not only a hazard. MRI relies on nuclear magnetic resonance, a radio receiver is tuned so its circuit resonates at the chosen station's frequency, and a musical instrument's body resonates to amplify the note being played.