Damping, driving, and resonance

Observation

Shove a playground swing once and walk away: the arc shrinks pass by pass until the swing hangs still. Stand behind it instead and give a light push each time it returns, timed to its own rhythm, and the arc grows alarmingly, far beyond anything one push could produce. Push at random moments and much of your effort cancels itself. Two facts sit in that scene. Free oscillation decays, so something is draining energy. And weak pushes, correctly timed, beat the drain. The swing analogy has a limit worth stating at the outset: a hand delivers a brief shove once per cycle, while the model below drives with a smoothly varying force at every instant. The timing logic carries over; the shape of the push does not.

The question this entry answers: what sets the rate of the decay, and why does the response to a periodic push depend so sharply on the push's frequency?

Variables

The displacement x is the distance from equilibrium and t is time. The mass m and stiffness k carry over from the undamped oscillator, whose natural angular frequency omega_0 = sqrt(k/m) is the rate the system picks for itself. Three quantities are new. The damping coefficient b is the drag force per unit of velocity, so its units are force divided by speed. The drive frequency omega is the angular frequency of the external periodic force, chosen by whoever is pushing, and F is that force's amplitude. The steady amplitude A is the displacement amplitude the driven system eventually settles into. One dimensionless ratio, written Q and built below from m, omega_0, and b, counts how many cycles the free motion survives; it earns its name in the derivation.

Model

Start from what free decay looks like. Displace a real oscillator and the envelope of its swing shrinks by the same fraction each cycle: an exponential fade. A drag force proportional to velocity, written -b x' where x' is the velocity dx/dt, is the simplest linear loss law, and it is the choice that produces exactly an exponential envelope, as the derivation shows. It is a modeling choice, and a testable one: sliding friction of constant magnitude gives a straight-line envelope instead, and drag growing with the square of the speed gives yet another shape, so the decay fit in the Build task either comes out exponential or convicts the model. Add an external periodic force F cos(omega t), write x'' for the acceleration d2x/dt2, and Newton's second law gives one equation holding the whole subject:

m x'' + b x' + k x = F cos(omega t)

Everything below unpacks this line: free decay from the case F = 0, the driven response from the general case.

Derivation

Undriven decay. Set F = 0 and try a cosine inside a shrinking envelope: x(t) = A_0 e^(-(b/2m) t) cos(omega_d t + phi), where A_0 is the starting amplitude, phi fixes where in the cycle the clock starts, and omega_d is the frequency of the decaying oscillation. Substituting shows this satisfies the equation provided omega_d^2 = omega_0^2 - (b/2m)^2. Two results fall out. The envelope decays at the rate b/2m, so heavier drag or lighter mass means faster fading, and the loss is a fixed fraction per cycle, which is the exponential envelope the model was chosen to reproduce. And the free oscillation runs at omega_d, a touch below omega_0, with a gap that shrinks as damping lightens. A natural yardstick for "light" is the ratio Q = m omega_0 / b: a short calculation with the envelope shows the amplitude falls to 1/e of its starting value after Q/pi full cycles, so Q directly counts how long the ringing lasts. A ratio that measures how good an oscillator is at holding its motion has earned its conventional name, the quality factor.

Driven steady state. Turn the drive on and wait several multiples of 2m/b for the leftover free motion to die away; what remains oscillates at the drive's frequency, whatever the system's own preference. Try x(t) = A cos(omega t - delta), where delta is the angle by which the response lags the drive. Substituting and collecting the cos(omega t) and sin(omega t) terms gives two equations in A and delta; squaring and adding eliminates delta and yields the amplitude response:

A(omega) = (F/m) / sqrt((omega_0^2 - omega^2)^2 + (b omega / m)^2)

Read it across the range. At omega near zero the denominator is close to omega_0^2 = k/m, so A is about F/k, the static stretch the same force would produce held steady. At high omega the denominator grows as omega^2 and A falls away: the mass cannot follow so fast a push. Near omega = omega_0 the term (omega_0^2 - omega^2)^2 collapses toward zero and the damping term alone holds the denominator up, giving A(omega_0) = F/(b omega_0), which rearranges to Q times F/k. The response at the natural frequency is Q times the static response, and the tall region has a width of roughly omega_0/Q. A tall narrow peak, earned by weak damping, is what the word resonance names.

One caveat the light-damping habit hides. Minimizing the full denominator puts the displacement peak at omega_peak^2 = omega_0^2 - b^2/(2m^2), slightly below omega_0; the velocity amplitude and the average power absorbed peak exactly at omega_0. At high Q the three frequencies omega_d, omega_0, and omega_peak crowd so close together that ordinary measurement cannot split them.

The energy view, worded carefully. Over one cycle the drive does net work only through the component of its force in phase with the velocity; the out-of-phase component adds and removes energy in equal shares. In steady state that per-cycle work exactly balances what damping drains, and the balance holds at every drive frequency. What distinguishes resonance is that there the lag delta reaches 90 degrees, the drive aligns fully with the velocity, and the pumping is at its largest.

Limits and boundary conditions

The equation assumes loss proportional to velocity; where the measured envelope is straight-line or otherwise non-exponential, the loss law is different and the formulas above do not apply as written. Small amplitude is assumed too, since the linear restoring force is itself a small-displacement law. A(omega) describes the steady state only; for several decay times after any change, the motion is a mixture of the fading free oscillation and the growing driven one. And there is a heavy-damping boundary: the denominator has an interior minimum, and hence A(omega) a peak, only when Q exceeds roughly 1/sqrt(2). Below that, the response falls monotonically from F/k with no peak anywhere. This boundary is what makes the "infinite amplitude" error precise: for any nonzero b the peak is finite, about Q times the static response, and for heavy enough damping there is no peak to speak of.

Common mistakes

Three errors recur. Treating resonance as demanding an exact frequency match: the response is a peaked band of width about omega_0/Q, so a drive merely near the natural frequency still builds a large amplitude, and at low Q the band is broad. Treating damping as a pure defect to be engineered away: added damping is a standard cure for a destructive resonance, since the peak height Q times F/k falls as b rises, and a car's shock absorbers work exactly this way. A skyscraper's tuned mass damper is a related but distinct device, a tuned counter-oscillator that absorbs the motion and carries damping, rather than pure added damping to the main structure. And expecting the resonant amplitude to grow without bound: the divergence belongs to the idealized b = 0 case, and any real loss caps the peak at a value the damping sets.

Build with it

Build or borrow one oscillator; a mass hanging on a spring serves. Stage one, decay: displace it, release, and record the amplitude cycle by cycle, on video if timing by eye is unreliable. Fit the envelope. If it is exponential, the velocity-proportional loss model has passed its test; read off the decay rate b/2m, and get Q by counting the cycles for the amplitude to fall to 1/e of its start and multiplying by pi. Stage two, resonance: name the three frequencies in play, the free-decay frequency omega_d you timed, the natural frequency omega_0 computed from it and Q, and the predicted displacement-peak frequency omega_peak, and state whether your Q is high enough that all three coincide within your measurement tolerance, which they do at high Q and measurably fail to do at low Q. Then drive the support through a range of frequencies, with a hand-moved frame, a small motor carrying an offset mass, or a speaker, and confirm that the steady amplitude peaks at the predicted frequency within a stated tolerance, 5 percent being reasonable. Stage three, the decision this capability exists for: pick one concrete change, either added damping (a vane in water, a magnet inducing eddy currents in a nearby plate) or a shifted mass or stiffness, and state in advance whether it will move the peak away from a stated drive frequency or cap the steady amplitude below a stated level, with predicted numbers. Make the change and re-measure. Success requires both parts: the original peak confirmed at the prediction within tolerance, and the chosen change producing its predicted effect.