Resonance

Resonance is the large swing of an oscillator pushed by a periodic force at a frequency close to its own. Here a mass hangs on a spring, a damper slows it, and a force $F_0\cos\omega t$ pushes it. The page records the swing cycle by cycle, shows when each push feeds energy in, and compares the result with the steady response at every drive frequency.

The drive frequency is $r = \omega/\omega_0$, displacement is in units of the static deflection $x_{\mathrm{st}} = F_0/k$, and time is in natural periods $T_0$. With damping, the motion from rest is a steady swing at the drive frequency plus a free response that dies away. The yellow points mark the largest $|x|$ in each drive cycle. They approach the steady amplitude as the free part fades, sometimes rising and falling on the way.

The strip under the record is the power $P = F v$. A push adds energy when it points the way the mass is moving. In the damped steady swing at $r = 1$ the mass lags the force by exactly a quarter cycle, so the force is in step with the velocity and every push feeds energy in. In a steady swing away from $r = 1$ the force works against the motion for part of each cycle, shown hatched. Far from $r = 1$ that part approaches half the cycle, and the work nearly cancels.

Why does the swing stop growing? Take a sinusoidal swing of amplitude $A$ that lags the force by a quarter cycle, as the steady swing at $r = 1$ does. In one cycle the force puts in $\pi F_0 A$, and the damper, with coefficient $c$, takes out $\pi c\omega A^2$. Input grows with $A$ and loss with $A^2$, so they balance at a single amplitude, $A = x_{\mathrm{st}}/(2\zeta)$. While the swing builds, the energy account shows each cycle's actual input, loss and change in stored energy.

Less damping makes the peak higher and narrower. With $\zeta > 0$ the mean input power is at least half its peak over a band of drive frequencies exactly $2\zeta$ wide, the half-power width marked on the chart. The displacement peaks below $r = 1$, at $r = \sqrt{1 - 2\zeta^2}$, as long as $0 < 2\zeta^2 < 1$.

With weak damping the free part oscillates at its own frequency, $\omega_d = \omega_0\sqrt{1 - \zeta^2}$. When that is close to the drive frequency but not equal, the two drift in and out of step, so the record rises and falls in beats. The beats fade as damping removes the free part. Without damping nothing removes it and the motion never settles. For $r \ne 1$ the curve then gives the amplitude of a periodic motion that the run does not approach; at $r = 1$ the swing grows without limit.