Beats and Resonance in a Footbridge

Push a structure at nearly its own rhythm and the motion swells and fades in slow waves called beats. Push at exactly its rhythm and each push adds energy: the swings grow without limit. That's why soldiers break step on bridges.

How it works

Mechanical vibrations

In one sentence

For $\omega \ne \omega_0$ , $y = \frac{F}{\omega_0^2 - \omega^2}(\cos\omega t - \cos\omega_0t)$ solves $y'' + \omega_0^2y = F\cos\omega t$ from rest; by a sum-to-product identity it equals $\frac{2F}{\omega_0^2 - \omega^2}\sin\frac{(\omega_0 - \omega)t}{2}\sin\frac{(\omega_0 + \omega)t}{2}$ .

General formula

{\htmlData{sym=3}{y}}'' + {\htmlData{sym=0}{\omega_0}}^2\,{\htmlData{sym=3}{y}} = {\htmlData{sym=2}{F}}\cos{\htmlData{sym=1}{\omega}} t, \;\; {\htmlData{sym=3}{y}}(0) = {\htmlData{sym=3}{y}}'(0) = 0 \;\Rightarrow\; {\htmlData{sym=3}{y}} = \frac{\htmlData{sym=2}{{F}}}{{\htmlData{sym=0}{\omega_0}}^2 - {\htmlData{sym=1}{\omega}}^2}\left(\cos{\htmlData{sym=1}{\omega}} t - \cos{\htmlData{sym=0}{\omega_0}} t\right)

The bridge's own rhythm and the pushes drift in and out of step. In step, every push adds energy; out of step, every push takes it away.

What the symbols mean

$\omega_0$
Natural frequency
The rhythm the bridge swings at by itself.
$\omega$
Forcing frequency
The rhythm of the pushes, like footsteps.
$F$
Push strength
How hard each push is, per unit mass.
$y$
Displacement
How far the bridge has moved from rest.

An undamped structure with natural frequency $\omega_0$ is pushed at frequency $\omega$ , starting from rest.
The solution is a difference of two cosines, the same thing as a fast wave inside a slow envelope: beats.
The envelope's height $2F/|\omega_0^2 - \omega^2|$ grows as the two frequencies approach each other.
At exactly $\omega = \omega_0$ the formula breaks down and the solution becomes $\frac{F}{2\omega_0}t\sin\omega_0t$ : resonance.

Learn the math

Mechanical VibrationsDiff Eq · Lesson 21, Second Order Differential Equations