BELIEVE   ME   NOT!    - -     A   SKEPTICs   GUIDE  

Resonance

No description of SHM would be complete without some discussion of the general phenomenon of resonance, which has many practical consequences that often seem very counterintuitive.^13.10 I will, however, overcome my zeal for demonstrating the versatility of Mathematics and stick to a simple qualitative description of resonance. Just this once.

The basic idea is like this: suppose some system exhibits all the requisite properties for SHM, namely a linear restoring ``force''   $Q = - k \, q$  and an inertial factor  $\mu$. Then if once set in motion it will oscillate forever at its ``resonant frequency''   $\omega = \sqrt{k \over \mu}$,  unless of course there is a ``damping force''   $D = - \kappa \mu q$  to dissipate the energy stored in the oscillation. As long as the damping is weak  [ $\kappa \ll \sqrt{k \over m}$],  any oscillations will persist for many periods. Now suppose the system is initially at rest, in equilibrium, ho hum. What does it take to ``get it going?''

The hard way is to give it a great whack to start it off with lots of kinetic energy, or a great tug to stretch the ``spring'' out until it has lots of potential energy, and then let nature take its course. The easy way is to give a tiny push to start up a small oscillation, then wait exactly one full period and give another tiny push to increase the amplitude a little, and so on. This works because the frequency  $\omega$  is independent of the amplitude  q<sub>0</sub>.  So if we ``drive'' the system at its natural ``resonant'' frequency  $\omega$, no matter how small the individual ``pushes'' are, we will slowly build up an arbitrarily large oscillation.^13.11

Such resonances often have dramatic results. A vivid example is the famous movie of the collapse of the Tacoma Narrows bridge, which had a torsional [twisting] resonance^13.12 that was excited by a steady breeze blowing past the bridge. The engineer in charge anticipated all the other more familiar resonances [of which there are many] and incorporated devices specifically designed to safely damp their oscillations, but forgot this one. As a result, the bridge developed huge twisting oscillations [mistakes like this are usually painfully obvious when it is too late to correct them] and tore itself apart.

A less spectacular example is the trick of getting yourself going on a playground swing by leaning back and forth with arms and legs in synchrony with the natural frequency of oscillation of the swing [a sort of pendulum]. If your kinesthetic memory is good enough you may recall that it is important to have the ``driving'' push exactly ${\pi \over 2}$ radians [a quarter cycle] ``out of phase'' with your velocity - i.e. you pull when you reach the motionless position at the top of your swing, if you want to achieve the maximum result. This has an elegant mathematical explanation, but I promised . . . .