down for many tens or hundreds of meters down the road, with your control over the car seriously compromised. For that reason, shock absorbers are strongly damped with a suitable fluid (basically a thick oil).

If the oil is too thick, however, the shock absorbers become overdamped. The car takes so long to come back to equilibrium after a bump that compresses them that one rides with one’s shocks constantly somewhat compressed. This reduces their e ectiveness and one feels “every bump in the road”, which is also not great for safe control.

Ideally, then, your car’s shocks should be barely underdamped. This will let the car bounce through equilibrium to where it is “almost” in equilibrium even faster than a perfectly critically damped shock and yet still rapidly damp to equilibrium right away, ready for the next shock.

So here’s how to test the shocks on your car. Push down (with all of your weight) on each of the corner fenders of your car, testing the shock on that corner. Release your weight suddenly so that it springs back up towards equilibrium.

If the car “bounces” once and then returns to equilibrium when you push down on a fender and suddenly release it, the shocks are good. If it bounces three or four time the shocks are too underdamped and dangerous as you could lose control after a big bump. If it doesn’t bounce up and back down at all at all and instead slowly oozes back up to level from below, it is overdamped and dangerous, as a succession of sharp bumps could leave your shocks still compressed and unable to absorb the impact of the last one and keep your tires still on the ground.

Damped oscillation is ubiquitous. Pendulums, once started, oscillate for only a while before coming to rest. Guitar strings, once plucked, damp down to quiet again quite rapidly. Charges in atoms can oscillate and give o light until the self-force exerted by their very radiation they emit damps the excitation. Cars need barely underdamped shock absorbers. Very tall buildings (“skyscrapers”) in a city usually have specially designed dampers in them as well to keep them from swaying too much in a strong wind. Houses are build with lots of damping forces in them to keep them quiet. Fully understanding damped (and eventually driven) oscillation is essential to many sciences as well as both mechanical and electrical engineering.

9.4: Damped, Driven Oscillation: Resonance

By and large, most of you who are reading this textbook directly experienced damped, driven oscillation long before you were five years old, in some cases as early as a few months old! This is the physics of the swing, among other things. Babies love swings – one of our sons was colicky when he was very young and would sometimes only be able to get some peace (so we could get some too!) when he was tucked into a wind-up swing. Humans of all ages seem to like a rhythmic swaying motion; children play on swings, adults rock in rocking chairs.

Damped, driven, oscillation is also key in another nearly ubiquitous aspect of human life – the clock. Nature provides us with a few “natural” clocks, the most prominent one being the diurnal clock associated with the rotation of the Earth, read from observing the orientation of the sun, moon, and night sky and translating it into a time.

The human body itself contains a number of clocks including the most accessible one, the heartbeat. Although the historical evidence suggests that the size of the second is derived from systematic divisions of the day according to numerological rules in the extremely distant past, surely it is no accident that the smallest common unit of everyday time almost precisely matches the human heartbeat. Unfortunately, the “normal” human heartbeat varies by a factor of around three as one moves from resting to heavy exercise, a range that is further increased by the abnormal heartbeat of people with cardiac insu ciency, cardioelectric abnormalities, or taking various drugs. It isn’t a very precise clock, in other words, although as it turns out it played a key role in the development of

tall building requires a careful consideration and damping of the natural frequencies of the structure. The Tacoma Bridge Disaster185 serves as a modern-times example of the consequences of failing to design for resonance. In more recent times part of the devastation caused by the Haiti Earthquake186 was caused by the lack of earthquake-proofing – protection against earthquake-driven resonances – in the cheap construction methods used in buildings of all sorts.

From all of this, it seems like establishing at the very least a semi-quantitative understanding of resonance is in order, and as usual math, physics or engineering students will need to go the extra 190080 barleycorn grain lengths and work through the math properly. To manage this, we need to begin with a model.

9.4.1: Harmonic Driving Forces

Let’s start by thinking about what you all know from learning to swing on a swing. If you just sit on a swing, nothing happens. You have to “pump” the swing. Pumping the swing is accomplished by pulling the ropes and shifting your weight at the highest point of each oscillation so that the force exerted by the rope no longer passes through your center of mass and hence can exert a torque in the current direction of rotation. You know from experience that this torque must be applied periodically at a frequency that matches the natural frequency of the swing and in phase with the motion of the swing in order to increase the amplitude of the swinging oscillation. If you simply jerk backwards and forwards with the same motions as those used to pump “right” but at the wrong (non-resonant) frequency or randomly, you don’t ever build up much amplitude. If you apply the torques with the wrong phase you will not manage to get the same amplitude that you’d get pumping in phase with the motion.

That’s really it, qualitatively. Resonance consists of driving an oscillator at its natural frequency, in phase with the motion, to achieve the greatest possible oscillation amplitude (ultimately limited by things like practical physical constraints and damping).

A pendulum, however, isn’t a good system for us to use as a quantitative model. For one thing, it isn’t really a harmonic oscillator – we automatically adjust our pumping to remain resonant as we swing closer and closer to the angle of π/2 where the swing chains become limp and we can no longer cheat some torque out of the combination of the pivot force and gravity, but the frequency itself starts to significantly change as the small angle approximation breaks down, and breaking it down is the point of swinging on a swing! Who wants to swing only through small angles!

The kind of force we exert in swinging a swing isn’t too great, either. It is hardly smooth – we really only pump at all very close to the top of our swinging motion (on both sides) – in between we just coast. We’d prefer instead to assume a periodic driving force that is mathematically relatively easy to treat.

These two things taken together more or less uniquely determine the best model for us to use to understand resonance. This is an underdamped mass on a spring being driven by an external harmonic driving force, all in one dimension:

In this expression, both the angular frequency ω and the amplitude of the applied force are free parameters. Note especially that ω is not necessarily the natural or shifted frequency of the mass on the spring – it can be anything, just as you can push your little brother or sister on a swing at

185Wikipedia: http://www.wikipedia.org/wiki/Tacoma Narrows Bridge (1940). Again, a marvelous article that contains a short clip of the bridge oscillating in resonance to collapse. Note that the resonance in question was due more to the driving of several wave resonances rather than a simple harmonic oscillator resonance, but the principle is exactly the same.

186Wikipedia: http://www.wikipedia.org/wiki/2010 Haiti earthquake.