Figure 123: Three curves: Underdamped (b/m = 2π) barely oscillates. T ′ is now clearly longer than T0. Critically damped (b/m = 4π) goes exponentially to zero in minimum time. Overdamped (b/m = 8π) goes to zero exponentiall, but much more slowly.

through zero at all.

Note that these inequalities and equalities that establish the critical boundary between oscillating and non-oscillating solutions involve the relative size of the inverse time constant associated with damping compared to the inverse time constant (times 2π) associated with oscillation. When the former (damping) is larger than the latter (oscillation), damping wins and the solution is overdamped. When it is smaller, oscillation wins and the solution is underdamped. When they are precisely equal, oscillation precisely disappears, k isn’t quite strong enough compared to b to give the mass enough momentum to make it across equilibrium to the other side. Keep this in mind for the next semester, where exactly the same relationship exists for LRC circuits, which exhibit damped simple harmonic oscillation that is precisely the same as that seen here for a linearly damped mass on a perfect spring.

Example 9.3.1: Car Shock Absorbers

This example isn’t something one can compute, it is something you experience nearly every day, at least if you drive or ride around in a car.

A car’s shock absorbers are there to reduce the “bumpiness” of a bumpy road. Shock absorbers are basically big powerful springs that carry your car suspended in equilibrium between the weight of the car and the spring force.

If your wheels bounce up over a ridge in the road, the shock absorber spring compresses, storing the energy from the “collision” briefly and then giving it back without the car itself reacting. However, if the spring is not damped, the subsequent motion of the car would be to bounce up and