Damped Harmonic OscillatorsIf you leave a pendulum swinging, it will eventually come to a stop because of friction and air resistance. Our omission of friction up to this point is justified because our results closely describe what actually happens when frictional forces are weak or when only a few oscillations are considered.

From energy conservation, we know that the amplitude of the motion must decrease. This decrease is approximately exponential, given by

x = Aoe^{-gt}cos(wt), where g = bt/2mwhere the constant

(gamma) depends on the frictional force and the mass. When t = 0, the exponential function is 1 so we obtain our previous result. As t approaches infinity, x goes to zero. Functionally, the exponential term squeezes the cosine function down to zero. The image below uses actual data from a spreadsheet program.g

This is typical of underdamped motion. There are two other kinds of damped motion:

Overdamped Motion- If the friction is too strong, the oscillator won't move, but if it is just a touch less, it will move slowly toward equilibrium and then stop.See Figure Below.

Critically Damped Motion- It corresponds to a damped spring that returns as quickly as possible to the equilibrium position without overshooting. When an engineer wants to make a system in which oscillations disappear in the least amount of time, they critically damp the system. (Think about shock absorbers in a car). See figure below:

Forced Oscillators and ResonanceIf instead of letting a system move in harmonic motion of its own accord, we push on it, different situations will occur. Sometimes a driving force will have little or no effect, while at other times having incredible effect

When the frequency of the driving force corresponds to the natural angular frequency, the situation of resonance is achieved. In this situation, a series of small pushes lead to large changes in amplitude.

If accidentally the frequency of a driving force matches the natural frequency of a system, the results can be disastrous. In 1940, the Tacoma Narrows Bridge in Washington state was accidentally designed with its natural frequency very close to the natural frequency of the wind currents. Soon after it was put in place it began to swing, farther and farther each time until it snapped.

It is recommended against soldiers marching across a bridge in step for a similar reason. Resonance also explains why some singers can break crystal glasses with certain notes. These notes correspond to the natural frequency of the crystalline structure of the glass.

For homework, try:

13.39, 13.41