### Damped Harmonic Oscillator Driven by a Square Wave

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__NOTES__:

1. The resonant frequency w_{o}
of the oscillator is
1.414, so the oscillations grow quickly at this driving frequency. Looking
simultaneously at the graphs of force and motion, it is clear that positive work
is being done on the system at all the right times.

2.
Students can be queried about other effective drive frequencies. Typical
answers are:

a.
2w_{o
}: Here, work that is done during positive motion is exactly
cancelled by work done during negative motion so the net energy input is 0.

b.
w_{o} / 2 : Once again, work done during
positive motion exactly cancels work done during negative motion so the net
energy input is 0.

c.
3w_{o
}: Same as above.

d.
w_{o} / 3 : This one works because the
first harmonic of a square wave at this frequency coincides with w_{o}.
Note that, compared to driving at the resonant frequency
w_{o}, the amplitude of
oscillations grows at 1/3 the rate because the amplitude of this harmonic is 1/3
as large as that of the fundamental. This behavior can also be understood in the
context of the work done: every 3 cycles, the force is applied during 2
positive 1/2 cycles and 1 negative 1/2 cycle, for a total of 1/3 the energy
input when driven at the resonant frequency.

e.
Students should now realize that the only effective drive frequencies are odd
fractions of w_{o} where the square wave
has a harmonic at the resonant frequency of the oscillator.

3.
A small amount of damping can be added to show how the work done by the
resistive force in real oscillators eventually balances the input energy,
yielding steady-state motion.

__Reference__: Jerry B. Marion and Stephen T. Thornton, *Classical
Dynamics of Particles and Systems* (4th edition, 1995) p. 137-140.

* This demonstration was written by Tim Gfroerer and scripted by Mario
Belloni.