Chapter 7
Forced Oscillations
Now suppose that we have a harmonic oscillator that is being driven, orforced. For example, imagine a
spring that has a massmattached to one end, and the other end is connected to a motor-driven piston that
moves back and forth. What happens in this case is that the motion of the oscillator is fairly complicated at
first, then settles down to a “steady-state” motion, where the oscillator oscillates at the same frequency as the
driving force.
Suppose we have a damped oscillator whose natural oscillation frequency is! 0 D
p
k=m, and the
oscillator is being driven by a force of of the formF.t/DF 0 sint, so the driving force has amplitudeF 0
and angular frequency. Then after the initial complicated motion has died out, the steady-state motion will
be an oscillatory motion with the same frequency as the driving force,
x.t/DAcos.tCı/: (7.1)HereAis the amplitude of the motion, which will depend on how far the driving frequencyis from the
natural frequency! 0 :
AD
F 0 =m
q
.^2 ! 02 /^2 C.b=m/^2: (7.2)
7.1 Resonance
Notice that in Eq. (7.2), the denomonator will be smallest whenD! 0 , so that the oscillator is being
driven at its natural frequency of oscillation. This situation is calledresonance, and can result in very large
oscillations. (Note that in Eq. (7.2) if the damping constantbD 0 andD! 0 , the denominator is zero and
amplitude becomes infinite!) We’re familiar with examples of resonance in everyday life: for example, an
opera singer who sings a loud, high note and is able to shatter a crystal goblet. Engineers have to be careful
in designing things like buildings, bridges, aircraft, spacecraft, etc. that the objects won’t be subjected to
being driven at one of the natural frequencies of oscillation of the object. Marching soldiers break step when
crossing a bridge, just in case the cadence of the march is at one of the natural frequencies of oscillation of
the bridge, which could cause the bridge to collapse.
Fig. 7.1 shows a plot of amplitude vs. forcing frequency for a typical forced oscillator. Resonance is
shown by the large increase in the amplitude of the forced oscillations whenD! 0. The smaller the
damping force, the larger the amplitude at resonance.