Chapter 6
Damped Oscillations
If you build a real simple harmonic oscillator by attaching a mass to a spring and letting it oscillate back and
forth, you’ll find that it doesn’t oscillate forever, as would be predicted by Eq. (5.2). Instead, the motion will
damp out due to frictional forces, and the oscillator will eventually stop oscillating.
We can model the damping forceFdas being proportional to the speedvof the oscillator:
FdDbv; (6.1)wherebis a damping constant (in units of kg/s). There are three different cases of damped motion:under-
damped,overdamped, andcritically damped. In the following discussion, the natural oscillation frequency
of the undamped oscillator is^1! 0 D
p
k=m.6.1 Underdamped
In the underdamped case, the damping constantbis small (b<2m! 0 ), and the oscillations gradually
decrease in amplitude. In this case, the motion will be described by
x.t/DAe.b=2m/tcos.!^0 tCı/; (6.2)whereAis the initial amplitude andıis the phase constant. The underdamped oscillator oscillates at a slower
frequency!^0 than if it were undamped, where!^0 is given by
!^0 D! 0
s
1 b
2m! 02
: (6.3)
Fig. 6.1 shows what the motion looks like: it is a cosine curve modulated by an overall exponentially decaying
“envelope”.
6.2 Overdamped
Now imagine that a simple harmonic oscillator is immersed in a thick liquid like honey. In this case the
damping constantbis large (specifically,b>2m! 0 ), and the motion is said to beoverdamped. If the mass is
(^1) The quantity! 0 is customarily pronounced “omega-nought”,noughtbeing an old-fashioned term forzero.