we did not reach this in class but I include it anyway because it might help you solve the homework problems due next Monday
Until now we have focused on the spontaneous oscillations of
mechanical systems with linear restoring forces. In many cases
we are in a situation where external conditions dictate a frequency
of oscillation. An example might be a car with unbalanced tires and
something which rattles inside. When driving at a velocity, v, the whole car shakes
at an angular frequency 
. As we vary the speed the
driving frequency thus changes and sometimes we can hear that
certain rattling noises become more pronounced at specific
speeds. This phenomenon is called resonance. When the frequency
of the driving force matches the natural frequency of oscillation
then the response in terms of the amplitude of oscillation
is largest and produces most noise.
We can account for this phenomenon analytically by adding a
time dependent driving force to our differential equation:
Using the analysis based on complex numbers which was
alluded to above it is simple to show that the amplitude resulting
from a fixed driving force, 
peaks for 
:
Features to note are that damping determines the width of the resonance peak. Specifically we have the Full Width at Half Maximum
Thus a weakly damped system has a sharp resonance.
Engineers have to build heavily damped systems which do not
have resonances in the range of frequencies corresponding to
expected periodic perturbations. So their job is not done
by ensuring static equilibrium. They also have to analyze
the dynamics of structures to make sure that resonant conditions will
not occur. As an example of a job gone wrong in this respect we
show a movie of the famous collapse of the Tacoma Narrows Bridge
in Tacoma Washington 1940.