We have been studying car suspensions in order to illustrate the role of
visualization in understanding the mathematics that describe a physical
situation. Let's step back and think about what we've been doing.
The problem has been to determine how a car bounces up an down in response to
regularly spaced bumps in a road. We were able to simplify the problem
somewhat because a car suspension is a linear system. (We asked you to take
the simplification on faith.)
In the end, our problem was that given
* the frequency at which bumps occur,
we wanted to determine, for a given car,
* the ratio of the height of each bounce to the height of each bump (the
gain), and
* the amount of time by which each bounce trails the bump that caused it (the
phase delay).
We did the mathematics, which may or may not have been very revealing to you.
We could have graphed the overall results for each kind of car, but this would
have required a three-dimensional graph (since there are two outputs and one
input), which would have been hard to interpret.
In the end, we took advantage of the fact that the gain and the phase delay are
independent of one another and used separate two-dimensional graphs to
understand each individually. These graphs have the advantages that they are
easy to understand and communicate the solution completely.
Plots of gain and phase delay as a function of frequency crop up in all fields
of engineering. For example, they are used to describe the behaviors of stereo
amplifiers and the way that bridges vibrate in response to traffic. In the
checkoff section, you will see how they can be used to understand one aspect
of the Earth's climate!