System Response

Things don't generally vibrate by themselves. A car, for example, vibrates in
response to the bumps in the road. Suppose that you are designing the
suspension system (springs and shock absorbers) of a car. Ideally, you'd like
your car to have a smooth ride no matter the pattern of bumps on the road.
That is, you'd like the car to have low-amplitude (or even no-amplitude)
vibrations under all driving conditions.

A car's suspension is a ``linear system.'' This means that the frequency with
which the car bounces up and down will always be equal to the frequency at
which the bumps in the road occur. For example, if you go over bumps at the
rate of five per second (in other words, at a frequency of 5 Hz), your car will
bounce up and down at a frequency of 5 Hz. There's nothing you can do about
it.

You can design the car's suspension to minimize the amplitude of
the bounce. But this job is not as straightforward as you might imagine. Any
suspension system that you design will have different behaviors at different
frequencies. You might, for example, design a suspension that produces moderate
bounces at low frequencies (bumps spaced far apart) and small bounces at
high frequencies (bumps spaced close together).

What all this means is that if you are designing a car suspension, you will be
interested in its gain. The gain of a system is a function of frequency,
and is the ratio between the amplitude of the response of the system being
vibrated and the amplitude of the system causing the vibration.

For example, suppose that you are driving a car along a road with regularly
spaced 12-inch high bumps. Suppose that the car has a gain of 1.0 at a
frequency of 1 Hz and a gain of 0.25 at a frequency of 10 Hz. This means that
if you hit one bump per second, the car will bounce up and down 12 inches per
vibration. (We obtain this by multiplying the size of the bumps by the gain.)
If you go ten times faster, the car will bounce up and down only three inches
per vibration. (Of course, it will be bouncing ten times more frequently.)

If you take more advanced classes in mechanical or electrical engineering,
you'll learn how to derive the gains for mechanical and electrical systems.
Since it is not our intention to teach you how to derive gains in this course,
we're going to give you some that we have already derived. You will then be
able to experiment with them.

Given below are functions, called transfer functions, that describe the
behavior of the suspensions of three kinds of cars: a full-size American car
from around 1980 (boat), a small off-road Japanese truck (truck), and an
expensive European touring sedan (sedan).

boat[s_]:=s -> (9*s + 53) / (27*s*s + 9*s + 53);

truck[s_]:=(160*s + 400) / (9*s*s + 160*s + 400);

sedan[s_]:=(2.143*(s*s+49))/((0.25*s*s+2.86*s+9.14)*
(0.25*s*s+2.104*s+11.488));

Each transfer function takes a complex number as its argument and computes a
complex number as its result.

Given a transfer function G, you can obtain the gain for a given frequency
f by taking the absolute value of G(f I). (Don't forget that I is
Mathematica's symbol for the square root of -1.) You may not be familiar with the
notion of the absolute value of a complex number. The important things to know
are that it is a real number and that it can be computed with the
function Abs[ ].

Graph the gain of each kind of car as a function of frequency, for frequencies
ranging from 0 to 20 Hz. Keep in mind that the gain shows you how the height
(amplitude) of the bumps will translate into the height of the bounces for
different frequencies of bumps. Since Mathematica automatically adjusts the range
of the vertical axis, the curves will be hard to compare unless all three are
graphed on the same plot. But since Mathematica doesn't label the curves, it'll be
hard to tell them apart unless you first plot them individually!