MATH 2250
Differential Equations and Linear Algebra
Fall Semester, 2008, University of Utah
Homework 9 Computing Assignment
This assignment refers to Example 5 in section 5.6 of your text book
(page 353), which describes a mathematical model for the oscillations of a
car traveling over a “washboard” road surface. Please read this example
before beginning. You should also look at problems 29 and 30 (page 359),
which further investigate the same model (and will give you a hint for what
answers you should get for the questions below).
The vertical motion of the car as it bounces up and down is modeled as a
mass resting on a spring. The car traveling over the undulations of the road
then applies a force to mass-spring system. Using Newton’s 2nd law, we can
derive the following differential equation.
mx00 + cx0 + kx = cy 0 + ky
In this equation, x(t) is the (unknown) vertical distance of the car away from
its equilibrium position, and y(t) describes the motion over the road surface.
We are assuming that the road surface undulates like a sine curve, so we
define
y(t) = a sin ωt
The circular frequency ω depends not only on the shape of the road, but also
on the velocity v of the car moving over it. We define
ω=
2πv
L
For the constants m, c, k, a, and T , use the values found in Example 5 in
the book. We are assuming now that the dashpot is connected, so use the
value of c given (do not set c = 0). You will be choosing the value of the
car’s velocity v.
Use Maple to solve the differential equation and plot the solution x(t)
for various values of v. Included with your plots should be the graph of the
road’s shape y(t) for reference. You should see that, for different velocities,
the car’s vertical motion could have a higher or a smaller amplitude than the
road surface itself.
Answer the following:
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1. Suppose that your car would be damaged if it bounces with an amplitude of more than 10 centimeters (0.1 meters). Find a velocity that
would cause a damaging motion and display the graph.
2. What happens if you drive very slowly over this surface (e.g. 10
miles/hour or less)? Graph an example and describe what you see.
3. What happens if you drive very fast over this surface (highway speed)?
Graph an example and describe what you see.
I have provided much of the Maple code that you will need to do the
above. It is posted on the class web site as HW9Guide.mws. You will have
to fill in the missing parts of the code to make it work, and then choose
different values of v to graph as described above.
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