U NIVERSITY OF C APE T OWN
MAM2040F Ordinary Differential Equations
TASK 3 – A S UZUKI S WIFT ON A WASHBOARD ROAD
Problem Description
Please submit your task report in a single PDF file via Gradescope before the deadline of 14 May
2024 at 11:59pm.
You are welcome to discuss both the numerical and analytical aspects of this task with other students in the class. If you do so, please ensure that you acknowledge this in your report! Please
cite any texts or material you use appropriately too.
Although you are free to discuss numerical and analytical aspects of this task with others,
as well as consult other sources if need be, all work presented in the task report must be
your own. Please ensure that you acknowledge discussion with others and cite sources used
appropriately.
Problem Description
The upward displacement y(t) of a car like a Suzuki Swift driving along a washboard road1 without
acceleration can be modelled as a driven, damped linear oscillator by the non-homogeneous ODE
mÿ + cẏ + ky = cẏR + kyR , with
2πvt
yR (t) = A sin
λ
(1)
,
(2)
where the function yR (t) describes the road’s height at the car’s position at any time t. The remaining variables in the problem are as follows:
• m is the car’s mass with value 1315kg.2
• k is the spring constant with value 8 × 104 Nm−1 .
• c is the damping coefficient with value 2 × 103 Nm−1 s.
1 This is a road with a profile described by the function f (x) = A sin
2 As per the Suzuki Swift’s gross vehicle weight specifications.
2πx
.
λ
• v is the car’s velocity, assumed constant.
• λ is the wavelength of the road with value 10m.
• A is the amplitude of the road with value 5cm.
Tasks
1. Solve eqn (1) for y(t) using any method that you are familiar with.
2. From your solution, obtain an expression for the amplitude of the car’s vibrations, A, as a
function of its speed, v. Plot the graph of A(v) versus v. Use your graph to estimate the speed
v? at which resonance occurs, and give the approximate value of A, A(v? ), at this speed.
3. Solve eqn (1) numerically for different values of v. Try and verify the curve of A(v) versus v
you obtained numerically.
Use the initial conditions y(0) = 0 = ẏ(0) and decide on a suitable time integration interval;
choose it in such a way that any transient behaviour in the solution dies out, leaving the
solution to reach a steady state.
Use any programming language of your choice.
4. Write up your results in the form of a short report using either MS Word or LATEX. Your
report should include the following:
(a) Your analytical solution to eqn (1), along with the relevant work you did to obtain it.
(b) A plot of A(v) versus v and what both v? and A(v? ) are – show the relevant work you
did to obtain this result analytically.
(c) Three plots of the numerical solution to eqn (1) for three different values of v each, and,
if you are able to:
(d) Include a plot that numerically verifies the original curve you obtained for A(v) versus
v.
(e) A short conclusion summarising your key results.
Please do NOT produce your computer codes, and do not describe the numerical scheme(s)
you’ve used to obtain your numerical solution.
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