Andrew Roth's Fourier Series/Laplace Transform Project

Problem Statement

Use a real life application of the Fourier Series or the Laplace Transform

Solution

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For my project I will plot a car's wheel running over the rumble strips on the side of the road using the Fourier Transform. This will take the form of a simple square wave as the tire will fall into the dip and bump back up to the surface of the the rumble strip. The Fourier Transform may be used because the action of the car running over the rumble strips can be considered periodic.

For simplicity. I will assume the height and width of each rumble strip is 2 inches with 2 inches gap between. I will also place the origin of the coordinate system in the middle of a rumble strip to give it even symmetry.

Recall that the equation for a Fourier Series is:

$a_0+{\mathrm{\Sigma }}_{n=1}^{\mathrm{\infty }}[a_{n}cos(\frac{2\pi nt}{T})+b_{n}sin(\frac{2\pi nt}{T})]$

By inspection, $a_0=1$ and $T=4$

Also by inspection since the "wave" has even symmetry, we know that $b_n=0$

Now we must find $a_n$. Recall that

$a_n=\frac{2}{T}\underset{0}{\overset{T}{\int }}f(t)cos(\frac{2\pi nt}{T})dt$

$=\frac{2}{T}\underset{\frac{-T}{2}}{\overset{\frac{T}{2}}{\int }}2cos(\frac{2\pi nt}{T})dt$

$=\frac{2}{\pi n}[sin(\frac{\pi n}{2})-sin(\frac{-\pi n}{2})]$

$=\frac{4sin(\frac{\pi n}{2})}{n\pi }$

So our Fourier Series is:

$1+{\mathrm{\Sigma }}_{n=1}^{\mathrm{\infty }}[\frac{4sin(\frac{\pi n}{2})}{n\pi }cos(\frac{2\pi nt}{T})]$

Octave Script

clear all;
close all;
t=-10:.01:10;
T=4
M=100   
sum1=1;
for n=1:2:M,
sum1 = sum1+(4*sin(pi*n/2)/(n*pi))*cos(2*pi*n*t/T);
end
plot(t,sum1,'b-')
title('Fourier Series Representation of a car moving over rumble strips on a road.')
xlabel('time (seconds)')
ylabel('position of tire')
grid on;
print("squarewave.png","-dpng")