Alex's Assignment

Alex's Fourier Project

I chose to do my LNA project on a spring mass and dampener system such as what you would find on an automobile. As shown on the picture we have a mass (M), spring (K), and dampener (D).

General Equations

The first thing I need to do is write down the general equations for the Fourier Series:

$\begin{array}{rl}x(t)& =x(t+T)=a_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n\cos (n{\omega }_{0}t)+b_n\sin (n{\omega }_{0}t)\\ {\omega }_0& =2\pi f_0\\ a_0& =\frac{1}{T}\underset{0}{\overset{T}{\int }}f(t)dt\\ a_n& =\frac{2}{T}\underset{0}{\overset{T}{\int }}f(t)\cos (n{\omega }_{0}t)dt\\ b_n& =\frac{2}{T}\underset{0}{\overset{T}{\int }}f(t)\sin (n{\omega }_{0}t)dt\\ \end{array}$

If odd function, $a_n=0$

If even function, $b_n=0$

Equations

Now that we have the basic equations for a Fourier Series we can begin to calculate for the spring dampener system. knowing that as the vehicle travels down a rough road it creates a displacement x we can equate the formula for the system. The forces apposing the displacement are:

The mass of the vehicle for one tire $\frac{1}{4}M$

The spring constant $Kx$

And the Dampening of the shock $Du$

Plugging in the forces into an Ordinary Differential Equation we get:

$M{u}^{\prime }+Du+Kx=0$