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 (c).

Equations

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

${\begin{aligned}x(t)&=x(t+T)=a_{0}+\sum _{n=1}^{\infty }a_{n}\cos(n\omega _{0}t)+b_{n}\sin(n\omega _{0}t)\\\omega _{0}&=2\pi f_{0}\\a_{0}&={\frac {1}{T}}\int _{0}^{T}f(t)\,dt\\a_{n}&={\frac {2}{T}}\int _{0}^{T}f(t)\cos(n\omega _{0}t)\,dt\\b_{n}&={\frac {2}{T}}\int _{0}^{T}f(t)\sin(n\omega _{0}t)\,dt\\\end{aligned}}$

If odd function, $a_{n}=0$

If even function, $b_{n}=0$

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Now that we have the basic equations for a Fourier Series we can begin to calculate for the spring dampener system.