Applications of Fourier series

Driving a mass-spring system with an arbitrary periodic force by solving one harmonic at a time and superposing the steady-state responses.

Notes on Diffy Qs: Differential Equations for Engineers, Applications of Fourier series

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4 exercises

Exercise 101

Let $F(t) = \sin(2\pi t) + 0.1 \cos(10 \pi t)$ . Find the steady periodic solution to $x'' + \sqrt{2}\, x = F(t)$ . Express your solution as a Fourier series.

2.5

Exercise 102

Let $F(t) = \sum_{n=1}^\infty e^{-n} \cos(2 n t)$ . Find the steady periodic solution to $x'' + 3 x = F(t)$ . Express your solution as a Fourier series.

2.5

Exercise 103

Let $F(t) = \lvert t \rvert$ for $-1 \leq t \leq 1$ extended periodically. Find the steady periodic solution to $x'' + \sqrt{3}\, x = F(t)$ . Express your solution as a series.

2.5

Exercise 104

Let $F(t) = \lvert t \rvert$ for $-1 \leq t \leq 1$ extended periodically. Find the steady periodic solution to $x'' + \pi^2 x = F(t)$ . Express your solution as a series.

2.5