Chapter 5.6, Problem 20P

In Problem 19 through 23, we explore the effect of some nonsinusoidal periodic forcing functions:

Consider the initial value problem

$y\text{'}\text{'}+0.1y\text{'}+y=f(t),$ $y(0)=0,$ $y\text{'}(0)=0.$

Where

$f(t)=1+2{\displaystyle \underset{k=1}{\overset{n}{\sum }}{(-1)}^k{u}_{k\pi }(t)}$

a) Find the solution of initial value problem.

b) Plot the graph of $f(t)$ and $y(t)$ on the same set of coordinate axes. Use a large enough value of $n$ and a long enough $t-$ interval so that the transient part of the solution has become negligible and the steady same state is clearly shown.

c) Estimate the amplitude and frequency of the steady –state part of the solution.

d) Compare the result of part (b) with those from Section 4.6 for a sinusoidally forced oscillator.