Chapter 5.6, Problem 23P

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

Consider the initial value problem

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

Where

$f(t)=1+2{\displaystyle \underset{k=1}{\overset{n}{\sum }}{(-1)}^k{u}_{11k/4}(t)}.$

Observe that this problem is identical to problem 19 excepts that the frequency of the forcing terms has been increased somewhat.

a) Find the solutions of this initial value problem.

b) Let $n\ge 33$ and plot $y(t)$ and $f(t)$ for $0\le t\le 90$ or longer on the same set of coordinate axes. Your plot should show a clearly recognizable beat.

c) From the graph in part (b), estimate the “slow period” and the “fast period” for this oscillator.

d) For a sinusoidally forced oscillator, it was shown in Section 4.6 that the “slow frequency” is given by $\left|\omega -{\omega }_0\right|/2,$ where ${\omega }_0$ is natural frequency of the system and $\omega$ is the forcing frequency. Similarly, the “fast frequency” is $(\omega -{\omega }_0)/2$. Use these expressions to calculate the “fast period” and the “slow period” for the oscillator in this problem. How well do the results compare with your estimate from part(c) ?