Chapter 4.10, Problem 5E

An undamped system is governed by

$\begin{array}{l}m\frac{d^{2}y}{d{t}^2}+ky=F_0\cos \gamma t;\\ y\left(0\right)=y^{\prime }\left(0\right)=0,\end{array}$

where $\gamma \ne \omega :=\sqrt{k/m}$.

a. Find the equation of motion of the system.

b. Use trigonometric identities to show that the solution can be written in the form

$y\left(t\right)=\frac{2F_0}{m\left({\omega }^2-{\gamma }^2\right)}\sin \left(\frac{\omega +\gamma }{2}t\right)\sin \left(\frac{\omega -\gamma }{2}t\right)$

c. When $\gamma$ is near $\omega$, then $\omega -\gamma$ is small, while $\omega +\gamma$ is relatively large compared with $\omega -\gamma$. Hence, $y\left(t\right)$ can be viewed as the product of a slowly varying sine function, $\sin \left[\left(\omega -\gamma \right)t/2\right]$, and a rapidly varying sine function, $\sin \left[\left(\omega +\gamma \right)t/2\right]$. The net effect is a sine function $y\left(t\right)$ with frequency $\left(\omega -\gamma \right)/4\pi$, which serves as the time-varying amplitude of a sine function with frequency $\left(\omega +\gamma \right)/4\pi$. This vibration phenomenon is referred as beats and is used in tuning stringed instruments. This same phenomenon in electronics is called amplitude modulation. To illustrate this phenomenon, sketch the curve $y\left(t\right)$ for $F_0=32$, $m=2$, $\omega =9$, and $\gamma =7$.