Chapter 4.5, Problem 42E

Forced Vibrations. As discussed in Section 4.1, a vibrating spring with damping that is under external force can be modeled by

$m{y}^{″}+b{y}^{\prime }+ky=g\left(t\right)\left(15\right)$

where $m>0$ is the mass of the spring system, $b>0$ is the damping constant, $k>0$ is the spring constant, $g\left(t\right)$ is the force on the system at time t, and $y\left(t\right)$ is the displacement from the equilibrium of the spring system at time t. Assume $b^2<4mk$.

a. Determine the form of the equation of motion for the spring system when $g\left(t\right)=\sin \beta t$ by finding a general solution to equation (15).

b. Discuss the long-term behavior of this system. [Hint: Consider what happens to the general solution obtained in part (a) as $t\to +\infty$]