PROBLEM 42E

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

where m > 0 is the mass of the spring system, b > 0 is the damping constant, k > 0 is the spring constant, g(t) is the force on the system at time t, and y(t) is the displacement from the equilibrium of the spring system at time t. Assume b2 < 4mk.

(a) Determine the form of the equation of motion for the spring system when 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

Solution:

Step 1:

In this problem, we have to determine the form of the equation of motion for the spring system.