In mechanical design one sometimes must determine system parameters even though information on the system forces is incomplete. For example, the differential equation governing the motion x(t)of an externally forced damped mass spring oscillator was shown in Section 4.1 (page 153) to bewhere f(t) is the external force; the other parameters are defined in Section 4.1. Assume the system is underdamped (b^2 < 4mk; see Section 4.9) and starts from rest [x(0) = 0, x'(0) = 0] and that the force is bounded: | f(t)| < = A for all t. (a) Show that the transforms X(s) and F(s) of x(t) and f(t) are related by X(s) = (b) Use the convolution theorem to derive the formulawhere(c) Show that the motion x(t) is bounded under these circumstances byand also by . (d) Suppose the mass m = 5 kg, the spring constant k = 3000 N/m, and the force is bounded by A = 10 N. What range of values for the damping constant b will ensure a displacement of 1 cm (0.01 m) or less?