Question

PLEASE SOLVE 4.11

Example 4.11 One Cycle of Cosine Function revisited Find the response of the system in Example 4.7 using the convolution integral. Let m = 1 kg and use (0) = 0 and (0)1 m/s

Answer 1

An undamped system is driven by the function, x + 16 (x) = F(t)/m = 1/m {cos 4t if 0 lessthanorequalto t lessthanorequalto pi 0 else where Let m = 1 kg ,x (0) = 0, x (0) = 1 m/s Therefore The equation of motion is given by x + 16 (x) = {cos 4(t) if 0 lessthanorequalto t lessthanorequalto pi 0 elsewhere The impulse response function with = 0 is g(t) = 1/m w_m sin w_n t u(t) For this problem, g(t) = 1/4 sin 4t u(t) The forced response is given by integral x_forced (t) = integral_0^t g(t - tau) F (tau) d tau = integral^t_0 1/4 sin (t - tau F(tau) d tau For t < pi, x_forced (t) = integral^t_0 1/4 sin 4 (t - tau) cos 4 tau d tau for tau < pi , x_forced (t) = integral^t_0 1/4 sin 4 (t - tau) cos 4 tau d tau = 1/8 t sin 4 t