Question

(a)Use Netwons Second Law of Motion to prove that the equation governing the forced damped harmonic oscillator (spring-mass system) is: mx"(t) + cx'(t) + kx(t) = F(t): (Explain what the constants m; c; k are and what the function F(t) is. Draw a picture of the system.)

(b)Assume m = 1; c = 0; k = 4; that F(t) = cos(2t); and that the object attached to the spring begins from the rest position. Find the position function using the method of undetermined coefficients and prove that the solution x(t) verifies limit(as t approaches infinity) |x(t)| = infinity

(c) Assume now that m = 1; c = 4; k = 13 and that F(t) = cos(t): Assume also that the object starts at rest with initial velocity x'(0) = 1: Solve the corresponding IVP and show that the solution x(t) decomposes as a sum x(t) = x_transient(t) + x_steadystate(t) to be identified and check that it satisfies: limit(as t approaches infinity) x_transient(t) = 0 and x_steadystate(t) is a combination of sin(t) and cos(t)

Answer 1

I hope I don't need to solve the arbitrary constants A and B in problem (b).