Question

Consider the following single degree of freedom mass-spring damper system with m=1 kg, c=3 N.s/m, and k=2N/m. The system is at rest when a force 5e^-3t is applied. By using the concept of the Lagrange Transform (using partial fractions), obtain the response, x(t) of the system.

Answer 1

mx = -cx -kx + f(t) mx + cx + kx = f(t) x + (c/M) x + (k/M) = f(t)/M M = 1 kg C = 3 Ns/M K = 2 N/m F = 5 times 10^-3 t C/M = 2 xi w_n k/m = 2 w_n^2 x + 2 xi w_n x + w_n^2 x = 1/m f(t) Impulse force at t = 0 rightarrow pair of step functions f(t) = F[u(t) -u(t -elementof)] elementof = small step x + 2 xi w_n x + w_n^2 x = 1/m F[u(t) -u(t -elementof)] L{x + 2 xi w_n x + w_n^2 x} = L[1/m {F(u(t) - u(t - elementof))} apply Laplace, s^2 x(s) + sy(0) - x' (0) + 2 xi w_n s x () - 2 xi w_n x (0) + w_n^2 x(s) = F/M [1/s - 1/s e^-elementof s] (s^2 + 2 xi w_n s +