Consider the following diagram where the input is force f(t).

~I m 1---f(t)

Translational Mechanical System

{a) Obtain a differential equation in x that describes the behavior of the system.

{b) Use the following information: x(O) = 1, x(O) = 0, m = 1, b = 4, k = 3 and f(t) = 9, use Laplace transforms to show that X ( s) _ s2 + 4s + 9

- s(s2 + 4s + 3) ·

(c) Find the system response x(t) and plot it.

{d) Use the IVT and FVT to check your plot.

(e) Given the following new information: x(O) = 0, x(O) = 0, m = 1, b = 2, k = 5 and f(t) = unit imp11lse, show that 1

X ( s) = s 2 + 2s + 5 Use the method of completing the square to find the response, x(t) and plot the system response.

Problem