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

(a) Obtain a differential equation in \(x\) that describes the system's behaviour.

(b) Use the following information: \(x(0)=1, \dot{x}(0)=0, m=1, b=4, k=3\) and \(f(t)=9\), use Laplace transforms to show that \[
X(s)=\frac{s^{2}+4 s+9}{s\left(s^{2}+4 s+3ight)}
\]

(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: \(x(0)=0, \dot{x}(0)=0, m=1, b=2, k=5\) and \(f(t) \equiv\) unit impulse, show that \[
X(s)=\frac{1}{s^{2}+2 s+5}
\]
Use the method of completing the square to find the response, \(x(t)\) and plot the system response.

Transcribed Image Text:

R(s) N(s) Y(s) G(s) D(s) H(s) Translational Mechanical System