Chapter 2, Problem 2.35P

a. Obtain the transfer functions $X\left(s\right)/F\left(s\right)$ and $Y\left(s\right)/F\left(s\right)$ for the following model:

$4\stackrel{\dot{}}{x}=y\stackrel{\dot{}}{y}=f\left(t\right)-3y-12x$

b. Compute $\tau$ , $\zeta$ , ${\omega }_n$ , and ${\omega }_d$ for the model.

c. If $f\left(t\right)=u_s\left(t\right)$ , will the responses $x\left(t\right)$ and $y\left(t\right)$ oscillate? If so, compute the radian oscillation frequency, and estimate how long it will take for the oscillations to disappear.

d. Suppose that $f\left(t\right)=u_s\left(t\right)$ and $x\left(0\right)=\stackrel{\dot{}}{x}\left(0\right)=0$ . Obtain $x\left(t\right)$ and $y\left(t\right)$ .