Consider the spring$-$mass$-$damper system defined by $($all parameters are in SI units$)$

$3x^{}+12\mathrm{.}47x^{}+75x=15cos5t$

Assume zero initial conditions. Complete following steps.

Part I

a$.$ Determine if the system is underdamped, critically damped or overdamped.

b$.$ Compute the steady$-$state response $x_p$

c$.$ Compute the transient response $x_n$

d$.$ Total response $x(t)=x_n+x_p$

f$.$ Plot $x(t)$

Part II

a$.$ Compute normalized magnitude $x(r)$ of the system to harmonic excitation $F_{0}cost$

b$.$ Find $x_{\text{p}\text{e}\text{a}\text{k}}$ and the corresponding $r_{\text{p}\text{e}\text{a}\text{k}}$ and ${}_{\text{p}\text{e}\text{a}\text{k}}$

c$.$ Plot $x(r)($hand sketch is not acceptable$)$