Problem $3$

Consider the vehicle modeled as a two$-$degree$-$of$-$freedom system that allows for bounce $x(t)$ and pitch motion $(t)$ as shown below figure.

$($a$)$ Derive the equation of motion.

$($b$)$ Write the equation of motion in matrix form.

Let reasonable values for a truck be $m=4,000kg,J_{cg}=2,560kg*m^2,c_1=c_2=2,000N*\frac{s}{m},k_1=k_2=20,000\frac{N}{m},l_1=0\mathrm{.}9m,l_2=1\mathrm{.}4m.$ With these values,

$($c$)$ Calculate the natural frequencies and mode shapes for the truck.

$($d$)$ Plot the mode shapes.

$($e$)$ Obtain modal matrix and modal damping ratios.

$($f$)$ Determine the responses of $x(t)$ and $(t)$ to the engine being shut off, which is modeled as an impulse moment applied to $(t)$ of $1,000N*m.$