Chapter 10, Problem 33P

The displacement $y\left(t\right)$ of the spring-mass system shown in Fig. P10.33 is measured from the equilibrium configuration of the spring, where the static deflection is $\delta =\frac{\stackrel{⃛}{m}\stackrel{\dot{}}{g}}{k}$.

If m =4kg,k = 1.5N/m, $\omega =0.5\text{rad/s}$ ,and a sinusoidal force of $0.25\cos \left(0.5t\right)$ is applied at t=0, the displacement $y\left(t\right)$ satisfies the second order differential equation

$4\ddot{y}\left(t\right)+1.5y\left(t\right)0.25\cos \left(0.5t\right).$ (a) Find the transient response $y_{tran}\left(t\right)$ of the system. What is the natural frequency of the response?

(b) Show that the steady-state solution is given by $y_{ss}\left(t\right)=0.5\cos \left(0.5t\right).$.

Show all steps.

(c) Assume the total solution is given by

$y\left(t\right)=c_3\cos \left(\sqrt{0.375}t\right)$ $+c_4\sin \left(\sqrt{0.375}t\right)+0.5\cos \left(0.5t\right)$ Given $y\left(0\right)=0.5$ and $\stackrel{\dot{}}{y}\left(0\right)=0$, show that $c_3=0$ and $c_4=0$. Show all steps.

(d) Given the total solution $y\left(t\right)=0.5\cos \left(0.5t\right)$, determine the minimum value of the deflection $y\left(t\right)$ and the that time it reaches that value.