Problem $3$ Consider the vertical pendulum of length $3$ where $=0\mathrm{.}5m$ shown in Fig. $3$ that can rotate about the pivot point $O$ with rotational viscous friction coefficient $b_r=24Nm\frac{s}{r}ad.$ The pendulum bar is massless but point masses of value $m=2kg$ are attached at the top$-$point B $,$ bottom$-$point $D$ and mid$-$point $C,$ as shown. A horizontal spring of stiffness $k=100\frac{N}{m}$ is attached at the pendulum's top point $B.$ A displacement input $z(t)$ at the free end point A of the spring sets the system to motion, as shown. Small angular displacements from the vertical equilibrium position of the pendulum are considered.

a$)$ Derive the transfer function $G(s)$ of the system from the displacement input $z(t)$ to the angular displacement output $(t)$ of the pendulum. Use and substitute the values of the system parameters $,m,k$ and $b_r$ as stated above in the calculation of $G(s).$

b$)$ Considering a displacement input $z(t)=A(1-e^{-t}).$ What should be the value of the displacement input constant $A($in m $)$ such that the steady$-$state angular displacement

c$)$ Calculate the complete angular displacement response $(t)$ of the pendulum to a displacement input $z(t)=A(1-e^{-t})$ where A has the value calculated in b