QUESTION $1$:

Consider the penclulum system model shown in the Figure $1.$ A spring and clamper cre affached at the midpoint of the pendutum arm as shown, The spring is unstretched when $=a.$ Assume no energy loss in the bearing at $O.$

Figure $1$

a$)$ Determine the equation of motion for the system in terms of $m,L,k,b,$ and $$ and linearize using the small angle approximation.

$[5$ Marks$]$

b$)$ if the pendulum mass, $m,$ is $2$ kg $,$ the length, $L,$ is $1$ m $,$ and the stittness, k $,$ is $122$ m $.$ calculate $n$ and then determine the value of b required to obtain $25\%$ damping.

$[5$ Marks$]$

c$)$ Develop a Simulink model to describe the system response $[$angular displacement$)$ of the system.

$[10$ Marks$]$