$1.($a$)$ Figure $1$ shows a rigid link in which its free end is connected to a linear spring and a linear damper. The length of the rigid link is $\backslash ($ L $\backslash )$ while the spring's constant and damping coefficient are $\backslash ($ k $\backslash )$ and $\backslash ($ c $\backslash ),$ respectively. To actuate this rigid link, we have mounted a motor at its revolute joint and its moment of inertia at the joint is given as $\backslash ($ J $\backslash ).$ The torque supplied by the motor is given as $\backslash (\backslash$tau $\backslash )$ while there is another horizontal force $\backslash ($ f $\backslash )$ which is applied at the centre of the rigid link. Assuming zero initial conditions, derive and draw out the block diagram components of this system in which the output of the system is the angular displacement generated by the rigid link $\backslash ((\backslash$theta$)\backslash ).$ We can assume that $\backslash (\backslash$theta $\backslash )$ is small and ignore gravity.

$(4$ marks$)$

$($b$)$ Determine the DC gain, natural frequency and damping ratio of the system.

$(3$ marks$)$

$($h$)$

Fig. $1.$ A rigid link that can be actuated by a motor, which is placed at its revolute joint. $($a$)$ The configuration of the system in which the rigid link's length is $\backslash ($ L $\backslash ).($b$)$ In addition to the effects of the linear spring and damper, a horizontal force and the motor torque are affecting the motions of the rigid link.