Question $9.[10$ Marks$]$

A mechanical system is shown in Figure $10.$ The system is made up of two masses $m_1$

and $m_2$ that are connected to each other via a pulley with a mass moment of inertia

$I_p.$ The system is configured such that the cables connecting $m_1$ and $m_2$ always

remain in tension. The base of the system is given a displacement $s(t)=$Ssin$t.$

Figure $10$: A mechanical system under base excitation

$($a$)$ The effect of gravity can be accounted for directly in this system without needing

to include the force due to gravity in our equations. Why is this possible and

what must be done to ensure the effect of gravity is accounted for?

$($b$)$ Draw free body diagrams for the the two masses and the pulley, omitting the

forces due to gravity and counteracting normal forces.

$($c$)$ What is the equation of motion defining the displacement of $m_1,x(t),$ in terms of

the stiffness of the two springs $k,$ the damping coefficient of the viscous damper

$c,$ the radii of the pulley $R,$ the masses $m_1$ and $m_2,$ the magnitude of the base

excitation $S,$ and the frequency of the base excitation $?$

$($d$)$ What is the magnitude of the steady$-$state oscillation of $x(t)?$ Give your answer

in terms of the same variables as part $($c$)$ above.