Question $2(10$ marks$)$

Figure $2$ shows a lifting apparatus comprising two pulleys connected by a flexible rope. A force

$f(t)$ applied to a handle of length L that is rigidly attached to Pulley $1$ raises $($and if the direction

of the force is reversed lowers$)$ the suspended mass m $.$ The displacement of mis measured by

the variable $y(t).$

The two pulleys are constrained to rotate about their centers on bearings. Pulley $1$

$($including the handle$)$ has an inertia $J_2$ about its center of rotation. Its radius is $r$ and its bearing

exhibits viscous damping $b.$ Similarly, Pulley $2$ has an inertia $J_2,$ a radius $r_2,$ and its bearing

exhibits viscous damping $b_2.$ The section of rope between Pulley $1$ and Pulley $2$ has stiffness $K_2,$

and that between Pulley $2$ and the mass $m$ has stiffness $K_1,$ The rope has zero mass.

Determine the transfer function $Y\frac{s}{F}(s).$