Problem $3$ Consider the motion transmission system shown in Fig $3$a that consists of a rack $R$ and a gear$-$lever component $GL.$ The rack $R$ has mass $m$ and moves horizontally without friction due to a horizontal force $f(t).$ The gear$-$lever component $GL$ rotates about the fixed center $O$ of the gear with radius $r$ due to the coupling with the rack. The moment of inertia of the gear$-$lever component is I. The lever part of the component $GL$ has length $l$ and its end point A is connected to a vertical spring with stiffness $k$ as shown. The system is initially at equilibrium such that the lever is horizontal.

i$)$ Find the transfer function $G(s)$ from the horizontal force $f(t)$ to the angular rotation $(t)$ of the gear$-$lever component about the point $O,$ with respect to the system parameters $m,k,$I,$r$ and $l.$

ii$)$ Find the time response $(t)$ of the angular rotation of the gear$-$lever component when the horizontal force $f(t)$ is a unit step input in terms of the system parameters $m,k,$I,$r$ and $l.$

iii$)$ Fig $3$ b shows the angular response $(t)$ for a unit step input force $f(t).$ Given the system parameter values of inertia $I$

Step Response

Fig $3$b: Angular response to a unit step input