(Solved): Problem 3 Consider the motion transmission system shown in Fig 3a that consists of a rack R and a g ...

Problem 3 Consider the motion transmission system shown in Fig 3a 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 $\mathbf{O}$ of the gear with radius $\mathbf{r}$ 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 $\mathrm{O}$, with respect to the system parameters $\mathbf{m},\mathbf{k},\mathbf{I},\mathbf{r}$ and $\mathbf{l}$. 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 $3\mathrm{b}$ shows the angular response $?(t)$ for a unit step input force $f(t)$. Given the system parameter values $\mathbf{m}=1\mathrm{kg},\mathbf{l}=0.4\mathrm{m}$, and $\mathbf{r}=0.1\mathrm{m}$, estimate the unknown values of the spring stiffness $\mathbf{k}$ and the moment of inertia $I$. Fig 3a: Coupled rack \& gear-lever system