Problem $3.$ Servomotor $[4$ points$]$

Figure P$2\mathrm{.}16.$ Schematic of a rotational$-$translational mechanical drive system.

$($from KGS$)$

The above diagram illustrates a mechanical drive system in which a servomotor $($with rotational inertia $J_1)$ drives a pinion gear $($with rotational inertia $J_2$ and pitch radius $R)$ that is meshed with a rack $($of mass $m).$ The rack is constrained to translation by a linear bearing with viscous friction $b,$ while the servomotor and pinion assembly is constrained to rotational motion by a set of bearings that have negligible friction. We can safely assume that the shaft between the motor and the pinion gear is rigid, but there is a significant compliance between the rack and the environment $($with stiffness $k).$ Please develop an input$-$output differential equation of the motion of this system with the motor torque $T_i$ as an input and the tension $F$ in the spring $k$ as an output.

$[$:$(J_1+J_2{R}^2)T_i^{}=R(\frac{J_1+J_2{R}^2}{R^2})$