A rack and pinion machine is typically in many mechanical applications. A rack has teeth on a long metal support that translates by a distance $x(t).$ A rack is constrained to move

proportional to the spur rotation by $x=R.$ This is similar to two spur gears in that a constraint force is generated by the teeth on the spur gear in contact with the teeth on the rack. Rack

translational speed must be the same as the spur gear tangential speed at radius $r.$ Assuming the gear is connected to a rigid shaft and inertia $J$ and viscous damper $($bearing$)B_t$ and shaft

spring $($softness$)K_t$ to ground with applied external torque $T(t)$ and rack translation is grounded by a spring $K$ and viscous damper $B,$ develop a transfer function from input torque $T$ to

output rack motion $x.$ Develop the model in the standard way by modeling the system. Create correct and complete free body diagrams. Solve Newton's laws. Take a Laplace transform.

Solve for the transfer function.