$(3)$ Motor system with gears, rack, linear spring & torsional damper

rotating shaft moments of inertia $I_1,I_2,$ rack mass $m,$ gear radii $r_1,r_2,$ rotational $($torsional$)$

damper $c_r$ attached to $2^{nd}$ shaft, linear spring $k$ attached to rack, shaft rotations ${}_1,{}_2$ and rack

position $x$ as shown, all zero with spring at rest. Input torque $(t)$ on shaft $1.$ Find the equation

of motion of the system in terms of ${}_1(t).$