Question $1(20$ marks$)$

Two gears are attached to a uniform shaft whose torsional rigidity is GJ$.$ The shaft is free to rotate in frictionless bearings, as shown below, but there is an external moment $T_i$ applied to each respective gear. Neglect the rotational inertia of the shaft, and write the two equations of motion in matrix format

$($a$)$ Using ${}_1$ and ${}_2,$ the absolute angles of rotation of the gears, as coordinates, determine the equations of motion of this system.

$($b$)$ Use as coordinate $q_1$ the absolute rotation $($i$.$e$.{}_1)$ of gear $1,$ and for coordinate $q_2$ the rotation of gear $2$ relative to gear $1($i$.$e$.,{}_2-{}_1).$