$(30$ pts$)[$Simscape$]$ The figure below presents a simplified model of an electrical drill, where: $V_a(t)$ is

the voltage applied to the drill's motor; N is the gear ratio of the drill's gearbox; $J_m$; $J_c$; and $J_d$

represent, respectively, the motor inertia, the drill chuck inertia, and the drill bit inertia $($which is

concentrated at its tip$)$; $K_m$ and $K_d$ represent the torsional stiffnesses ofthe motor shaft and drill bit,

respectively; and damping coefficients $B_c$ and $B_d$ represent, respectively, viscous damping due to

bearings acting on the drill chuck and viscous damping between the drill bit and the material being

cut. Additional interactions between the drill bit and the material, including the effects of friction and

chip removal, are represented by the load torque $T_{\text{L}\text{O}\text{A}\text{D}}(t)$ :

a$.(10$ pts$)$ Derive the complete governing equations of motion for this system. For each

governing equation, indicate which variable's dynamics the equation represents.

b$.(10$ pts$)$ Assuming the motor inertia $J_m$ is negligible, and the motor shaft is rigid, obtain a set

of governing equations that relate the applied voltage $V_a(t)$ and load torque $T_{\text{L}\text{O}\text{A}\text{D}}(t)$ to the

angular velocity ${}_d^{}(t)$ of the drill's tip.

c$.(10$ pts$)$ Develop a physical model of this system in Simscape, with sensors to measure the

drill tip's angle, ${}_d(t)$ and the current in the motor, $i_a(t).$