Problem $2(5$pts$)$:

A valve$-$controlled motor is used with a constant pressure source, $P_S,$ to control the speed of a load with inertia $J_1$ and a torque $T_1$ and linear damper $B_1$ acting on it$,$ as shown above. The lumped fluid compliance $C_t$ arises from the compressibility of the fluid under pressure between the control valve and the working parts of the hydraulic motor. The valve is a variable nonlinear fluid resistor in the form of a variable$-$area orifice and follows the equation.

$Q_v=A(t)C_d(\frac{2}{}{)}^{0\mathrm{.}5}\frac{P_{12}}{|P_{12}{|}^{0\mathrm{.}5}}$

The shaft can be considered stiff, so the motor inertia $J_m$ can be lumped with the load and the motor damping can be lumped with the load damping. Assume no leakage.

Draw the free$-$body diagram of the mechanical portion of the system.

Write the necessary and sufficient set of equation for the system

Rearrange the set of equations to produce the output relating ${}_1(t)$ to the input $A(t)($Linearize if need be$)$