Problem $2(25$ pts$)$

An electric motor drives a fixed$-$displacement hydraulic pump, which affects the time$-$rate of pressure $P$ in a cylinder. The linearized dynamics for this Hydromechanical system are

DC Motor Circuit: $L{I}^{}+RI=e_{in}(t)-K_b,$

Motor$/$Pump Dynamics: $J{}^{}+b_{mp}=K_{m}I-d_p(P-P_{\text{a}\text{t}\text{m}}),$

Hydraulic Cylinder: $P^{}=\frac{}{V_0}(d_p-A(z^{})),$

Mechanical Load: $m{z}^{}+b{z}^{}=PA-P_{\text{a}\text{t}\text{m}}A.$

where $e_{in}(t)$ is voltage input to the motor; I,$L,R,$ and $K_B$ are the current, inductance, resistance, and back$-$emf constant of the motor; $$ is the angular velocity of the motor. $K_m$ is the motor$-$torque constant; $$ is the fluid bulk modulus; $V_0$ is the nominal cylinder volume; $d_p$ is the pump volumetric displacement; $A$ is the hydraulic cylinder piston area; $m$ is the mass of the mechanical load; $b$ is the translational friction coefficient; $z$ is the position of the mass; and $P_{\text{a}\text{t}\text{m}}$ is atmospheric pressure.

The output variables are mechanical position $z,$ cylinder pressure $P,$ and motor$/$pump velocity $.$ The input variables are voltage $e_{in}(t)$ and atmospheric pressure $P_{\text{a}\text{t}\text{m}}.$ Obtain a complete state$-$space model $($i$.$e$.,$ a set of state equations and a set of output equations$).$ Express the state$-$space model into vector$-$matrix forms.

The system parameters are:

Piston $+$ mechanical load mass $m=20kg,$

Viscous friction $b=2000N-\frac{s}{m},$

Piston area $A=5{10}^{-4}{m}^2,$

Fluid bulk modulus $=6\mathrm{.}8{10}^{8}Pa,$

Nominal cylinder volume $V_0=3{10}^{-4}{m}^3,$

Pump volumetric displacement $d_p=1\mathrm{.}7{10}^{-7}\frac{m^3}{r}ad,$

Motor$-$torque constant $K_m=0\mathrm{.}6N-\frac{m}{A},$

Back$-$emf constant $K_b=0\mathrm{.}6V-\frac{s}{r}ad,$

Motor inductance $L=0\mathrm{.}004H,$

Motor resistance $R=0\mathrm{.}5,$

Motor $+$ pump inertia $J=1\mathrm{.}6{10}^{-3}kg-m^2,$

Motor $+$ pump friction $b_{mp}=2{10}^{-4}N-m-\frac{s}{r}ad,$

Atmospheric pressure $P_{\text{a}\text{t}\text{m}}=1\mathrm{.}0133{10}^{5}Pa.$

At time $t=0,$ all states $($except cylinder pressure $P)$ are zeros $($i$.$e$.,$ zero stored energy$).$ The initial hydraulic pressure is equal to atmospheric pressure $P_{\text{a}\text{t}\text{m}}.$

The input voltage is a $1-$second pulse:

$e_{in}(t)=\{\begin{array}{cc}25V,& 0t1s\\ 0,& t>1s\end{array}$

Use MATLAB's Isim command to simulate the hydromechanical system's response to a $1-$second pulse input. Plot the following system responses for a total simulation time of $2$ seconds:

Motor$/$pump angular velocity $(t)($in rpm$),$

Cylinder pressure $P(t),$

Mechanical load position z$($t$)($in cm$).$

$4$

Determine:

The constant angular velocity of the motor$/$pump $($in rpm$)$ during the pulse voltage input,

The final position of the mechanical load $($in cm $)$ at $t=2s.$

Problem $2(25$ pts$)$

An electric motor drives a fixed$-$displacement hydraulic pump, which affects the time$-$rate of pressure $P$ in a$,$ cylinder. The linearized dynamics for this Hydromechanical system are

DC Motor Circuit: $L{I}^{}+RI=e_{in}(t)-K_b,$

Motor$/$Pump Dynamics: $J{}^{}+b_{mp}=K_{m}I-d_p(P-P_{\text{a}\text{t}\text{m}}),$

Hydraulic Cylinder: $P^{}=\frac{}{V_0}(d_p-A(z^{})),$

Mechanical Load: $m{z}^{}+b{z}^{}=PA-P_{\text{a}\text{t}\text{m}}A.$

where $e_{in}(t)$ is voltage input to the motor; I,$L,R,$ and $K_B$ are the current, inductance, resistance, and back$-$emf constant of the motor; $$ is the angular velocity of the motor. $K_m$ is the motor$-$torque constant; $$ is the fluid bulk modulus; $V$