Problem $3(25\%)$

Fig. $3($a$)$ shows a mechatronic system for moving the rack and pinion a motor. The symbols are as follows:

Motor subsystem:

$L$ : inductance, $C$ : capacitance, $R$ : resistance, J : inertia, $k_b$ : back EMF constant, $k_a$ : motor torque constant, $b_m$ : damping coefficient of the motor shaft, ${}_m$ : Motor torque, ${}_L$ : applied torque, $e_a$ : input voltage, $e_b$ : back EMF,$$ : angular velocity. $E_a,E_b,,T_m,T_L$ are Laplace Transform of $e_a,e_b,,{}_m,{}_L.$

Rack and pinion subsystem:

$m$ : rack mass, $b$ : damping coefficient, $k$ : spring coefficient, $r_p$ : radius of the pinion, $x$ : rack displacement, $v$ : rack velocity,,${}_L$ : torque delivered by the pinion. $V,T_L$ are Laplace Transform of v and ${}_L.$

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Fig. $3$

$($a$)$ For the motor subsystem as shown in Fig. $3($b$),$

i$.$ Simplify the three components into one impedance $Z.$

ii$.$ Express the motor system in terms of $J,b_m,K_a,K_b,{}_m,{}_L,Z,I_a,E_a,E_b,$

iii. Draw the block diagram for the motor subsystem.

$($b$)$ For the rack and pinion system in Fig. $3($c$),$ the mass and rotational inertia of the pinion can be neglected, and no$-$slip occurs between the rack and pinion. Determine the transfer function $\frac{V(s)}{T_L(s)}.$

$($c$)$ Draw the block diagram for the whole system with input voltage $E_a$ as the input and $V$ as the output.