Consider the electromechanical system shown in Figure 6.49a. It consists of a cart of mass \(m\) moving without slipping on a ground track. The cart is equipped with an armature-controlled DC motor, which is coupled to a rack and pinion mechanism to convert the rotational motion to translation and to create the driving force \(f\) for the system. Figure \(6.49 \mathrm{~b}\) shows the equivalent electric circuit and the mechanical model of the DC motor, where \(r\) is the radius of the motor gear. The torque and the back emf constants of the motor are \(K_{t}\) and \(K_{e}\), respectively.

a. Derive the differential equation of the system relating the cart position \(x\) and the applied voltage \(v_{\mathrm{a}}\).

b. Determine the transfer function \(X(s) / V_{\mathrm{a}}(s)\) by using the differential equation obtained in Part (a). Assume that all initial conditions are zero.

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Sensor gear m Motor gear + Track Va V Motor gear Ra La 0000 + B eb tm (a) FIGURE 6.49 (b) Problem 3. (a) A DC motor-driven cart and (b) the equivalent electric circuit and the mechanical model of the DC motor.