Consider the electromechanical system shown in Figure 6 49a It consists of a cart of mass m moving

Consider the electromechanical system shown in Figure 6 49a It consists of a cart of mass m moving

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 6.49b 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 and Ke, respectively.

(a) Derive the differential equation of the system relating the cart position x and the applied voltage Va.
(b) Determine the transfer function X(s)/Va(s) by using the differential equation obtained in Part (a). Assume that all initial conditions are zero.