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Consider the electromechanical dynamic system shown in Figure 1(a). It consists of a cart of mass m moving without slipping on a linear 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 for the system. Figure 1(b) shows the simplified equivalent electric circuit and the mechanical model of the DC motor, where r is the radius of the motor gear (which in this case is also a wheel of the cart). Ra is the resistance of the armature circuit. The torque and the back emf constants of the motor are K, and Kb, respectively. Neglect the armature inductance and the mass moment of inertia of the motor. The driving force fis related to the torque by f -Tm/r. Tm is the torque generated by the motor. Motor Sensor Motor gear ear Track FIGURE 1, (a) physical system and (b) equivalent electrical circuit and simplified mechanical model of DC motor. NOTE: The 'Motor gear' on the right is one of the wheels of the cart. Part 1 DC-Motor-Driven Cart Dynamic System Modeling and Analysis (30 Points) Due April 12th a) Derive the differential equations of the system relating the cart position x(t) and the applied b) Determine the transfer function X(s)/Va (s). Assume that all initial conditions are zero voltage va(t) c) Assume the applied voltage va(t) as the input and the cart position x as the output. Determine the state-space form of the system Consider the electromechanical dynamic system shown in Figure 1(a). It consists of a cart of mass m moving without slipping on a linear 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 for the system. Figure 1(b) shows the simplified equivalent electric circuit and the mechanical model of the DC motor, where r is the radius of the motor gear (which in this case is also a wheel of the cart). Ra is the resistance of the armature circuit. The torque and the back emf constants of the motor are K, and Kb, respectively. Neglect the armature inductance and the mass moment of inertia of the motor. The driving force fis related to the torque by f -Tm/r. Tm is the torque generated by the motor. Motor Sensor Motor gear ear Track FIGURE 1, (a) physical system and (b) equivalent electrical circuit and simplified mechanical model of DC motor. NOTE: The 'Motor gear' on the right is one of the wheels of the cart. Part 1 DC-Motor-Driven Cart Dynamic System Modeling and Analysis (30 Points) Due April 12th a) Derive the differential equations of the system relating the cart position x(t) and the applied b) Determine the transfer function X(s)/Va (s). Assume that all initial conditions are zero voltage va(t) c) Assume the applied voltage va(t) as the input and the cart position x as the output. Determine the state-space form of the system
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