Question

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 Kt and Kp, respectively. Neglect the armature inductance and the mass moment of inertia of the motor. The driving force fis related to the torque by fTm/r. Tm is the torque generated by the motor Motor Sensor Motor gear car 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 voltage va (t) b) Determine the transfer function X(s)/Va (s). Assume that all initial conditions are zero. 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. d) Construct Simulink block diagrams to represent this electromechanical cart system based on equations found in part a) and part b) respectively. [For part a) consider the equation after you find the Laplace Transform.] Find the corresponding step responses x(t) of the system and compare them. You can assign any appropriate parameter values to variables such as m, Ra, Kb, and K, in your simulation.

This assignment is for my Engr dynamics systems class.

Answer 1

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