va​ - applied voltage vg​ - generator voltage (back EMF) ia​ - armature current ω - armature speed tm​ - motor torque Ra

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va​ - applied voltage vg​ - generator voltage (back EMF) ia​ - armature current ω - armature speed tm​ - motor torque Ra​-armature resistance J-armature inertia b-armature damping Km​ - motor constant Kg​ - generator constant The figure shows an experimental test stand used to quantify the response characteristics of differing rotors. The input to the system is the applied voltage va​ which drives the DC motor. It is assumed that vg​=Ks​ω and tm​=Km​ia​. The test stand has a total length of L and is assumed to have negligible mass. The rotor of mass m is attached to the end of the test stand via a movable collar attachment. The rotor produces a thrust, T, that is dependent only on the armature speed. Assume that the system is in static equilibrium when the test stand is in the horizontal orientation. A small angle assumption for the test stand is valid as long as the absolute value of the test stand orientation remains less than 20∘. bt​=2.0 N−s/m (a). First, find the transfer function relating the armature speed Ω(s) to the applied voltage Va​(s) and the transfer function relating the orientation of the test stand Φ(s) to the thrust produced by the rotor T(s). Assume small angles of motion.
(b). The rotor is spun up at various armature speeds. Each speed is held until the system reaches equilibrium, then the displacement of the spring, d is measured by hand. The following table shows the results of the procedure. Assuming that there is a linear relationship between the angular rate of the motor and the thrust produced by the rotor, find the least squares solution that relates the thrust produced to the angular rate of the motor (cT​=ωT​). (Use Matlab to solve the least square problem based on the given data)
(c). Using the static gain cT​; draw the block diagram of the system. The input should be the applied voltage and the output should be the orientation of the test stand. Label the intermediate signals. (d). It is desired to tests the system's response to a step input voltage of 10 V (the maximum voltage for this particular DC motor). Given that Km​=1000 N−m/A,​Kr​=0.001 V−s/rad,Ra​=10Ω, b=0.9 N−m/(rad/s), and J=0.1 kg−m2, plot the step response using MATLAB functions tfO and stepO Comment on the validity of our system model given the results. If any assumptions were broken, suggest a simple modification to the mechanical system such that the assumption is satisfied with the same input voltage. Plot the modified system's step response. (HINT: consider moving the position of the rotor) (e). It is now desired to command test stand orientations using feedback control. Assume that we have a sensor that measures the test stand. orientation Φ(s) with unity gain (1.e. it measures the test stand orientation exactly). Also assume that we have a proportional controller that takes the test stand orientation error as an input and produces the applied voltage to the DC motor. First, derive the closedloop transfer function Φ(s)/Φre​(s). Next, find a proportional gain, kp​, such that the rise-time of the system is less than 0.26sec, overshoot is less than 60%, and settling time is less than 5.5 sec. (Use the unmodified system). You can use MATLAB's feedbackO function and stepinfoO function to determine the rise-time, overshoot, and settling time. Start with kp​=1 and change the gain until the requirements are satisfied Plot the resultant step response.