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Unformatted text preview: 2.14/2.140 Problem Set 5 Assigned: Fri. March 16, 2007 Due: Thurs. March 22, 2007, in class Reading: Notes Chapter 5 and 6. The following problems are assigned to both 2.14 and 2.140 students. Problem 1 This problem revisits the paint stirrer considered in Quiz 1. A brushed DC motor is used to stir a bucket of paint as shown in the figure below. The motor has a torque constant K [Nm/A] and the motor plus stirrer inertia is J [kg-m 2 ]. The motor is considered ideal in that it has no inductance and no resistance; the only resistance in the electrical circuit is R as shown. The rotor is rigidly linked to the stirrer, which has no ﬂexibility; the combined assembly has a rotational velocity Ω rad/sec. The effect of the paint on the stirrer is modeled as a rotational damper B [Nms/rad]. a) Solve for the transfer function G p ( s ) = Ω( s ) /V i ( s ) in terms of the system parameters. Now set the parameter values as K = 0 . 5 Nm/A, R = 5 Ω, B = 0 . 15 Nms/rad, and J = 0 . 2 kgm 2 . For these parameter values, make a hand sketch of the Bode plot for G p . b) Now, let the system input V i be driven by a PI controller of the form G c ( s ) = K p 1 + K i . s 1 This system has a speed reference input Ω r and output Ω as shown in the block diagram below Choose the controller gain values K p and K i to set the loop crossover frequency ω c = 10 rad/sec, with a phase margin φ m = 45 ◦ . Show your calculations. You should be able to accomplish this design using hand-sketched Bode plots, and then using Matlab for confir- mation. c) Create a Simulink simulation of the loop with the controller implemented as shown in the block diagram above. For now, let the integrator have an unbounded output (leave the integrator limit output box unchecked). Set your simulation to use the variable step solver with a max step size limit of 0.01 sec, and use the ode45 solver. Note that the standard simulation window has a simulation step time of 10 sec. You may want to adjust this value. Let the input reference Ω r take a step from zero to 10 rad/sec at t = 0, from initial rest conditions. Run this as a Simulink simulation and record and plot the responses V i ( t ) and Ω( t ). What is the maxiumum value of the V i during this transient? What is the steady-state value...
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.14 taught by Professor Davidtrumper during the Spring '07 term at MIT.
- Spring '07