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Unformatted text preview: = . MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Problem Set #5 Solution Posted: Friday, Oct. 19, 07 1. Modify the Laplacedomain KVL equation (2) to include the inductance. Argue that (4), (6), (910) need not be modified in this system, and thus (11) remains valid. Answer: When we include the inductance L in our model, we have to take into account the voltage drop across the inductor in the KVL equation. V s ( s ) = I ( s ) R + I ( s ) Ls + K v ( s ) . In the mechanical part, the torque generated by the motor drives the same inertias and has to overcome the same friction; that is, the mechanical part doesnt change. Therefore (4), (6), (9-10) need not be modified and (11) remains valid. 2. Substitute (11) to your new KVL equation from question 1 and derive the open loop transfer function V ( s ) /V s ( s ). Answer: Inserting (11) to the new KVL relation in question 1, we obtain J + r 2 M D + r 2 f v V ( s ) V s ( s ) = s + ( R + Ls ) V ( s ) + K v rK m rK m r = J + r 2 M Ls 2 + ( J + r 2 M ) R + ( D + r 2 f v ) L s + D + r 2 f v R + K v V ( s ) . rK m rK m rK m r The transfer function is rK m V ( s ) L ( J + r 2 M ) V s ( s ) 2 R D + r 2 f v ( D + r 2 f v ) R + K m K v s + + s + L J + r 2 M L ( J + r 2 M ) 3. Substitute L = 1 H and the remaining numerical values from the Supplement, and find the poles of your transfer function. You should find that this system is underdamped. Answer: Using numerical values given in the supplement, the transfer function is simplified to V ( s ) . 3162 /L = . V s ( s ) s 2 + (1 /L + 1) s + 2 /L 1 Amplitude 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Here L = 1 H, so the transfer function is V ( s ) . 3162 = . V s ( s ) s 2 + 2 s + 2 Poles: p 1 = 1 + j , p 2 = 1 j . Natural frequency n = 2 rad/s. Damping ratio = 2 / (2 n ) = 2 / (2 2) = 1 / 2 = 0 . 7071 < 1: underdamped. 4. Use the LTI tool to plot the step response of the openloop system. Compute the steadystate analytically using the transfer function from question 3 and verify that it matches the LTI tools prediction. Answer: Step Response 1 2 3 4 5 6 Time (sec) From the final value theorem, 1 . 3162 . 3162 v = lim s = = 0 . 1581 . s s s 2 + 2 s + 2 2 5. Use the plot from question 4 to estimate the rise time of the system. How does it compare with the rise time of the openloop system from the Supplement ( i.e. , the identical system but without the inductance.) Answer: From the step response plot, T r 1 . 5 (s). (The normalized rise time method in Nise textbook gives you the same result.) The rising time of the inductancefree system is 1.1 sec from the following step response; (It also can be computed...
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This note was uploaded on 02/23/2012 for the course MECHANICAL 2.004 taught by Professor Derekrowell during the Fall '08 term at MIT.
- Fall '08