sol04 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Problem Set #4 Solution Posted: Friday, Oct. 5, ’07 1. (a) Problem 21(a) from Nise textbook, Chapter 2 (page 113). (b) After you find the transfer function, locate the zeros and poles, draw them on the s –plane, and determine the type of the response of the system (undamped, underdamped, critically damped, or overdamped). a. Answer: We choose current i 1 ( t )and i 2 ( t ) as shown in the figure. V i V 1 I 1 ( s )= 10 6 5 × 10 5 + s V o V 1 I 2 ( s )= 10 6 10 5 + s and I 2 ( s )= I 1 ( s ) . Also V 1 = 0 because the positive terminal is connected to the ground. The negative terminal should have the same potential. Arranging the above equations, we obtain the transfer function given by V o ( s ) 1 ( s + 10) = . V i ( s ) 5 ( s +2) b. Answer: Pole: -2, Zero: -10. It is a first order system since has only one pole. It also has one zero. 1
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± 0 Imaginary Axis Pole−Zero Map Step Response 1 −0.2 −0.3 0.5 −0.4 Amplitude −0.5 −0.6 −0.7 −0.5 −0.8 −0.9 −12 −1 −10 −8 −6 Real Axis −4 −2 0 0 −1 0.5 1 1.5 Time (sec) 2 2.5 3 (Drawing the step response is not required) 2. Problem 25 from Nise textbook, Chapter 4 (page 236). Answer: From the free body diagram, x = f ( t ) 33 x 15 ˙ x. In the Laplace domain, the equation of motion is (3 s 2 +15 s + 33) X ( s )= F ( s ) . The transfer function is X ( s ) 1 / 3 = . F ( s ) s 2 +5 s +11 (For the following items, refer to the lecture note 7 p.26) The natural frequency ω n = 11 rad/s. The damping ratio ζ =5 / 2 n =0 . 7538. %OS = exp ζπ × 100=2 .7% 1 ζ 2 T s ζω 4 n =1 . 6 sec. T p = π =1 . 4415 sec. ω n 1 ζ 2 To compute the rise time approximately, we can use the normalized rise time method in section 4.5 of Nise textbook (pp.196). Since the damping
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ratio ζ 0 . 75, the normalized rise time will be 2.2965 (s). If we divide it by ω n , the rise time
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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.

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sol04 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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