lecture_31 - MIT OpenCourseWare http:/ocw.mit.edu 2.004...

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MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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H ( s ) u ( t ) = A s i n ( w t + q ) y ( t ) = A | H ( j w ) | s i n ( t q Ð H ( j ) ) s s L i n e a r S y s t e m | H ( s ) | j M I | H ( j ) | i s a " s l i c e " o f t h e | H ( s ) | s u r f a c e a l o n g t h e i m a g i n a r y a x i s o f t h e s - p l a n e . Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 31 1 Reading: Nise: 10.1 Class Handout: Frequency Response and Bode Plots 1 Sinusoidal Frequency Response (continued) In Lecture 30 we saw that the steady-state response of a linear system with transfer function H ( s ) to a sinusoidal input u ( t ) = A sin( ωt + θ ) is y ss ( t ) = A H ( ) sin ( ωt + θ + H ( )) | | where H ( ) = { H ( ) } + ± 2 { H ( ) } | | ² 2 H ( ) = arctan ± ±{ H ( ) } ² ²{ H ( ) } We note that H ( ) = H ( s ) s = , | that is H ( ) is H ( s ) evaluated along the imaginary axis of the s -plane. 1 copyright c D.Rowell 2008 31–1
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The Frequency Response of Systems with Zeros If a system has a transfer function N ( s ) H ( s ) = D ( s ) the frequency response function is N ( ) H ( ) = . D ( ) For complex a and b , a/b = a / b and ( a/b ) = b , so that | | | | | | a N ( ) 2 { N ( ) } + ± 2 { N ( ) } (1) | H ( ) | = | | D ( ) | | = 2 { D ( ) } + ± 2 { D ( ) } H ( ) = N ( ) D ( ) = arctan ± ±{ N ( ) } ² arctan ± ±{ D ( ) } ² (2) { N ( ) } { D ( ) } Example 1 Find and plot the frequency response of s + 5 H ( s ) = s + 10 The frequency response function is + 5 H ( ) = + 10 and N ( ) ω 2 + 25 | H ( ) | = | | D ( ) | | = ω 2 + 100 H ( ) = N ( ) D ( ) = arctan ³ ω ´ arctan ³ ω ´ 5 10 The following MATLAB commands were used to plot the frequency response: w=0:.2:100; sys=zpk(-5,-10,1) y=freqresp(sys,w); plot(w,squeeze(abs(y)))
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This note was uploaded on 02/23/2012 for the course MECHANICAL 2.004 taught by Professor Derekrowell during the Spring '08 term at MIT.

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lecture_31 - MIT OpenCourseWare http:/ocw.mit.edu 2.004...

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