lecture_30

# lecture_30 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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Unformatted text preview: 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 . A- A T = t A s i n ( B ) A s i n ( M t + B ) 2 F M Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 30 1 Reading: Nise: 10.1 • 1 Sinusoidal Frequency Response 1.1 Definitions Consider a sinusoidal waveform f ( t ) = A sin( ωt + φ ) where A is the amplitude (in appropriate units) ω is the angular frequency (rad/s) φ is the phase (rad) In addition we can define T the period T = 2 π/ω (s) f the frequency, ( f = 1 /T = ω/ 2 π ) (Hz) 1 copyright c D.Rowell 2008 30–1 F ( t ) t I n p u t t o m a s s m F ( t ) f r i c t i o n B R e s p o n s e s t e a d y - s t a t e r e s p o n s e t r a n s i e n t r e s p o n s e v ( t ) m v ( t ) m The Euler Formulas: We will frequently need the Euler formulas e jωt = cos ( ωt ) + j sin( ωt ) e − jωt = cos ( ωt ) − j sin( ωt ) or conversely cos( ωt ) = 1 e jωt + e − jωt 2 sin( ωt ) = 1 e jωt − e − jωt 2 j 1.2 The Steady-State Sinusoidal Response Assume a system, such as shown above, is excited by a sinusoidal input. The total response will have two components a) a transient component, and a steady-state component y ( t ) = y h ( t ) + y p ( t ) . We define the steady-state component as the particular solution y p ( t ). Let the system dif- ferential equation be d n y d n − 1 y dy d m u d m − 1 u...
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lecture_30 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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