MIT18_03S10_c16

MIT18_03S10_c16 - 18.03 Class 16, March 10, 2010 Frequency...

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18.03 Class 16 , March 10, 2010 Frequency response [1] Variation of parameters [2] Summary of complex gain [3] Second order frequency response [4: Supplement] First order frequency response [5: Supplement] Other systems [not done in lecture] [1] I promised on Monday to show you what you can do if A is not constant in p(D)x = A e^{rt} . Example: 3x" + 8x' + 6x = (t^2 + 1) e^{-t} . Now B is not constant. Try for a solution of the form x_p = u e^{-t} for some u . This is what led us to the ERF; but now u is allowed to be nonconstant. This is called *variation of parameters*. 6 ] x_p = u e^{-t} 8 ] x_p' = (u' - u) e^{-t} 3 ] x_p" = (u" - u' - u' + u) e^{-t} ------------------------------------------- (t^2 + 1) e^{-t} = (3u" + 2u' + u) e^{-t} Cancel the e^{-t} : 3u" + 2u' + u = t^2 + 1 This is solvable by undetermined coefficients. In fact, by an incredible stroke of luck, we have already solved it! u_p = t^2 - 4t + 3 so x_p = u_p e^{-t} = ( t^2 - 4t + 3 ) e^{-t} This method will always replace the given equation with another on in which the right hand side is the same as before but without the e^{rt} . [2] Complex gain summary: General stuff (independent of order!) Input: y = A cos(omega t). Complex input: y_cx = A e^{i omega t} z_p = exponential system response to input y_cx Complex gain: z_p = H(omega) y_cx Polar: H(omega) = |H(omega)| e^{- i phi} z_p = |H(omega)| e^{- i phi} A e^{i omega t} = |H(omega)| A e^{i(omega t - phi)} Real part: x_p = |H(omega)| A cos(omega t - phi)
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Gain: g(omega) = |H(omega)| Phase lag: phi(omega) = \phi = - Arg(H(omega)) [3] I want to go back to the spring/mass/dashpot system and think some more about what we can learn about the system response. | ______ | | | ________________| |---\/\/\/\/\/------| |---------======= |------- | |______| ----------------| | | | | | |----------> |---------> | x | y m x" + bx' + kx = by' (*) Input signal: y = A cos(omega t) System response: x I demonstrated the Mathlet Amplitude and Phase: Second Order II. It has
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This note was uploaded on 01/18/2012 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c16 - 18.03 Class 16, March 10, 2010 Frequency...

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