Review Problem 4 solution

Review Problem 4 solution - 2 .5 2 1 2 5 1 2 X s G s s s s...

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4. What is the response of x to a unit impulse in ( ) u t ? (This is the impulse response, ( ) g t .) This solution can and should be found in two alternative ways, by taking the inverse Laplace transform of ( ) G s and by taking the time derivative of the solution obtained in 3. The results should be the same. METHOD 1: For this problem take ( ) 0 0 x = and ( ) 0 0 x = d . The Laplace Transform Table shows the pair ( ) t δ and 1 . From problem 1, ( ) ( ) ( ) 0 0 2 2 2 2 5 2 5 U s s x x X s s s s s + + = + + + + + d . Substituting the zero value initial conditions and noting that the input is ( ) 1 U s = gives the impulse response, in frequency space, ( ) ( ) ( ) ( ) ( ) 2 2
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Unformatted text preview: 2 .5 2 1 2 5 1 2 X s G s s s s = = = + + + + . From Laplace Transform Table: note the transform pair sin at e t ω-and ( ) 2 2 s a + + . Using the equivalences 1 a = and 2 = , gives: ( ) ( ) .5 sin 2 t x t g t e t-= = METHOD 2: From Problem 3, ( ) ( ) ( ) .2 .2cos2 .1sin 2 t x t h t e t t-= =-+ . Using the derivative approach ( ) ( ) ( ) ( ) ( ) .2 .2cos2 .1sin 2 .2cos2 .1sin 2 .4sin 2 .2cos2 t t t d e t t dh t x t e t t e t t dt dt--- -+ = = = +--+ Reducing the final expression gives ( ) .5 sin 2 t x t e t-=...
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This note was uploaded on 02/29/2008 for the course ME 242 taught by Professor Perreira during the Spring '08 term at Lehigh University .

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