hw1_SOL

hw1_SOL - MAE 171a Homework Problem Set 1 Prof. Schwartz...

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Unformatted text preview: MAE 171a Homework Problem Set 1 Prof. Schwartz Due: June 28th 1. Find the Laplace Transform of (a) f ( t ) = e- . 7 t cos(2 t ) Using transform number 15 ( L bracketleftbig e- at cos t bracketrightbig = s + a ( s + a ) 2 + 2 ) L bracketleftbig e- . 7 t cos(2 t ) bracketrightbig = s + 0 . 7 ( s + 0 . 7) 2 + 4 = s + 0 . 7 s 2 + 1 . 4 s + 4 . 49 (b) f ( t ) = t 3 e- at Using transform number 17 ( L bracketleftbig t n e- at bracketrightbig = n ! ( s + a ) n +1 ) L bracketleftbig t 3 e- at bracketrightbig = 6 ( s + a ) 4 (c) f ( t ) = t 1 ( t ) + ( t- 1) 1 ( t- 1)- 2 t 1 ( t- 1) where 1 ( t ) is the unit step function. First, rewrite f ( t ) so that f ( t ) = t 1 ( t )- ( t- 1) 1 ( t- 1)- 2 1 ( t- 1). Then it follows that L [ f ( t )] = 1 s 2- e- s s 2- 2 e- s s = 1- e- s- 2 se- s s 2 2. Find the Inverse Laplace Transform of (a) F ( s ) = 1 s ( s +2) = R 1 s + R 2 s +2 R 1 = lim s 1 s + 2 = 1 2 R 2 = lim s - 2 1 s =- 1 2 thus f ( t ) = 1 ( t ) 2 (1- e- 2 t ). This is a typical step response to a first order system. Note the effectiveness of the Final Value Theorem....
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hw1_SOL - MAE 171a Homework Problem Set 1 Prof. Schwartz...

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