ME 130_F08 Key to Exam2

# ME 130_F08 Key to - ME130 APPLIED ENGINEERING ANALYSIS Fall 2008 Section 02 Key to Exam#2#1 Find the Laplace transform of the following function by

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1 ME130 – APPLIED ENGINEERING ANALYSIS Fall 2008 – Section 02 Key to Exam #2 #1. Find the Laplace transform of the following function by direct integration [] tt st st st 00 0 st 22 0 ee 1 sinh ( t ) Note : sinh ( t ) 2 L 1 sinh ( t) (1 sinh( t))e dt e dt e sinh ( t)dt 11 e , ( Note, the second term sss s is obtained usin g the int egration ω− ω ∞∞ −−   ω =     = = + ω ωω =− + = + −ω ∫∫ () st t t st t(s ) ) ) ) by parts method) Or sinh( t))e dt e e e dt s2 ed t t 1 1 s2 ( s ) ( 1 1 1 s2s s s ω ω −+ ω ω ω = + =+    ω ω  ω s Using the same approach, one can get Laplace transform of f(t) = 1 + cosh( ω t), and the result is 1s L1 cosh( t) ss = +

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2 #2. Use Appendix 1 (text) to find the function f(t) corresponding to the Laplace transform (a) [] 2 3s 1 Lf( t ) F ( s ) s4 s 5 + == (b) 2 2 s2 s 1 t ) F ( s ) (s 1)(s 2s 3) ++ +− Part (a) () 2 t5 t 2 t 3 t3 t 2t A B F(s) s 4 s5 s1s5 s1 s5 Solving for A and B: 18 A , and B Usin g Appendix 1, 33 1 8 f(t) e e e e e 3 3 The above result can be expressed as 7 e 3cosh(3t) sinh(3t) 3 −− = + + +  =+= +   =+ Part (b) 22 2 2 2 tt 3 t s 1 s 1 s 1 F(s) s1( s3 ) ABC s1 31 Solving for A, B, and C: A 1, B , and C 44 So, usin g Appendix 1 , te e e = −+ + + = +
3 #3. Using Laplace transform method, determine the solution of the following initial value problems. That is, find y (t). (a) 2 2 dy 5 6 y 12, with y(0) 0, and y'(0) 0 dt dt ++= = = (b) 2 2 2 y 2 t , w i t h y ( 0 )0 , a n d y ' ( 0 dt dt ++ = = = Part (a) Take Laplace transform of the equation, [] () ( ) 2 2 2 2 2 2t L5 6 y L 1 2 dt dt 12 s Y(s) sy(0) y'(0) 5 sY(s) y(0) 6Y(s) s 12 12 12 Y(s) s 5s 6 Y(s) ss ( s 5 s 6 ) s s 2 s 3 AB C , solving for A, B, and C: 2 s 3 A2 ,B 6 , a n d C4 So, usin g Appendix 1, y(t) 2 6e  =   −− + + = = = = + + =+ + == = =− + 3t 4e

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## This note was uploaded on 09/08/2010 for the course ME 130 at San Jose State University .

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ME 130_F08 Key to - ME130 APPLIED ENGINEERING ANALYSIS Fall 2008 Section 02 Key to Exam#2#1 Find the Laplace transform of the following function by

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