exam01_soln

exam01_soln - PROBLEM NO. 1: (30%) SOLUTION The equations...

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PROBLEM NO. 1: (30%) SOLUTION The equations of motion for a system are known to be the following set of three, first- order differential equations: ! x 1 + x 1 ! x 2 = u t ( ) ! x 2 + 2 x 2 ! x 1 ! x 3 = 0 ! x 3 + 3 x 3 ! x 2 = 0 with u t ( ) being the input and x 1 , x 2 and x 3 being the outputs. For this system, a) determine the transfer function relating the input u and the output variable x 2 . Write this transfer function as a single rational fraction of the Laplace transform variable s . b) determine the corresponding single input-output differential equation relating the input u and the output variable x 2 . SOLUTION Taking the LT of the system equations with zero IC’s: s + 1 ( ) X 1 s ( ) ! X 2 s ( ) = U s ( ) " X 1 s ( ) = X 2 s ( ) + U s ( ) # $ % / s + 1 ( ) s + 2 ( ) X 2 s ( ) ! X 1 s ( ) ! X 3 s ( ) = 0 s + 3 ( ) X 3 s ( ) ! X 2 s ( ) = 0 " X 3 s ( ) = X 2 s ( ) / s + 3 ( ) Substituting the first and third equation into the second: s + 2 ( ) X 2 s ( ) ! X
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exam01_soln - PROBLEM NO. 1: (30%) SOLUTION The equations...

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