# ex1s - /s(b[20 pts Find the step response of the system...

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ME 451 Examination #1 February 15, 2008

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1. Consider the mechanical system shown below with input f ( t ) and output y ( t ) . f ( t ) y ( t ) M B K (a) [20 pts] Find the transfer function from f ( t ) to y ( t ) . (b) [15 pts] Find the transfer function from f ( t ) to a ( t ) = ¨ y ( t ) . (a) M ¨ y ( t ) + B ˙ y ( t ) + Ky ( t ) = f ( t ) ( Ms 2 + Bs + K ) Y ( s ) = F ( s ) Y ( s ) F ( s ) = 1 Ms 2 + Bs + K (b) a ( t ) = ¨ y ( t ) A ( s ) = s 2 Y ( s ) A ( s ) F ( s ) = s 2 Ms 2 + Bs + K 1
2. Consider the system shown below. r ( t ) y ( t ) u ( t ) F ( s ) G ( s ) H ( s ) + + - (a) [15 pts] Find the transfer function from r ( t ) to y ( t ) . (b) [15 pts] Find the transfer function from r ( t ) to u ( t ) . (a) Y ( s ) = F ( s ) R ( s ) + G ( s ) U ( s ) and U ( s ) = R ( s ) - H ( s ) Y ( s ) Y ( s ) = F ( s ) R ( s ) + G ( s )[ R ( s ) - H ( s ) Y ( s )] [1 + G ( s ) H ( s )] Y ( s ) = [ F ( s ) + G ( s )] R ( s ) Y ( s ) R ( s ) = F ( s ) + G ( s ) 1 + G ( s ) H ( s ) (b) Y ( s ) = F ( s ) R ( s ) + G ( s ) U ( s ) and U ( s ) = R ( s ) - H ( s ) Y ( s ) U ( s ) = R ( s ) - H ( s )[ F ( s ) R ( s ) + G ( s ) U ( s )] [1 + G ( s ) H ( s )] U ( s ) = [1 - F ( s ) H ( s )] R ( s ) U ( s ) R ( s ) = 1 - F ( s ) H ( s ) 1 + G ( s ) H ( s ) 2

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3. Consider the system shown below with input r ( t ) and output y ( t ) . r ( t ) y ( t ) C ( s ) G ( s ) + - (a) [15 pts] Find the step response of the system when C ( s ) = 1 and G ( s ) = 1
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Unformatted text preview: /s . (b) [20 pts] Find the step response of the system when C ( s ) = 5 and G ( s ) = 1 / [ s ( s + 2)] . (a) Y ( s ) R ( s ) = C ( s ) G ( s ) 1 + C ( s ) G ( s ) ⇒ Y ( s ) = 1 s ( s + 1) = 1 s +-1 s + 1 ⇒ y ( t ) = [1-exp(-t )] u ( t ) (b) Y ( s ) R ( s ) = C ( s ) G ( s ) 1 + C ( s ) G ( s ) ⇒ Y ( s ) = 5 s ( s 2 + 2 s + 5) = 1 s +-( s + 1) ( s + 1) 2 + 4 +-1 ( s + 1) 2 + 4 ⇒ y ( t ) = [1-exp(-t ) cos(2 t )-1 / 2 exp(-t ) sin(2 t )] u ( t ) ⇒ y ( t ) = { 1-√ 5 / 2 exp(-t ) sin[2 t + arctan(2)] } u ( t ) 3...
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ex1s - /s(b[20 pts Find the step response of the system...

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