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exam02_soln - PROBLEM NO 1(40 Name Consider the following...

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PROBLEM NO. 1 (40%) Name Consider the following third-order, input-output differential equation: !!! y + 110 !! y + 1000 ! y = 100 u t ( ) + 1000 ! u t ( ) where u t ( ) is the INPUT and y t ( ) is the OUTPUT. a) Determine the transfer function G s ( ) = Y s ( ) / U s ( ) . b) Determine the poles and zeros of G s ( ) . c) Suppose we consider an input of u t ( ) = sin ! t . On the axes provided on a following page, construct the straight-line approximations for the dB MAGNITUDE of the transfer function for steady-state response. In this work: clearly indicate the break frequencies. clearly indicate the contributions of the elemental transfer functions to the overall magnitude plot. In particular, show the roll-off rates for each elemental transfer function in your plot. use a STRAIGHT EDGE in constructing this response curve. d) Determine the expression for the exact steady-state response y ss t ( ) corresponding to ! = 5 rad / sec . Taking LT of I/O equation: s 3 + 110 s 2 + 1000 s ! " # $ Y s ( ) = 1000 s + 100 [ ] U s ( ) % G s ( ) = Y s ( ) U s ( ) = 1000 s + 100 s 3 + 110 s 2 + 1000 s = 1000 s + 100 s s + 10 ( ) s + 100 ( ) = 100 10 ( ) 100 ( ) 10 s + 1 ( ) s 0.1 s + 1 ( ) 0.01 s + 1 ( ) = 0.1 ( ) 1 + s / 0.1 ( ) 1 s !
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