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Control ENG HW_Part_34

# Control ENG HW_Part_34 - t = t= 1 2 8.8 Verify that for X(t...

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= 2 1 ξ π - = t t 8.8 Verify that for X(t) =A sin ω t, for a second order system, ( ) ( ) ) sin( 2 ) ( 1 ) ( 2 2 2 φ ω ξτ ω + + - = t t A t Y 2 1 ) ( 1 2 tan ωτ ξωτ φ - - = - ) 1 2 ( 1 ) ( ) ( 2 2 2 2 + + + = s s s A s Y ξτ τ ω - + - + + + - = ) ( ) ( ) ( ) ( 2 1 1 1 1 1 2 s s D s s C j s B j s A A s Y ω ω τ ω Now as t B t A t Y t ω ω sin cos ) ( , 11 11 + = Where 1 1 11 B A A + = ) ( 1 1 11 B A j B - = to determine order the in j j s put B A ω ω - = , , 1 1 ) )( ( 2 2 1 1 s j s j j A - - - = ω ω ω ) )( ( 2 2 1 1 s j s j j B + + = ω ω ω - - - + + = ) )( ( 1 ) )( ( 1 2 2 1 2 1 11 s j s j j s j s j A ω ω ω ω ω

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+ + + + + - - + - - - = ) )( ( ) ( ) ( 2 2 2 2 2 2 1 2 1 2 1 2 2 1 2 1 2 11 ω ω ω ω ω ω ω ω ω s s s s js js s s js js j A + + + = ) )( ( ) ( 2 2 2 2 2 1 2 1 11 ω ω s s s s A similarly + + - = ) )( ( ) ( 2 2 2 2 2 1 2 2 1 11 ω ω ω ω s s s s B using 2 2 1 2 1 1 2 τ τ ξ = - = + s s s s = 2 2 2 2 2 2 2 2 1 ) 1 2 ( 2 2 4 τ ξ τ τ ξ - = - = + s s + - + - = 4 2 2 2 4 2 11 ) 1 2 ( 2 1 2 ω ξ τ ω τ τ ξ τ ω A A 2 2 2 2 3 2 1 2 +
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