Control ENG HW_Part_34

Control ENG HW_Part_34 - πt = t= 1−ξ 2 8.8 Verify that...

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Unformatted text preview: πt = t= 1−ξ 2 8.8 Verify that for X(t) =A sin ωt, for a second order system, A Y (t ) = (1 − (ωt ) ) + (2ξτ ) 22 φ = − tan −1 sin(ωt + φ ) 2 2ξωτ 1 − (ωτ ) 2 Y ( s) = A 1 2 22 ( s + ω ) (τ s + 2ξτs + 1) Y ( s) = Aω A1 B1 C1 D1 + + + τ 2 ( s − jω ) s + jω ( s − s1 ) ( s − s2 ) 2 Now as t → ∞, Y (t ) = A11 cos ωt + B11 sin ωt Where A11 = A1 + B1 B11 = j( A1 − B1 ) to determine A1 , B1 put s = jω ,− jω in the order A1 = −j j B1 = 2ω ( jω − s1 )( jω − s2 ) 2ω ( jω + s1 )( jω + s2 ) A11 = j 1 1 ( s + jω )( s + jω ) − ( jω − s )( jω − s ) 2ω 1 2 1 2 A11 = j ( −ω 2 − js1ω − js2ω + s1 s2 ) − ( −ω 2 + js1ω + js2ω + s1 s2 ) 2 2 2ω ( s1 + ω 2 )( s2 + ω 2 ) ( s1 + s2 ) ( s1 s2 − ω 2 ) 11 A11 = 2 similarly B = 2 2 2 2 2 2 2 ( s1 + ω )( s2 + ω ) ω ( s1 + ω )( s2 + ω ) using s1 + s2 = 2 2 = s1 + s2 = − 2ξ 4ξ 2 τ2 τ − 2 τ2 s1 s2 = 1 τ2 2(2ξ 2 − 1) = τ2 2ξ − Aω τ A11 = 2 2 τ 1 2ω 2 4 + 2 ( 2ξ − 1) + ω τ 4 τ − 2 Aωξ = τ3 2 2 1 2ξω ω − 2 + τ τ Aϖ 11 and B = τ 1 ω τ 2 = = 2 − 2 Aωξτ (1 − (ωτ ) 2 ) 2 + ( 2ξωτ ) 2 2 2 2ξω −ω2 + τ A(1 − (ωτ ) 2 ) (1 − (ωτ ) 2 ) 2 + ( 2ξωτ ) 2 1 2 2 −ω τ ...
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This note was uploaded on 11/13/2011 for the course COP 4355 taught by Professor Koslov during the Spring '10 term at University of Florida.

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Control ENG HW_Part_34 - πt = t= 1−ξ 2 8.8 Verify that...

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