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Unformatted text preview: Solutions for Test 2, ’09 1 Problem . why Explain a) ∫ ∫ = = t T t d t g F T t u d t g F t x τ τ τ τ τ τ ) ( ) ( ) ( ) ( ) ( ) ( ) ( T t u F kx x m = + : to according excited ICs, zero with system SDOF Undamped . , For ) ( ) ( = ⋅ = < ∫ t d t g t x T t τ τ . , For ∫ ∫ ∫ ⋅ = ⋅ + ⋅ = t T t T T d t g F d t g F d t g t x T t τ τ τ τ τ τ τ τ ) ( ) ( ) ( ) ( ) ( ) ( these. of both gives ∫ = t T d t g F T t u t x τ τ τ ) ( ) ( ) ( ) ( Solutions for Test 2, ’09 1 Problem . find , From b) ) ( ) ( ) ( ) ( ) ( t x d t g F T t u t x t T ∫ = τ τ τ ) ( T t u F kx x m = + : to according excited ICs, zero with system SDOF Undamped ∫ ∫ = = t n n t d t m F T t u d t g F T t u t x ) ( sin ) ( ) ( ) ( ) ( ) ( τ τ ϖ ϖ τ τ τ t T n n t m F T t u  = ) ( cos ) ( 2 τ ϖ ϖ [ ] ) ( cos 1 ) ( T t k F T t u n = ϖ Solutions for Test 2, ’09 2 Problem . transform Laplace the Find a) ) ( s F ) ( T t u F kx x m = + : to according excited ICs, zero with system SDOF Undamped s e F s F T t u F t F sT = → = ) ( ) ( ) ( . transform Laplace the Find b) ) ( s X ) ( 1 ) ( ) ( ) ( 2 2 2 2 2 n sT n n sT s s e m F k ms s e F s F s G s X ϖ ϖ ϖ + = + = = . response the Find c) ) ( t x [ ] ) ( ) ( cos 1 ) ( T t u T t k F t x n = ϖ Solutions for Test 2, ’09 3 Problem modes. the Sketch modes. and s frequencie natural the Find =  + 3 1 1 3 2 1 2 1 x x L T x x m m : EOMs with system 2DOF Undamped ) 4 )( 2 ( ) 1 9 6 ( 3 1 1 3 3 3 ) det( 2 2 2 2 2 2 2 2 = = + = = = = γ γ γ γ ϖ γ γ γ ϖ ϖ ϖ L T L T T mL L T m L T L T L T m L T M K ) ( mL T mL T T mL 4 2 4 , 2 2 2 2 1 2 = = → = = ϖ ϖ ϖ γ , Solutions for Test 2, ’09 3 Problem modes. the Sketch modes. and s frequencie natural the Find =  + 3 1 1 3 2 1 2 1 x x L T x x m m : EOMs with system 2DOF Undamped  = → =  = = → =  = 1 1 1 1 1 1 ) ( 1 1 1 1 1 1 ) ( 2 22 12 2 2 2 2 1 21 11 1 2 1 1 : For : For u u u u u u u u M K u u M K ϖ ϖ Solutions for Test 2, ’09 4 Problem . and responses Find . : Modes . , : s frequencie Natural ) ( ) ( 1 1 5 . 1 2 5 2 1 2 1 2 1 t x t x U m k m k  = = = ϖ ϖ =  +...
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This note was uploaded on 09/18/2011 for the course ASE 365 taught by Professor Staff during the Spring '10 term at University of Texas at Austin.
 Spring '10
 Staff

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