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Unformatted text preview: Beams L x t x f t x v x EI t x v x A < < = = ′ ′ ′ ′ + , ) , ( ) ) , ( ) ( ( ) , ( ) ( ρ : beam a of vibration (flexural) transverse free for PDE L x t F x V x EI t F x V x A t F x V t x v < < = ′ ′ ′ ′ + = , ) ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) , ( ρ : becomes PDE the so : solution ) (separable s synchronou Assume C x V x A x V x EI t F t F = ′ ′ ′ ′ = ) ( ) ( ) ) ( ) ( ( ) ( ) ( ρ : Rearrange 2 2 ) ( ) ( ) ) ( ) ( ( ) ( ) ( ϖ ρ ϖ = ′ ′ ′ ′ = = < = x V x A x V x EI t F t F C C CF F : get we so , Let . have must we from Again, : in ODE an gives x x V x A x V x EI t F t F 2 ) ( ) ( ) ) ( ) ( ( ) ( ) ( ϖ ρ = ′ ′ ′ ′ = vibration. of modes and s frequencie natural obtain to solved be can which problem, eigenvalue al differenti beam the represents end), each at (two BCs four with combined ODE, This ϖ L x x V x A x V x EI < < = ′ ′ ′ ′ ), ( ) ( ) ) ( ) ( ( 2 ρ ϖ L x x V x A x V x EI < < = ′ ′ ′ ′ , ) ( ) ( ) ) ( ) ( ( 2 ρ ϖ : rearranged Or, constant. are and which in cases handle can We ) ( ) ( x A x EI ρ L x V EI A V A EI < < = ′ ′ ′ ′ , 2 ρ ϖ ρ : becomes ODE , , constant With L x V V EI A < < = ′ ′ ′ ′ ≡ , 4 2 4 β ρ ϖ β : is ODE so , Define . : n eq' stic Characteri β β β β β ± ± = → = + = , ) )( ( 2 2 2 2 4 4 i s s s s . Then x x x i x i e A e A e A e A x V β β β β + + + = 4 3 2 1 ) ( . and , that Recall x x e x i x e x x i β β β β β β sinh cosh sin cos ± = ± = ± ± . Then x C x C x C x C x A A x A A x A A i x A A x V β β β β β β β β sinh cosh sin cos sinh ) ( cosh ) ( sin ) ( cos ) ( ) ( 4 3 2 1 4 3 4 3 2 1 2 1 + + + = + + + + + = : case fixed Fixed V(x ) ) essential. are BCs 4 (All : BCs ) ( , ) ( , ) ( , ) ( = ′ = = ′ = L V L V V V x C x C x C x C x V β β β β sinh cosh sin cos ) ( 4 3 2 1 + + + = : solution general to BCs essential apply DEVP, For 1 3 3 1 : ) ( C C C C V = → = + = ) implies ( ) ( : ) ( 2 4 4 2 = = = → = + = ′ ϖ β β C C C C V ) sinh (sin ) cosh (cos ) ( 2 1 x x C x x C x V β β β β + = : fixed end Left ) sinh...
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 Spring '10
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