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Unformatted text preview: Beams L x t x f t x v x EI t x v x A < < = = + , ) , ( ) ) , ( ) ( ( ) , ( ) ( : 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 < < = + = , ) ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) , ( : becomes PDE the so : solution ) (separable s synchronou Assume C x V x A x V x EI t F t F =  = ) ( ) ( ) ) ( ) ( ( ) ( ) ( : Rearrange 2 2 ) ( ) ( ) ) ( ) ( ( ) ( ) (  =  = = < = 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 ) ( ) ( ) ) ( ) ( ( ) ( ) (  =  = 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 L x x V x A x V x EI < < = ), ( ) ( ) ) ( ) ( ( 2 L x x V x A x V x EI < < = , ) ( ) ( ) ) ( ) ( ( 2 : rearranged Or, constant. are and which in cases handle can We ) ( ) ( x A x EI L x V EI A V A EI < < = , 2 : becomes ODE , , constant With L x V V EI A < < = , 4 2 4 : is ODE so , Define . : n eq' stic Characteri = = + = , ) )( ( 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  + + + = 4 3 2 1 ) ( . and , that Recall x x e x i x e x x i 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 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 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 = = = = + = C C C C V ) sinh (sin ) cosh (cos ) ( 2 1 x x C x x C x V  + = : fixed end Left ) sinh...
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 Spring '10
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