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Unformatted text preview: Axial vibration problems ) ( ) ( ) , ( ) ( t F x U t x u u EA u A = ← = ′ ′ : vars. of n sep' try : PDE ρ 2 ) ( ) ( ϖ ρ ρ = ′ ′ = → ′ ′ = AU U EA F F F U EA F AU 2 2 2 = + ′ ′ → = + ′ ′ → = ′ ′ U U U E U U A U EA x β ρ ϖ ρ ϖ : ODE . , , for BCs need : Solution β β β 2 1 2 1 sin cos ) ( C C x C x C x U + = constant) (need : ODE Get A AU U EA x ρ ϖ 2 ) ( = ′ ′ : ODE the satisfies solution general This U A U EA x C x C x U ρ ϖ β β 2 2 1 sin cos ) ( = ′ ′ + = ( 29 ) sin cos ( ) ( ) ( ) sin cos ( 2 1 2 2 2 1 2 x C x C A E U A x C x C EA U EA β β β ρ ϖ β β β + = = + = ′ ′ . and about something us tell and s, ' the determine conditions boundary The 2 1 C C β : case fixed Fixed U(x ) ) essential" " or kinematic, are BCs (both : BCs ) ( , ) ( = = L U U case. string fixed fixed the to analogous is case This x C x U C U x C x C x U β β β sin ) ( ) ( sin cos ) ( 1 2 1 = → = = → + = 2 2 2 2 ) ( sin ) ( L E L E r L L L U r r r r r ρ β ρ β ϖ λ π β β = = = → = → = → = , 2 , 1 , sin ) sin( ) ( , 2 = = = = r L x r C L x L C x U L E r r r r r r π β ρ π ϖ : case free Fixed O bj2 6 1 x C x U C U x C x C x U β β β sin ) ( ) ( sin cos ) ( 1 2 1 = → = = → + = O bj2 6 3 , 2 , 1 , 2 ) 1 2 ( sin ) sin( ) ( = = = r L x r C L x L C x U r r r r π β U(x ) : case free Free BCs) natural" " or kinetic (two : BCs ) ( , ) ( = ′ = ′ L U EA U EA x C x U C U x C x C x U β β β β cos ) ( ) ( sin cos ) ( 2 2 1 = → = = ′ → + = , 2 , 1 , , sin ) ( 2 = = → = → = = ′ r L E r r L L EAC L U EA r r ρ π ϖ π β β β : , 2 , 1 , , cos ) cos( ) ( = = = r L x r C L x L C x U r r r r π β U(x) : case mass Fixed BC) (essential : BCs ) ( = U x C x U C U x C x C x U β β β sin ) ( ) ( sin cos ) ( 1 2 1 = → = = → + = U(x ) m L C AL E L EAC β ρ α ρ β β β sin ) ( cos 2 = L L L EAC β α β β ρ β = cot sin ) ( : by divide s, ' , Cancel equation. stic characteri this of roots Find BC) (natural : ) ( ) ( 2 L mU L U EA u m u EA L x L x ϖ = ′ = ′ = = ) is (so : Let AL m AL m ρ α ρ α ) ( = L...
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 Spring '10
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