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ASE 365 - Lecture 38

# ASE 365 - Lecture 38 - Continuous systems Boundary value...

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Continuous systems ICs boundary) at FBD or (obvious, BCs interior) in element of (FBD PDE : problem value Boundary terms BC maybe terms, interior include operators M & K operators M & K w.r.t. orthogonal are ions Eigenfunct modes s, frequencie natural EVP al Differenti variables of separation solution s Synchronou : response Free - - EOMs modal get to ity orthogonal use : response Forced

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: response harmonic beam Cantilever - ) of (part : PDE F K M = + < < - = + ] [ ] [ 0 , ) ( 0 v v L x e a x F v EI v A t i ϖ δ ρ F0ei ωt a L-a ( 29 ( 29 ( 29 F K M ), ( ) ( )] ( [ ), ( ) ( )] ( [ ), ( 1 1 x V t x V x V t x V x V s r r r s r r r s = + = = η η t i s s t i s s s s e a V F t N e m m ϖ ϖ η ϖ ϖ ) ( ) ( ) ( 0 2 2 = = + - - + + - - = ) sinh (sin sinh sin cosh cos cosh cos ) ( x x L L L L x x C x V r r r r r r r r r r β β β β β β β β = = - = 1 2 2 0 ) ( ) , ( , ) ( ) ( s t i s s s s s s e x V t x v m a V F ϖ η ϖ ϖ η
Continuous systems—approximation response. and modes, s, frequencie natural exact found have and BCs) simple section, - cross uniform with members (single systems simple at looking been ve We' : e cooperativ so not are structures Real members multiple of assembly flexure) torsion, (axial, n deformatio of modes different sections - cross nonuniform conditions continuity and boundary difficult etc. ate. Approxim what? then option, an not are solutions exact When

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]) [ , ( ]) [ , ( ) ( ] [ ] [ 2 2 U U U U x U U U M K M K ϖ ϖ = = : product inner take to integrate , by DEVP Multiply ( 29 ( 29 ( 29 ( 29 ) ( )] ( [ ), ( )] ( [ ), ( ) ( 2 x u x u x u x u x u x u R ϖ = M K : quotient s Rayleigh' Obj303
( 29 ( 29 . so , , that so normalized - mass are modes If ) ( min ) ( 1 ) ( 2 1 2 1 2 2 2 1 2 2 1 2 2 2 x u R x u R m x u r r r r r r r r r r r = = = = ϖ ϖ η η ϖ ϖ ϖ η η ϖ ( 29 ( 29 wins. lowest The system). simpler a of modes gravity, under nt displaceme static (e.g., functions admissible some for evaluating by ed approximat be can eigenpair first the Then ) ( ) ( x u R x u R ( 29 later... More . of values stationary for solving and function, admissible an is where letting by ed approximat be can eigenpairs Other ) ( ) ( ) ( ) ( 1 x u R x q x x u j j n j j ψ ψ = =

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( 29 : ion eigenfunct first the resembles that function trial admissible an using by e approximat can We ) ( ) ( min ) ( 2 1 x x u R x u ψ ϖ = ( 29 L x x dx x A dx x EA x R L L = ) ( , ) ( ) ( ) ( 2 0 2 0 ψ ψ ρ ψ ψ ( 29 ( 29 ( 29 2 2 2 2 1 2 2 2 0 0 2 0 2 0 4674 . 2 2 3 3 / 1 1 1 1 ) ( L E L E L E L E dx x A dx EA dx L x A dx L EA x R L L L L ρ ρ π ϖ ρ ρ ρ ρ ψ = = = = = u(x )
. since BC, end free the satisfies : better do can We 2 2 2 1 ) ( 2 1 ) ( L x L

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ASE 365 - Lecture 38 - Continuous systems Boundary value...

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