ASE 365 - Lecture 37 - Review for test 3 equations. s...

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Unformatted text preview: Review for test 3 equations. s Lagrange' analysis. modal by response c nonharmoni MDOF modes. body rigid with Systems etc. ion, normalizat mass ion, ion/expans decomposit Modal . at force apply : example string Plucked a x F = F L-a a x ) ( a x F v T- = - : nt displaceme initial For L a L F C a L F L C L v C v ) ( ) ( ) ( ) ( 1 1 2-- = =- + = = = : BCs ) ( ) ( ) ( 2 1 1 a x u a x F C x C Tv a x Fu C v T-- + + =-- + = - : Integrate ---- = ) ( ) ( ) ( ) ( a x u a x L x a L T F x v : nt displaceme Initial ( 29 ( 29 )] ( [ , 1 , )] ( [ , 1 = = = x v V m x v V m r r r r r r M M : ICs Modal , 2 , 1 , sin 2 ) ( 1 = = = r L x r AL x V m r r : Set , 2 , 1 , sin 2 ) ( ) ( ) ( sin 2 ) ( 2 2 = = ---- = r L a r r L AL T F dx a x u a x L x a L T F A L x r AL L r Then nt. displaceme initial linear piecewise the of tion representa series" Fourier " this for that Note 2 1 ) ( r r t r AL T L x r L a r T r FL t r AL T L a r r L AL T F L x r AL t x V t x v r r r r r r 2 1 2 2 2 2 2 1 1 cos sin sin 2 cos sin 2 sin 2 cos ) ( ) , ( = = = = = = : Response . at force apply : example beam" Plucked " a x F = F L-a a x ) ( a x F v EI- = : nt displaceme initial For ) ( , ) ( 2 = = = = C v EI C v : BCs end Left ) ( 6 3 a x Fu C v EI- + = : Integrate modes. same supported Simply - ) ( ) ( 2 6 2 3 a x u a x F C x C v EI-- + + = ) ( 2 ) ( 2 3 2 1 2 2 3 a x u a x F C x C x C v EI-- + + + = ) ( 6 ) ( 3 1 2 2 3 3 a x u a x F C x C x C x C EIv-- + + + + = : have we , With 2 = = C C F L- a a x ) ( ) ( 6 3 a x u a x F x C v EI-- + = ) ( 6 ) ( 3 1 3 3 a x u a x F x C x C EIv-- + + = L a L a L Fa C a L F L C L a L F L v L a L F C a L F L C L v EI 6 ) 2 )( ( 6 ) ( 6 ) ( ) ( 6 ) ( ) ( 6 ) ( 1 3 1 2 3 3-- = =- + +-- =-- = =- + = : BCs end Right ( 29 ( 29 < <- + +- < <-- +- = L x a L a a L xa aL x a x EIL F a x a L a L xa L a x EIL F x v x v , ) 2 ( 3 6 , ) 2 )( ( ) ( 6 ) ( ) (...
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ASE 365 - Lecture 37 - Review for test 3 equations. s...

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