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ASE 365 - Lecture 36

ASE 365 - Lecture 36 - Example U(x s s s r r r AU U EA AU U...

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Unformatted text preview: : Example U(x) s s s r r r AU U EA AU U EA ρ ϖ ρ ϖ 2 2 ) ( ) ( = ′ ′- = ′ ′- dx AU U dx U EA U U EA U dx U EA U dx AU U dx U EA U U EA U dx U EA U s L r s s L r L s r s L r r L s r r L s L r s r L s ρ ϖ ρ ϖ ∫ ∫ ∫ ∫ ∫ ∫ = ′ ′ + ′- = ′ ′- = ′ ′ + ′- = ′ ′- 2 2 ) ( ) ( L L L L L U k U m U EA u k u m u EA- = ′ →-- = ′ 2 ) ( ϖ : BC k m dx AU U dx U EA U U k U U m U dx AU U dx U EA U U k U U m U s L r s s L r L s r L s s r r L s r r L s L r s L r r s ρ ϖ ϖ ρ ϖ ϖ ∫ ∫ ∫ ∫ = ′ ′ + +- = ′ ′ + +- 2 2 2 2 + = + ′ ′ + = + ′ ′ ∫ ∫ ∫ ∫ L s r s L r s L s r s L r L r s r L s r L r s r L s U m U dx AU U U k U dx U EA U U m U dx AU U U k U dx U EA U ρ ϖ ρ ϖ 2 2 ( 29 ( 29 ( 29 ( 29 ] [ , ] [ , ] [ , ] [ , 2 2 s r s s r r s r r s U U U U U U U U M K M K ϖ ϖ = = : form) symmetric (in is This rs r r rs r L s r s L r rs r L r s r L s m k U k U dx U EA U m U m U dx AU U δ ϖ δ δ ρ 2 = = + ′ ′ = + ∫ ∫ : relations ity Orthogonal Continuous systems—response F K M = + )] , ( [ )] , ( [ t x u t x u : problem value Boundary " " , admissible any For : theorem" Expansion " ) ( ) ( ) ( ) ( lim ) , ( ) , ( 1 1 t x U t x U t x u t x u r r r r n r r n η η ∑ ∑ ∞ = = ∞ → = = F K M = + ∑ ∑ ∞ = ∞ = 1 1 ) ( )] ( [ ) ( )] ( [ r r r r r r t x U t x U η η : becomes BVP ( 29 ( 29 ( 29 . and then , let we If rs r r rs r s r rs r s r r r r m k U U m U U U U m δ ϖ δ δ 2 ] [ , ] [ , ] [ , = = = ≡ K M M ( 29 ( 29 ( 29 F K M , ) ( ] [ , ) ( ] [ , ) , ( 1 1 s r r r s r r r s s U t U U t U U U = + ⋅ ∑ ∑ ∞ = ∞ = η η : Take , 2 , 1 ), ( ) ( ) ( 2 = = + s t N t m t m s s s s s s η ϖ η : EOMs Modal ICs. modal need we : EOMs modal For , 2 , 1 ), ( ) ( ) ( 2 = = + s t N t m t m s s s s s s η ϖ η ∑ ∑ ∞ = ∞ = = = = = 1 1 ) ( ) ( ) , ( ) ( ) ( ) , ( r r r r r r x U x u x u x U x u x u η η , : profiles velocity and nt displaceme initial from are These ( 29 ( 29 ( 29 ( 29 1 1 1 1 ] [ , )] ( [ , ] [ , )] ( [ , s s r r sr s r r r s s s s r r sr s r r r s s...
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ASE 365 - Lecture 36 - Example U(x s s s r r r AU U EA AU U...

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