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

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Unformatted text preview: : Example U(x) s s s r r r AU U EA AU U EA 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 = + - = - = + - = - 2 2 ) ( ) ( L L L L L U k U m U EA u k u m u EA- = -- = 2 ) ( : 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 = + +- = + +- 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 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 = = : 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 2 = = + = + : relations ity Orthogonal Continuous systemsresponse 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 = = = = F K M = + = = 1 1 ) ( )] ( [ ) ( )] ( [ r r r r r r t x U t x U : 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 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 = + = = : Take , 2 , 1 ), ( ) ( ) ( 2 = = + s t N t m t m s s s s s s : 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 = = = = = = 1 1 ) ( ) ( ) , ( ) ( ) ( ) , ( r r r r r r x U x u x u x U x u x u , : 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...

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