ASE 365 - Lecture 41

# ASE 365 - Lecture 41 - Assumed-modes method be" s...

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Unformatted text preview: Assumed-modes method be?" s ' lowest the would what etc.), cubic, quadratic, (or linear were If " asks, approach Ritz- Rayleigh The ϖ ) ( x u ) ( ) ( ) , ( 1 t q x t x u i n i i ∑ = ≈ ψ : ion Approximat coordinate d generalize function, admissible = = ) ( ) ( t q x i i ψ s. ' the governing EOMs derive to used are equations s Lagrange' Then . and , , into d substitute is of ion approximat assumed" " this method, modes- assumed the In ) ( ) , ( t q W V T t x u i nc δ (BVP). problem response the solving for method, tion discretiza series another is method modes- assumed The ( 29 [ ] ( 29 ) Ritz!- Rayleigh for as same the is ( M M t q t q m t q t q x x t q x t q x t x u t x u T T j i n i n j ij j i n i n j j i j n j j i n i i q q 2 1 ) ( ) ( 2 1 ) ( ) ( ) ( ), ( 2 1 ) ( ) ( ), ( ) ( 2 1 )] , ( [ ), , ( 2 1 1 1 1 1 1 1 ≈ = ≈ ≈ = ∑∑ ∑∑ ∑ ∑ = = = = = = ψ ψ ψ ψ M M M ( 29 [ ] ( 29 ) Ritz!- Rayleigh for as same the is ( K K t q t q k t q t q x x t q x t q x t x u t x u V T j i n i n j ij j i n i n j j i j n j j i n i i q q 2 1 ) ( ) ( 2 1 ) ( ) ( ) ( ), ( 2 1 ) ( ) ( ), ( ) ( 2 1 )] , ( [ ), , ( 2 1 1 1 1 1 1 1 ≈ = ≈ ≈ = ∑∑ ∑∑ ∑ ∑ = = = = = = ψ ψ ψ ψ K K K ( 29 ( 29 nc T i n i nc i i n i i i n i i nc t q t Q t q x t q x t x u W Q q δ δ δ ψ δ ψ δ δ ≈ = ≈ ≈ = ∑ ∑ ∑ = = = ) ( ) ( ) ( ), ( ), ( ) ( ), , ( 1 , 1 1 F F F nc nc T nc T T K M W K V M T Q q q Q q q q q q = + = = = : EOMs the give equations s Lagrange' , and , , With δ δ 2 1 2 1 DEVP. the addresses also method modes- assumed the So identical. is ion eigensolut e approximat modes- assumed the so Ritz,- Rayleigh from one the as same the is which , problem eigenvalue algebraic the give EOMs vibration, free For q q M K λ = . , for solved be to ODEs of system a are EOMs The n i t q n K M i nc , , 1 ) ( = = + Q q q ? and get we can How . and from conditions initial need We ) ( ) ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) , ( 1 1 i i i n i i i n i i q q q x x u x u q x x u x u ∑ ∑ = = ≈ = ≈ = ψ ψ [ ] ( 29 ( 29 ) ( ) ( ) ( ] [ , ) ( ) ( , ) ( ) ( ) ( ), ( : ) ( ) ( 1 1 1 q ⋅ = ≈ = ≈ ≡ ∑ ∑ ∑ ∫ = = = ) of row ( and between product inner mass" " the at Look M i q m q q x dx...
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ASE 365 - Lecture 41 - Assumed-modes method be" s...

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