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ASE 365 - Lecture 40

# ASE 365 - Lecture 40 - 29 ion eigenfunct first the...

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Unformatted text preview: ( 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 2 ψ ψ ρ ψ ψ ( 29 ( 29 ( 29 2 2 2 2 1 2 2 2 2 2 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 L x L x L x L x x- =- = ′ - = ψ ψ ( 29 ( 29 ( 29 2 2 2 2 1 2 2 2 4 4 1 3 2 2 2 2 2 2 2 2 1 2 2 4674 . 2 2 5 . 2 15 / 2 3 / 1 20 / 1 4 / 1 3 / 1 3 / 1 1 1 2 ) ( L E L E L E L E L E dx x Lx x L A dx x Lx L EA dx L x Lx A dx L x L EA x R L L L L ρ ρ π ϖ ρ ρ ρ ρ ρ ψ = = ≈ = = +- +- = +- +- = - - ≡ ∫ ∫ ∫ ∫ u(x ) ( 29 ). and (over functions quadratic admissible all over minimize and let better, even Or, 2 1 2 2 1 2 2 1 1 ) ( ) ( ) ( ) ( q q x u R q L x q L x q x q x x u + = + = ψ ψ ( 29 . and : Result 2 2 2 2 2 1 2 , 4527 . ) ( 4674 . 2 2 4860 . 2 ) ( min 2 1 - = = = ≈ = L x L x x u L E L E L E x u R q q ρ ρ π ϖ ρ . set ly, equivalent or, Set How? , 2 2 1 1 2 1 = ∂ ∂ + ∂ ∂ = = ∂ ∂ = ∂ ∂ q q R q q R R q R q R δ δ δ u(x ) ij n i i j j j j ij ij i q x x U M K n R m k x n ∑ = ≈ • = = • • • 1 2 ) ( ) ( , { ) ( ψ ϖ λ λ δ ψ ion eigenfunct e approximat ing correspond the and e approximat an gives } eigenpair each then EVP the solving by subspace D- in set , ts coefficien mass & stiffness calculate functions admissible choose : method Ritz- Rayleigh the by ion eigensolut e approximat an For q q q . need we , For - = = ∂ ∂- ∂ ∂ = ∂ ∂ ∑ ∑ = = j n j ij j n j ij i i i q m R q k q D R q N q R 1 1 ) ( 2 ) ( q q problem. eigenvalue algebraic d generalize familiar the , or , of row the gives This q q q q q q M R K M R K i th ) ( ) ( = =- ( 29 ( 29 1 1 ] [ , 1 ] [ , , , 1 ) ( 1 1 + + = = = ≡ •- + = = ′ ′ = ≡ • = = • ∫ ∫ ∫ ∫-- j i AL dx L x L x A dx...
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ASE 365 - Lecture 40 - 29 ion eigenfunct first the...

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