NumericalMethodsChaper04

NumericalMethodsChaper04 - 1 Numerical Methods Chapter 4...

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Unformatted text preview: 1 Numerical Methods - Chapter 4 – Eigenvalue Problem } { } ]{ [ } ]{ [ = + x K x M & & Note: For structural and finite element problems ] [ M and ] [ K are symmetric. Try the solution: ) cos( ) ( ϕ ω + = t X t x i i ⇒ ) sin( ) ( ϕ ω ω + − = t X t x i i & ⇒ ) cos( ) ( 2 ϕ ω ω + − = t X t x i i & & ⇒ ) cos( } { )} ( { ϕ ω + = t X t x ) sin( } { )} ( { ϕ ω ω + − = t X t x & ) cos( } { )} ( { 2 ϕ ω ω + − = t X t x & & ⇒ } { ) cos( } ]{ [ ) cos( } ]{ [ 2 F t X K t X M = + + + − ϕ ω ϕ ω ω ⇒ } ]{ [ } ]{ [ 2 = + − X K X M ω ⇒ ( ) } { ] [ ] [ = − X M K λ ⇒ } ]){ [ ] [ ] ([ 2 1 = − − X I K M ω ⇒ ( ) { } } { ] [ ] [ = − X I D λ ⇔ } { } ]{ [ X X D λ = where ] [ ] [ ] [ 1 K M D − = and 2 ω λ = ⇒ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 3 2 1 3 2 1 3 33 32 31 2 23 22 21 1 13 12 11 M M O M M M L L L L M O M M M L L L n nn n n n n n n X X X X d d d d d d d d d d d d d d d d λ λ λ λ Note: ] [ ] [ ] [ 1 K M D − = is in general not symmetric. ⇒ } { } { = X trivial solution or ( ) ] [ ] [ det = − I D λ The eigenvalues are the roots of the polynomial given by: ( ) ] [ ] [ det = − I D λ n λ λ λ λ L 3 2 1 , , where 1 1 λ ω = , 2 2 λ ω = , 3 3 λ ω = … n n λ ω = Eigenvector Extraction Solve: ( ) { } } { ] [ ] [ ) ( = − i i X I D λ i = 1 to n ⇒ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ) ( ) ( 3 ) ( 2 ) ( 1 3 2 1 3 33 32 31 2 23 22 21 1 13 12 11 M M O M M M L L L L M O M M M L L L i n i i i i i i i nn n n n n n n X X X X d d d d d d d d d d d d d d d d λ λ λ λ For each eigenvalue i λ there corresponds an eigenvector } { ) ( i X . 2 Suppose we found the eigenvalues i λ of matrix ] [ D and we want to find the corresponding eigenvectors. ⇒ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − ) ( ) ( 3 ) ( 2 ) ( 1 3 2 1 3 33 32 31 2 23 22 21 1 13 12 11 M M L M O M M M L L L i n i i i i nn n n n n i n i n i X X X X d d d d d d d d d d d d d d d d λ λ λ λ ⇒ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′...
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This note was uploaded on 10/16/2011 for the course MECHANICAL 581 taught by Professor Wasfy during the Fall '11 term at IUPUI.

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NumericalMethodsChaper04 - 1 Numerical Methods Chapter 4...

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