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CA-AA242B-Ch6 - AA242B MECHANICAL VIBRATIONS AA242B...

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AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Solution Methods for the Generalized Eigenvalue Problem These slides are based on the recommended textbook: M. G´ eradin and D. Rixen, “Mechanical Vibrations: Theory and Applications to Structural Dynamics,” Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Outline 1 Eigenvector Iteration Methods 2 Subspace Construction Methods AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Eigenvector Iteration Methods Introduction Generalized eigenvalue problem associated with an n -degree-of-freedom system Kx = ω 2 Mx n eigenvalues 0 ω 2 1 ω 2 2 ≤ · · · ≤ ω 2 n and n associated eigenvectors x 1 , x 2 , · · · , x n AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Eigenvector Iteration Methods Introduction Dynamic flexibility matrix consider a system that has no rigid-body modes K is non-singular construct the dynamic flexibility matrix defined as D = K - 1 M the associated eigenvalue problem is Kx = ω 2 Mx Dx = λ x with λ = 1 ω 2 eigenvectors are same as for the generalized eigenproblem Kx = ω 2 Mx eigenvalues are given by λ i = 1 ω 2 i λ 1 λ 2 ≥ · · · ≥ λ n drawbacks of working with D building cost is O ( n 3 ) non-symmetric matrix and as such must be stored completely AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Eigenvector Iteration Methods The Power Algorithm Determination of the fundamental eigenmode an iterative solution to Dx = λ x can be obtained by considering the iteration z p +1 = Dz p proof: expand z 0 in the basis of the eigenmodes of the system z 0 = n X i =1 α i x i Then, at the p -th iteration, z p = D p z 0 = n X i =1 α i D p x i = n X i =1 α i λ p i x i = λ p 1 α 1 x 1 + n X i =2 α i λ i λ 1 « p x i ! Recall that λ 1 λ 2 ≥ · · · ≥ λ n and assume that λ 1 > λ 2 = z p λ p 1 α 1 x 1 k z p +1 k k z p k λ 1 as p → ∞ AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Eigenvector Iteration Methods The Power Algorithm Convergence analysis p , z p = λ p 1 x 1 + λ p 2 x 2 + · · · = z T p +1 e j z T p e j = λ p +1 1 x T 1 e j + λ p +1 2 x T 2 e j + · · · λ p 1 x T 1 e j + λ p 2 x T 2 e j + · · · = λ p +1 1 ( x 1 + r p +1 x
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