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Unformatted text preview: 2.092/2.093 — Finite Element Analysis of Solids & Fluids I Fall ‘09 Lecture 21- Solution of the Generalized Eigenvalue Problem Prof. K. J. Bathe MIT OpenCourseWare Reading assignment: Chapters 10 and 11 MU ¨ + KU = R (1) Aside: M could have zero masses. Then we use Gauss elimination on K to remove zero-mass DOFs, but we denote the final matrix still as K . Then, in free vibrations: MU ¨ + KU = (2) where now M and K are assumed to be positive definite matrices, i.e. U ˜ T MU ˜ > 0, U ˜ T KU ˜ > for any U ˜ = 0. Then, we obtain the eigenvalue problem Kφ = λ Mφ Kφ i = λ i Mφ i (A) → where < λ 1 λ 2 ≤ . . . ≤ λ n . ≤ φ 1 φ 2 φ n Recall: φ T i Mφ j = δ ij φ T i Kφ j = ω i 2 δ ij = λ i δ ij The Case of Multiple Eigenvalues Assume λ 1 = λ 2 < λ 3 , i.e. λ 1 has a multiplicity of 2 ( m = 2), φ 1 and φ 2 are two eigenvectors for λ 1 and λ 2 , and φ 1 = φ 2 . Then, we have K α φ 1 = λ 1 M α φ 1 ( α : any constant) (3) K β φ 2 = λ 1 M β φ 2 ( β : any constant) (4) Hence, K ( α φ...
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- Fall '09
- Finite Element Analysis, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix, ki, generalized eigenvalue problem