MIT2_092F09_lec23

MIT2_092F09_lec23 - 2.092/2.093 — Finite Element Analysis...

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Unformatted text preview: 2.092/2.093 — Finite Element Analysis of Solids & Fluids I Fall ‘09 Lecture 23- Solution of Kφ = λ M φ Prof. K. J. Bathe MIT OpenCourseWare Reading assignment: Chapters 10, 11 We have the solutions < λ 1 λ 2 ≤ . . . ≤ λ n . Recall that: ≤ φ 1 φ 2 φ n Kφ i = λ i Mφ i (1) In summary, a necessary and sufficient condition for φ i is that Eq. (1) is satisfied. The orthogonality conditions are not sufficient, unless q = n . In other words, vectors exist which are K- and M-orthogonal, but are not eigenvectors of the problem. Φ = φ 1 . . . φ n (2) ⎡ ⎤ λ 1 zeros Φ T M Φ = I ; Φ T K Φ = Λ = ⎢ ⎣ . . . ⎥ ⎦ (3) zeros λ n Assume we have an n × q matrix P which gives us P T MP = I ; P T KP = A diagonal matrix q × q q × q → Is a ii necessarily equal to λ i ? ⎡ ⎤ a 11 zeros ⎢ ⎥ ⎣ a 22 ⎦ . zeros . . If q = n , then A = Λ , P = Φ with some need for rearranging. If q < n , then P may contain eigenvectors (but not necessarily), and A may contain...
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MIT2_092F09_lec23 - 2.092/2.093 — Finite Element Analysis...

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