MIT2_092F09_lec16

# MIT2_092F09_lec16 - 2.092/2.093 Finite Element Analysis of...

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2.092/2.093 Finite Element Analysis of Solids & Fluids I Fall ‘09 Lecture 16 - Solution of Dynamic Equilibrium Equations, cont’d Prof. K. J. Bathe MIT OpenCourseWare Reading assignment: Sections 9.1-9.3 Recall from our last lecture the general dynamic equilibrium equation and initial conditions: 0 MU ¨ + CU ˙ + KU = R ( t ) ; 0 U , U ˙ (1) This equation can be solved by: Mode superposition Direct integration Mode Superposition i = ω i 2 i (2) The ω i 2 are the eigenvalues and φ i are the eigenvectors for this system. Solve for ω i 2 , φ i : ω 2 ω 2 . . . ω 2 1 2 n 0 ���� ���� ���� for φ 1 for φ 2 for φ n where each φ i refers to a mode shape. Aside: Consider, picking “a” φ , = α ˜ (3) 1 where α is a nonzero scalar. Obviously, K α φ = ˜ = R . If φ ˜ is an eigenvector, then the load R 1 obtained using φ ˜ gives us back the vector φ ˜ (now scaled by α ). We also used orthonormality to establish that: φ T i j = δ ij φ T i j = ω i 2 δ ij The definition of the Rayleigh quotient is φ T ρ ( φ ) = φ T where φ can be any vector. So, we have ρ ( φ i ) = ω i 2 ω 1 2 ρ ( φ i ) ω n 2 Recall that the strain energy for any vector φ is 1 2 φ T . Thus, the strain energy corresponding

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