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Unformatted text preview: 2.092/2.093 Finite Element Analysis of Solids & Fluids I Fall 09 Lecture 18- Modeling for Dynamic Analysis & Solution Prof. K. J. Bathe MIT OpenCourseWare From last lecture, MU + CU + KU = R ( t ) ; U , U (1) KU = F I , the internal force calculated from the element stresses. Mode Superposition The mode superposition method transforms the displacements so as to decouple the governing equation (1). Thus, consider: n U = i x i (2) i =1 We start with the general solution, where i is an eigenvector. Then Eq. (1) becomes x i + 2 i i x i + i 2 x i = i T R = r i ( i = 1 , . . . , n ) (3) For damping, assume a diagonal C matrix: T C = . . . zeros 2 i i . zeros . . = 1 . . . n The initial conditions are x i = i T M U , x i = T i M U . We consider and solve n such single-DOF systems: The mass m is 1, and the stiffness is i 2 . In Eq. (1) we have fully coupled equations. By performing the transformation, we obtain n decoupled equations. r i can be a complicated function of time....
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- Fall '09
- Finite Element Analysis