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MIT2_092F09_lec18 - 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 18 - Modeling for Dynamic Analysis & Solution Prof. K. J. Bathe MIT OpenCourseWare From last lecture, MU ¨ + CU ˙ + KU = R ( t ) ; 0 U , 0 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 = Σ φ i x i (2) i =1 We start with the general solution, where φ i is an eigenvector. Then Eq. (1) becomes x ¨ i + 2 ξ i ω i x ˙ i + ω i 2 x i = φ i T R = r i ( i = 1 , . . . , n ) (3) For damping, assume a diagonal C matrix: Φ T C Φ = . . . zeros 2 ξ i ω i . zeros . . Φ = φ 1 . . . φ n 0 The initial conditions are x i = φ i T M 0 U , 0 x ˙ i = φ T i M 0 U ˙ . We consider and solve n such single-DOF systems: The mass m is 1, and the stiffness is ω 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. Direct Integration In direct integration, no transformation is performed. I. Explicit Method: Central Difference Method The explicit method evaluates Eq. (1) at time t to obtain the solution at time t
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