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MIT2_092F09_lec19 - 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 19 - Modeling for Dynamic Analysis & Solution, cont’d Prof. K. J. Bathe MIT OpenCourseWare In the last lecture, we discussed two methods for solving this general system: 0 MU ¨ + CU ˙ + KU = R ( t ) ; 0 U , U ˙ I. Mode Superposition p U = Σ φ i x i ( t ) i =1 So far, we considered the case where p = n . II. Direct Integration We concluded that implicit methods are only of interest if they are unconditionally stable. How many modes do we need to include in the mode superposition method? (What should we select for p ?) We consider three cases: I. Initial Conditions 0 U = α 1 φ 1 , 0 U ˙ = 0 , R = 0 The response is only in φ 1 . Therefore, if the initial conditions “use” certain mode shapes only, the response caused by these initial conditions will only be in these certain mode shapes. II. Loads, Spatial Distribution 0 R = α 1 0 U = U ˙ = 0 Again, the response is only in φ 1 and the conclusion of section I holds here as well.
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