MIT2_092F09_hw7 - 2.092/2.093 FINITE ELEMENT ANALYSIS OF...

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2.092/2.093± F INITE E LEMENT A NALYSIS OF S OLIDS AND F LUIDS F ALL 2009± Homework 7 Instructor: Prof. K. J. Bathe Assigned: Session 16 TA: Seounghyun Ham Due: Session 19 Problem 1 (20 points): Consider the two degree of freedom system 1 0 u �� 1 4 1 u 1 10 0 0 ⎢ ⎥ + ⎢ ⎥ = ⎢ ⎥ ; U = 0 ; U = 0 0 2 u �� 2 1 4 u 2 0 Perform a mode superposition solution using only the lowest frequency and corresponding mode shape. Problem 2 (10 points): Consider the four-node finite element shown with its boundary conditions. Assume that Gauss elimination is performed in the usual order for u 1 , u 2 , …, and so on, i.e., from the lowest to the highest degree of freedom number. Determine for cases 1 and 2 whether any zero diagonal element will be encountered in the elimination process, and if so, at what stage of the solution this will be the case. u 3 u 1 u 2 u 4 u 5 u 1 u 2 u 4 u 5 u 3 u 6 Case 1 Case 2
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Problem 3 (10 points): Consider the single degree of freedom equation. �� 0 -12 0 2U+8U=0
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