HW9 - ρ = ∂ ∂ = ∂ ∂ 1 2 2 2 2 2 E c t u c x u a...

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MCE 561 Computational Methods in Solid Mechanics Homework Assignment 9 Due April 18, 2011 1. The two, linear-element solution to Example 6.2.1 was expressed by n n U U h t h h t h h t h h t h U U h t h h t h h t h h t h α - - α - + α - + α - - = α + α - α - α + + 3 2 1 3 2 ) 1 ( 3 1 ) 1 ( 6 1 ) 1 ( 6 1 ) 1 ( 2 3 2 3 1 6 1 6 1 2 3 2 Note that the critical time step for this problem was determined as 0631 . 0 = critical t . Using MATLAB (see example code in online class notes) or similar software, evaluate and plot the time behavior of ) ( ) , 1 ( 3 t U t u = for the following three cases: 060 . 0 , 065 . 0 , 0 = = α t and 065 . 0 , 5 . 0 = = α t . You can refer to Figure 6.2.3 in the text for the expected results. 2. One dimensional wave propagation down an elastic bar is governed by the equation
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Unformatted text preview: ρ = ∂ ∂ = ∂ ∂ / , 1 2 2 2 2 2 E c t u c x u a.) Develop the finite element equation based on the Newmark time stepping scheme. b.) Set up a two- linear element model of the problem (0 < x < L ) ) , ( , sin ) , ( , ) , ( ) , ( = π = = = x u L x x u t L u t u Use equal size elements with h = L /2, 3 / 1 2 , 2 / 1 = β = γ = α , and Δ t = h/c , develop a time stepping relation for u 2 n+1 . Check on the appropriateness of the time step. Later I will show you a MATLAB evaluation of this time stepping scheme and a comparison with the exact solution....
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This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.

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