A bar is free to move in the horizontal direction as shown. At t=0, displacement of the free end is U and velocity vanis

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A Bar Is Free To Move In The Horizontal Direction As Shown At T 0 Displacement Of The Free End Is U And Velocity Vanis 1 (95.86 KiB) Viewed 105 times

A bar is free to move in the horizontal direction as shown. At t=0, displacement of the free end is U and velocity vanishes. Use the Finite Element Method on a regular spatial grid with i e {0,1} and the Discontinuous-Galerkin method with step size At to find the displacement and velocity of the free end at t = At. Cross-sectional area A, density p of the material, and Young's modulus E of the material are constants. P A,P, E х L k Ar (ai1–2a; +di+1)+F; + f'Ar=m* (ä;=1+4ä; +ä;+1) i e{1,2,...,n-1} Ar 6 Ar k Ar -(a1 - 20)+Fo+f' :-m'-(2äo +ä])=0 or do = do, Ar 2 Ar k Ar (an-1-an)+F, + f' Ar 2 + n m' (2än +än-1)=0 or an = an, 6 - d; -8i = 0 and à; -h; = =0. 1) Simplify to get kuj + müj = 0 2 a a g 2) = [6–30² 6-0² 12+a+ -6c2 6-3a = {a} where a= k -ΔΙ m à At άΔΙ (ata) | Ath) i-1