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# hw2A_98F - Solution of Homework#2 1998 Fall MEAM 501...

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Solution of Homework #2, 1998 Fall MEAM 501 Analytical Methods in Mechanics and Mechanical Engineering One-dimensional Finite Element Method and Related Eigenvalue Problems Let us consider axial vibration of an elastic bar, whose length is L while the axial rigidity is EA, shown in Fig. 1: u(t,x) x f(t,x) k L k R EA(x) Figure 1 Vibration of an Elastic Bar in the Axial Direction Suppose that the left and right end points are supported by two discrete springs whose spring constant is given by k L and k R . The equation of motion of this elastic bar is written as ( 29 L in f x u EA x t u A , 0 2 2 + = r where r is the mass density, and the boundary condition is written by . 0 L x at u k x u EA and x at u k x u EA R L = - = = - = - We shall apply the weighted residual method that is constructed by the finite element method to derive a discrete system of the axial vibration problem. To this end, let the domain (0,L) be decomposed into N E number of finite elements W e , e = 1, .... ,N E , and let each finite element consist of M number nodes in which the axial displacement is assumed to be a M-1 degree polynomial:

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( 29 ( 29 ( 29 ( 29 ( 29 { } ( 29 ( 29 e M e e M M j j j e t u t u s N s N s N t u x t u Nu = = = = : ..... , 1 1 1 ( 29 ( 29 ( 29 { } e e e M j j j e x x s N s N s N x x Nx = = = = 3 1 3 1 1 : ..... ( 29 C M j k k k j k j x x x x s N = - - = 1 1 2 3 4 M s=-1 s=+1 x 1 x 2 x 3 x 4 x M element e = (x 1 ,x M ) Figure X A Finite Element W e = (x 1 , x M ) Noting that the weighted residual formulation of the equation of the motion and the boundary condition may be represented by the integral form ( 29 ( 29 ( 29 ( 29 w fwdx L t w L t u k t w t u k dx x w x u EA w t u A L R L L 2200 = + + + , , , 0 , 0 , 0 0 2 2 r that is ( 29 ( 29 ( 29 ( 29 w fwdx L t w L t u k t w t u k dx x w x u EA w t u A e e E e N e R L N e 2200 = + + + = = , , , 0 , 0 , 1 1 2 2 r