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Unformatted text preview: Solution of Homework #2, 1998 Fall MEAM 501 Analytical Methods in Mechanics and Mechanical Engineering Onedimensional 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 , 2 2 + ∂ ∂ ∂ ∂ = ∂ ∂ r where r is the mass density, and the boundary condition is written by . 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 M1 degree polynomial: ( 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 Ω 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 = + + ∂ ∂ ∂ ∂ + ∂ ∂ ∫ ∫ , , , , , 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 = + + ∂ ∂ ∂ ∂ + ∂ ∂ ∑ ∫ ∑ ∫ = Ω = Ω , , , , , 1 1 2 2 r the finite element approximation of the solution u and weighting function w in each finite element W e using the Lagrange polynomial, yields the following discrete problem w dx fN w w u k w u k dxu x N x N EA dt u d dx N AN w e e E E E e e N e M i i i e M N M N R L N e M i M j j e j i j e j i i e 2200 = + + ∂ ∂ ∂ ∂ + ∑∑ ∫ ∑∑∑ ∫ ∫ = = Ω = = = Ω Ω , 1 1 1 1 1 1 1 1 1 2 2 r Here we have applied the finite element approximation of the weighting function w: ( 29 ( 29 e M i i i e s N w x w Nw = = ∑ = 1 in a finite element W e . Matrices [ ] ∫ Ω = = e dx N AN m m j i ij ij e r , M and [ ] ∫ Ω = = e dx dx dN dx dN EA k k j i ij ij e , K are called the element mass and stiffness matrices, respectively. Modifying the elementare called the element mass and stiffness matrices, respectively....
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 Fall '09

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