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

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Unformatted text preview: 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 , 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 M-1 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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hw2A_98F - Solution of Homework#2 1998 Fall MEAM 501...

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