This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Eigenvalues and Eigenvectors : Basis for Modal Analysis September 22, 1998 In the first model problem, we have consider vibration of an elastic string: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , , , , , , 2 2 2 2 = = = = = + = L t u t u x u x u x v x t u L in f ku x u T t u m Here a distributed elastic foundation is assumed to be connected on the elastic string spanned by a tensile force T , and u and v are the initial displacement and velocity at the initial time t = 0. Applying the weighted residual method with the finite number of trial functions and test functions ( 29 ( 29 { } x x i j y f , , a discrete problem ( 29 ( 29 2 2 , v u u u f Ku u M = = = + dt d dt d where ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 = = = = = = + = + = = = L i i i L i i i L i i i L i j i j i j i j ij L i j i j ij dx x x v v v dx x x u u u dx x x t f f f dx x x k dx d dx d T k dx d dx d T k dx x x m m m , , , , , , , , y y y y y y y f y f y f y f y f y f We shall now consider a homogeneous problem that the right hand side, the applied distributed force f is zero: Ku u M = + 2 2 dt d and we shall consider a stationary problem with harmonic motion such that (29 1 , = = i e t t i x u w where is a frequency of the harmonic motion in time, and x is independent of time t . Substitution of the harmonic motion into the equation of motion, we have 2 , w l l = = + Kx Mx that is 2 , w l l = = Mx Kx . When m = n in the discrete problem, this is called a generalized eigenvalue problem , where is an eigenvalue and x is an associated eigenvector of K and M . If we can decompose the mass matrix into the following form T LL M = where L is nonsingular in the sense that its inverse L1 exists, then x L Mx L x L KL L x LL Mx Kx T T T T l l l l = = = = 1 1 Putting x L y T = we can convert the generalized eigenvalue problem to the form of the standard eigenvalue problem ( 29 y Ay y y L K L l l = = , 1 1 T . In this case, if M and K are symmetric, then A is also symmetric. That is, the transformation x L y T = of the eigenvector, yields a symmetric eigenvalue problem. However, M1 K need not be symmetric even for both symmetric M and K , and then the eigenvalue problem x Kx M l = 1 must be solved only by a unsymmetric eigenvalue solver. Suppose that we have n number of eigenvalues, 1 ,...., n , and n number of linearly independent eigenvectors, x 1 ,...., x n , of the generalized eigenvalue problem: n i i i i ,....., 2 , 1 , = = Mx Kx l . Applying the GramSchmidt orthogonalization process with respect to a given matrix M of n number of linearly independent vectors z 1 ,....., z n : ( 29 ( 29 ( 29 n i i i i i i j j i j i i ,...., 2 , , , , , , 1 1 1 1 1 1 = = = =  = z M z z x x Mz x z z Mz z z x n number of linearly independent eigenvectors x 1 ,...., x n , can be orthonomalized with respect to the matrix M ( i.e. ( 29 ) , ij j i d = Mx x , and then they can span the n dimensional space...
View Full
Document
 Fall '09

Click to edit the document details