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Eigenvalue problem

  1. Apr 14, 2013 #1
    Eigenvalue problem after galerkin

    1. The problem statement, all variables and given/known data

    i am working on vibration of cylindrical shell analysis, after solving the equations of motion by galerkin method , i reach to an eigenvalue problem this way :

    { p^2*C1+p*C2+C3 } * X=0

    C1,C2,C3 are all square matrices of order n*n and all are known ones, and X is a vector of unknown coeffs.

    now for nontrivial solution, the determinant of the coefficient matrix must vanish:

    det(p^2*C1+p*C2+C3)=0

    which should be reduced to a standard eigenvalue problem by transferring to state-space as:

    det(M-p*I)=0

    where:

    M=[-(C1^-1)*C2 , I ; -(C1^-1)*C3 , 0]

    where I is the identity matrix of order (n*n) and M is a (2n*2n) matrix.






    2. Relevant equations



    3. The attempt at a solution

    now i must find the eigenvalues of matrix M which is expected to be the values of p

    also there is an alternative way for finding p , which is solving det(p^2*C1+p*C2+C3) in terms of p and getting the roots ,

    my problem is that these two methods don't give exactly the same answers , they're similar but some p values are different , for example after soving the det(p^2*C1+p*C2+C3) the result is :

    p=
    0.1204 + 370.8*i
    - 4.264 - 394.2*i
    - 8.648 + 370.8*i
    - 4.264 + 506.2*i
    - 4.264 - 506.2*i
    - 4.264 + 394.2*i
    0.1204 - 370.8*i
    - 4.264 - 443.6*i
    - 4.264 + 443.6*i
    - 8.648 - 370.8*i

    but when i get eigenvalue of M matrix in matlab , i see:

    p=
    1.0e+02 *

    -0.0426 + 3.6016i
    -0.0426 + 5.0844i
    -0.0426 - 3.6016i
    -0.0426 + 3.9915i
    -0.0426 - 5.0844i
    -0.0426 + 4.4363i
    -0.0426 + 3.7287i
    -0.0426 - 3.9915i
    -0.0426 - 4.4363i
    -0.0426 - 3.7287i


    in matlab one time i write eig(M) to find eigenvalues , and one time i do this way:

    solve(det(p^2*C1+p*C2+C3) , p)


    also all p values are expected to be complex and they are . the values taken by two methods are simillar in some values but are totally different in others.

    i don't know why these two are not the same !

    PLEASE HELP me !! thanks in advance
     
    Last edited: Apr 14, 2013
  2. jcsd
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