# Eigenvalue problem

1. Apr 14, 2013

### pm1366

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 !