Solving Eigenvalue Problem After Galerkin

[pm1366]

Eigenvalue problem after galerkin

Homework Statement

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.

Homework Equations

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

 

[KataKoniK]

it is important to understand the underlying principles and assumptions in any mathematical method used for analysis. In this case, the Galerkin method is a popular approach for solving partial differential equations by approximating the solution as a linear combination of basis functions. However, it is important to note that this method can lead to a system of equations that may not have a unique solution.

In the case of your eigenvalue problem, it is important to carefully check the assumptions made in the Galerkin method and the resulting equations. It is possible that there may be some errors in the derivation or implementation of the method, leading to the discrepancies in the solutions obtained by the two methods. It may also be helpful to check the condition number of your coefficient matrix to ensure that it is well-conditioned and the solutions obtained are accurate.

Furthermore, it may be beneficial to compare your results with other numerical methods or analytical solutions, if available, to verify the accuracy of your solutions. it is important to carefully validate and verify any numerical methods used for analysis to ensure the reliability of your results.