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pm1366
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Eigenvalue problem after galerkin
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.
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
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
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