Hello everyone,
I attempted to perform an orthogonality assessment on the Eigenmodes subsequent to conducting modal analysis on some structure.
The analysis was conducted using ANSYS 22R1 software. An example of an orthogonality check was performed using APDL commands. The check is deemed successful when the product of [(transpose) Phi M Phi] results in the identity matrix. In this equation, Phi represents the modal matrix of the specified (n) modes, and M denotes the mass matrix.
According to most textbooks, vectors are considered orthogonal when their dot products equal zero. Consequently, the dot product of each mode (vector) with the others in the Phi matrix should yield an identity matrix. I attempted to do the task by employing the
load Phi_MMF.txt
data = zeros(203490,1);
for r=1:203490
data(r,1)=Phi_MMF(r,1); %transforming from MMF form to common matrix form
end
size(data)
modes = reshape(data,5814, 35); %the modal matrix of first 35 modes
MODES=modes';
% Initialize a matrix to store the results
orthogonality_matrix = zeros(35, 35);
% Loop to check orthogonality for all pairs of columns
for i = 1:35
for j = i:35
% Calculate the dot product between column i and column j
dot_product = dot(MODES(:, i),MODES(:, j));
orthogonality_matrix(i, j) = dot_product;
end
end
% Display the orthogonality matrix
disp("Orthogonality Matrix:");
disp(orthogonality_matrix);
I am uncertain about the distinction between two rules and would appreciate insight from any fellow who have encountered the rule [(transpose) Phi M Phi ] as a means of verifying orthogonality in any academic literature.
Regards
X^T K X = diag{k_i} and X^T M X = diag{m_i}, where k_i and m_i denotes modal stiffness and modal mass.
Now you can scale your eigenvectors, such that X^T M X = I is an identity matrix and X^T K X = \diag{\omega_i}^2. \omega_i = 2\pi f_i
There is no direct orthogonality of modes, unless your mass matrix is an identity matrix.
Nils Wagner
I am so grateful for your help,
This statement is highly coherent and logical. Is it possible to examine the Modal Assurance Criterion (MAC) by directly utilizing the eigenvectors (represented by the columns of the MODES matrix in my code) and substituting them into this equation?
(l A^T B l ^2)/ [(A^T A) (B^T B)]
where A and B are any two eigenvectors of MODES matrix
Regards
Mohamed A M Salem Can you share your Ansys model in .cdb format?
The MAC matrix can be easily generated.
Mohamed A M Salem Here you go. All you need is the free education edition of PERMAS. See https://www.intes.de/k_permas/overview/academic_license
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