I have a discrete object with n nodes and M mode shapes evaluated at each node (therefore a matrix 3xnxM). I would like to find the mass normalized mode shapes so that generalized mass matrix is the identity matrix?
Your modes diagonalize your mass matrix (they should anyway), so that outside round-off errors,
Phi^T M Phi = diag(µj)
We call the values in the diagonal the modal masses, you only need to scale each mode shape to set the modal mass to one by dividing by 1/sqrt(µ_j).
Once you have your mass, simply do
Phi = Phi * diag(sqrt( diag(Phi'*(M*Phi)).^(-1) ) )
if your mass matrix is M and your modal basis Phi. The result will be scaled to modal masses of 1. You can further check that the new projection Phi^T M Phi gives the identity.
Hi, thanks for your answer this is indeed what I am trying to do but I was wondering how to normalize this matrix Phi. At the moment I have a 3D matrix and I am not sure how this normalization is carried out?
Your modes diagonalize your mass matrix (they should anyway), so that outside round-off errors,
Phi^T M Phi = diag(µj)
We call the values in the diagonal the modal masses, you only need to scale each mode shape to set the modal mass to one by dividing by 1/sqrt(µ_j).
Once you have your mass, simply do
Phi = Phi * diag(sqrt( diag(Phi'*(M*Phi)).^(-1) ) )
if your mass matrix is M and your modal basis Phi. The result will be scaled to modal masses of 1. You can further check that the new projection Phi^T M Phi gives the identity.
Mode shapes are mass normalised in order to remove the need to know the generalised (modal) mass
For my case, it results in the same medal matrix, because Mn is an identity matrix; and if I have to divide by the square root, mode by mode; essentially is the same.
Any comments, please?
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