I was attending the Dynamic Analysis Seminar (16-18 april 2012)in Houston. I would like to know the meaning of the mass participation factor in page 4-27 (seminar Notes). All the literature available uses the "mode participation factor" (It is mentioned in this page of Seminar Notes too) but not the "mass participation factor".
Is the "mass participation factor" the "effective modal mass" mentioned in many Dynamics analysis books? In this case, the formula shown on page 4-27 doesn´t match with the dynamic theory in other books
For the other hand, the seminar notes in page 4-59 state that the multiplication of Φ K Φ-1 results in a "rotated" stiffness matrix (diagonal matrix). I think the only way to get a diagonal matrix is through the multiplication of ΦT K Φ. If I rotate the coordinate axes I don´t get a diagonal matrix.
Is there anything wrong in the seminar Notes? Can anyone help me with this?

Regards,
Edgar Rincon

By algebra, in circumstances defined by theory, diagonalization of endomorphisms and corresponding matrices can be performed using eigenvectors (corresponding to eigenvalues).
This is the kernel of that theory, it is not a simple rotation of coordinates.

Regards.

Maybe these pages need a math improvement, but I’m in doubt this will be helpful for an engineer. By the other hand, I'm in doubt that defining a "rotation" really gives an intuitive explanation.

A diagonal matrix may offer computational and storage advantages in simulations. That’s why one practical way in computation is to force the math to diagonal matrices.

How? By a "specialized" way: in algebra there is a "spectral theorem" which states that if a matrix A is symmetric, you can find an orthogonal matrix P such that P^(-1)*A*P is a diagonal matrix with entries the eigenvalues. That orthogonal matrix is made by eigenvectors. We are interested to construct matrix A as symmetric (and nature is helping us...) in order to get the benefit of this theorem.

So the eigenvalues and eigenvectors are not exotic stuff (as appears in university math courses…), they are powerful tools in computation. Specific algorithms are used to find eigenvalues/ eigenvectors. The physical interpretation of the eigenvalues and eigenvectors (which come up from solving the system) are that they represent the frequencies and corresponding mode shapes. I think this statement would clarify why we perform the modal analysis, why we need to analyze "enough" modes and ... what is the "meaning" of stress results from modal analysis in terms of reported quantities.

I would add that eigenvectors are orthogonal and there is a simple theorem saying the inverse of an orthogonal matrix is its transpose. That’s why in some books you find the transpose matrix of eigenvectors instead of the inverse of that matrix.

But in the end is not more than a specific way to perform matrices calculations.

Best regards.

Edgar,

I agree with your concern.

Regarding the expression for mass participation factor, i.e. mode participation factor( = PHI TRANSPOSE X MASS MATRIX X INFLUENCE VECTOR DIVIDED BY PHI TRANPOSE TIMES MASS MATRIX TIMES PHI) times PHI, in my opinion is not correct and such an expression is what I have not seen anywhere. In many text books, mode and mass participation factor are used alternatively and mass participation is used for seismic loading and mode participation is used for general dynamic loading.Instead of this expression for mass partcipation factor, the seminar notes should show the expression for modal mass ( = square of the numerator in the expression for mode participation factor divided by the same numerator)and this should be included as an output parameter in CAESAR II.

Regarding the last part, yes, transpose can be replaced with inverse if it is a orthogonal matrix but the orthogonality is w.r.t Mass and Stiffness matrices.

Hope this helps.

Regards