Are the modal participation factors bounded for shock response spectrum analysis?

Question

Recently dug myself into the theory of shock response spectrum analysis, but one thing is not clear for me.

The theory says that the peak response of the structure can be calculated as the product of the participation factor and the pertaining point of the shock response spectrum.

The participation factor is calculated as: $$ \Gamma_{kj}=\phi_{j}^{T}\textbf{M}\textbf{1}_k $$ where $\phi$ is the eigenvector, $\textbf{M}$ is the mass matrix and $\textbf{1}$ is the corresponding rigid body motion vector.

Since the partcipation factor calculation includes the eigenvactors, which are not exact as they are scaleable, it would mean, that the response of the strcture would depend on the eigenvector normalization.

So my question would be that does it mean that it is assumed that for this type of analysis the eigenvectors are mass normalized and if so are they bounded to a specific value?

Reference: https://www.comsol.co.in/multiphysics/response-spectrum-analysis

Answer 1

In the section "The Multiple DOF System" the Comsol document says

It has been assumed that the mass matrix normalization of the eigenmodes is used and that the damping matrix can be diagonalized by the eigenmodes.

Eigenvectors are assumed to be mass normalized in any mathematical derivation using them, unless somebody wants to be deliberately perverse.

The sensible way to define modal participation factors makes them dimensionless quantities. For mass normalized vectors, $\phi_j^{-1}\mathbf{M}\phi_j = 1$ so $\phi_j^{-1}\mathbf{M}$ has the dimension 1/length, and that is multiplied by a length (the rigid body motion vector) in your formula.

Note the participation factors can be negative, because the sign of a mass normalized eigenvector is arbitary even though its magnitude is fixed by the normalization.