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

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?

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.