# How to calculate the relative contribution of each mode in seismic dynamic mode analysis?

In structural dynamic mode analysis, one would have to carry out modal analysis in order to get different modes that contribute to the response of the seismic ( or wind) force. Here is a nice summary.

In a large structural model, there are so many modes, and we can't take them all. So we can only take the first few prominent modes ( with low frequencies), assuming that the modes will grow less and less important as the frequency becomes higher and higher.

Then the issue arises: how to judge the relative importance of different modes ( whatever the term 'importance' may mean?) I find that the often-cited mode participation factor

$\Gamma=\frac{\sum M \phi}{\sum\phi^T M \phi}$

doesn't work, because it depends on how we normalize the eigenvector, as one can readily see from the above formula.

Is there any other factor that allows us to judge whether a mode is more important or not?

• In your links, Wikipedia's formula for the mode participation factor is just wrong. Your other link (vibrationdata.com) gives the correct formula, which doesn't depend on how you normalize $\phi$.. – alephzero Mar 10 '16 at 21:02

You should be normalizing your eigenvector so that the generalized mass matrix (defined by $\hat{m} = \phi^T M \phi$) is the identity matrix, and therefore, the generalized mass of the rth mode (defined by $M_r = \phi^T_r M \phi_r$) has the value of 1. This should give you consistent modal participation factors.

The modal mass participation ratio is widely used as the metric to determine the relative significance of modes in a modal response spectrum analysis. It has even been codified in many of the codes of record. For example, ASCE 7-05 (7-10 is at home, sorry), Section 12.9.1 states that,

The analysis shall include a sufficient number of nodes to obtain a combined modal mass of at least 90 percent of the actual mass in each of the orthogonal horizontal directions of the response considered in the model.

And to elaborate on what I think may be an underlying question in your query (and the code stipulation), you calculate total effective mass as, $$M_{eff} = \Sigma\Bigg[\frac{\Gamma_i^2}{\hat{m}_{ii}} \Bigg]$$

If $M_{eff}$ is greater than 90% of your overall system mass, you've considered an appropriate number of modes.

Likewise, the effective mass participation of each mode may be used as a guide to determine the relative "importance" of each node, $$m_{eff,i} = \frac{\Gamma_i^2}{\hat{m}_{ii}}$$

Dynamics of Structures by Chopra is a really good reference if you are attempting to do this all by hand.

Edit: Check out this pretty good discussion and examples written by Tom Irvine.this Oops, this was your original link!

• Your linked discussion and example is the same one that was linked in the original question. ;) – hazzey Mar 10 '16 at 17:43
• .....ha, too many browser windows open. – William S. Godfrey- S.E. Mar 10 '16 at 18:36