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How does 'mass participation' relate to the effective mass of
a particular modal shape? Are they analogous?

I've noticed that if the modal is unrestrained, the rigid body
modes will account for almost all of the mass participation, and
the total in each direction sums to nearly 1, but when the model is
restrained the sums become much less. I figure this is because the
entire body moves when unrestrained, and so this will account for a
large percentage of mass participation, and therefore make other
modes seem negligible. Am I right to assume that if I were to
compute many more modal shapes (100s, say) for the restrained model
that the sum of the mass participation in each direction would
indeed add up to 1 (or close to it)?

Mass participation factor is indicative of how much of the
structure's mass is taking part in a certain vibration mode. You
are right in your appreciation of the unrestrained model and how it
correlates to rigid body modes which have a mass participation
close to 1. Also at very high frequencies the MPF should be close
to one, but will never reach 1.