Participation Factor And Effective Mass

The modal mass for mode is defined as

The direction information can be extracted using the participation factor. The participation factor for mode in the direction is given by

where the vector is a rigid body vector in the direction. The effective mass is similar but defined as

The rigid body vectors are defined as

So a rigid body vector for a rotation of about global z can be defined as

The effective mass in a rotated axis system can be calculated from the participation factors and effective masses.

So

The sum of the effective mass in any given direction over all the modes is the total unrestrained mass. Staring with the definition of effective mass

The rigid body vector can be written as

So the term in the numerator of the effective mass becomes so

Also the total mass and so

So the sum of the effective masses over all the modes is the total unrestrained mass.