Generalized modal mass

 

In modal analysis theory, the concept of modal mass, aka generalized modal mass, appears in most formulations.

In SCIA Engineer, the calculation protocol of eigen frequencies displays a table of so-called relative modal masses. Those give information about the participation of each mode in GCS directions X, Y and Z when seismic action is applied to the structure. Similar values are provided for rotational components, i.e. relative modal mass inertia values around GCS axes. Relative modal masses and inertias are not the same as modal masses. For more information, see "Modal mass vs Relative modal mass".

Definition

Modal masses, aka generalized modal masses, are related to the overall kinetic energy of each mode. They depend on the chosen normalization method.

where

is the generalized modal mass for the i-th mode

is the i-th mode shape vector

is the mass matrix of the structure

By definition, mode shape vectors , aka eigenvectors, are dimensionless - they have no units. In SCIA Engineer, they are presented with the same units as displacements for convenience, as it allows easily scaling the display of the results and their presentation in results tables. Consequently, generalized mass values have the same units as mass, i.e. [kg].

Scaling the mode shapes

Mathematically, mode shapes are eigenvectors. They can be scaled by any scaling factor and still remain, from a mathematical point of view, the same eigenvector.

There are different normalization approaches. The two most common ones are

Normalization on modal mass

In SCIA Engineer, the normalization of each mode shape φi is defined in such a way, that its generalized modal mass is equal to 1:

The main benefit of that approach is, that it allows simplifying many parts of subsequent theoretical developments. As the modal mass is equal to 1, it will simply disappear from the equations.

Normalization on maximum displacement

The other common scaling approach for mode shape scaling is setting the maximum displacement of a given mode to 1. That approach is not supported in SCIA Engineer.

To obtain a displacement-normalized mode shape from SCIA Engineer, multiply all displacement and rotation values of a given mode shape by the factor

where xi is the maximum displacement of the mode shape obtained in SCIA Engineer.

However, when doing so, the corresponding generalized modal mass for that re-scaled mode shape will also be modified. The re-scaled modal mass is:

Practical use of the generalized modal mass

For practical application in dynamic modal analysis, a mode shape vector must always be associated with the corresponding generalized modal mass.

In essence, the generalized modal mass allows to use eigenvectors in equations regardless of the method used for normalization.

For instance, considering an arbitrary modal scaling factor ci, the scaled modal participation factor can be written as:

and the scaled modal effective mass can then be written as:

which, conveniently, does not depend on the scaling factor.

All end results of a seismic spectral analysis are independent from the chosen normalization (effective masses, internal forces, displacements...). However, not all related quantities that appear during the calculation are (e.g. mode shapes, modal participation factors).