Modal mass vs. Relative modal mass

 

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.

Modal mass

Modal masses, aka generalized modal masses, are related to the overall kinetic energy of each mode. They depend on the chosen normalization method, but not on the direction.

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

In SCIA Engineer, the normalization of each mode shape φ_i is defined in such a way, that the generalized modal mass is equal to 1 (see Normalization of mode shapes below):

Modal masses, aka generalized modal masses, are always equal to 1 in SCIA Engineer because of the normalization that is used.

Relative modal mass

Relative modal masses are direction dependent. They reflect the contribution of each mode to the base shear of the structure in a given direction. They are not dependent on the normalization.

where

is the direction vector of the considered seismic action; it has the same number of values as mode shape vectors; each value is 1 in the considered direction k and 0 in other directions

is the modal participation factor of the i-th mode in direction k; thanks to the normalization used in SCIA Engineer, the formula can be simplified as shown

is the seismic modal mass of the i-th mode in direction k; again, thanks to the normalization used in SCIA Engineer, the formula can be simplified as shown

is the total moving mass of the structure in direction k (displayed in the calculation protocol as sum of masses - moving mass)

is the relative modal mass of the i-th mode in direction k, as displayed in the table of relative modal masses in the calculation protocol of eigen values

Normalization of mode shapes

The resolution of eigenvalue problems, such as the calculation of eigenmodes, produces mode shapes and associated eigenvalues (in dynamic modal analysis, those correspond to eigenfrequency values).

A mode shape is, in essence, non-dimensional, which means that it can be scaled in arbitrary way. Only the relative values within a mode shape are relevant (i.e. the ratio between the values).

To avoid result inconsistencies, mode shapes are normalized before being delivered to the user. Various techniques exist.

In SCIA Engineer, normalization based on the modal mass is used, because it allows a great deal of simplifications in the formulation of equations for the resolution of dynamic problems.

Consider a raw, non-normalized mode shape vector i:

The generalized modal mass for the raw mode shape is:

Then, applying the following scaling to the raw mode shape:

Therefore, the generalized mass of the normalized mode shape is:

Note: a mode shape is dimensionless, i.e. it has no units. Consequently, the units of generalized modal masses are also mass units.

Note: although mode shapes are dimensionless, they are presented with units in SCIA Engineer for better readability.