The mass related information (mass, center of mass, and mass moments of inertia) is printed out in a mass summary for all analyses that include mass.
Depending on the model, mass related information is calculated using one of two different methods. If the model is three-dimensional, a precise computation is performed, as detailed in Precise Calculation of Mass Related Information. For all other cases, a lumped calculation is performed, as described in Lumped Calculation of Mass Related Information along with its limitations.
The mass summary by element type is always based on the basic calculation.
The mass summary is calculated accumulating each element contribution. It does not reflect the boundary conditions, coupled degrees of freedom (CP), constraint equations (CE, CERIG, RBE3), and multipoint constraint approach in contact elements with pilot node.
The mass summary is based on unscaled mass properties (see MASCALE command).
The total rigid body mass properties with respect to the origin is given by:
 | (14–288) |
where:
| [Mrig] is the total rigid body mass properties of the model. |
| N is the number of elements. |
| [Di] is a matrix containing the six rigid body motion vectors of the ith element. See Equation 15–159 for more information about these vectors. |
| [Mi] is the mass matrix of the ith element. |
The total rigid body mass matrix can be partitioned as follows:
 | (14–289) |
where:
| [Mt] is the translational mass matrix. |
| [Mtr] and [Mrt] are the coupled translational/rotational mass matrices. |
| [Mr] is the rotational mass matrix and contains the mass moments of inertia. |
The translational mass principal characteristics are obtained from the eigensolution of matrix [Mt]. The eigenvalues are the principal masses and the eigenvectors are the mass principal directions:
 | (14–290) |
where:
| MX, MY, and MZ are the principal masses. |
| [Φ] is the matrix of the translational mass eigenvectors representing the mass principal directions. |
If the principal masses are not equal, the center of mass location with respect to the principal axes is not unique. These locations are computed as:
 | (14–291) |
 | (14–292) |
 | (14–293) |
where:
| cdmX, cdmY, and cdmZ are the center of mass locations with respect to the mass principal axes. |
|
If the principal masses are equal, the center of mass location is unique:
 | (14–294) |
where:
| M = MX = MY = MZ is the mass of the model. |
The mass moments of inertia with respect to the mass principal axes are calculated as:
 | (14–295) |
where:
|
The inertia principal characteristics are obtained from the
eigensolution of matrix 
. The eigenvalues are the principal moments of inertia
and the eigenvectors are the moment of inertia principal directions.
This precise mass calculation is not used for:
These cases use the lumped approximation outlined in Lumped Calculation of Mass Related Information.
The computation of the mass moments and products of inertia, as well as the model center of mass, is described in this section. This approach assumes that the mass is lumped at the center of each element. The model center of mass is computed as:
 | (14–296) |
 | (14–297) |
 | (14–298) |
where typical terms are:
| Xc = X coordinate of model center of mass (output as XC) |
 |
 |
| ρ = element density, based on average element temperature |
| Vi = volume of element i |
 |
| {No} = vector of element shape functions, evaluated at the origin of the element coordinate system |
| {Xi} = global X coordinates of the nodes of element i |
|
The moments and products of inertia with respect to the origin are:
 | (14–299) |
 | (14–300) |
 | (14–301) |
 | (14–302) |
 | (14–303) |
 | (14–304) |
where typical terms are:
| Ixx = mass moment of inertia about the X axis through the model center of mass (output as IXX) |
| Ixy = mass product of inertia with respect to the X and Y axes through the model center of mass (output as IXY) |
Equation 14–299 and Equation 14–301 are adjusted for axisymmetric elements.
The moments and products of inertia with respect to the model center of mass (the components of the inertia tensor) are:
 | (14–305) |
 | (14–306) |
 | (14–307) |
 | (14–308) |
 | (14–309) |
 | (14–310) |
where typical terms are:
 = mass moment of inertia about the X axis through the model center
of mass (output as IXX) |
 = mass product of inertia with respect to the X and Y axes through
the model center of mass (output as IXY) |
The above mass calculations are not intended to be precise for all situations, but rather have been programmed for speed. It may be seen from the above development that only the mass (mi) and the center of mass (Xi, Yi, and Zi) of each element are included. Effects that are not considered are:
The mass being different in different directions.
The presence of rotational inertia terms.
The mixture of axisymmetric elements with non-axisymmetric elements (can cause negative moments of inertia).
Tapered thicknesses.
Offsets used with beams and shells.
Trapezoidal-shaped elements.
The generalized plane strain option of PLANE182 - 2-D 4-Node Structural Solid and PLANE183 - 2-D 8-Node Structural Solid. (When these are present, the center of mass and moment calculations are completely bypassed.)
Thus, if these effects are important, a separate analysis can be performed using inertia relief to find more precise center of mass and moments of inertia (using IRLF,-1). Inertia relief logic uses the element mass matrices directly; however, its center of mass calculations also do not include the effects of offsets.
It should be emphasized that the computations for displacements, stresses, reactions, etc. are correct with none of the above approximations.
The center of mass and mass moment of inertia calculations for keypoints, lines, areas, and volumes (accessed by KSUM, LSUM, ASUM, VSUM, and *GET commands) use equations similar to Equation 14–296 through Equation 14–310 with the following changes:
Only selected solid model entities are included.
Lines, areas, and volumes are approximated by numerically integrating to account for rotary inertias.
Keypoints are assumed to be unit masses without rotary inertia.
Lines are assumed to have unit mass per unit length.
Each area uses the thickness as:
(14–311)
where:
t = thickness Each area or volume is assumed to have density as:
(14–312)
where:
ρ = density
Composite material elements presume the element material number (defined with the MAT command).