The following modal analysis topics are available:
Valid for structural and fluid degrees of freedom (DOFs). Electrical and thermal DOFs may be present in the coupled field mode-frequency analysis using structural DOFs.
The structure has constant stiffness and mass effects.
There is no damping, unless the damped eigensolver (MODOPT,DAMP or MODOPT,QRDAMP) is selected.
The structure has no time varying forces, displacements, pressures, or temperatures applied (free vibration).
This analysis type (accessed with ANTYPE,MODAL) is used for natural frequency and mode shape determination. The equation of motion for an undamped system, expressed in matrix notation using the above assumptions is:
 | (15–49) |
Note that 
, the structure stiffness matrix, may include prestress effects
(PSTRES,ON). For a discussion of the damped eigensolver
(MODOPT,DAMP or MODOPT,QRDAMP) see Eigenvalue and Eigenvector Extraction.
For a linear system, free vibrations will be harmonic of the form:
 | (15–50) |
where:
 = eigenvector representing the mode shape of the ith natural frequency |
 = ith natural circular frequency (radians per unit
time) |
 = time |
Thus, Equation 15–49 becomes:
 | (15–51) |
This equality is satisfied if either 
or if the determinant of 
is zero. The first option is the trivial one and, therefore, is not of
interest. Thus, the second one gives the solution:
 | (15–52) |
This is an eigenvalue problem which may be solved for up to n values of 
and n eigenvectors 
which satisfy Equation 15–51 where n is the number of
DOFs. The eigenvalue and eigenvector extraction techniques are discussed in Eigenvalue and Eigenvector Extraction.
Rather than outputting the natural circular frequencies 
, the natural frequencies (
) are output; where:
 | (15–53) |
where:
 = ith natural frequency (cycles per unit time) |
If normalization of each eigenvector 
to the mass matrix is selected (MODOPT,,,,,,OFF):
 | (15–54) |
If normalization of each eigenvector 
to 1.0 is selected (MODOPT,,,,,,ON), 
is normalized such that its largest component is 1.0 (unity).
The participation factors for a given direction are defined as:
 | (15–55) |
where:
 = participation factor for the ith mode |
 = eigenvector normalized using Equation 15–54
|
 = vector describing the direction (see Equation 15–171) |
The participation factors are independent of the normalization method
(Nrmkey on the MODOPT command has no
effect).
If the unsymmetric eigensolver is used (MODOPT,UNSYM), the left eigenvectors, if available, are used to calculate the participation factors.
Details about effective mass and cumulative mass fraction can be found in Effective Mass and Cumulative Mass Fraction.
