## Eigen Analysis

When performing a static analysis, it is possible to evaluate the eigenvalues of the system and associated eigenvectors. The step control parameter data section defines the four parameters that control this eigen analysis.

1. The requested number of eigenvalues, neig, (Default value: neig = 0) controls the eigenvalue extraction process. If neig = 0, the eigenvalue analysis is not performed; if neig > 0, neig natural vibration frequencies for small amplitude motion about the present equilibrium position will evaluated. All requested eigenvalues are calculated to machine accuracy, and the corresponding eigenvectors are computed as well.
2. The flag eigpflag (Default value: eigpflag = 0) controls printout during the eigenvalue extraction process: if eigpflag = 1 large output files can result.
3. When computing the eigenvalues, the gyroscopic terms are neglected, (default: @GYROSCOPIC_TERMS = NO). If the keyword @GYROSCOPIC_TERMS = YES is used, the gyroscopic terms will be taken into account. Note that in that case, the eigenvalues and associated eigenvectors could be complex numbers.
4. It is often the case that the stiffness matrix describing the static behavior of multibody systems is singular. Indeed, the stiffness matrix associated with a system presenting rigid body modes is singular. It is possible to remove these singularities by adding fictitious spring connections during the solution process. If those fictitious springs are present in the stiffness matrix during the eigenanalysis procedure, the eigenvalues will be affected by the stiffness constants of these fictitious springs.
5. To overcome this problem, the singularity of the stiffness is removed using a different procedure when performing the eigenanalysis. The eigenproblem is expressed as

[K - ω2M] u = 0,

where K and M are the system stiffness and mass matrices, respectively, ω the natural frequency, and u the corresponding eigenvector. If the stiffness matrix is singular, the following simple manipulation is performed

[(K + ρM) - (ω2 + ρ) M] u = 0,

where ρ is the eigenspectrum shift factor. The two eigenproblem have identical eigenvectors, and the spectrum of the second problem is simply shifted by eigenspectrum shift factor. Note that matrix K + ρM is not singular, and hence, the eigenspectrum of the second problem is evaluated easily.