The equilibrium equation for a structure performing free vibration appears as the eigenvalue problem
where K is the stiffness matrix of the structure and M is the mass matrix. Damping is neglected. The solution of the eigenvalue problem yields n eigenvalues l, where n is the number of degrees of freedom. The vector x is the eigenvector corresponding to the eigenvalue.
The eigenvalue problem is solved using a matrix method called the Lanczos method. Not all eigenvalues are required -- only a small number of the lowest eigenvalues are normally calculated.
The natural frequency 
follows directly from the eigenvalue 
.
In order to run a normal modes analysis, an EIGRL bulk data entry needs to be given because it defines the number of modes to be extracted. The EIGRL card needs to be referenced by a METHOD statement in a SUBCASE in the subcase information section.
The Lanczos eigensolver used in OptiStruct provides two different ways of solving the problems. If the eigenvalue range is defined on EIGRL has no upper bound and less than 50 modes, the faster method is applied. It is not necessary to define boundary conditions using an SPC statement. If no boundary conditions are applied, a zero eigenvalue is computed for each rigid body degree of freedom of the model.
It is possible to request the computation of residual vectors in conjunction with a normal modes analysis. Residual vectors are static displacements ortho-normalized with the eigenvectors to be used in an external frequency response analysis. In order to get this output, users have to define degrees of freedom using USET, USET1. The degrees of freedom are then used to define loads in the unit load method to compute the residual vectors. RESVEC = YES needs to be defined in the normal modes subcase. Boundary conditions can be defined using SPC or inertia relief analysis and need to be applied.