The free vibration equations of equilibrium are of the form:

The generalized eigenvalue problem can be formulated by assuming a periodic solution to the above problem:

in which K is the global stiffness matrix and M is the lumped diagonal building mass matrix.

The eigenvalue extraction consists of determining the lowest frequencies (ω, which also corresponds to highest periods) and mode shapes (Φ) of the structure. The fundamental structural periods in the global X and Y directions can be used in the generation of equivalent static seismic loads.

The subspace iteration method (Bathe, 1996) is used to extract these quantities. The method is mostly effective in an eigenvalue problem where lowest frequencies (i.e., highest periods) are of interest while higher frequencies are not. Therefore, the method is particularly focused on obtaining the lowest frequencies instead of obtain all frequencies. The subspace iteration method basically consists of setting starting trial iteration vectors and using them to form a reduced subspace of the problem, which is conveniently solved by generalized Jacobi method. The eigenvalues and eigenvectors (dynamic mode shapes) are determined and iterations carried out until convergence is achieved. A check is carried out to verify that the required eigenvalues and corresponding eigenvectors are evaluated.

Referring to rigid diaphragm concept, building masses are lumped (concentrated) at rigid diaphragm mass centers. This leads to a modeling configuration where only a few nodes have mass and vast majority of the nodes are massless. RAM Frame applies a static condensation method to remove these massless degrees of freedom from the solution set since they do not have any effect in calculating dynamic properties of structures (i.e., periods of structures and eigenvectors). Hence, the reduced system after static condensation is solved by the subspace iteration method. For pseudo-flexible diaphragms, diaphragm masses are distributed among contributing frames in the diaphragm, based on user defined frame participation ratios. And finally for Semirigid diaphragms, spatial distribution of diaphragm mass is directly taken into account in such a way that computed mass over each diaphragm is distributed among diaphragm nodes (generated when meshing diaphragms) and then the system solved for eigenvalues with distributed masses.

Calculated dynamic properties of structures are reported in "Periods and Modes" report. In the current implementation, calculated mode shapes are normalized with respect to the building mass matrix. Thus, the following equation is always satisfied:

where I is identity matrix. In literature, there exist different normalization techniques. It should be noted, however, that analysis results do not change based on the chosen normalization method. The above method is chosen because of its numerical advantages in computer implementations.

As seen in the previous equation, the mode Shapes values are units-dependent due to the linearization method. A special note is added to the end of the report to provide some guidance regarding mode shapes and their relationship to the selected units.