2.3.1.      Static (Guyan) Reduction

The simplest TAM procedure uses the Guyan reduction method [1]. This is the default method in Nastran. The eigenvalue problem for the FEM can be written as follows:

                                    (2.3.1-1)

If we simply neglect the mass terms, the upper partition of (2.3.1-1) can be solved:

                                                                (2.3.1-2)

The transformation matrix from the FEM DOF to the TAM DOF is:

                                                         (2.3.1-3)

or

                                                                                                    (2.3.1-4)

In cases where multiple Guyan reductions are required it is sometimes more efficient to decompose the stiffness matrix and calculate the transformation matrix using the following identity:

                                               (2.3.1.5)

where

                                     

This identity is used in the fast iterative residual kinetic energy (IRKE) algorithm described in Section 3.4.3, and a Genetic Algorithm for selecting A-set DOF that will be discussed in sections 2.5.8 and 3.4.8.

The reduced stiffness and mass matrices can now be formed using the original FEM matrices and the transformation matrix G:

                                                           (2.3.1.6)

The Guyan reduction method makes the assumption that there are no forces on the omitted DOF. This is not an accurate assumption if any of the non-instrumented DOF have mass. Errors due to neglecting mass at omitted DOF are the cause of the typical frequency and mode shape errors introduced by Guyan reduction. While it is usually not possible to include all DOF with mass, care should be taken to include those DOF with large mass.