2.7.           Mode Shape Orthgonalization

A common assumption in evaluating modal data with respect to a FEM, is that the mass distribution of the FEM is accurate. In this case the measured mode shapes should be orthogonal to each other with respect to the analytical mass matrix as measured by (2.6-1). However, the self-orthogonality of the measured modes is never exact, and it is sometimes desirable to transform the modes so that they are orthogonal with respect to the analytical mass matrix [19]. A set of orthogonalized modes can be calculated as follows:

                                              (2.7-2)

where

= Square root of self orthogonality matrix such that

The square root of the self-orthogonality matrix (TO) can be easily calculated using a singular value decomposition (SVD)[10].

The test mode shapes can either be orthogonalized with respect to a reduced mass matrix, or they can be expanded to the full FEM and orthogonalized with respect to the full FEM matrix. If the orthogonalization is performed using the reduced mass matrix the results will depend on the reduction method, while if it performed using the full matrix the results will depend on the expansion method.