2.8.           Mode Shape Expansion

Another means of comparing test and analysis mode shapes is to expand the measured mode shapes up to the full FEM. The expansion can be performed using a number of different methods. The simplest method uses the same shapes used to reduce the mass matrix. These can be Guyan, dynamic, IRS, modal or hybrid TAM reduction shapes. However, two other methods are also available with TAMKIT. The first is a dynamic expansion method. The dynamic expansion is similar to the dynamic TAM, but the expansion shapes are recalculated for each test mode using the measured frequency of that mode. The equation for dynamic expansion is:

                   (2.8-1)

Another method is the System Equivalent Reduction Expansion Process (SEREP) [14]. This method is very similar to the modal TAM, except that a consistent transformation is applied to the instrumented DOF. This means that the mode shape coefficients on the expanded mode shapes () are different than the actual measured mode shape coefficients (). This is sometimes called a “smoothed” SEREP, while the modal TAM is an “unsmoothed” SEREP. The equations for SEREP are as follows:

          (2.8-2)

where:

           - Measured test mode shapes on A-set DOF

           - FEM mode shapes on A-set DOF

          - FEM mode shapes on O-set DOF

           - Expanded test mode shapes on O-set DOF

           - Expanded test mode shapes on A-set DOF

The primary purpose of expanding the measured mode shapes to the full FEM is to visualize the modes on the same mesh as the analysis modes. However, there are a number of other advantages. As well as calculating the mode shapes, other measures such as grid point forces, elemental forces and stresses, elemental strain and kinetic energy and modal effective mass can all be calculated from the expanded modes. The modal effective mass (MEM) is a particularly useful measure of the amount of mass associated with motion of a mode and can be applied to both the test and the analysis modes. The equations for MEM are given in Section 2.2.2, and can be applied to the test mode shapes using the analytical mass matrix.

The expanded test modes can also be compared with the FEM modes using orthogonality measures based on the full rather than the reduced mass matrix. This can result in more accurate estimates of cross-orthogonality since more accurate expansion methods such as dynamic expansion can be applied. This is particularly powerful when the comparison between FEM and TAM modes is unsatisfactory. The cross-orthogonality and MAC calculations for the expanded modes use the same equations presented in Section 2.4, but are applied to the unreduced (G-set) mode shapes and mass matrix.