2.9.1.      Residual Forces

Given a FEM and a set of measured modal data, there is one residual force vector associated with each measured mode shape. It is defined as follows:

                              (2.9.1-1)

For a number of test modes this can be expressed as follows:

                          (2.9.1-2)

Each row of the residual force matrix corresponds to a single DOF from the FEM, and each column corresponds to a single test mode. Rows with large residual forces indicate that large errors exist in that location for that test mode, while rows with small residual forces indicate that small errors exist in that location for that test mode. If the residual force vectors indicate a particular region of the model across a number of modes, this is a strong indication that this region will need to be modified in order to match the measured modal data.

Since it is not practical to measure all DOF in the FEM, there are two ways to calculate residual forces. The first is to use the full FEM mass and stiffness matrices and a set of expanded test mode shapes. The expansion can be performed using any of the methods described in the previous chapter, but dynamic expansion is the preferred approach since the residual forces are only non-zero on measured DOF. The second is to use the A-set reduced mass and stiffness matrices and only the measured partition of the test mode shapes. Whether residual forces are calculated using dynamically expanded mode shapes or reduced mass and stiffness matrices, they are non-zero only on the measured DOF. This implies that the ability of the residual force method to localize an error is limited by the amount of instrumentation in that region of the structure.