Whether residual forces are calculated using expanded test mode shapes or reduced mass and stiffness matrices, it is possible to use Minimum Rank Perturbation Theory (MRPT) to calculate a minimum rank change to the stiffness matrix that will exactly match the measured modes [17, 18]. The MRPT stiffness matrix is given as follows:
(2.9.2-1)
where:
(2.9.2-2)
The inner term in Equation (2.9.2-2) is only symmetric in the case where the modes
have been orthogonalized. In the case where the modes have not been orthogonalized,
the symmetric part is taken so that the perturbed stiffness matrix is
symmetric.
If the MRPT stiffness perturbation is calculated using reduced matrices it is clearly non-zero only on the measured DOF. If it is calculated using full matrices and dynamically expanded mode shapes it is also only non-zero on the measured DOF since the equations for dynamic expansion enforce zero residual forces on the unmeasured DOF. In both these cases, the computations required to calculate the stiffness perturbation are trivial. It is not practical to calculate the MRPT stiffness perturbation using other expansion methods since it will, in general, be a fully populated matrix on all DOF of the FEM.
The MRPT stiffness matrix will exactly match all the measured test modes, but it cannot be related directly to physical parameters in the model. For that reason, it is most useful as an error localization technique since it identifies measured DOF that require the greatest modification to the stiffness matrix to match the measured data. It can be viewed as a method of viewing the residual force data.