26 FEM Calibration with FRF Damping Equalization 271 Here we use a modal reduction scheme to create a surrogate model used by the calibration procedure. The eigenmodes of the corresponding undamped system at the nominal parameter configuration, belonging to all eigenvalues in a frequency range that significantly overlaps the frequency range of interest, are then used for reduction. To save computational effort, that reduction basis is kept within one full calibration cycle and thus not modified as the parameter settings vary during iterations. Let the eigenvalue problem formulated at the nominal parameter setting p0 be K.p0/T DM.p0/T (26.23a) Ddiag.!n/8 !low !n !high (26.23b) Then the reduced mass and stiffness matrices at any parameter settingpof the reduced model are M.p/ DT T M.p/T (26.24a) K.p/ DT T K.p/T (26.24b) and, in particular at the nominal configuration M0 .p/ DT T M.p0/T (26.25a) K0 .p/ DT T K.p0/T (26.25b) Since the transformation matrix T is invariant to parameter changes, the gradients of the reduced order matrices with respect to the ith parameter pi are M;i DdM=dpi ˇ ˇ ˇ pDp0 DT T dM=dpi ˇ ˇ ˇ pDp0 T (26.26) and K;i DdK=dpi ˇ ˇ ˇ pDp0 DT T dK=dpi ˇ ˇ ˇ pDp0 T (26.27) The gradients of the full size FEM are computed by a finite difference approximation scheme in the numeric examples below. With finite difference calculation general parameterization is allowed and the need of source code access to the FE code can be eliminated. A surrogate model that is linear in the parameters is taken as the first order expansion of the Taylor series of the reduced order model as QM.p/ DM0 C np X iD1 .pi pi;0/M;i (26.28a) QK.p/ DK0 C np X iD1 .pi pi;0/K;i (26.28b) with pi,0 being the ith parameter at the nominal setting. It can be observed that the surrogate model approximation error increases with parameter variation fromp0 for two reasons. The first is that the reduction basis T is kept constant and the second is that the [M(p), K(p)] model itself is not necessarily linear in the parameters. One notes that, once the reduced order model and its gradients are established from the full size FE mode, no further evaluation of the FE model is required. That leads to very fast computations. With the state transformation xDTz, the reduced order surrogate model fits well into the state-space setting of the system transfer function as given by Eqs. 26.3a, 26.3b and 26.5.
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