270 T.J. Abrahamsson and D.C. Kammer and —n D Re.œn/=Im.œn/8Im.œn/<0 (26.14b) In the process of damping equalization, the real parts of the poles are perturbed such that the damping is made equal for all modes. The modal dampings are then set to a single fixed value —0, i.e. —n D—0 (26.15) The effect of such damping equalization is that the oscillatory imaginary part of the poles are preserved and the real damping part is modified such that the perturbed system poles are now œn WD —0Im.œn/=iIm.œn/8Im.œn/>0 (26.16a) and œn WD—0Im.œn/=iIm.œn/8Im.œn/<0 (26.16b) and the modified state-space realization is given by Eqs. 26.12a and 26.13. This in turn gives us a modified transfer function for the experimental model, such that the transfer function used for calibration with damping equalization is H X DCX i!I A 1 X 1BCD (26.20) At this stage it should be obvious that the application of the system identification procedure on the raw test data HX raw has led us to a mathematical model which we can evaluate for any frequency. In particular it means that we can use the equal log-frequency increments as detailed in Sect. 26.2.1 for transfer function evaluation. In addition to that, we are also able to make fictitious modifications of the system under test. A particularly useful such modification is that we can adjust the system damping level, leaving stiffness and inertia properties intact, such that all system modal damping are set equal. The model calibration of the FE model can then be made towards this fictitious experimental model for calibration of parameters that relate to mass and stiffness only. For the FE based system representation, the modal damping allows for a simple representation. For a system with given mass and stiffness matrices MandKwe have the viscous damping matrixVtobe [21] V DMXdiag.mn/ 1diag.2— 0mn!n/diag.mn/ 1X T M (26.21) with eigenfrequencies !n, modal masses mn, and the modal matrixXgiven by the undamped system’s eigenvalue problem KXDMXdiag.!n/ (26.22a) diag.mn/ DX T MX (26.22b) In a calibration procedure we are then able to search for the mass and stiffness related parameters p of the FE model fK(p), M(p)g that render the transfer function HA given by Eqs. 26.3a, 26.3b and 26.5 and that let the criterion function of Eqs. 26.6 and 26.7 to be minimal. One should note that the discrete frequencies used to evaluate Eq. 26.7 does not have to match the discrete frequencies used in testing. 26.2.3 Surrogate Modeling To be practical, FRF based model calibration requires rapid calculations of the frequency responses of the model as the parameter settings change in the iterative search for minimum deviation to test data. For most industry size FEMs in use, sufficiently rapid such calculations are infeasible without model reduction. On the other hand, model reduction may also take significant time. Therefore, a balance needs to be struck between the time spent for model reduction and the accuracy loss imposed by the reduction process.
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