7 Techniques for Synthesizing FRFs from Analytical Models 77 viscous damper can be modeled using the spring element connector and specifying the damping values, which is referred as element damping in Eq. (7.6). 7.4 Frequency Response Assurance Criteria (FRAC) Any two frequency response functions representing the same input-output relationship can be compared using a metric known as the Frequency Response Assurance Criterion (FRAC). Once the FRFs are synthesized from FE model, they can be compared with the measured FRFs and the FRAC can be computed to understand the strength of correlation. For example, the FRAC for two FRFs, measured FRFHpq.!/ and analytical FRF OHpq.!/, representing the relation between output DOF p and input DOF q, can be computed using the expression shown in Eq. (7.7). The frequency resolution of the analytical FRF data and measured FRF data should match. FRACpq D ˇ ˇ ˇ P!2 !D!1 Hpq.!/ OH pq.!/ˇ ˇ ˇ 2 P!2 !D!1 Hpq.!/H pq.!/ P!2 !D!1 OHpq.!/ OH pq.!/ (7.7) 7.5 Experimental Example A FE model of a rectangular steel plate structure of dimensions 0:86 0:57 0:0063m(3400 22:500 :2500) was developed using ANSYS Workbench R14.5. A cold rolled steel rectangular plate structure was fabricated (ED2:05 1011 Pa.2:9734 107 psi/, D0:29and D7850kg=m3 .0:2836lb=in3/), with 160points marked on a 0:05 0:05m(200 200) grid. Each of these 160points were impacted and FRFs were measured at 21reference locations using uniaxial accelerometers. The system mass, damping and stiffness matrices were obtained from the FE model. A named selection of points corresponding to the impact locations (which also included the sensor locations) was created. The displacement for the named selection represents the modal coefficients. The analytical FRFs were generated using the full space method from the analytical model. 7.5.1 FRF Comparison The FE model was calibrated to the match measured modal frequencies and modal vectors. The FE model was validated by comparing analytical FRFs from the model with the measured FRFs. In addition, the results obtained for two perturbed mass configurations with unconstrained boundaries were compared with the predictions from the updated model to check its robustness. Figure 7.1 shows a comparison of the driving-point and cross-point FRFs for the rectangular plate. 7.5.2 Effect of Structural Damping The structural damping does not affect the modal frequency. Its effect is only to limit the response at resonance. The effect of structural damping on the plate was studied. The FRF for various values of loss coefficient values are plotted in Fig. 7.2. It is evident that the modal frequencies do not change with increasing values, but the amplitude of the response at resonance is reduced. 7.6 Conclusions • Modal correlation, updating, validation and calibration are important tasks in order to reduce uncertainty in predictions from a FE model. Various metrics to perform modal correlation are available. The Frequency Response Assurance Criterion (FRAC) aims to quantify the strength of correlation between measured and synthesize FRFs. It is particularly advantageous to correlate FRFs as it works directly on raw input-output relations.
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