132 Nicholas Pomianek et al. Table 1 Model properties and bounds set during optimization. Property Lower bound Upper bound Units η 0.005 0.005 ηj 0 0.03 E 200 200 GPa Ej 0 20000 GPa ρ 7800 7800 kg/m3 ρj 7000 11000 kg/m3 Fig. 3 The effective joint region is defined as any part of the structure that includes a frictional interface, dimension in mm. finite element FRF is used as the cost function for a particle swarm optimization process. The EJR properties ηj, Ej, andρj are the decision variables for this optimization process. Upon each iteration of the optimization, the EJR element properties are updated, the FEA model is evaluated, and the degree of similarity to the experimental dataset is calculated. The degree of similarity was calculated using a handful of different FRF similarity metrics from the literature. The most straightforward metric tested is the root mean squared percent error (RMSPE) between the two FRFs being compared [9]. This is a simplistic approach that does not leverage the modal information contained in FRFs but is valuable as a point to compare against more sophisticated techniques. The R2 value of a linear fit (ILR2) between the imaginary parts of each FRF serves as another metric focused on phase [9]. The FRF scaling factor (FRFSF) criterion is another alternative focused on the magnitude differences between FRFs. FRFSF= P f2 f=f1 | X(f)| Pf2 f=f1 | Y(f)| (5) In this equation, Xis the reference FRF, Y is the comparison FRF, and f1 and f2 are the lower and upper bounds of the frequency range of interest. The frequency response assurance criterion (FRAC) is a well established inner-product approach that was used as a baseline to compare to more recent approaches [10]. FRAC= P f2 f=f1 X(f)Y∗(f) 2 Pf2 f=f1 X(f)X∗(f) Pf2 f=f1 Y(f)Y∗(f) (6) Both FRAC and the modified frequency response assurance criterion (MFRAC), which scales FRAC by the ratio of the minimum to maximum powers of each FRF, are investigated. As an alternative to FRAC, the mean cross signature scale factor (CSF) was also applied due to its improved sensitivity to damping. [9]. CSF= f2 Xf=f1 2|X∗(f)Y(f)| (X∗(f)X(f))+(Y∗(f)Y(f)) 1 Nf (7)
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