108 H. R. Kramer et al. The one notable exception is the third resonant frequency. Here, the response from the weighted beam is much greater than from the unweighted beam. When there is no mass, the higher frequency modes dampen out quickly, leading to this result. Next, the phase of the above FRFs was studied. The phase of the FRFs was unwrapped for clarity and then log shifted using the LFS method. The resultant curves shared similar features but were separated by a shift of . 2π which can be seen in Fig. 14.7. This shift exemplifies the usefulness of the LFS and PSM methods. While LFS succeeds in aligning resonant frequencies, it still preserves the unique features of each curve. PSM was then applied to the shifted and unshifted curves. Studying the resulting curves reveals that before LFS there are more jumps between 0 and 1. This makes the signals more difficult to compare and masks the similar features. After applying LFS, the PSM curve still has several jumps between 0 and 1, but they make more sense intuitively. Aligning these features allows for a more direct comparison of the shape of the curves by mitigating the differences caused by the frequency shift. This information is still preserved, however, in the value of the lag factor. The lag factor describes how much the frequency of the comparison FRF must be shifted to align with the reference FRF. This value is derived by maximizing the cross-correlation between the two FRFs, with further details described in [4]. 14.4 Conclusions This paper demonstrated the advantages of using the LFS method in conjunction with existing FRF similarity metrics. An Euler-Bernoulli fixed-free beam was simulated with different length parameters to generate similar FRF signals. An experimental beam was studied on a vertical shaker with varying tip masses to validate the LFS method on physical system responses. Similarity metrics were applied to the resulting FRFs of these systems with and without LFS to gain insight into its advantages. When LFS was applied with the similarity metrics, the results showed a greater correlation between the FRFs. Aligning the peaks of the comparison FRF to the reference FRF leads to more intuitive results while quantifying the shift in the frequency domain. Selecting a suitable window size reduced large peaks in simulation while still preserving important feature information. We find these methods presented in [4] and the proposed windowing method to be beneficial for comparing similar FRFs. Acknowledgments This research is supported by the Delivery Environments program under the Office of Engineering and Technology Maturation at Los Alamos National Laboratory (LA-UR-22-31189). Appendi x Table 14.1 Simulation parameters Parameter Symbol Value Unit Young’s modulus E 10 11 N/m 2 Density ρ 10 3 kg/m 3 Area moment of inertia I 10−12 m 4 Cross-sectional area A 10− 5 m 2 Beam length L 1.0 m Measurement location xa 1.0 m Excitation location xh 1.0 m Damping ζ 10−3
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