70 M. Dorosti et al. Table 8.2 Correlations in the frequency range of interest before/after IPZ model updating using G1 and G2 Gn;1 vs. Ge;1 Gu r;1 vs. Ge;1 Gn;2 vs. Ge;2 Gu r;2 vs. Ge;2 Gn;3 vs. Ge;3 Gu r;3 vs. Ge;3 XS 0:025 0:999 0:025 0:991 0:026 0:669 XA 0:004 0:994 0:004 0:993 0:004 0:994 Table 8.3 Correlations in the frequency range of interest before and after IPZ model updating using only G1 Gn;1 vs. Ge;1 Gu;1 vs. Ge;1 Gn;3 vs. Ge;3 Gu;3 vs. Ge;3 XS 0:025 0:946 0:026 0:336 XA 0:004 0:950 0:004 0:954 Furthermore, the shape (XS) and amplitude (XA) correlation measures [6], listed in Table 8.2, show significant improvement after IPZ model updating not only for the FRFs which are used in the model updating (G1 and G2), but also for the unmeasured FRF (G3). Correlation measures for the case where only G1 is included in the model updating process is listed in Table 8.3. The results indicate that correlations for both G1 and G3 are higher when an extra FRF G2 is used in the model updating process. This means the reduced FE model is indeed better improved, when more FRFs are included the model updating process. 8.4 Conclusion In this paper, the IPZ model updating technique is extended to enable the use of multiple FRFs so that more information of eigenmodes can be used. The IPZ model updating using multiple FRFs is verified based on a simulated experiment of a pinned-sliding beam. It was shown that using more FRF measurements, from different actuator/sensor configurations, helps to better improve the reduced FE model toward the (simulated) real system. After IPZ model updating, the numerical FRFs match very well with the experimental FRFs. This not only holds for the two experimental FRFs which were used in the model updating procedure, but also for the third FRF related to an unmeasured performance variable. Although the method seems promising, there are still unresolved issues in the proposed model updating method which need further investigation: (1) a proof of convergence to a local minimum, although the convergence behavior shown in Fig. 8.3 is promising, a rigorous proof is lacking and (2) after model updating the stiffness matrix shows a small amount of asymmetry. References 1. Yang, J., Ouyang, H., Zhang, J.-F.: A new method of updating mass and stiffness matrices simultaneously with no spillover. J. Vib. Control 22(5), 1181–1189 (2016) 2. Mottershead, J.E., Link, M., Friswell, M.I.: The sensitivity method in finite element model updating: a tutorial. Mech. Syst. Signal Process. 25, 2275–2296 (2011) 3. Jaishi, B., Ren, W.-X.: Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimisation technique. Mech. Syst. Signal Process. 21, 2295–2317 (2007) 4. Dorosti, M., Fey, R., Heertjes, M., van de Wal, M., Nijmeijer, H.: Finite element model reduction and model updating of structures for control. In: 19th World Congress The International Federation of Automatic Control, Cape Town (2014) 5. Arora, V., Adhikari, S., Friswell, M.: FRF-based finite element model updating method for non-viscous and non-proportional damped system. In: International Conference on Structural Engineering Dynamics, Lagos (2015) 6. Dorosti, M., Heck, F., Fey, R., Heertjes, M., van de Wal, M., Nijmeijer, H.: Frequency response sensitivity model updating using generic parameters. In: American Control Conference (ACC) (2016) 7. Arora, V.: Constrained antiresonance frequencies based model updating method for better matching of FRFs. Inverse Prob. Sci. Eng. 22(6), 873–888 (2014) 8. D’Ambrogio, W., Fregolent, A.: Results obtained by minimising natural frequency and antiresonance errors of a beam model. Mech. Syst. Signal Process. 17(1), 29–37 (2003) 9. de Kraker, B.: A Numerical-Experimental Approach in Structural Dynamics. Shaker, Maastricht (2013)
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