PSI - Issue 17

978 H. Lopes et al. / Procedia Structural Integrity 17 (2019) 971–978 Lopes et al./ Structural Integrity Procedia 00 (2019) 000 – 000 the rotations in the direction is ʹʹͲͻš͸ͳͻ and for the rotations in the direction is ʹͳ͵ͺš͸ͻͳ . The transitions between white and black in the phase maps are called fringes and represent curves with the same modal rotation level. The distribution and number of fringes are defined according to the complexity of the modal rotation and its amplitude, respectively. The filtered phase maps are obtained by applying the sine/cosine average filter to the unfiltered phase maps. By unwrapping the filtered phase maps and applying Eqs (9) or (10) one obtains the modal rotations. One can clearly observe that the modal rotations in Figure 3 are very similar to those in Figure 5(c) and 6(c). Furthermore, with the purpose of validating the identification of the material constants with the present approach in a quantitively way, a numerical comparative analysis between the numerical and experimental modal rotations was carried-out by computing the modal assurance criterion (MAC). For perform this comparison, the numerical modal rotations were interpolated to obtain the same spatial resolution as the one of the experimental modal rotations. Table 1. MAC values between experimental and numerical modal rotations. Mode ( ) ͳ ʹ ͵ Ͷ ͷ ͸ ͹ ͺ ͻ ͳͲ ͳͳ ͳʹ ͳ͵ ͳͶ MAC ( (n ) , ( ) ) ͲǤͻ͹ʹ ͳǤͲͲͲ ͲǤͺͻͷ ͳǤͲͲͲ ͲǤͻ͹͹ ͲǤͻͻͺ ͲǤͻͻ͹ ͲǤͻ͹Ͳ ͲǤͻͻͶ ͲǤͻͶ͵ ͲǤͻ͵ͺ ͲǤͻͻ͸ ͲǤͻ͹͹ ͲǤͻͻͳ MAC ( (n ) , ( ) ) ͳǤͲͲͲ ͲǤͻͻ͹ ͲǤͺͶͺ ͲǤͻͻ͸ ͲǤͻͻ͵ ͲǤͻͻͶ ͲǤͻͻʹ ͲǤͻʹͻ ͲǤͺ͹ͳ ͲǤͻͶʹ ͲǤͻ͹͸ ͲǤͺ͸ͻ ͲǤͻ͹ͷ ͲǤͻͺʹ As one sees in Table 2, the MAC values are very closed to one, indicating that the modal rotations from the finite element model are very similar to those experimentally measured. These results prove the validity of the method used in this word to identify materials constants of laminated composite plates. 4. Conclusions The results in this paper show that the use of vibration characteristics, along with the solution of an optimization problem, allows the identification of materials constants of laminated composite plates. It was found that the three optimization solvers give similar results, although the particle swarm algorithm gives the lowest value of the objective function. The discrepancy between numerical and experimental natural frequencies does not exceed ͲǤ͵͵ %, allowing to say that the finite element model results can accurately represents the dynamic behavior of the laminated. Furthermore, the comparative analysis of the numerical and experimental modal rotations, measured with shearography, leads to a MAC average value of ͲǤͻ͸ͷ , which is very close to unity. Acknowledgements This publication is supported by LAETA, Project UID/EMS/50022/2019 through IDMEC and INEGI/UP, Portugal. References Aebischer, H.A., Waldner, S., 1999. Simple and effective method for filtering speckle-interferometric phase fringe patterns. Optics Communications 162, 205–210. Araújo, A. L., Soares, C. M., de Freitas, M. J., 1996. Characterization of material parameters of composite plate specimens using optimization and experimental vibration data. Composites Part B: Engineering, 27B, 185–191. dos Santos, J.V.A., Lopes, H., 2018. Damage Localization Based on Modal Response Measured with Shearography, in Vibration Based Structural Health Monitoring Methods, In: Nobari, A. S., Aliabadi, F. M. H. (Eds), World Scientific Publishing, London, UK, pp. 141–172. Ghiglia, D., Pritt, M., 1998, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, Wiley, New York, USA. Katunin, A., Gnatowski, A., 2012. Influence of heating rate on evolution of dynamic properties of polymeric laminates. Plastics Rubber and Composites 41(6), 233–239. Kreis, T., 2005, Handbook of Holographic Interferometry: Optical and Digital Methods, WileyVCH, Weinheim, Germany. Lago, S, Brignolo, S, Cuccaro, R, Musacchio, C, Albo, P. A. G., Tarizzo, P., 2014. Application of acoustic methods for a non-destructive evaluation of the elastic properties of several typologies of materials, Applied Acoustics, 75, 10–16. Tam, J.H., Chao, O.Z., Ismail, Z., Ang, B.C., Khoo, S.Y., 2016. Identification of material properties of composite materials using non-destructive vibrational evaluation approaches: A review, Mechanics of Advanced Materials and Structures 24(12), 971–986. Wooh, S.C., Daniel, I.M., 1991. Nondestructive Determination of Elastic Constants of Composite Materials. In: Thompson, D.O., Chimenti, D.E. (Eds) Review of Progress in Quantitative Nondestructive Evaluation. Springer, Boston, MA, USA. 8