PSI - Issue 17

H. Lopes et al. / Procedia Structural Integrity 17 (2019) 971–978 Lopes et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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the natural frequencies of the plate from the well-defined peaks. A total of fourteen natural frequencies are present in the band [ Ͳ – ͳǤ͸ kHz].

(a)

(b)

Fig. 1. Extraction of natural frequencies: (a) Experimental setup, (b) Magnitude of four mobility functions, using the estimator H 1 .

2.3. Identification of properties

The natural frequencies, modal displacements and rotations were computed with the Finite Element Method (FEM), using a Kirchhoff non-conforming element of four nodes and three degrees of freedom per node, being the domain discretized in ͹ͷšʹ͵ elements. The orthotropic material properties were found by solving an optimization problem, using (1) particle swarm, (2) genetic and (3) pattern search algorithms. These are some of the most common algorithms of the family of derivative-free methods, which have the advantage of dealing with problems were the derivatives cannot be obtained or are difficult to obtain, such as the case where the functions are discontinuous or noisy. The major disadvantages of this kind of algorithms is that they are in general much slower than the gradient based algorithms for large problems and, depending on the initial values, they not always find the same solution. The objective function for the optimization problem was formulated as the sum of absolute differences between the numerical circular natural frequencies (n) , obtained with FEM, and the circular natural frequencies measured experimentally (e) : min , , , = ∑| (e) − (n) | =1 (1) which is subject to 10 GPa ≤ ≤ 50 GPa (2) 10 GPa ≤ ≤ 50 GPa (3) 1 GPa ≤ ≤ 20 GPa (4) 0.05 ≤ ≤ 0.4 (5) where nf are the total number of modes considered for the analysis and , , and are the material constants to identify. To obtain a reliable and meaningful solution, ʹͲ tests were carried out for each algorithm. The lower and upper bounds in the material constants were defined taking into consideration values from a three-point bending test presented in a previous study (Katunin and Gnatowski (2012)). It should also mention that for each run the starting value of the design variables of the pattern search algorithm was obtained by generating random numbers in the feasible domain. The stopping criteria was set to ͳͲ Ǧ͸ for the change in position and mesh size in the pattern search algorithm and to ͳͲ Ǧ͸ in the relative change of the objective function in the particle swarm and genetic algorithms. Figure 2 show the convergence behavior if each one of the three algorithms used to solve the problem. The lowest value of the objective function, defined in Eq. (1), is ͵ͷǤʹͳ , obtained with particle swarm algorithm after ͳ͸ͻ iterations and using a swarm size of ͳͲͲ . This solution corresponds to the orthotropic material properties = 27.17 GPa, = 31.28 GPa, = 6.4 GPa and = 0.1659 . Table 1 lists the natural frequencies experimentally measured, the corresponding natural frequencies obtained with the finite element model defined with the identified material