PSI - Issue 36

O.L. Derkach et al. / Procedia Structural Integrity 36 (2022) 71–78 O.L. Derkach et al. / Structural Integrity Procedia 00 (2021) 000 – 000

74 4

derivatives; [ T ] is the matrix of coordinates’ transformation, which is used for obtaining the effective elastic coefficients in relations (2) and (3) at the given angle of fibers θ :

2 q n n q 2 0 0 0 2 0 0 0 2 0 0 1 0 0 0 ; qn qn  −    2 2

         

  T

cos , 

sin . 

q

n

(5)

=

=

=

     

0 0 0 0 0 0

0 0

q n n q

−

2 q n −

2

0 0 0

qn qn −

The process of integration was made by the volume of the finite element Ω e in the determination of the mass and stiffness matrices in Eq. (4). To verify the validity of the developed FE model of the composite beam, the comparison of calculated principal frequencies of flexural vibrations as a function of the angle θ with the experimental data obtained in Ghoneam (1995) was performed for the beam with the following parameters: l = 300 mm and cross-section of b × h = 23 × 3 mm, as well as the following material properties of E-glass polyester: E 11 = 36.75 GPa, E 22 = 6.67 GPa, G 12 = 3.40 GPa; μ 12 = 0.33, ρ = 1750 kg/m 3 . As seen from Fig. 2, the dependence of the principal frequency ( f 0 = p 0 /2π) of flexural vibrations for intact cantilever beam FE model with the regular mesh size of 2.5 mm on the fibers ’ orientation θ correlates with the experimental data well. Noteworthy is that the peculiar feature of the proposed calculation model is the possibility to conduct the investigations for more complex physical dependencies of the composite material properties on the parameters of its structure and, respectively, matrix coefficients in Eq. (2). Thus, the determination of the natural frequencies of vibrations in structural elements made of polymer matrix composite can be based on the experimentally obtained physical relations of energy dissipation in the materials, which consider the decrement dependence on the amplitude and vibration frequency. It is a significant advantage in analyzing the vibration stress level of the structural elements as compared with the existing commercial FE software. Moreover, the presented model can be used to analyze the passive and active damping of vibrations of the structural elements made of composite materials as have been done in Derkach et al. (2020) and Yershov et al. (2020).

Fig. 2. Comparison of the calculated principal frequencies of flexural vibrations of the intact beam for different angle of the fibers obtained using FE model (solid line) with the experimental data (■) obtained in Ghoneam (1995).