PSI - Issue 50

D.A. Oshmarin et al. / Procedia Structural Integrity 50 (2023) 212–219 Oshmarin D.A., Iurlova N.A., Sevodina N.V. / Structural Integrity Procedia 00 (2019) 000 – 000

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3. An object under study Here we consider a cantilever rectangular plate with dimensions: length l 1 = 500 mm, width b 1 = 60 mm, thickness h 1 = 2 mm (fig. 1). A piezoelectric element having length l p = 50 mm, width b p = 20 mm and thickness h p = 0.3 mm is attached to its surface. The upper and lower surfaces of the piezoelectric element are completely electroded. The position of the piezoelectric element is determined by its center of mass which is located on the plate along the symmetry axis at the distance of 55 mm from the clamped edge. The plate is made of the viscoelastic material which dynamic behavior is described by the complex dynamic shear and the bulk moduli. The real parts of the complex moduli are correspondingly equal to 8 Re 6.71 10 G Pa   and 10 Re 3.33 10 B Pa   . The imaginary parts are chosen as follows: 7 Im 6.71 10 G Pa   and 9 Im 3.33 10 B Pa   . These material properties correspond to ones that have reinforced polyurethane brand Ellastolan. Within the current research we assume that the components of complex moduli do not depend on frequency of vibrations in a frequency range under study. Specific density of the viscoelastic material is 1190 el   kg/m 3 .

Fig. 1. Computational sketch for a cantilever plate with attached piezoelectric element.

The piezoelectric element made of piezoelectric ceramics CTS-19 is polarized along z axis and has the following material properties: 10 11 22 10.9 10 C C    Pa, 10 13 23 5.4 10 C C    Pa, 10 12 6.1 10 C   Pa, 10 33 9.3 10 C   Pa, 10 44 55 66 2.4 10 C C C     Pa, C/m 2 , 33 14.9   C/m 2 , 51 42 10.6     m 2 , 9 11 22 8.2 10 э э     F/m, 9 33 8.4 10 э    F/m,  p = 7500 kg/m 3 . The electrodes of piezoelectric element supplied with an electric signal with prescribed characteristics of potential difference and electric current. 4. Mechanical response of the system to applied loading Let's consider the first three bending modes of vibrations of such a plate. These modes are realized at frequencies of 1.962 Hz, 12.006 Hz and 32.956 Hz correspondingly. We apply the following options of loading conditions: kinematic (components of displacement vector varying according to the harmonic law applied to the clamped end of the plate); force (a concentrated force varying according to the harmonic law is applied to the free end of the plate); potential difference or current applied to the electrode surfaces of the piezoelectric element. In the numerical implementation, both variants of mechanical loading are set using boundary conditions of the form (5), and variants of electrical loading are set using boundary conditions of the form (6). Figure 2 shows in logarithmic coordinates the dependences of the magnitude of the mechanical response of the plate on the magnitude of kinematic loading (Fig.3.a), force loading (Fig.3b), potential difference applied to the piezoelectric element (Fig.3c) and electric current (Fig.3d) for the three vibration modes under consideration. Here the blue line corresponds to the first mode, the red line corresponds to the second one and the green line corresponds to the third mode. A ratio z stat u u is accepted as a quantity that characterizes the mechanical response of the system. Here z u is maximal value of amplitude of component z u of displacement vector at the free end of the plate under forced