Parametric Vibrations of a Hinged Thermoviscoelastic Rectangular Piezoelectric Plate with Shear Strains and Dissipative
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International Applied Mechanics, Vol. 56, No. 3, May, 2020
PARAMETRIC VIBRATIONS OF A HINGED THERMOVISCOELASTIC RECTANGULAR PIEZOELECTRIC PLATE WITH SHEAR STRAINS AND DISSIPATIVE HEATING TAKEN INTO ACCOUNT* V. G. Karnaukhov1*, V. I. Kozlov1, and T. V. Karnaukhova2
We presented the model of parametric vibrations of the hinged rectangular thermoviscoelastic piezoelectric plate taking into account the shear strains and dissipative heating. A solution to this problem is reduced to the classic Mathieu equation. We studied the effect of the temperature of dissipative heating on the parametric vibrations of the plate. Keywords: viscoelastic piezoelectric plate, parametric vibrations, dissipative heating, shear strains, Mathieu equation Introduction. A large number of publications are devoted to the forced vibrations of thin-walled elements made of passive (without piezoelectric effect) and active (with piezoelectric effect) materials [1–4]. A number of studies on the forced vibrations of thin-walled elements, taking into account the effect of dissipative heating were published in [5–15]. However, there are very few published studies of the parametric vibrations of structural elements made of piezoelectric materials. There are no studies of parametric vibrations taking into account the coupling of the electromechanical and temperature fields, such as the dissipative heating caused by hysteresis losses in an inelastic material. Meanwhile, at a certain value of the amplitude of the harmonic load, the temperature of dissipative heating can reach the point of material degradation when the active material loses the piezoelectric effect and becomes passive [8–15]. In this case, it becomes impossible to cause parametric vibrations in an element made of such a material by applying a harmonic potential difference. We will call such a load critical. For a passive material, the critical load is the melting point or temperature at which the performance of the structure deteriorates. To determine the critical load on a piezoelectric element, it is necessary to solve the related problem of thermoelectromechanics for various amplitudes of harmonic load and find the temperature amplitude when the dissipative heating becomes equal to the Curie point. In this case, a specific type of thermal destruction of an inelastic rectangular plate takes place, when it is not divided into parts, but, as indicated above, ceases to fulfill its functional purpose due to the transformation of the active material of the plate into a passive one. The purpose of this article is to obtain a simple formula for the critical load. Here we will use the basic relations and notation of the article [16]. 1. Problem Formulation and Solution. We consider a three-layer rectangular plate with an inner passive layer of thickness h0 and two external identical piezoactive layers of thickness h1 . The total thickness of the plate is H = h0 + 2h1 . In
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P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterova St., Kyiv, Ukraine 03057; *
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