Analysis of the general equations of the transverse vibration of a piecewise uniform viscoelastic plate


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Abstract

This article discusses the analysis of the general equations of the transverse vibration of a piecewise homogeneous viscoelastic plate obtained in the “Oscillation of inlayer plates of constant thickness” [1]. In the present work on the basis of a mathematical method, the approached theory of fluctuation of the two-layer plates, based on plate consideration as three dimensional body, on exact statement of a three dimensional mathematical regional problem of fluctuation is stood at the external efforts causing cross-section fluctuations. The general equations of fluctuations of piecewise homogeneous viscoelastic plates of the constant thickness, described in work [1], are difficult on structure and contain derivatives of any order on coordinates x, y and time t and consequently are not suitable for the decision of applied problems and carrying out of engineering calculations. For the decision of applied problems instead of the general equations it is expedient to use confidants who include this or that final order on derivatives. The classical equations of cross-section fluctuation of a plate contain derivatives not above 4th order, and for piecewise homogeneous or two-layer plates the elementary approached equation of fluctuation is the equation of the sixth order. On the basis of the analytical decision of a problem the general and approached decisions of a problem are under construction, are deduced the equation of fluctuation of piecewise homogeneous two-layer plates taking into account rigid contact on border between layers, and also taking into account mechanical and rheological properties of a material of a plate. The received theoretical results for the decision of dynamic problems of cross-section fluctuation of piecewise homogeneous two-layer plates of a constant thickness taking into account viscous properties of their material allow to count more precisely the is intense-deformed status of plates at non-stationary external loadings.

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The general equations of oscillation of piecewise homo-geneous viscoelastic plates of constant thickness, described in [1], are complex in structure and contain derivatives of any order with respect to x, y coordinates and time t, and, therefore, are not suitable for solving applied problems and performing engineering calculations. To solve applied problems, instead of general equations, it is advisable to use approximate ones that include one or another finite order in derivatives. The classical equations of transverse vibration of a plate contain derivatives of no higher than 4th order, and for piecewise homogeneous or two-layer plates, the simplest approximate equation of vibration is a sixth order equation. If in the operators (1.3.8) given in [1] we restrict ourselves to the first two terms, then from equation (1.3.11) L1(W2) = F1(x, y, t), where are the operators L1 and F1(x, y, t) equal to: we obtain the approximate integral-differential equation (1) where are the operators Qj and F1(x, y, t) equal to: If the plate is homogeneous, and W - is the transverse displacement of the points of the “middle” surface - the plane of the plate, then in this case the dependences are satisfied N0 = N1; M0 = M1; P2 = 1; h0 = h1; C0 = C1; D0 = D1. and equation (1) goes into equation (4) where on the left is the product of two operators: the first describes the process of longitudinal oscillation, and the second describes the transverse vibration. The approximate equation from the general equation (1.3.12) given in [1] is introduced similarly and we obtain for (5) where are the operators Qj and F1(x, y, t) equal to: Despite the fact that equation (1) is approximate, it is quite complicated. The operators (2) contain all parameters and operators characterizing both the mechanical and rheological properties of the piecewise homogeneous plate material and its geometric dimensions. Approximate equation (1) is simplified in particular cases when solving specific oscillation problems. For example, operators (2) are greatly simplified when the Poisson ratios of both components are constant, or when the thicknesses of both components are equal, and so on. For example, if h0 = h1 and ν0 = ν1, then the operators Qj in (6) have the form: The sixth order operator in equation (1) can also be represented as the product of second and fourth order operators if the plate is elastic and the coefficients Qj connected by addiction Q2Q4Q7 = Q1Q5Q7 + Q3Q4Q6. For a two-layer elastic plate with given parameters of its components, relation (7) gives a 10th order algebraic equation with respect to the relation h2/h1, the sixth-order operator in (1) can be represented as the product of two lower-order operators if the coefficients Qj and Aj linked by dependencies Q1 = A1A2; Q2 = A1A4 + A2A3; Q3 = A2A4; Q4 = A1A5; Q5 = A2A5; Q6 = A1A6; Q7 = A2A6. Findings 1. The study of vibrations of piecewise-homogeneous plates in an accurate three-dimensional formulation allows us to derive the general and approximate equations of vibration of such plates based on them without using any hypotheses. 2. It is shown that the simplest approximate equation of vibration of a two-layer plate is a sixth-order equation with respect to derivatives describing its longitudinal-transverse vibration. 3. For an elastic two-layer plate, the sixth-order operator splits into the product of the second-longitudinal and fourth-order transverse-wave operators if the thicknesses of the plate components satisfy the derived equation containing the parameters of these components. 4. Formulas are obtained for determining displacements and stresses through the sought-for functions at any point of a two-layer plate.
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About the authors

Mmatmatisa L. Jalilov

Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-Khorazmiy

Email: mamatiso2015@yandex.ru
Cand. Sci. (Eng.); Head at the Department “Computer Systems” Fergana, Republic of Uzbekistan

Rustam Kh. Rakhimov

Institute of Materials Science, SPA “Physics-Sun”, Academy of Science of Uzbekistan

Email: rustam-shsul@yandex.com
Dr. Sci. (Eng.); Head at the Laboratory No. 1 Tashkent, Republic of Uzbekistan

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