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The dynamic stability problem of viscoelastic orthotropic and isotropic plates is considered in a geometrically nonlinear formulation using the generalized Timoshenko theory. The problem is solved by the Bubnov-Galerkin procedure combined with a numerical method based on quadrature formulas. The effect of viscoelastic and inhomogeneous properties of the material on the dynamic stability of a plate is discussed.Introduction. The use of new composite materials in the design and development of strong, light-weight, and reliable structures calls for improved mechanical models of deformable solids and mathematical models for structure calculations taking into account the actual properties of structural materials. Numerous experimental studies have shown that most composite materials possess pronounced viscoelastic properties [2-6] and are inhomogeneous [6,7].The classical Kirchhoff-Love model is effective in solving some applied problems, but in most cases it fails to give adequate solutions [8]. This is true primarily for calculations of the dynamic stability of viscoelastic shells made of composite materials with heterogeneous anisotropic structure [3,6,9]. This formulation of elastic problems was considered in [7,8,[10][11][12][13], where, however, only some properties of structural materials were taken into account.In a number of papers, the viscoelastic properties of materials were taken into account only in the shear directions (see, e.g., [3]). In previous calculations, exponential kernels were used as relaxation kernels but they cannot provide an adequate description of the real processes occurring in shells and plates at the initial time [14]. The choice of exponential kernels in the calculations is not random. By differentiation, the resulting systems of integrodifferential equations were reduced to high-order ordinary differential equations, which, in most cases, were solved by the Runge-Kutta numerical method. Previously existing methods were not applicable for solving these problems with weakly singular kernels of the type of Koltunov, Rzhanitsyn, Abel, and Rabotnov kernels.The numerical method developed in [1] using quadrature formulas has made it possible to solve systems of nonlinear integrodifferential equations with singular kernels. This method provides a high accuracy of calculations, is universal, and can be used to solve a wide class of dynamic problems of the theory of viscoelasticity. The results of [2,9] obtained by this method agree well with experimental data.It is worth noting that in contrast to the isotropic formulation of the dynamic problems of viscoelastic systems, where the integrodifferential equations contain only one relaxation kernel with three different rheological viscosity parameters, the orthotropic formulation includes seven different kernels and the number of different rheological parameters increases to 21, which leads to very intricate calculations.The objective of the present paper is to study the dynamic stability of viscoelastic isotropic and orthotropic plates using va...

We discuss the problem of the dynamic stability of a viscoelastic cylindrical panel with concentrated masses in a geometrically nonlinear formulation that is based on the Kirchhoff-Love hypothesis. The effect of the action of concentrated masses is introduced into the equation of motion of a cylindrical panel using the Dirac δ-function. The problem is solved by the Bubnov-Galerkin method based on a polynomial approximation of deflections together with a numerical method based on the use of quadrature formulas. The choice of the Koltunov-Rzhanitsyn singular kernel is justified. Comparisons between the results obtained from different theories are presented. The Bubnov-Galerkin method convergence is investigated for all problems. The effect of the material viscoelastic properties and concentrated masses on the process of the dynamic stability of a cylindrical panel is shown.Introduction. During the intense development of the modern industry, a reduction in the materials consumption of machine structures is one of the main problems of the mechanical and civil engineering. For the purpose of material saving, the need arises to manufacture thin-walled structures. The thinner is the element and the more flexible it is, the more strongly its susceptibility to buckling and loss of stability is manifested. The latter is accompanied by a catastrophic development of deformations and, as a rule, by a structural failure. From this standpoint, in the production of lightweight, durable and reliable structures, it is reasonable to use the materials which make it possible not only to improve their operating characteristics but, in a number of cases, to create the structures unfeasible with traditional materials. Here, the calculation procedure and structural design involving the consideration of their actual properties are rather complicated. Today, the development of efficient solution algorithms for nonlinear problems of dynamic stability of shells, panels and plates is the most pressing issue.Plates, panels or shells with objects fixed as additional masses have found wide use due to high viscoelastic and strength properties. In the design of structural elements, the prediction of their dynamic characteristics depending on the shape, mass distribution, viscoelastic properties of the material, etc. is an urgent problem.Longitudinal and transverse ribs, cover straps, fasteners, device assemblies and subassemblies act mainly as additional masses [1,2]. In a theoretical consideration of this kind of problems, it is sometimes convenient to interpret connected elements as additional masses rigidly attached to systems and concentrated at points.Works dealing with vibrations and dynamic stability of elastic systems with concentrated masses are known [3][4][5][6][7][8] where problems were solved in a linear formulation. Thus, for example, work [9] deals with the investigations on the nonlinear vibrations and dynamic stability of elastic cylindrical panels and shells carrying concentrated masses. Similar problems of vibrations and dynamic...

The present work discusses the problem of dynamic stability of a viscoelastic circular cylindrical shell, according to revised Timoshenko theory, with an account of shear deformation and rotatory inertia in the geometrically nonlinear statement. Proceeding by Bubnov-Galerkin method in combination with a numerical method based on the quadrature formula the problem is reduced to a solution of a system of nonlinear integro-differential equations with singular kernel of relaxation. For a wide range of variation of physical mechanical and geometrical parameters, the dynamic behavior of the shell is studied. The influence of viscoelastic properties of the material on the dynamical stability of the circular cylindrical shell is shown. Results obtained using different theories are compared.