Ташкентский финансовый институт; к.ф.-м.н., доцент Д.А. Ходжаев, Ташкентский институт ирригации и мелиорации Аннотация. В работе приводятся численный метод и алгоритм решения задач динамики вязкоупругих тонкостенных элементов конструкций переменной толщины. Уравнения движения относительно прогибов описываются интегро-дифференциальными уравнениями (ИДУ) в частных производных. При помощи метода Бубнова-Галеркина, основанного на многочленной аппроксимации прогибов, задача сводится к исследованию системы обыкновенных ИДУ, где независимой переменной является время. Система ИДУ решается предложенным численным методом, на основе которого описан алгоритм численного решения и создана программа на алгоритмическом языке Delphi. Исследование нелинейных колебаний тонкостенных элементов конструкции с учетом переменной толщины в геометрической нелинейной постановке позволило выявить ряд механических эффектов. В зависимости от физико-механических и геометрических параметров рассмотренных вязкоупругих тонкостенных элементов конструкций даны рекомендации по использованию жесткости системы. Ключевые слова: тонкостенные конструкции; переменная толщина; вязкоупругость, неоднородность; метод Бубнова-Галеркина; интегро-дифференциальные уравнения Введение В прикладных задачах механики деформируемых систем приходится встречаться с процессами, при описании которых необходимо оперировать имеющими разрывы величинами, различными по своему физико-механическому содержанию. В последнее время стало появляться все больше работ, посвященных результатам исследований критического состояния, колебаний и напряженно-деформированного состояния (НДС) конструкций с физико-механическими особенностями разрывного типа, т. е. конструкций cо ступенчато-переменной толщиной, с армированиями, неоднородностями структуры, местными включениями в виде сосредоточенных масс и отверстий, либо пониженной жесткости в виде ребра с учетом изотропных и анизотропных свойств материала [1-7].
The problems of oscillations of a viscoelastic cylindrical panel with concentrated masses are investigated, based on the Kirchhoff-Love hypothesis in the geometrically nonlinear statement. The effect of the action of concentrated masses is introduced into the equation of motion of the cylindrical panel using the δ function. To solve integro-differential equations of nonlinear problems of the dynamics of viscoelastic systems, a numerical method is suggested. With the Bubnov–Galerkin method, based on a polynomial approximation of the deflection, in combination with the suggested numerical method, the problems of nonlinear oscillation of a viscoelastic cylindrical panel with concentrated masses were solved. Bubnov–Galerkin’s convergence was studied in all problems. The influence of the viscoelastic properties of the material and concentrated masses on the process of oscillations of a cylindrical panel is shown.
In modern engineering and construction, thin-walled plates and shells of variable thickness, subjected to various static and dynamic loads, are widely used as structural elements. Advances in the technology of manufacturing thin-walled structural elements of any shape made it possible to produce structures with predetermined patterns of thickness variation. Calculations of strength, vibration and stability of such structures play an important role in design of modern apparatuses, machines and structures. The paper considers nonlinear vibrations of viscoelastic orthotropic cylindrical panels of variable thickness under periodic loads. The equation of motion for cylindrical panels is based on the Kirchhoff-Love hypothesis in a geometrically nonlinear statement. Using the Bubnov-Galerkin method, based on a polynomial approximation of deflections, the problem is reduced to the study of a system of ordinary integro-differential partial differential equations, where time is an independent variable. The solution to the resulting system is found by a numerical method based on the feature elimination in the Koltunov-Rzhanitsyn kernel used in the calculations. The behavior of a cylindrical panel with a wide range of changes in physico-mechanical and geometrical parameters is investigated.
The paper is devoted to improving the theory of bending and vibrations of three-layer plates with transverse compressible filler and thin outer bearing layers. For the outer layers, the Kirchhoff-Love hypothesis is accepted and the motion of their points is described by the equations of the theory of thin plates relative to forces and moments. Unlike bearing layers, a filler is considered as a three-dimensional body that does not obey any simplifying hypotheses. The equations of the bimoment theory of thick plates with respect to forces, moments and bimoments, created in the framework of the three-dimensional theory of elasticity, taking into account the nonlinearity of the distribution law of displacements and stresses over the thickness, are taken as the equations of motion of the filler. Expressions of forces, moments, and bimoments in the layers, as well as boundary conditions at the edges of a three-layer plate with respect to force factors, are given. In the conjugate zones of the layers, the complete contact conditions for the continuity of displacements and stresses are set. An example is considered and numerical results are obtained.
Geometrically nonlinear mathematical model of the problem of parametric oscillations of a viscoelastic orthotropic plate of variable thickness is developed using the classical Kirchhoff-Love hypothesis. The technique of the nonlinear problem solution by applying the Bubnov-Galerkin method at polynomial approximation of displacements (and deflection) and a numerical method that uses quadrature formula are proposed. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. Parametric oscillations of viscoelastic orthotropic plates of variable thickness under the effect of an external load are investigated. The effect on the domain of dynamic instability of geometric nonlinearity, viscoelastic properties of material, as well as other physical-mechanical and geometric parameters and factors are taken into account. The results obtained are in good agreement with the results and data of other authors.
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