Abstract
In this paper, a mathematical model of complex oscillations of a flexible micropolar cylindrical mesh structure is constructed. Equations are written in displacements. Geometric nonlinearity is taken into account according to the Theodore von Karman model. A non-classical continual model of a panel based on a Cosserat medium with constrained particle rotation (pseudocontinuum) is considered. It is assumed that the fields of displacements and rotations are not independent. An additional independent material parameter of length associated with a symmetric tensor by a rotation gradient is introduced into consideration. The equations of motion of a panel element, the boundary and initial conditions are obtained from the Ostrogradsky – Hamilton variational principle based on the Kirchhoff – Love’s kinematic hypotheses. It is assumed that the cylindrical panel consists of n families of edges of the same material, each of which is characterized by an inclination angle relative to the positive direction of the axis directed along the length of the panel and the distance between adjacent edges. The material is isotropic, elastic and obeys Hooke’s law. To homogenize the rib system over the panel surface, the G. I. Pshenichnov continuous model is used. The dissipative mechanical system is considered. The differential problem in partial derivatives is reduced to an ordinary differential problem with respect to spatial coordinates by the Bubnov – Galerkin method in higher approximations. The Cauchy problem is solved by the Runge – Kutta method of the 4th order of accuracy. Using the establishment method, a study of grid geometry influence and taking account of micropolar theory on the behavior of a grid plate consisting of two families of mutually perpendicular edges was conducted.
Highlights
It is assumed that the cylindrical panel consists of n families of edges of the same material, each of which is characterized by an inclination angle relative to the positive direction of the axis directed along the length of the panel and the distance between adjacent edges
The differential problem in partial derivatives is reduced to an ordinary differential problem with respect to spatial coordinates by the Bubnov – Galerkin method in higher approximations
A study of grid geometry influence and taking account of micropolar theory on the behavior of a grid plate consisting of two families of mutually perpendicular edges was conducted
Summary
В работе построена математическая модель сложных колебаний гибкой микрополярной цилиндрической панели сетчатой структуры. Рассматривается неклассическая континуальная модель панели на основе среды Коссера со стесненным вращением частиц (псевдоконтинуум). Граничные и начальные условия получены из вариационного принципа Остроградского – Гамильтона на основании кинематических гипотез Кирхгофа – Лява. Предполагается, что цилиндрическая панель состоит из n семейств ребер одного материала, каждое из которых характеризуется углом наклона относительно положительного направления оси, направленной по длине панели, и расстоянием между соседними ребрами. Для гомогонизации системы ребер по поверхности панели применяется континуальная модель Г. В качестве примера проведено исследование влияния геометрии сетки и учета микрополярной теории на поведение сетчатой панели, состоящей из двух семейств взаимно перпендикулярных ребер. Ключевые слова: цилиндрическая панель, микрополярная теория, сетчатая структура, модель Кирхгофа – Лява, метод Бубнова – Галеркина, метод установления, континуальная модель Г.
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