Chaotic dynamics of the size-dependent non-linear micro-beam model

Krysko, Anton V.; Awrejcewicz, Jan; Павлов, С. П.; Zhigalov, Maxim V.; Krysko, Vadim A.

doi:10.1016/j.cnsns.2017.02.015

Cited by 49 publications

(14 citation statements)

References 26 publications

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“…), the approximation of beam deflection by third-order polynomials is associated with the names of Pelekh and Sheremetev, whereas in Western countries, this theory is referred to as the Reddy-Levinson theory. Motivated by the briefly described and documented historical reasons, the authors of the present study [23] suggested using the name 'Sheremetev-Pelekh-Reddy-Levinson (SPRL) theory' for this mathematical model. Among the known algorithms for calculating physically nonlinear systems, there is no universal one, i.e., the efficiency of a particular method depends mainly on the type and parameters of nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials

et al. 2021

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In this work, mathematical models of physically nonlinear plates and beams made from multimodulus materials are constructed. Our considerations are based on the 3D deformation theory of plasticity, the von Mises plasticity criterion and the method of variable parameters of the theory of elasticity developed by Birger. The proposed theory and computational algorithm enable for solving problems of three types of boundary conditions, edge conditions and arbitrary lateral load distribution. The problem is solved by the finite element method (FEM), and its convergence and the reliability of the results are investigated. Based on numerical experiments, the influence of multimodulus characteristics of the material of the beam and the plate on their stress–strain states under the action of transverse loads is illustrated and discussed.

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Section: Introductionmentioning

confidence: 99%

Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials

et al. 2021

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“…In the body of beam 1 a certain arbitrary reference curve 0  z is fixed; the OX goes along main curvature of the reference curve, whereas the axis OZ is directed to the reference curvature center. In the given coordinates the beam structure as a 2D object  is defined as For construction of a Pelekh-Sheremetev beam mathematical problem, we will formulate the hypothesis [4,6] (the third approximation hypothesis): -cross sections don't remain flat and perpendicular to the deformed axis of a beam; -a turn and a normal curvature for a beam we will set in a form:…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The study of nonlinear dynamics and contact interaction of beam structures is a very important issue at the present stage development of science. In the Russian and foreign scientific literature we can find a large number of works devoted to the study of Euler-Bernoulli [1], Timoshenko [2], Peleh-Sheremetev [3,4] beams models, but there are no papers on the contact interaction of beams of third-order. A third-order mechanics model is used in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Chaotic dynamics of two beams described by the kinematic hypothesis of the third approximation in the case of small clearance

Салтыкова

Папкова

Кrysko

2018

J. Phys.: Conf. Ser.

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Abstract. The chaotic dynamics of the contact interaction of two beams described by the thirdapproximation hypothesis (the Pelekh--Sheremetyev model) are studied in the paper. There is a small clearance between the beams. One of the beams is subjected to the action of a transversal harmonic load. Contact pressure is determined by Kantor's method. The mathematical model of the beam structure taking into account the geometric nonlinearity and contact interaction. The system of partial differential equations reduces to the ordinary differential equations by the finite differences method with the approximation of the second order. The resulting system is solved by the Runge-Kutta methods of various orders. The chaotic vibrations of two beams were investigated by the methods of nonlinear dynamics. The reliability of the obtained results was grounded. The Lyapunov exponents are calculated by three different algorithms -Kantz, Wolf, Rosenstein. The 2D and 3D phase portraits and the Fourier power spectra, the Poincaré pseudo section were constructed. The phenomenon of frequency synchronization has been detected.Keywords -Contact interaction, 3d approximation beam, nonlinear dynamics, finite difference method, Runge-Kutta methods, geometric nonlinearity. IntroductionBeams and beam structures are widely used in modern industry, machine building, rocket engineering, etc. Often such structures are subjected to various external dynamic influences. The study of nonlinear dynamics and contact interaction of beam structures is a very important issue at the present stage development of science. In the Russian and foreign scientific literature we can find a large number of works devoted to the study of Euler-Bernoulli [1], Timoshenko [2], Peleh-Sheremetev [3, 4] beams models, but there are no papers on the contact interaction of beams of third-order. A third-order mechanics model is used in this paper. The hypothesis of the third approximation is commonly called the Reddy hypothesis [5] in the foreign literature. But this theory was first described in [6]. It was shown in [4]. The Pelekh-Sheremetev hypothesis makes it possible to maximize the model to a threedimensional and to obtain the most accurate results. In 1964 two Ukrainian scientists M.P. Sheremetev and B.L. Pelekh [6] proposed a third-order theory. This theory was applied to the calculation of beams of plates and shells in publications [7]. Twenty-seven years later, in the works of Levinson [8] and Reddy [5,9], this model was reopened. Thus, the following situation has developed. In the Russian scientific press and in publications of scientists from several countries of Eastern Europe, the approximation of the deflection function by a polynomial of the third degree is associated with the names of M.P. Sheremetev and B.L. Pelekh. In the other works this theory is called the ReddyLevinson theory. We will adhere to the historical name -the Pelekh-Sheremetev model. The equations describing the dynamic processes are very difficult, so it is not possible to find an analytica...

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“…На основе теории пластин Кирхгофа и нелинейных деформаций фон Кармана получены нелинейные зависящие от размера поперечные и плоские уравнения движения. В работах [11], [12], [13] построены математические модели и изучено контактное взаимодействие нескольких балок. Целью исследования настоящей работы является построение математической модели и создание методов изучения контактного взаимодействия нанопластин и нанобалки в зависимости от размерно-зависимого параметра.…”

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Nonlinear Dynamics of the Contact Interaction of a Three-Layer Plate-Beam Nanostructure in a White Noise Field

Yakovleva

Кrysko

Krysko

2018

DSMM

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Аннотация. В работе впервые построена математическая модель контактного взаимодействия многослойной наноструктуры, состоящей из двух нанопластин и нанобалки между ними с малыми зазорами. Для описания размерно-зависимых эффектов данной наноструктуры применена модифицированная моментная теория. Верхний и нижний слои представляют собой нанопластины, подчиняющиеся кинематической гипотезе Кирхгофа, а средний слой-нанобалку Эйлера-Бернулли. Контактное взаимодействие учитывается по модели Б.Я. Кантора. Нанопластины и нанобалка изотропные, упругие и соединены они через краевые условия. В работе изучено влияние величины зазора между слоями и шумового поля. Для решения и анализа этих конструктивно-нелинейных задач применяются методы качественной теории дифференциальных уравнений, вейвлет-анализ, методы анализа знака старшего показателя Ляпунова. Система дифференциальных уравнений сводится к задаче Коши методом Бубнова-Галеркина в высших приближениях и методом конечных разностей с аппроксимацией 0(h 2) и 0(h 4) по пространственной координате. Далее задача Коши решается методами Рунге-Кутты 4-го, 6-го, 8-го порядка точности по времени. Анализ показал, что величина зазора существенно влияет на контактное взаимодействие элементов многослойной наносистемы и на характер их сложных колебаний. Также наличие шумового поля вовлекает в контактное взаимодействие элементы, которые находились в покое при прежних значениях остальных параметров. Ключевые слова: контактное взаимодействие наноструктуры, нелинейные колебания, аддитивный белый шум, балочно-пластинчатая наноструктура.

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Chaotic dynamics of the size-dependent non-linear micro-beam model

Cited by 49 publications

References 26 publications

Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials

Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials

Chaotic dynamics of two beams described by the kinematic hypothesis of the third approximation in the case of small clearance

Nonlinear Dynamics of the Contact Interaction of a Three-Layer Plate-Beam Nanostructure in a White Noise Field

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