Visualization of Transition's Scenarios from Harmonic to Chaotic Flexible Nonlinear-elastic Nano Beam's Oscillations

Вадим Крысько, V. Krysko, Ирина Папкова, I. Papkova, Екатерина Крылова, E. Krylova, Антон Крысько, A. Krysko
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引用次数: 1

Abstract

In this study, a mathematical model of the nonlinear vibrations of a nano-beam under the action of a sign-variable load and an additive white noise was constructed and visualized. The beam is heterogeneous, isotropic, elastic. The physical nonlinearity of the nano-beam was taken into account. The dependence of stress intensity on deformations intensity for aluminum was taken into account. Geometric non-linearity according to Theodore von Karman’s theory was applied. The equations of motion, the boundary and initial conditions of the Hamilton-Ostrogradski principle with regard to the modified couple stress theory were obtained. The system of nonlinear partial differential equations to the Cauchy problem by the method of finite differences was reduced. The Cauchy problem by the finite-difference method in the time coordinate was solved. The Birger variable method was used. Data visualization is carried out from the standpoint of the qualitative theory of differential equations and nonlinear dynamics were carried out. Using a wide range of tools visualization allowed to established that the transition from ordered vibrations to chaos is carried out according to the scenario of Ruelle-Takens-Newhouse. With an increase of the size-dependent parameter, the zone of steady and regular vibrations increases. The transition from regular to chaotic vibrations is accompanied by a tough dynamic loss of stability. The proposed method is universal and can be extended to solve a wide class of various problems of mechanics of shells.
柔性非线性弹性纳米梁振荡从谐波到混沌过渡场景的可视化
本文建立了变号载荷和加性白噪声作用下纳米梁非线性振动的数学模型并进行了可视化。梁是不均匀的,各向同性的,弹性的。考虑了纳米束的物理非线性。考虑了铝的应力强度与变形强度的关系。几何非线性根据西奥多·冯·卡门的理论进行了应用。得到了修正偶应力理论下Hamilton-Ostrogradski原理的运动方程、边界和初始条件。用有限差分法将非线性偏微分方程组简化为柯西问题。用有限差分法在时间坐标上求解柯西问题。采用Birger变量法。从微分方程定性理论和非线性动力学的角度对数据进行可视化处理。使用广泛的可视化工具,可以根据Ruelle-Takens-Newhouse的场景确定从有序振动到混沌的过渡。随着尺寸相关参数的增大,稳定和规则振动区域增大。从规则振动到混沌振动的转变伴随着剧烈的动态稳定性损失。所提出的方法具有通用性,可推广到求解各种壳体力学问题。
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