一类柱面几何非线性系统的动力学行为研究

发布时间：2018-04-14 13:24

本文选题：分岔和混沌 + 柱面几何非线性　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：单摆是一类典型的具有柱面特征的几何非线性系统,在非线性动力学中扮演着极为重要的角色。近年来,一类含有几何非线性特征的弹簧质量系统SD振子的提出,由于其原创性得到业界人士的广泛关注,此类系统能够准确还原几何非线性系统的本质非线性特征。本研究旨在继承和发展以SD振子为核心的几何非线性原创理论,为工程中出现柱面几何非线性提供原始的理论依据。柱面几何非线性系统即具有柱面特征的几何非线性系统,是由两类典型的几何非线性力学模型传统摆和SD振子耦合而成,具有无理型和三角函数型耦合的强非线性特征,展现了光滑和不连续的多稳态动力学行为。本文通过建立一系列具有柱面特征的几何非线性动力学模型,提出针对该系统的解析研究方法,完整展现系统的全局动力学行为,精确描述其局部动力学特征;基于非线性动力学研究方法,揭示柱面几何非线性系统的分岔和混沌等动力学现象,厘清柱面几何非线性动力系统的全局分岔等大振幅运动规律以及小振幅运动下的共振响应等局部动力学特征。主要内容和成果如下:首先,基于SD振子构建了旋转SD振子力学模型,旋转SD振子是由水平匀质圆盘与一对端点固定的斜拉弹簧铰接而成,是一类典型的柱面几何非线性系统。在自由振动系统中,发现不连续系统具有连接标准鞍点和非标准鞍点的类异宿轨道;基于Hamilton函数得到类异宿轨道解析表达式,厘清了类异宿轨道具有的极限特征及其对应的运动规律。在强迫振动系统中,借助Melnikov方法数值分析了光滑系统中两类同宿轨道对应的混沌阈值;针对类异宿轨道提出广义Melnikov方法,通过不等式放缩解析得到不连续系统的混沌阈值,应用数值模拟验证了混沌阈值的准确性。其次,建立了倒摆和旋转SD振子耦合的旋转摆力学模型,旋转摆系统为典型的具有双稳态特征的柱面几何非线性系统。由于传统近似方法无法描述柱面几何非线性系统呈现的全局动力学行为即旋转运动,我们通过旋转摆系统的平衡点解析解形式及其分岔特性“双叉形分岔”引入柱面近似系统刻画其全局特征;基于平衡点稳定性和相同叉形分岔点建立了原系统和柱面近似系统参数之间的同胚映射;研究发现柱面近似系统成功描述了原系统的单摆,倒摆和双稳态等动力学行为且近似系统的奇异闭轨有解析解。在线性阻尼和简谐外激励联合作用下,解析得到柱面近似系统混沌振动判据的表达式,借助数值模拟验证了理论分析的准确性。通过相同参数下理论分析和数值对比,发现原系统和近似系统具有相似的拓扑动力学行为,尤其在分岔图,周期解,混沌吸引子结构,混沌数值特征等方面具有较高的吻合度。再者,为呈现柱面几何非线性旋转摆系统的分岔和极限环特性,引入柱面非线性周期扰动项,建立了双稳态柱面几何非线性非保守系统。随扰动参数变化,非保守系统经历了丰富的动力学分岔,如叉形分岔、同异宿轨道转迁、Hopf分岔、同宿轨道分岔、类同宿轨道分岔、周期轨道的鞍结分岔、Hopf-同宿轨道分岔、Hopf-周期轨道的鞍结分岔等。基于分岔理论和数值计算,厘清了非保守系统所有静态分岔和动态分岔的形成和演化过程以及给出对应分岔区间和分岔曲线上的极限环类型、个数、位置及其稳定性。揭示两次周期轨道的鞍结分岔的形成过程即半稳定极限环的形成过程,分别是由同宿轨道和平衡点特性引起;值得注意光滑非保守系统由于经历两次周期轨道的鞍结分岔导致出现5个极限环;发现第二类同宿轨道破裂产生两个对称的柱面极限环。最后,提出了具有双稳态特征的柱面几何非线性参激摆系统,该参激系统展现传统参激摆和SD振子的耦合动力学特性。在小振幅运动下,新参激系统可简化为标准的马休方程和SD振子方程的耦合,简化系统保留原系统光滑和不连续的动力学特性。借助平均法得到了参激振动系统的幅频响应关系,并用数值方法给出了光滑和不连续系统相应周期解的分布,发现数值结果与理论分析吻合。本研究刻画了该系统的局部动力学特征,克服了传统截断方法非光滑的局限性。基于理论分析和数值验证相结合的方法,研究参激系统的全局动力学特性,揭示了柱面几何非线性参激系统的复杂周期和混沌动力学现象,如摆动周期解,旋转周期解,摆动-旋转周期解,摆动混沌解,旋转混沌解,摆动-旋转混沌解以及共存周期解和混沌解等。此外,简单分析了三稳态柱面几何非线性旋转摆系统的自由振动,发现其具有丰富的静态分岔和非标准奇点,如超临界叉形分岔、亚临界叉形分岔、鞍结分岔、同异宿轨道转迁、尖点、类鞍点、切鞍点等。
[Abstract]:The pendulum is a typical nonlinear system with cylindrical characteristics, nonlinear dynamics plays a very important role. In recent years, the SD oscillator spring mass system comprising a class of nonlinear characteristics of the original, because of its widely concern in the industry, the system can accurately restore the geometric nonlinear characteristics of nature nonlinear system. The purpose of this study is to inherit and develop the original geometric nonlinear theory with SD oscillator as the core, to provide a theoretical basis for the original cylindrical geometry nonlinear engineering. Cylindrical geometry of non geometric nonlinear system linear system with cylindrical feature, is composed of two kinds of geometric nonlinear mechanics model of typical pendulum and SD coupled oscillator, nonlinear characteristic with irrational type and trigonometric function type coupling, revealed the steady-state kinetics for smooth and discontinuous as . through the establishment of a series of cylindrical feature geometric nonlinear dynamic model, puts forward the analytical approach of the system, complete show global dynamics of the system, the accurate description of the local dynamics; nonlinear dynamics research methods based on geometric nonlinear system in exposing the bifurcation and chaos of the cylinder, clarify the local dynamics characteristics the amplitude of the motion law of cylindrical geometry of nonlinear dynamical systems and global bifurcation of small amplitude motion under the resonance response. The main contents and results are as follows: firstly, based on constructed rotating SD oscillator model in SD oscillator, SD oscillator is composed of a rotating disk with a uniform level of endpoint cable-stayed spring hinges a cylindrical geometry is a typical nonlinear system. In the free vibration system, found that the discontinuous system has connected standard and non-standard saddle point Quasi class heteroclinic orbits of saddle point; Hamilton function class of heteroclinic orbits based on analytical expressions, clarify the motion characteristics and the corresponding class limit heteroclinic orbits with forced vibration. In the system, using Melnikov numerical method of chaotic threshold corresponding smooth systems with two types of persistent track analysis for different classes; the orbits of generalized Melnikov method, the inequality analytic chaos threshold of discontinuous systems, numerical simulation is used to verify the accuracy of the chaos threshold. Secondly, established the mechanical model of the inverted pendulum swing rotation and rotation of the SD oscillator coupling, rotating pendulum system for cylindrical geometry nonlinear bistable system has the typical characteristics. Because the traditional approximate method to describe the global dynamic behavior shows that the geometric nonlinear system for cylindrical rotary motion, we balance the pendulum system through the rotation point of the analytical form And "double bifurcation pitchfork bifurcation into cylindrical approximation system to describe the global characteristics of the original system; establish the stability of equilibrium and the same fork bifurcation point and cylindrical approximation based on homeomorphism mapping between the system parameters; the study found that the cylinder approximate system successfully describes the original system of the pendulum, inverted pendulum and bistable dynamics behavior and approximation the singular closed orbit analytic solution. The combined effect of incentive in linear damping and external harmonic, the analytic expression of a cylindrical approximation system of chaotic vibration criterion, using numerical simulation to verify the accuracy of theoretical analysis. Under the same parameters through theoretical analysis and numerical comparison, found that the original system and the approximate system with topological dynamics similarity in particular, bifurcation, periodic solution, structure of chaotic attractor, chaos has a high degree of agreement of numerical characteristics. Furthermore, to present several cylinder What non bifurcation and limit cycle characteristics of linear rotating pendulum system, the introduction of cylindrical nonlinear periodic perturbations, a non conservative system cylindrical geometric nonlinear bistability. With perturbation parameters, non conservative system has rich dynamics, such as pitchfork bifurcation, homoclinic and heteroclinic transition, Hopf bifurcation, homoclinic orbit bifurcation. Class of homoclinic orbit bifurcation, saddle node bifurcation of Hopf- periodic orbits, homoclinic orbit bifurcation, Hopf- periodic orbits of saddle node bifurcation bifurcation theory and numerical calculation. Based on clarifying the types of limit cycles of non conservative system formed all the static bifurcation and dynamic bifurcation and the evolution process and the corresponding bifurcation and bifurcation curves on the interval the number, location and stability. The formation process of the semi stable limit cycle of the saddle node bifurcation reveals two cycle orbits, are caused by homoclinic orbits and equilibrium characteristics; Note the smooth non conservative system for saddle node bifurcation through two periodic orbits lead to 5 limit cycles; found second types of homoclinic orbits rupture limit cycles to produce two cylindrical symmetry. Finally, put forward with bistable characteristics of cylindrical geometry nonlinear parametrically excited pendulum system, the parametric system show the coupling dynamics of the traditional parametrically excited pendulum and SD oscillator. In small amplitude motion, coupled parametric system can be simplified to the standard Mathieu equation and SD oscillator equation, simplified system retains the dynamical characteristics of the original system is smooth and discontinuous. The average method to get the vibration of frequency response relationship. The distribution of smooth and non periodic solutions of the corresponding continuous system is given by numerical method. The numerical results are consistent with theoretical analysis. This study describes the local dynamic characteristics of the system, to overcome the traditional cutting method of non light The limitations of the slide. Methods of theoretical analysis and numerical verification based on the combination of the global dynamics of a parametrically excited system, reveals the complex periodic and chaotic phenomenon of cylindrical geometric nonlinear parametric system, such as oscillating periodic solutions, periodic solutions of rotation, swing - rotation period solution and chaotic solution swing, rotation of the chaotic solution swing, rotation and chaotic solutions coexisting periodic solutions and chaotic solutions. In addition, a simple analysis of the free vibration of cylindrical geometry nonlinear steady-state three rotating pendulum system, we find that it has rich static bifurcation and non standard singularities, such as supercritical subcritical pitchfork bifurcation, pitchfork bifurcation, saddle node bifurcation, homoclinic and heteroclinic orbit transition, sharp point, saddle point, saddle point cut.

【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O322

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