SD振子及其摩擦作用下的动力学行为研究

发布时间：2018-05-07 00:09

本文选题：几何非线性摩擦 + 自激SD振子　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：SD振子是具有光滑与不连续两种不同特征的非线性动力学系统,是物理、机械工程、航空航天等领域内一类典型的描述几何大变形、大位移的几何非线性动力学体系的基础和核心,长期以来一直深受国内外专家的广泛关注。随着科学技术的迅速发展,在工程力学、机械动力学、地质力学和结构力学等领域内由摩擦导致的几何非线性问题也日益突出,但由于受传统非线性理论的局限性,摩擦理论研究缺乏系统性,工程中参数的非定常性、建模时弱非线性项被忽略等因素的影响,非线性摩擦动力学研究仍处于一个艰难的起步阶段。本文通过SD振子的动力学研究,并结合传统的摩擦理论构建了一类在摩擦作用下SD振子的几何非线性摩擦动力学系统,为深入探讨几何非线性摩擦动力学研究和响应机理奠定了重要理论基础。本文采用理论分析和数值计算相结合的方法,深入地研究了SD振子及其摩擦作用下的一系列复杂非线性动力学现象,为工程应用研究提供重要的理论依据,主要内容与成果如下:研究SD振子的周期运动。展现SD振子的基本动力学特性并分析其自由振动的周期特性。借助完全椭圆积分并提出一个拓展平均法给出周期激励作用下SD振子的主共振响应方程,避免采用截断方法对无理非线性的处理方法,其结果适用于SD振子的光滑和不连续两种阶段。应用Lyapunov稳定性理论分析系统周期解的稳定性,并用数值方法给出相应周期解的分布情况,与理论结果得以印证。分析干摩擦作用下SD振子的平衡态问题。利用库伦锥理论和微分包含理论分别给出该摩擦系统的平衡态分布特征。应用Lyapunov稳定性理论和拉萨尔不变理论判断出双曲平衡点集的不稳定性和非双曲平衡点集的稳定性。利用Filippov稳定滑动模式理论,解释双曲平衡点集拥有一个“宽”的稳定流形和一个源于其边界鞍点的不稳定流形。应用数值模拟分析系统平衡点集的分岔以及相应的动力学行为变化过程。基于传统的传送带理论和SD振子的基础上构建一个具有几何非线性特征的摩擦自激振动系统,简称自激SD振子。在此系统中,摩擦力采用库伦摩擦力数学模型,通过分析该自激系统的平衡点以及动力学分岔,展现系统的多粘附区域、双曲结构变迁和摩擦导致非对称等非线性动力学特征。在线性阻尼和周期激励的扰动下,利用数值方法揭示系统的多粘滑周期解、多粘滑混沌解以及周期解与混沌解共存等动力学行为。解析探讨周期外激励作用下自激SD振子的粘滑混沌动力学行为。通过分析该自激系统在粘附模式和滑移模式下的能量方程描述系统能量输入或耗散行为。利用Melnikov方法对受迫自激SD振子进行混沌行为进行解析预测,由于该自激系统的非对称性,给出系统的两个混沌阈值条件,并应用数值模拟方法验证这些理论结果的可靠性。分析在Stribeck摩擦特性条件下自激SD振子的复杂分岔行为。利用Filippov理论方法研究系统的双切点分岔,并用相图描述双切点分岔的动力学行为。随着传送带速度变化,系统表现出Filippov意义下的滑动同宿分岔。在粘性阻尼扰动下,分析系统的平衡点及其稳定性,判断系统发生亚临界Hopf分岔,并给出Hopf分岔发生条件。利用规范型理论方法解析验证该Hopf分岔条件以及分岔类型,并用动力学相图展现系统发生的亚临界Hopf分岔和擦边分岔等动力学行为。
[Abstract]:SD oscillator is a nonlinear dynamic system with two different features of smoothness and discontinuity. It is the foundation and core of a typical geometric nonlinear dynamic system describing large deformation and large displacement in the fields of physics, mechanical engineering and aerospace, and has been widely concerned by experts at home and abroad for a long time. The rapid development of the geometric nonlinear problems caused by friction in the fields of engineering mechanics, mechanical dynamics, geomechanics and structural mechanics is also increasingly prominent. However, due to the limitations of the traditional nonlinear theory, the research of friction theory is lack of systematicness, the uncertainty of the parameters in the engineering, and the neglecting of the weak nonlinear term in the modeling. The study of nonlinear friction dynamics is still in a difficult starting stage. In this paper, a geometric nonlinear friction dynamic system of SD oscillator under friction is constructed through the dynamics study of SD oscillator and the traditional friction theory, which lays the foundation for the study of geometric nonlinear friction dynamics and response mechanism. In this paper, a series of complex nonlinear dynamic phenomena of SD oscillator and its friction are studied in this paper by combining theoretical analysis with numerical calculation. It provides important theoretical basis for the research of engineering application. The main contents and results are as follows: studying the periodic motion of the SD oscillator and showing the basic of the SD oscillator. The dynamic characteristics and the periodic characteristics of the free vibration are analyzed. With the help of the complete ellipse integral and an extended mean method, the main resonance response equation of the SD oscillator under the periodic excitation is presented, and the treatment method of the unrational nonlinearity is avoided by the truncation method. The result is suitable for the smooth and discontinuous two stages of the SD oscillator. The application of Lyapun The stability of the periodic solution of the system is analyzed by the OV stability theory, and the distribution of the corresponding periodic solution is given by the numerical method. The equilibrium state of the SD oscillator under dry friction is analyzed. The equilibrium distribution characteristics of the friction system are given by the Kulun cone theory and the differential inclusion theory. The stability of the system is applied to the stability of the system. The stability of the set of hyperbolic equilibrium points and the stability of the set of non hyperbolic equilibrium points is judged by the theory of sex and the lasalal invariant theory. By using the Filippov stable sliding mode theory, it is explained that the set of hyperbolic equilibrium points has a "wide" stable manifold and an unstable manifold derived from the saddle point of its boundary. The bifurcation of the set and the corresponding dynamic behavior change process. Based on the traditional conveyor belt theory and the SD oscillator, a friction self excited vibration system with geometric nonlinearity is constructed, for short, the self excited SD oscillator. In this system, the friction force adopts the mathematical model of the Kulun friction force, by analyzing the equilibrium point of the self excited system. And dynamic bifurcation, showing the multi adhesion region of the system, the hyperbolic structure change and friction lead to the asymmetric and other nonlinear dynamic characteristics. Under the perturbation of linear damping and periodic excitation, the numerical method is used to reveal the multi stick slip periodic solution, the multi stick slip chaotic solution and the coexistence of the periodic solution and the chaotic solution. The viscous and slippery chaotic dynamic behavior of a self excited SD oscillator under periodic excitations is analyzed. The energy input or dissipation behavior of the system is described by analyzing the energy equation of the self excited system under the mode of adhesion and slip. The chaotic behavior of the forced self excited SD oscillator is predicted by the Melnikov method, due to the asymmetry of the self excited system. The two chaotic threshold conditions of the system are given, and the reliability of these theoretical results is verified by numerical simulation. The complex bifurcation behavior of the self excited SD oscillator under the Stribeck friction characteristic is analyzed. The bifurcation of the double tangent point of the system is studied by the Filippov theory method, and the dynamic behavior of the double tangent bifurcation is described with the phase diagram. The system shows the sliding homoclinic bifurcation in the sense of Filippov. Under the viscous damping disturbance, the equilibrium point and stability of the system are analyzed. The subcritical Hopf bifurcation is determined and the condition of the Hopf bifurcation is given. The condition of the Hopf bifurcation and the type of bifurcation are analyzed by the standard theory method. The mechanical phase diagram shows the dynamical behaviors of subcritical Hopf bifurcation and edge bifurcation.

【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】：O313.5;O19

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