轮对系统的Hopf分岔研究

针对轮对系统中的非线性动力学问题, 本文基于Hopf分岔代数判据得到考虑陀螺效应的轮对系统Hopf分岔点解析表达式, 即轮对系统蛇形失稳的线性临界速度解析表达式. 基于分岔理论得到轮对系统的第一、第二Lyapunov系数表达式, 并结合打靶法分别得到不同纵向刚度下, 考虑陀螺效应与不考虑陀螺效应的轮对系统分岔图. 通过对比有无陀螺效应的轮对系统分岔图发现, 在同一纵向刚度下, 考虑陀螺效应的轮对系统线性临界速度和非线性临界速度均大于不考虑陀螺效应的轮对系统, 即陀螺效应可以提高轮对系统的运动稳定性. 基于Bautin分岔理论, 以纵向刚度和纵向速度作为参数, 分别得到考虑陀螺效应和不考虑陀螺效应的轮对系统, 从亚临界Hopf分岔到超临界Hopf分岔, 再从超临界Hopf分岔到亚临界Hopf分岔的迁移机理拓扑图. 通过对比有、无陀螺效应的轮对系统Bautin分岔拓扑图发现, 陀螺效应将改变轮对系统的退化Hopf分岔点, 但对于轮对系统Bautin分岔拓扑图的影响不大.      

Hopf分岔的代数判据及其在车辆动力学中的应用

利用Hurwitz行列式,给出平衡点失稳而发生Hopf分岔的代数判定准则和计算方法,这一方法将Hopf分岔点的求解转化为一个非线性方程的求解问题,从而克服了以前方法在计算Hopf分岔点时,对于参数的每一次变化通过求特征根并判定特征根的实部是否为零的庞大工作量。应用这一方法,我们进行了非线性车辆系统蛇行运动稳定性的研究,得到了轮对系统发生蛇行运动的临界速度的解析表达式。     

轮对非线性动力学系统蛇行运动的解析解

在对铁路车辆系统的极限环幅值和非线性临界速度进行分析时通常采用数值方法, 不便于研究其随系统参数的变化规律. 轮对系统保留了影响车辆系统动力学性能的几个关键要素: 如轮轨几何非线性约束、轮轨接触蠕滑关系和悬挂系统等, 可以反映铁路车辆系统蛇行运动的本质特性. 轮对系统自由度少、参数少, 可以采用解析方法进行分析. 本文选取合适的特征量把轮对非线性动力学方程无量纲化, 得到了带有小参数的两自由度微分方程; 采用多尺度方法对该方程进行了解析求解; 给出了轮对系统极限环幅值的解析表达式并对其稳定性进行了判定; 给出了轮对系统的分岔速度解析表达式, 并进而获得系统的非线性临界速度的解析表达式. 在对得到的解析解用数值结果进行验证后, 用得到的解析解进行了系统参数影响分析. 传统的分岔图计算方法(如降速法、路径跟踪法等)需对微分方程进行大量数值积分计算方可求解系统的非线性临界速度值, 而通过本文获得的解析表达式可直接给出系统的非线性临界速度值和极限环幅值, 便于研究轮对系统动力学特性随参数的变化规律,进行快速方案比对和筛选, 为转向架结构优化设计提供参考.      

三自由度齿轮系统的双参数分岔与全局特性研究

以三自由度齿轮系统为研究对象,通过构造参数平面内不同运动类型的边界线算法,得到了系统在参数平面内的分岔曲线。为了判断分岔曲线的分岔类型,构造了三自由度齿轮系统Poincaré映射的Jacobi矩阵及Floquet乘子算法。结合系统的分岔图、最大Lyapunov指数图(TLE)、相图、Poincaré映射图和Floquet理论,讨论了双参数平面上系统的分岔特性以及参数平面内系统动力学特性的演变,并利用胞映射法对系统随啮合频率变化下的全局动力学特性进行了研究。结果表明:系统在参数平面k-ξ33内存在倍化分岔曲线、鞍结分岔曲线、Hopf分岔曲线等;阻尼系数越大,综合误差越小,系统运动越稳定;鞍结分岔对系统的全局稳定性影响较大,而Hopf分岔对系统的全局稳定性影响较小。研究结果可为齿轮系统设计和参数选择提供理论依据,研究方法也适用于其它非线性系统的双参数分岔分析。     

通过分岔控制改变神经元的兴奋性类型

提出一种通过分岔控制改变神经元兴奋性类型的方法.采用一个基于washout滤波器的动态反馈控制实现对一个二维的Hindmarsh-Rose类的模型神经元的分岔动力学控制.这一模型神经元从静息态到峰放电态跨越一个不变圆上鞍结分岔(saddle-node on invariant circle,SNIC),呈现出第一类兴奋性.在该SNIC分岔前所期望的参数值处产生一个Hopf分岔,然后通过选择适当的控制器参数调节Hopf分岔的临界性.这样,模型神经元就呈现为第二类兴奋性,因此神经元兴奋性就从第一类改变成第二类.在这个控制器中,线性控制增益决定着Hopf分岔的位置,而非线性增益决定着Hopf分岔的临界性.     

Duffing系统在双参数平面上的动力学特性分析

将单参数最大Lyapunov指数的计算推广到双参数平面上,数值计算Duffing系统在双参数平面上的最大Lyapunov指数,得到系统在参数平面上周期运动、混沌运动、各种分岔曲线的参数区域;结合系统单参数分岔图、相图、庞加莱截面图讨论了系统在参数平面上的分岔混沌过程以及阻尼系数对系统双参数特性的影响。结果表明:在双参数平面上系统出现了周期跳跃、周期倍化分岔、叉式分岔等复杂的分岔曲线,而且这些分岔曲线随阻尼系数的增加不断发生着复杂变化;得到系统在以往单参数分岔过程中很少出现的分岔曲线相交、嵌套、演变等特殊现象;阻尼系数对系统双参数耦合动力学特性有重要的影响。本文对工程中其它多参数系统的参数耦合特性的研究具有一定的参考价值。     

考虑间隙反馈控制时滞的磁浮车辆稳定性研究

常导磁吸型(EMS)磁悬浮列车在悬浮控制中的每个环节,时滞是不可避免的,当时滞超过一定程度后,系统有可能失稳.本文针对EMS磁浮列车控制环节的临界时滞与车辆参数(如运行速度、反馈控制增益、导轨参数和悬挂参数)的关系开展研究.建立了磁浮车辆/导轨耦合动力学模型,车辆包含1节车辆和4个磁浮架,考虑车辆的10个自由度,每个磁浮架上包含4个悬浮电磁铁.导轨模拟为一系列简支Bernoulli-Euler梁,采用模态叠加法对导轨振动方程进行求解.采用传统线性电磁力模型实现车辆和轨道的耦合.采用比例-微分控制算法对电磁铁电流进行反馈控制,实现车辆稳定悬浮,并假设时滞均发生在控制环节,且只考虑间隙反馈控制环节的时滞.采用四阶龙格库塔法对耦合系统动力学方程进行求解,编写了数值仿真程序,计算得到车辆导轨耦合系统在考虑间隙反馈控制时滞时的响应.将系统运动发散时的时滞大小视为临界时滞,开展了参数规律影响分析.通过分析,给出了提高时滞条件下车辆稳定性的方法,包括增大导轨的弯曲刚度和阻尼比,减小间隙反馈控制增益并增大速度反馈控制增益,以及增大二系悬挂阻尼.      

考虑间隙反馈控制时滞的磁浮车辆稳定性研究

常导磁吸型(EMS)磁悬浮列车在悬浮控制中的每个环节,时滞是不可避免的,当时滞超过一定程度后,系统有可能失稳.本文针对EMS磁浮列车控制环节的临界时滞与车辆参数(如运行速度、反馈控制增益、导轨参数和悬挂参数)的关系开展研究.建立了磁浮车辆/导轨耦合动力学模型,车辆包含1节车辆和4个磁浮架,考虑车辆的10个自由度,每个磁浮架上包含4个悬浮电磁铁.导轨模拟为一系列简支Bernoulli-Euler梁,采用模态叠加法对导轨振动方程进行求解.采用传统线性电磁力模型实现车辆和轨道的耦合.采用比例–微分控制算法对电磁铁电流进行反馈控制,实现车辆稳定悬浮,并假设时滞均发生在控制环节,且只考虑间隙反馈控制环节的时滞.采用四阶龙格库塔法对耦合系统动力学方程进行求解,编写了数值仿真程序,计算得到车辆导轨耦合系统在考虑间隙反馈控制时滞时的响应.将系统运动发散时的时滞大小视为临界时滞,开展了参数规律影响分析.通过分析,给出了提高时滞条件下车辆稳定性的方法,包括增大导轨的弯曲刚度和阻尼比,减小间隙反馈控制增益并增大速度反馈控制增益,以及增大二系悬挂阻尼.     

两自由度机械臂λ^2=1下的Hopf分岔与混沌研究

研究了一类周期系数力学系统因周期运动失稳而产生Hopf分岔及混沌问题.首先根据拉格朗日方程给出了该力学系统的运动微分方程,并确定其周期运动的具有周期系数的扰动运动微分方程,再根据Floquet理论建立了其给定周期运动的Poincaré映射,根据该系统的特征矩阵有一对复共轭特征值从-1处穿越单位圆情况,分析该Poincaré映射不动点失稳后将发生次谐分岔、Hopf分岔、倍周期分岔,而多次倍周期分岔将导致混沌.并用数值计算加以验证.结果表明,随着分岔参数的变化,系统的周期运动可通过次谐分岔形成周期2运动,进而发生Hopf分岔形成拟周期运动,并再次经次谐分岔、倍周期分岔形成混沌运动.     

机电耦联系统余维3动态分岔研究

以r_{sl}, r_{f}以及x_{c}为分岔参数，对具有串补电容的单 机无穷大电力系统的失稳振荡问题，运用动态分岔理论进行了研究. 对系统同时出现有3对 纯虚根特征值的一类多参数高余维分岔情况，运用中心流行方法降维后得到约化方程，对此 强非线性约化方程的求解难点，运用多参数稳定性理论、谐波平衡法、归一化技术和Normal Form方法，得到了系统的解析解. 由分析得知，系统会出现3种Hopf分岔情况、二维环面 情况，以及三维环面分岔解，甚至会出现四维环面，或者更高维的环面分岔. 详细讨论 了系统各种分岔解的稳定性条件和稳定区域，并作了详细的数值分析加以验证.     

Effect of Lateral Linear Stiffness on Nonlinear Characteristics of Hunting Motion of a Railway Wheelset

Railway wheelset experiences the problem of hunting above a critical speed, which is a kind of self-excited oscillation. At the critical speed, it is known that the system undergoes a subcritical Hopf bifurcation. Therefore, for clarifying the nonlinear characteristics of hunting it is very important to detect, for example, the nonlinear forces in the wheelset due to the creep forces acting between the wheels and rails, and the nonlinear component of the resorting forces by the suspensions. However, it is impossible to determine each force quantitatively. In the present paper, it is first shown, by using the center manifold theory and the method of normal form, that the nonlinear characteristics of the bifurcation in a wheelset model with two degrees of freedom are governed by a single parameter, hence each nonlinear force need not be detected when examining the nonlinear characteristics. Also, a method of determining the governing parameter from experimentally observed radiuses of the unstable limit cycle is proposed. Next, we experimentally investigate the variation of the parameter due to the presence of linear spring suspensions in the lateral direction and discuss the variation of the nonlinear characteristics of the hunting motion, which depends on the lateral stiffness. As a result, the improvement of the stability of the wheelset against the disturbance by the linear spring suspensions is clarified.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

Hopf Bifurcation and Hunting Behavior in a Rail Wheelset with Flange Contact

An analytical investigation of Hopf bifurcation and hunting behavior of a rail wheelset with nonlinear primary yaw dampers and wheel-rail contact forces is presented. This study is intended to complement earlier studies by True et al., where they investigated the nonlinearities stemming from creep-creep force saturation and nonlinear contacts between a realistic wheel and rail profile. The results indicate that the nonlinearities in the primary suspension and flange contact contribute significantly to the hunting behavior. Both the critical speed and the nature of bifurcation are affected by the nonlinear elements. Further, the results show that in some cases, the critical hunting speed from the nonlinear analysis is less than the critical speed from a linear analysis. This indicates that a linear analysis could predict operational speeds that in actuality include hunting.     

Bifurcation of a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force

This paper presents the bifurcation behaviors of a modified railway wheelset model to explore its instability mechanisms of hunting motion. Equivalent conicity data measured from China high-speed railway vehicle are used to modify the wheelset model. Firstly, the relationships between longitudinal stiffness, lateral stiffness, equivalent conicity and critical speed are taken into account by calculating the real parts of the eigenvalues of the Jacobian matrix and Hurwitz criterion for the corresponding linear model. Secondly, measured equivalent conicity data are fitted by a nonlinear function of the lateral displacement rather than are considered as a constant as usual. Nonlinear wheel–rail force function is used to describe the wheel–rail contact force. Based on these modifications, a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force is set up, and then, some instability mechanisms of China high-speed train vehicle are investigated based on Hopf bifurcation, fold (limit point) bifurcation of cycles, cusp bifurcation of cycles, Neimark–Sacker bifurcation of cycles and 1:1 resonance. In particular, fold bifurcation of cycles can produce a vast effect on the hunting motion of the modified wheelset model. One of the main reasons leading to hunting motion is due to the fold bifurcation structure of cycles, in which stable limit cycles and unstable limit cycles may coincide, and multiple nested limit cycles appear on a side of fold bifurcation curve of cycles. Unstable hunting motion mainly depends on the coexistence of equilibria and limit cycles and their positions; if the most outward limit cycle is stable, then the motion of high-speed vehicle should be safe in a reasonable range. Otherwise, if the initial values are chosen near the most outward unstable limit cycle or the system is perturbed by noises, the high-speed vehicle will take place unstable hunting motion and even lead to serious train derailment events. Therefore, in order to control hunting motions, it may be the easiest way in theory to guarantee the coexistence of the inner stable equilibrium and the most outward stable limit cycle in a wheelset system.

     

BIFURCATIONS OF AIRFOIL IN INCOMPRESSIBLE FLOW

Bifurcations of an airfoil with nonlinear pitching stiffness in incompressible flow are investigated. The pitching spring is regarded as a spring with cubic stiffness. The motion equations of the airfoil are written as the four dimensional one order differential equations. Taking air speed and the linear part of pitching stiffness as the parameters, the analytic solutions of the critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed. By means of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcations are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcation are found. Some numerical simulation results are given.     

Supercritical and subcritical Hopf bifurcation and limit cycle oscillations of an airfoil with cubic nonlinearity in supersonic\hypersonic flow

In this paper, the Hopf bifurcations and limit cycle oscillations (LCOs) of an airfoil with cubic nonlinearity in supersonic\hypersonic flow are investigated. The harmonic balance method and multivariable Floquet theory are applied to analyze the LCOs of the airfoil. Four distinct cases of the LCOs response are detected in this system: (I) supercritical Hopf bifurcation, (II) a single subcritical Hopf bifurcation, (III) two subcritical Hopf bifurcations, and (IV) no Hopf bifurcation. Furthermore, the parameter variations domains separating the supercritical and subcritical Hopf bifurcations are presented using singularity theory.     

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.