铁道客车系统横向运动对称/不对称分岔行为与混沌研究
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摘要
本文以铁道客车系统为研究对象,系统地研究了该系统横向运动的对称／不对称分岔行为和混沌运动等非线性动力学行为。研究表明：由于各自由度之间内在的耦合性和非线性轮轨接触力等因素,不管是理想平直轨道上,还是非线性轮轨接触关系下直线轨道上运行的车辆系统,又或者是曲线轨道运行的不对称车辆系统,都有可能出现不对称的分岔行为和混沌运动。
第1章从铁道车辆系统横向运动稳定性、分岔与混沌、不对称车辆系统等方面的理论研究和工程应用背景出发,综述相关的研究成果,国内外的最新发展动态和存在的主要问题,并阐述本论文拟开展的相关研究工作。
第2章研究对称的转向架系统运行于理想平直轨道上的对称／不对称分岔行为和混沌运动。运用延续算法得到了系统Hopf分岔点和分岔后的周期解分支,据此确定了车辆系统的线性和非线性临界速度。同时,为了反映系统运动关于轨道中心线的对称／不对称状态,提出“合成分岔图”的构造方法,并利用该方法全面分析了对称转向架系统的实际运动形式和对称状态。研究表明,系统存在着大量的对称与不对称运动形式,包括简单的单周期运动、倍周期运动、混沌运动以及夹杂其间的若干多周期运动窗口。
第3章研究转向架系统稳态曲线轨道运行时不对称的分岔行为及其表征。由于整个系统关于轨道中心线是不对称的,因此系统表现出完全不对称的运动状态。为了准确反映曲线轨道运行时左右轮轮缘与钢轨的接触情况和运动关于轨道中心线的不对称特点,提出曲线轨道“最大-最小值”分岔图的构造方法,并应用该方法对稳态曲线轨道上转向架系统的运动形式、对称状态和轮缘接触情况进行了分析与讨论。研究表明,稳态曲线轨道运行的不对称转向架系统不仅平衡位置是偏离轨道中心线的,而且系统的非线性临界速度也要比直线轨道对称转向架系统的非线性临界速度低,且曲线轨道非线性临界速度随着曲率半径的减小而降低。
第4章研究对称的非线性轮轨接触关系下转向架系统的对称／不对称性分岔行为和混沌运动,所讨论的转向架系统仍应该是关于轨道中心线对称的。对临界速度的分析表明,由非线性轮轨接触关系得到的非线性临界速度要比由线性轮轨接触几何关系得到的非线性临界速度小一些,计算结果是偏于安全的。同时,通过部分参数对车辆临界速度的影响曲线分析后发现,由非线性轮轨接触关系得到的结果与实际车辆运行和实验情况吻合较好,因此建议在车辆动力学行为的分析中,尽量考虑非线性轮轨接触关系,由此得到的结果作为车辆设计、实验及运行的依据会更安全和更合理。此外,采用“升-降速”方法对转向架系统分岔行为的研究发现,系统由亚临界Hopf分岔引起的跳跃现象不突出,同样会存在不对称的运动形式。
第5章研究Vermeulen-Johnson蠕滑力和分段线性函数表达的轮缘力作用下整车模型直线轨道运行的对称／不对称分岔行为,并对系统运动对称性破坏的规律进行探讨。对临界速度的研究表明,由整车系统得到的非线性临界速度比由转向架系统得到的非线性临界速度低一些,计算结果偏于安全。对整车系统分岔行为和混沌运动的研究则表明,系统存在多个对称／不对称周期运动、多个对称／不对称拟周期运动、多个对称／不对称混沌运动并存的非线性动力学现象。此外,综合应用速度缓慢上升和缓慢下降的方法,通过数值手段探讨多自由度带有碰撞约束的对称车辆系统对称性破坏的规律。发现系统可能通过跳跃现象而失去原有的对称性并突变为不对称的运动,也有可能通过音叉分岔反复的经历对称性破坏和对称性恢复的过程并最终进入不对称的运动状态。
This dissertation takes the railway passenger car system as research object, and studies such nonlinear dynamical behaviors as the symmetric/asymmetric bifurcation and chaos of lateral motion in vehicle system. The research shows that, due to some reasons such as the coupling of each degrees of freedom and nonlinear wheel/rail interaction forces, it is likely that some asymmetric bifurcation behaviors and chaotic motions occur in the vehicle running on the ideal straight and perfect track, or the vehicle running on the track with nonlinear wheel/rail contact relation or the asymmetric vehicle in curves.
Based on the theoretic researches and engineering applications on the lateral motion stability, bifurcation and chaos and asymmetric vehicle system, Chapter One surveys the related achievements, recent development home and abroad, and the main problems in this area and illustrates the preparing work at the end of the chapter as well.
Chapter Two studies the symmetric/asymmetric bifurcation behaviors and chaotic motions in a symmetric truck system running on the ideal straight and perfect track. The Hopf bifurcation point and the periodic solution branches are obtained with continuation method, in which the linear and nonlinear critical speeds are easily determined. Meanwhile,'the resultant bifurcation'method is put forward in order to display the possible symmetric/asymmetric dynamic features around track centerline, and the method is applied to analyze the real motion form and the state of symmetry in truck system. Research results show that there exists lots of symmetric/asymmetric motion forms, including the single period motion, period-doubling motion, chaotic motion and several period windows among them.
Chapter Three focuses on the asymmetric bifurcation behaviors and their characteristics of a railway truck travelling on a curve with constant radius and superelevation. Because of the asymmetries of the whole system, the motion becomes fully asymmetric. To show the flange contact relation between wheels and rails and the asymmetric motion around track centerline, the'max-min amplitude bifurcation'method is brought forward, and the method is employed to construct the bifurcation diagram to show the motion form, the state of symmetry and the flange contact relation. It is shown not only the equilibrium position is off-centered around track centerline, but also the nonlinear critical speed is lower than the corresponding critical speed in straight track and the critical speed in curves is lowered with the decreasing of the curve radius.
Chapter Four researches on the symmetric/asymmetric bifurcation and chaos of the truck model with symmetric and nonlinear wheel/rail contact relation. The system is symmetric around track centerline. Analysis on the critical speed shows that the nonlinear critical speed with nonlinear wheel/rail contact relation is lower than the ones with linear wheel/rail contact geometry relation. Thus the system is safe. Meanwhile, it is found that the results with nonlinear wheel/rail contact relation are more accordant with the results in operation and experiment through analysis of the influence of some parameters on the critical speed. Thus it is advised that considering the nonlinear wheel/rail contact relation as much as possible to study the dynamic features in railway vehicle dynamics. It is more convicing and more reasonable under this circumstance when the findings is used as the basis of design, experiment and operation. Moreover, analysis on bifurcation behaviors of the truck sytem with'increasing-decreasing speed' method shows that the jump from sub-critical bifurcation is not strikingly obvious in the system and the asymmetric motion forms also exist as well.
Chapter Five represents studies on the symmetric/asymmetric bifurcation behaviors of a four-axle railway passenger car running on straight track with Vermeulen-Johnson creep force laws and flange force given by a piecewise linear function. The rule of symmetry-breaking is also discussed in this context. Research on the critical speed shows that the nonlinear critical speed in vehicle system is lower than the result in truck system, thus the results tend to be more convincing accordingly. Research on bifurcation and chaos shows that there exist several nonlinear dynamical phenomena, such as the coexistence of many symmetric/asymmetric periodic motions, quasi-periodic motions and chaotic motions. Furthermore, the study discusses the rule of symmetry-breaking of the vehicle system by applying the ramping method with slowly decreasing speed combined with the ramping method with slowly increasing speed. The results show that the motion may break symmetry though jump and becomes asymmetric, or undergo symmetry-breaking and symmetry-restoring processes repeatedly through many pitchfork bifurcations and finally the moiton becomes asymmetric.
第1章从铁道车辆系统横向运动稳定性、分岔与混沌、不对称车辆系统等方面的理论研究和工程应用背景出发,综述相关的研究成果,国内外的最新发展动态和存在的主要问题,并阐述本论文拟开展的相关研究工作。
第2章研究对称的转向架系统运行于理想平直轨道上的对称／不对称分岔行为和混沌运动。运用延续算法得到了系统Hopf分岔点和分岔后的周期解分支,据此确定了车辆系统的线性和非线性临界速度。同时,为了反映系统运动关于轨道中心线的对称／不对称状态,提出“合成分岔图”的构造方法,并利用该方法全面分析了对称转向架系统的实际运动形式和对称状态。研究表明,系统存在着大量的对称与不对称运动形式,包括简单的单周期运动、倍周期运动、混沌运动以及夹杂其间的若干多周期运动窗口。
第3章研究转向架系统稳态曲线轨道运行时不对称的分岔行为及其表征。由于整个系统关于轨道中心线是不对称的,因此系统表现出完全不对称的运动状态。为了准确反映曲线轨道运行时左右轮轮缘与钢轨的接触情况和运动关于轨道中心线的不对称特点,提出曲线轨道“最大-最小值”分岔图的构造方法,并应用该方法对稳态曲线轨道上转向架系统的运动形式、对称状态和轮缘接触情况进行了分析与讨论。研究表明,稳态曲线轨道运行的不对称转向架系统不仅平衡位置是偏离轨道中心线的,而且系统的非线性临界速度也要比直线轨道对称转向架系统的非线性临界速度低,且曲线轨道非线性临界速度随着曲率半径的减小而降低。
第4章研究对称的非线性轮轨接触关系下转向架系统的对称／不对称性分岔行为和混沌运动,所讨论的转向架系统仍应该是关于轨道中心线对称的。对临界速度的分析表明,由非线性轮轨接触关系得到的非线性临界速度要比由线性轮轨接触几何关系得到的非线性临界速度小一些,计算结果是偏于安全的。同时,通过部分参数对车辆临界速度的影响曲线分析后发现,由非线性轮轨接触关系得到的结果与实际车辆运行和实验情况吻合较好,因此建议在车辆动力学行为的分析中,尽量考虑非线性轮轨接触关系,由此得到的结果作为车辆设计、实验及运行的依据会更安全和更合理。此外,采用“升-降速”方法对转向架系统分岔行为的研究发现,系统由亚临界Hopf分岔引起的跳跃现象不突出,同样会存在不对称的运动形式。
第5章研究Vermeulen-Johnson蠕滑力和分段线性函数表达的轮缘力作用下整车模型直线轨道运行的对称／不对称分岔行为,并对系统运动对称性破坏的规律进行探讨。对临界速度的研究表明,由整车系统得到的非线性临界速度比由转向架系统得到的非线性临界速度低一些,计算结果偏于安全。对整车系统分岔行为和混沌运动的研究则表明,系统存在多个对称／不对称周期运动、多个对称／不对称拟周期运动、多个对称／不对称混沌运动并存的非线性动力学现象。此外,综合应用速度缓慢上升和缓慢下降的方法,通过数值手段探讨多自由度带有碰撞约束的对称车辆系统对称性破坏的规律。发现系统可能通过跳跃现象而失去原有的对称性并突变为不对称的运动,也有可能通过音叉分岔反复的经历对称性破坏和对称性恢复的过程并最终进入不对称的运动状态。
This dissertation takes the railway passenger car system as research object, and studies such nonlinear dynamical behaviors as the symmetric/asymmetric bifurcation and chaos of lateral motion in vehicle system. The research shows that, due to some reasons such as the coupling of each degrees of freedom and nonlinear wheel/rail interaction forces, it is likely that some asymmetric bifurcation behaviors and chaotic motions occur in the vehicle running on the ideal straight and perfect track, or the vehicle running on the track with nonlinear wheel/rail contact relation or the asymmetric vehicle in curves.
Based on the theoretic researches and engineering applications on the lateral motion stability, bifurcation and chaos and asymmetric vehicle system, Chapter One surveys the related achievements, recent development home and abroad, and the main problems in this area and illustrates the preparing work at the end of the chapter as well.
Chapter Two studies the symmetric/asymmetric bifurcation behaviors and chaotic motions in a symmetric truck system running on the ideal straight and perfect track. The Hopf bifurcation point and the periodic solution branches are obtained with continuation method, in which the linear and nonlinear critical speeds are easily determined. Meanwhile,'the resultant bifurcation'method is put forward in order to display the possible symmetric/asymmetric dynamic features around track centerline, and the method is applied to analyze the real motion form and the state of symmetry in truck system. Research results show that there exists lots of symmetric/asymmetric motion forms, including the single period motion, period-doubling motion, chaotic motion and several period windows among them.
Chapter Three focuses on the asymmetric bifurcation behaviors and their characteristics of a railway truck travelling on a curve with constant radius and superelevation. Because of the asymmetries of the whole system, the motion becomes fully asymmetric. To show the flange contact relation between wheels and rails and the asymmetric motion around track centerline, the'max-min amplitude bifurcation'method is brought forward, and the method is employed to construct the bifurcation diagram to show the motion form, the state of symmetry and the flange contact relation. It is shown not only the equilibrium position is off-centered around track centerline, but also the nonlinear critical speed is lower than the corresponding critical speed in straight track and the critical speed in curves is lowered with the decreasing of the curve radius.
Chapter Four researches on the symmetric/asymmetric bifurcation and chaos of the truck model with symmetric and nonlinear wheel/rail contact relation. The system is symmetric around track centerline. Analysis on the critical speed shows that the nonlinear critical speed with nonlinear wheel/rail contact relation is lower than the ones with linear wheel/rail contact geometry relation. Thus the system is safe. Meanwhile, it is found that the results with nonlinear wheel/rail contact relation are more accordant with the results in operation and experiment through analysis of the influence of some parameters on the critical speed. Thus it is advised that considering the nonlinear wheel/rail contact relation as much as possible to study the dynamic features in railway vehicle dynamics. It is more convicing and more reasonable under this circumstance when the findings is used as the basis of design, experiment and operation. Moreover, analysis on bifurcation behaviors of the truck sytem with'increasing-decreasing speed' method shows that the jump from sub-critical bifurcation is not strikingly obvious in the system and the asymmetric motion forms also exist as well.
Chapter Five represents studies on the symmetric/asymmetric bifurcation behaviors of a four-axle railway passenger car running on straight track with Vermeulen-Johnson creep force laws and flange force given by a piecewise linear function. The rule of symmetry-breaking is also discussed in this context. Research on the critical speed shows that the nonlinear critical speed in vehicle system is lower than the result in truck system, thus the results tend to be more convincing accordingly. Research on bifurcation and chaos shows that there exist several nonlinear dynamical phenomena, such as the coexistence of many symmetric/asymmetric periodic motions, quasi-periodic motions and chaotic motions. Furthermore, the study discusses the rule of symmetry-breaking of the vehicle system by applying the ramping method with slowly decreasing speed combined with the ramping method with slowly increasing speed. The results show that the motion may break symmetry though jump and becomes asymmetric, or undergo symmetry-breaking and symmetry-restoring processes repeatedly through many pitchfork bifurcations and finally the moiton becomes asymmetric.
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