汽车螺旋锥齿轮传动耦合非线性振动研究
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摘要
螺旋锥齿轮包括弧齿锥齿轮和准双曲面齿轮是汽车和工程机械传动系统的重要组成部分，其工作性能对整个传动系统有着至关重要的影响。随着汽车车速和功率的提高，汽车螺旋锥齿轮传动正朝着高速、重载方向发展，其动力学行为的研究便成为国内外学者所关注的课题，由于我国汽车齿轮振动噪声普遍比国外产品严重，因此汽车螺旋锥齿轮耦合非线性动力学行为的研究更是工程实际中亟待解决的重要课题之一。
由于润滑和安装的需要，以及制造、加工、安装等过程中出现的误差和齿轮传动过程中的磨损，使得齿轮啮合不可避免地出现齿侧间隙，再加上轮齿刚度的时变性，弧齿锥齿轮和准双曲面齿轮传动系统便成为一种参数激励且具有时变啮合刚度的间隙非线性系统。弧齿锥齿轮和准双曲面齿轮传动系统的耦合非线性振动特性的研究，不仅对于汽车螺旋锥齿轮的动态设计具有重要的实用价值，而且具有重要的理论意义。
本文基于国家自然科学基金项目“高速汽车传动三维多点接触碰撞动力学仿真及物理模拟”（批准号：50075088）的资助，围绕汽车螺旋锥齿轮耦合非线性动力学问题进行了相关研究。在弧齿锥齿轮和准双曲面齿轮传动系统的耦合振动、三维冲击动力接触非线性特性和含时变啮合刚度和齿侧间隙的非线性振动特性等方面的研究取得了成果，论文的主要内容包括：
1. 建立了弧齿锥齿轮和准双曲面齿轮轮齿有限元分析模型，并结合动力接触有限元混合算法分析了轮齿的动态啮合性能。
2. 利用集中参数法建立了弧齿锥齿轮和准双曲面齿轮传动系统多自由度的弯－扭－轴－摆耦合振动动力学模型和分析模型。模型中考虑了传动轴和轴承的弹性变形以及齿轮的啮合刚度激励、误差激励和啮合冲击激励。在此基础上建立了包括齿轮传动系统和结构系统的完整齿轮系统分析模型，对齿轮系统的动态响应进行了数值仿真分析。
3. 根据哈密尔顿原理推导了冲击动力接触问题的有限元运动微分方程，利用三维冲击动力接触有限元混合法编制了三维冲击动力接触问题的有限元程序，并对弧齿锥齿轮和准双曲面齿轮的三维冲击动力接触特性进行了分析计算。
4. 对弧齿锥齿轮和准双曲面齿轮传动间隙非线性系统进行三维冲击动力接触数值模拟，分析了齿侧间隙对轮齿受初速冲击、突加载荷冲击时齿面冲击接触力的影响，通过数值模拟，给出了齿面冲击接触力的变化规律。
5. 建立了弧齿锥齿轮和准双曲面齿轮的间隙非线性动力学模型和变参数、弯
扭耦合、包含多元非线性函数的多自由度非线性微分方程组，通过坐标变换，推导了齿轮系统无量纲非线性统一微分方程组，使方程组中的弹性恢复力项全部是统一形式的1元间隙非线性函数，便于求解和推广。
6. 采用5阶变步长自适应Runge-Kutta数值积分方法和Matlab软件的Simulink仿真工具箱对系统的无量纲统一微分方程进行了数值仿真分析，得到系统的单周期简谐响应、多周期次谐响应、拟周期响应和混沌响应等多种稳态响应结果。并结合响应的时间历程、相平面图、Poincaré映射图和FFT频谱图对得到的各类响应进行了详细的分析和比较。
7. 设计了准双曲面齿轮系统动态测试实验台，进行了试验模态测试和动态响应测试，实验结果与理论计算结果基本吻合。
本文在以下四个方面取得了创新性成果。
(1) 建立了弧齿锥齿轮和准双曲面齿轮系统包括传动系统和结构系统的完整的分析模型；
(2) 建立了弧齿锥齿轮和准双曲面齿轮系统动力分析模型并进行了动态响应分析；
(3) 进行了弧齿锥齿轮和准双曲面齿轮三维冲击动力接触特性分析，研究了有关参数对齿面冲击力的影响；
(4) 进行了弧齿锥齿轮和准双曲面齿轮含齿侧间隙、时变啮合刚度和传动误差时的非线性动力学分析。
It is the important component part of automobile and engineering machine transmission system that Gleason helical bevel gear including spiral bevel gear and hypoid gear, its operating characteristic has the most important influence for entire transmission system. Along with the raising of automobile speed and power, automobile spiral bevel gear and hypoid gear transmission are developing toward high speed and heavy load, The issues relating to its dynamics behavior have brought a great concern to scholars all over the world. Since the vibration and noise of the automobile gear of our country is generally more serious than foreign product, therefore the research of coupled nonlinear dynamics behavior of the automobile spiral gear is one of the most important topics that must be resolved urgently in the engineering application.The gearing mesh is bound to have some backlash, which may be designed to provide better lubrication and installation or due to manufacturing, machining and installation error and wear during gear transmission. Taking into account of the time variant of the meshing stiffness, spiral bevel gear and hypoid gear transmission system will be a non-linear system with backlash, which has parametric excitation and the variant of the meshing stiffness. Researching on the coupling non-linear vibration property of spiral bevel gear and hypoid gear transmission system will be given rise to not only important practical value but also important theoretical meaning for the dynamical design of automobile Gleason helical bevel gear.This thesis based on the national fund project of natural science "3-D multipoint impact contact dynamic simulation and physical analogue of the automobile high speed transmission "( Authorize number: 50075088). The focus of this thesis is on the coupling nonlinear dynamics problem of the automobile Gleason helical bevel gear. Some progress and accomplishment have been made on the key issues such as coupling vibration of spiral bevel gear and hypoid gear, 3-D dynamic contact-impact and the non-linear vibration behavior with time-varying stiffness and backlash. The major works of this thesis are listed as follow: 1. The finite element model of the spiral bevel gear and hypoid gear tooth is established, the dynamic meshing property is analyzed combining with the mixed finite element method for contact problems. 2. The coupled Lateral-torsional-axial-pendulant vibration dynamics model andnalysis model with multiple degrees of freedom of the spiral bevel gear and hypoid gear are established. Elastic deformation of the shaft and bearing, and the excitation of the time-varying stiffness, deviation and meshing impact are considered in the model. The analysis model of the integrity gear system including the gear transmission system and the structure system is established and the dynamic response of the gear system is analyzed using the numerical simulation method. 3. The finite element kinematical differential equation relating to the dynamic contact-impact problem is derived from Hamilton theory. The finite element program of 3-D dynamic contact-impact problem is developed making use of the mixed finite element method, and used to analyze the 3-D dynamic contact-impact characteristic of the spiral bevel gear and hypoid gear. 4. The 3-D dynamic contact-impact numerical simulation method is used to simulate non-linear characteristics of the spiral bevel gear and hypoid gear with backlash. Then the influence of the backlash to the impact force is analyzed either the initial speed or the sudden load is applied to driving gear. The changing rule of the contact-impact force is also by this numerical method. 5. The non-linear dynamics model with backlash is established and the multi-degree- of-freedom differential equations with variable parameter, translational-torsional coupling and multi-element non-linear function also are gotten. In term of coordination transform, a new set of uniform non-linear dimensionless differential equations of the gear system is constructed. So all of the elastic res
由于润滑和安装的需要，以及制造、加工、安装等过程中出现的误差和齿轮传动过程中的磨损，使得齿轮啮合不可避免地出现齿侧间隙，再加上轮齿刚度的时变性，弧齿锥齿轮和准双曲面齿轮传动系统便成为一种参数激励且具有时变啮合刚度的间隙非线性系统。弧齿锥齿轮和准双曲面齿轮传动系统的耦合非线性振动特性的研究，不仅对于汽车螺旋锥齿轮的动态设计具有重要的实用价值，而且具有重要的理论意义。
本文基于国家自然科学基金项目“高速汽车传动三维多点接触碰撞动力学仿真及物理模拟”（批准号：50075088）的资助，围绕汽车螺旋锥齿轮耦合非线性动力学问题进行了相关研究。在弧齿锥齿轮和准双曲面齿轮传动系统的耦合振动、三维冲击动力接触非线性特性和含时变啮合刚度和齿侧间隙的非线性振动特性等方面的研究取得了成果，论文的主要内容包括：
1. 建立了弧齿锥齿轮和准双曲面齿轮轮齿有限元分析模型，并结合动力接触有限元混合算法分析了轮齿的动态啮合性能。
2. 利用集中参数法建立了弧齿锥齿轮和准双曲面齿轮传动系统多自由度的弯－扭－轴－摆耦合振动动力学模型和分析模型。模型中考虑了传动轴和轴承的弹性变形以及齿轮的啮合刚度激励、误差激励和啮合冲击激励。在此基础上建立了包括齿轮传动系统和结构系统的完整齿轮系统分析模型，对齿轮系统的动态响应进行了数值仿真分析。
3. 根据哈密尔顿原理推导了冲击动力接触问题的有限元运动微分方程，利用三维冲击动力接触有限元混合法编制了三维冲击动力接触问题的有限元程序，并对弧齿锥齿轮和准双曲面齿轮的三维冲击动力接触特性进行了分析计算。
4. 对弧齿锥齿轮和准双曲面齿轮传动间隙非线性系统进行三维冲击动力接触数值模拟，分析了齿侧间隙对轮齿受初速冲击、突加载荷冲击时齿面冲击接触力的影响，通过数值模拟，给出了齿面冲击接触力的变化规律。
5. 建立了弧齿锥齿轮和准双曲面齿轮的间隙非线性动力学模型和变参数、弯
扭耦合、包含多元非线性函数的多自由度非线性微分方程组，通过坐标变换，推导了齿轮系统无量纲非线性统一微分方程组，使方程组中的弹性恢复力项全部是统一形式的1元间隙非线性函数，便于求解和推广。
6. 采用5阶变步长自适应Runge-Kutta数值积分方法和Matlab软件的Simulink仿真工具箱对系统的无量纲统一微分方程进行了数值仿真分析，得到系统的单周期简谐响应、多周期次谐响应、拟周期响应和混沌响应等多种稳态响应结果。并结合响应的时间历程、相平面图、Poincaré映射图和FFT频谱图对得到的各类响应进行了详细的分析和比较。
7. 设计了准双曲面齿轮系统动态测试实验台，进行了试验模态测试和动态响应测试，实验结果与理论计算结果基本吻合。
本文在以下四个方面取得了创新性成果。
(1) 建立了弧齿锥齿轮和准双曲面齿轮系统包括传动系统和结构系统的完整的分析模型；
(2) 建立了弧齿锥齿轮和准双曲面齿轮系统动力分析模型并进行了动态响应分析；
(3) 进行了弧齿锥齿轮和准双曲面齿轮三维冲击动力接触特性分析，研究了有关参数对齿面冲击力的影响；
(4) 进行了弧齿锥齿轮和准双曲面齿轮含齿侧间隙、时变啮合刚度和传动误差时的非线性动力学分析。
It is the important component part of automobile and engineering machine transmission system that Gleason helical bevel gear including spiral bevel gear and hypoid gear, its operating characteristic has the most important influence for entire transmission system. Along with the raising of automobile speed and power, automobile spiral bevel gear and hypoid gear transmission are developing toward high speed and heavy load, The issues relating to its dynamics behavior have brought a great concern to scholars all over the world. Since the vibration and noise of the automobile gear of our country is generally more serious than foreign product, therefore the research of coupled nonlinear dynamics behavior of the automobile spiral gear is one of the most important topics that must be resolved urgently in the engineering application.The gearing mesh is bound to have some backlash, which may be designed to provide better lubrication and installation or due to manufacturing, machining and installation error and wear during gear transmission. Taking into account of the time variant of the meshing stiffness, spiral bevel gear and hypoid gear transmission system will be a non-linear system with backlash, which has parametric excitation and the variant of the meshing stiffness. Researching on the coupling non-linear vibration property of spiral bevel gear and hypoid gear transmission system will be given rise to not only important practical value but also important theoretical meaning for the dynamical design of automobile Gleason helical bevel gear.This thesis based on the national fund project of natural science "3-D multipoint impact contact dynamic simulation and physical analogue of the automobile high speed transmission "( Authorize number: 50075088). The focus of this thesis is on the coupling nonlinear dynamics problem of the automobile Gleason helical bevel gear. Some progress and accomplishment have been made on the key issues such as coupling vibration of spiral bevel gear and hypoid gear, 3-D dynamic contact-impact and the non-linear vibration behavior with time-varying stiffness and backlash. The major works of this thesis are listed as follow: 1. The finite element model of the spiral bevel gear and hypoid gear tooth is established, the dynamic meshing property is analyzed combining with the mixed finite element method for contact problems. 2. The coupled Lateral-torsional-axial-pendulant vibration dynamics model and
引文
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