【作者】 刘洋；

【导师】 焦映厚；

【作者基本信息】 哈尔滨工业大学 ， 机械工程， 2023， 硕士

【摘要】 齿轮传动在人类社会各行各业中应用广泛,其运行是否平稳对整个系统至关重要。一对齿轮副进行工作时,为了防止受热膨胀导致卡死,相邻两齿间必然会预留间隙。间隙的存在会使得齿轮出现拍击的现象,导致系统出现一系列非线性行为。本文利用连续和不连续动力系统理论对含间隙齿轮传动系统非线性动力学行为进行研究,为含间隙的齿轮系统动力学研究提供了借鉴。介绍了连续和不连续动力系统理论的发展过程和研究进展,并详细阐述了两种求解方法的基本内容。连续动力系统理论广义谐波平衡法(the Generalized Harmonic Balance Method,GHB)是研究连续系统非线性动力学的重要方法之一,本文推导了广义谐波平衡法求解连续动力系统的详细过程,对周期解析解稳定性判断做出了定义。引出了不连续动力系统的相关定义,介绍了不连续动力系统流转换理论,并对其核心理论“G函数”进行推导,得出了流在不连续边界处的转换条件。利用连续动力系统广义谐波平衡法获得了一类单自由度齿轮非线性动力系统的近似解析解,该动力系统可以用于分析含间隙齿轮动力学模型,并且解析解的精度可以通过提高谐波项数来控制。分析了系统动态传递误差(DTE)对系统稳定性和分岔的影响,基于广义谐波平衡法获得了谐波幅值-动态传递误差幅值特性图,通过特征值分析,详细讨论了系统周期运动的稳定区间和分岔点。分析了系统存在的稳定和不稳定周期-1运动,并给出了系统运动时域波形图和相图,以及谐波幅值、谐波相位图。发现系统解的稳定和不稳定分支交叉点处存在霍普夫分岔,这些分岔会导致系统周期运动的拓扑结构发生改变。建立了考虑轴承支撑刚度、支撑阻尼和啮合刚度、啮合阻尼的二自由度振-冲模型。基于不连续动力系统理论,将齿轮运动的相平面分为三个部分:齿面啮合运动域、自由运动域和齿背啮合运动域。引入全局映射和局部映射动力学方法,描述了齿轮从冲击到啮合再到冲击和啮合的过程。通过数值分析研究了不同恢复系数对齿轮冲击啮合运动的影响。结果表明,齿隙引起的擦边分岔将导致系统的复杂映射结构甚至混沌,并且恢复系数可以直接影响碰撞、啮合的过程。引入啮合刚度和恢复系数可以合理地表征齿轮啮合过程中的弹性变形和能量损失,为将不连续动力系统理论应用于更复杂的多自由度柔性接触齿轮传动系统提供了理论模型。为有效地解释齿轮传动系统中的碰撞-啮合机理,建立了考虑时变啮合刚度、碰撞恢复系数、齿侧间隙的齿轮系统扭转振动动力学模型。利用不连续动力系统理论求解了该分段线性动力学方程,获得了系统关于齿轮转速频谱瀑布图。此外,分析了不同驱动力下启动阶段首次稳定啮合前的碰撞次数和时间规律。结果发现,在转速上升的过程中,齿轮系统时而存在着脱啮现象,尤其在某些区间出现了脱啮共振带,合理控制转速范围能够有效避免齿轮出现脱啮行为从而引发的振动噪声。更多还原

【Abstract】 Gear transmission is widely used in various industries in human society,and the stability of its operation is crucial for the entire system.When a pair of gears work,a backlash must be reserved between adjacent teeth to prevent jamming caused by thermal expansion.The existence of the backlash can cause impact between gears,leading to a series of nonlinear behaviors in the system.In this thesis,the nonlinear dynamical behavior of gear transmission systems with backlash are studied by the theory of continuous and discontinuous dynamical systems,providing a reference for the dynamic analysis of gear systems with backlash.The development and research progress of the theory of continuous and discontinuous dynamical systems are presented,and the basic contents of the two kinds of method are elaborated in detail.The generalized harmonic balance method(GHB)of continuous dynamical systems is one of the important method to study the nonlinear dynamics of continuous systems.The detailed process of solving periodic analytical solutions of continuous dynamical systems by GHB is derived,and the stability of periodic analytical solutions is defined.The definition of discontinuous dynamical systems is introduced,and the theory of flow switchability in discontinuous dynamical systems is presented,and the core theory "G function" is derived,and the switching conditions of the flow at the discontinuous boundary are obtained.An approximate analytical solution of a single degree-of-freedom(SDOF)gear nonlinear dynamical system is obtained using the GHB of the theory of continuous dynamical systems.The dynamical system can be used to analyze the gear dynamical model with backlash,and the accuracy of the analytical solution can be controlled by the number of harmonic terms.The influence of the dynamic transmission error(DTE)of the system on the stability and bifurcation of the system is analyzed.The characteristic diagram of harmonic amplitude-amplitude of dynamic transmission error is obtained based on the GHB,and the stable interval and bifurcation points of the system periodic motion are discussed in detail through eigenvalue analysis.The stable and unstable periodic-1 motion of the system is analyzed,and the time-histories displacement and trajectories of the periodic motions,as well as the harmonic amplitude and phase diagram,are given to illustrate the details of the motion.It is found that there are Hopf bifurcations at the cross point of the stable and unstable branches of the solutions,which will cause changes in the topological structure of the system periodic motion.A two-degree-of-freedom oscillator model with spring and damping elements is established.Based on the theory of discontinuous dynamical systems,the phase plane of gear motion is divided into three parts: the domain of tooth surface meshing motion,the domain of free motion and the domain of tooth back meshing motion.The global mapping and local mapping dynamical methods are introduced to accurately describe the process of gear impact,meshing,and impact and meshing.The influence of different restitution coefficients on gear impact-meshing motion is studied through numerical simulation.The results show that the grazing bifurcation caused by backlash will lead to the complex mapping structure of the system,even chaos.The restitution coefficient directly affects the impact-meshing behavior.The introduction of meshing stiffness and restitution coefficient can reasonably characterize the elastic deformation and energy loss during gear meshing,providing a theoretical model for applying the theory of discontinuous dynamical systems to more complex multi-degree-of-freedom flexible contact gear transmission systems.In order to effectively explain the impact-meshing mechanism in gear transmission systems,a four-degree-of-freedom gear-rotor dynamical model with variable meshing stiffness,gear backlash,and restitution coefficient is established.The segmented linear dynamical equation of the system is solved by the theory of discontinuous dynamical systems,and the waterfall diagram of frequency spectrum of the system with respect to the gear rotation speed is obtained.It is found that the gear system sometimes experiences disengagement during the process of increasing rotation speed,especially in certain intervals where disengagement resonance bands appear.Effectively controlling the range of rotation speed can avoid gear disengagement behavior and vibration noise.更多还原