环式少齿差行星齿轮传动的非线性动力学研究
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摘要
环式少齿差行星齿轮传动是一种由连杆机构和齿轮机构组合而成的传动装置。具有结构简单、传动比大,承载能力强等特点,但振动大、噪声高、发热严重一直是其存在的突出问题,这也在一定程度上影响了该传动的进一步推广和应用。现有动力学方面的研究仅局限于线性模型,而该类传动中不可避免地存在着间隙(如运动副间隙、齿侧间隙、轴承间隙)等非线性因素,这些因素将使该传动系统表现出非线性特征,为了全面反映该类传动的动力学行为特征,有必要对此类传动的非线性动力学问题进行研究。本文研究的目的就是为了解决环式少齿差行星齿轮传动非线性动力学方面的问题,以提高其动力学性能。主要内容包括：
以目前应用最广泛的三环齿轮传动作为研究对象。采用子结构综合法,将系统分为输入轴、支承轴、内齿板、输出齿轮及轴共四个子系统。在考虑齿轮时变啮合刚度、综合啮合误差以及齿侧间隙的情况下,采用集中质量法,建立了三环齿轮传动系统的耦合非线性动力学模型。模型中考虑了各轴弹性变形、行星轴承弹性变形、输出轴支承轴承弹性变形、内齿板的横向和纵向位移以及输出轴横向和纵向位移。通过合适的坐标变换消除了刚体位移,并进行无量纲化处理,使方程易于求解。本文的模型建立方法同样适用于其它环式少齿差行星齿轮传动。
研究了三环齿轮传动系统的参数振动问题。采用多尺度法,研究了环齿轮传动系统在齿轮内部激励以及外部激励作用下的主共振、超谐共振和组合共振及其稳定性等问题,给出了各种共振响应的一次近似定常解表达式,并推导了此定常解稳定性存在的条件。
采用谐波平衡法求解得到了三环齿轮传动系统的稳态响应。分析了综合啮合误差和时变啮合刚度对三环齿轮传动系统非线性频域特性的影响。
采用Runge-Kutta数值方法和MATLAB软件对系统在不同激励频率下的动态响应进行数值仿真。得到了系统由倍周期通向混沌、拟周期通向混沌的道路。结合相图、Poincare映射图等对系统的单周期、多周期、拟周期和混沌响应进行分析和比较。
为了验证非线性动力学模型的正确性,对三环齿轮传动系统进行了两个动态实验：固有频率的测试以及动态响应的测试。
根据齿轮平动的原理,设计了一种能实现完全动平衡的新型内平动齿轮传动装置。利用ANSYS软件对外齿板进行应力分析,结果表明其强度满足要求。同时,对这种传动进行了动力学分析,求出系统的固有频率以及主要零部件的动载荷,结果表明,载荷变化幅值较小,传动较平稳。
Ring planetary gear transmission with small tooth number difference is a drive which is composed of linkage mechanism and gear mechanism. It has simple structure, large speed-reduction ratio, high load capability. But Vibration, loud noise and severe fever are the outstanding problems, and they restrict its further extension and application in some degree. The present research on dynamics is restricted in linear model, but this transmission inevitably exist nonlinear factor such as clearance on movement pair, gear pair, bearing and so on. This factor will have nonlinear characteristic on this transmission. In order to fully reflect the dynamic behavior of this transmission, it is necessary to further study on the dynamics of such kind of system. The research purpose of this dissertation is to solve the problems of nonlinear dynamics on ring planetary gear transmission with small tooth number difference in order to improve the dynamic properties. The main contents are as follows:
Three-ring transmission being widely used is taken as research object, Subsystem synthesis is adopted and the system is divided into four subsystem including input shaft, supporting shaft, internal gear board, output gear and shaft. On the premise of considering time-varying mesh stiffness, gear meshing errors and gear backlash, Lumped mass method is adopted and a coupling nonlinear vibration model of three-ring transmission is established. The factors including the elasticity of all shaft, the elasticity of planetary bearing on all shaft, the elasticity of bearing on output shaft, horizontal and vertical on internal board and output shaft are considered. The rigid body movement is eliminated by use of suitable coordinate transformation, and the equations are convenient to be solved by use of dimensionless process. This model is applied to other ring planetary gear transmission with small tooth number difference.
Parameter vibration of the three-ring transmission system is researched. Multiple scale method is used to research the primary properties, ultraharmonic resonance properties, combination resonance properties and stabilities. The first-order stationary solutions are obtained under every condition respectively. The stability conditions of the stationary solutions are given at the same time in this dissertation.
Harmonic balance method is used to solve the stable response of the three-ring transmission system.The influence of gear meshing errors and time varying mesh stiffness on nonlinear frequency domain characteristic of the three-ring transmission system is analyzed.
Runge-Kutta numerical integration algorithm and MATLAB has been employed to simulate the dynamic responses on the different exciting frequency. A period doubling bifurcation route to chaos, and quasi-periodic bifurcation route to chaos have been identified. Various types of response such as Single-periodic, multi-periodic, quasi-periodic and chaotic response is analyzed and compared by use of phase plane plots, Poincare maps an so on.
The experimental researches on the dynamic properties of the three-ring transmission are made. The test of natural frequency and dynamic response are made in order to verify the rationality of the theoretical studies.
According to the principle of parallel gear, A new kind of internal parallel moving gears transmission with completely dynamic balance is designed. The stress of external toothed plate is carried out by ANSYS software, and the result shows that it can meet the requirement of strength. Dynamic analysis is researched and the natural frequency and dynamic load on main spare parts are obtained. The result shows that the load variation of amplitude is small and the transmission is relatively stable.
以目前应用最广泛的三环齿轮传动作为研究对象。采用子结构综合法,将系统分为输入轴、支承轴、内齿板、输出齿轮及轴共四个子系统。在考虑齿轮时变啮合刚度、综合啮合误差以及齿侧间隙的情况下,采用集中质量法,建立了三环齿轮传动系统的耦合非线性动力学模型。模型中考虑了各轴弹性变形、行星轴承弹性变形、输出轴支承轴承弹性变形、内齿板的横向和纵向位移以及输出轴横向和纵向位移。通过合适的坐标变换消除了刚体位移,并进行无量纲化处理,使方程易于求解。本文的模型建立方法同样适用于其它环式少齿差行星齿轮传动。
研究了三环齿轮传动系统的参数振动问题。采用多尺度法,研究了环齿轮传动系统在齿轮内部激励以及外部激励作用下的主共振、超谐共振和组合共振及其稳定性等问题,给出了各种共振响应的一次近似定常解表达式,并推导了此定常解稳定性存在的条件。
采用谐波平衡法求解得到了三环齿轮传动系统的稳态响应。分析了综合啮合误差和时变啮合刚度对三环齿轮传动系统非线性频域特性的影响。
采用Runge-Kutta数值方法和MATLAB软件对系统在不同激励频率下的动态响应进行数值仿真。得到了系统由倍周期通向混沌、拟周期通向混沌的道路。结合相图、Poincare映射图等对系统的单周期、多周期、拟周期和混沌响应进行分析和比较。
为了验证非线性动力学模型的正确性,对三环齿轮传动系统进行了两个动态实验：固有频率的测试以及动态响应的测试。
根据齿轮平动的原理,设计了一种能实现完全动平衡的新型内平动齿轮传动装置。利用ANSYS软件对外齿板进行应力分析,结果表明其强度满足要求。同时,对这种传动进行了动力学分析,求出系统的固有频率以及主要零部件的动载荷,结果表明,载荷变化幅值较小,传动较平稳。
Ring planetary gear transmission with small tooth number difference is a drive which is composed of linkage mechanism and gear mechanism. It has simple structure, large speed-reduction ratio, high load capability. But Vibration, loud noise and severe fever are the outstanding problems, and they restrict its further extension and application in some degree. The present research on dynamics is restricted in linear model, but this transmission inevitably exist nonlinear factor such as clearance on movement pair, gear pair, bearing and so on. This factor will have nonlinear characteristic on this transmission. In order to fully reflect the dynamic behavior of this transmission, it is necessary to further study on the dynamics of such kind of system. The research purpose of this dissertation is to solve the problems of nonlinear dynamics on ring planetary gear transmission with small tooth number difference in order to improve the dynamic properties. The main contents are as follows:
Three-ring transmission being widely used is taken as research object, Subsystem synthesis is adopted and the system is divided into four subsystem including input shaft, supporting shaft, internal gear board, output gear and shaft. On the premise of considering time-varying mesh stiffness, gear meshing errors and gear backlash, Lumped mass method is adopted and a coupling nonlinear vibration model of three-ring transmission is established. The factors including the elasticity of all shaft, the elasticity of planetary bearing on all shaft, the elasticity of bearing on output shaft, horizontal and vertical on internal board and output shaft are considered. The rigid body movement is eliminated by use of suitable coordinate transformation, and the equations are convenient to be solved by use of dimensionless process. This model is applied to other ring planetary gear transmission with small tooth number difference.
Parameter vibration of the three-ring transmission system is researched. Multiple scale method is used to research the primary properties, ultraharmonic resonance properties, combination resonance properties and stabilities. The first-order stationary solutions are obtained under every condition respectively. The stability conditions of the stationary solutions are given at the same time in this dissertation.
Harmonic balance method is used to solve the stable response of the three-ring transmission system.The influence of gear meshing errors and time varying mesh stiffness on nonlinear frequency domain characteristic of the three-ring transmission system is analyzed.
Runge-Kutta numerical integration algorithm and MATLAB has been employed to simulate the dynamic responses on the different exciting frequency. A period doubling bifurcation route to chaos, and quasi-periodic bifurcation route to chaos have been identified. Various types of response such as Single-periodic, multi-periodic, quasi-periodic and chaotic response is analyzed and compared by use of phase plane plots, Poincare maps an so on.
The experimental researches on the dynamic properties of the three-ring transmission are made. The test of natural frequency and dynamic response are made in order to verify the rationality of the theoretical studies.
According to the principle of parallel gear, A new kind of internal parallel moving gears transmission with completely dynamic balance is designed. The stress of external toothed plate is carried out by ANSYS software, and the result shows that it can meet the requirement of strength. Dynamic analysis is researched and the natural frequency and dynamic load on main spare parts are obtained. The result shows that the load variation of amplitude is small and the transmission is relatively stable.
引文
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