齿轮振动可靠性与修形减振策略研究
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摘要
齿轮传动系统在现代化的工业进程中有着无可替代的作用,在电力、能源、交通、国防和航空领域都得到了广泛的应用。发动机中的齿轮传动系统长期在高速、重载的条件下工作,同时必须满足低噪音、轻量化、高可靠性、长寿命的要求。因此,齿轮的振动可靠性和减小振动的齿廓修形研究对齿轮的设计具有重要意义。目前考虑振动的齿轮设计研究主要表现在齿轮固有特性和动态响应的研究方面,没有考虑齿轮参数的误差等随机因素的影响,而用概率描述随机性才更准确。因此有必要将可靠性理论引入齿轮的设计工作中,从振动的角度开展高速、重载齿轮的可靠性研究,并应用优化方法进行齿廓修形减振分析,这对提高齿轮的工作性能、可靠性都有重大的意义。
本研究从工程应用的角度出发,以直齿圆柱齿轮为研究对象,将遗传算法、修形理论、随机参数振动、非线性振动、动态可靠性理论等先进的理论和技术相结合,对齿轮系统的传递误差可靠性、非线性振动可靠性、结构频率可靠性,齿廓修形减振等多个方面进行了系统的研究,主要研究工作如下：
(1)基于齿轮系统非线性振动模型,通过数值逐步积分法把随机参数的概率特征反映在系统的随机响应中,并将随机过程理论引入振动可靠性的计算中来,以单位啮合周期内振幅超差作为失效准则,形成了一种基于随机过程跨越分析方法的非线性随机参数振动可靠度计算方法。该方法针对齿轮非线性振动的复杂性,采用数值求解并直接对振动响应的概率分布进行评估,避免了近似算法的误差,可直接应用在工程设计中。
(2)为了寻找减振效果最好的修形参数,依靠参数化有限元建模模拟修形齿轮的啮合过程,并引入遗传算法,以减小齿轮啮合传递误差的波动作为目标,对直齿圆柱齿轮齿廓修形参数进行了高精度的优化设计,通过对设计结果进行减振效果分析,说明该方法准确,有效,能大幅度的减小齿轮的扭转振动。
(3)为了揭示齿轮系统振动非线性行为的本质,避免混沌振动状态,在考虑传递误差、时变啮合刚度、齿侧间隙的基础上,建立单间隙以及多间隙的齿轮系统非线性动力学模型,用数值方法对模型进行了求解,分析了系统的非线性动力学特性,从相图和Poincare图上定性判断了混沌性态,并计算了当频率比变化时系统的分岔图和Lyapunov指数图,利用Lyapunov指数定量识别了混沌振动,判断了混沌振动区域,为齿轮非线性振动的可靠性分析奠定了基础。
(4)为了研究修形参数对齿轮啮合传递误差可靠性影响,用响应面法获取齿轮副随机参数啮合传递误差的极限状态函数,并利用Monte-Carlo可靠性及可靠性敏感度分析方法获取传递误差振幅不超差的可靠度及可靠性敏感度。
(5)基于ANSYS软件的参数化设计语言APDL(ANSYS Parametric Design Language)分别建立了渐开线齿廓、齿廓直线修形和齿廓抛物线修形时的齿轮啮合参数化有限元模型。可以准确模拟具有随机参数的齿轮副的啮合过程,并对齿轮时变啮合刚度和齿轮传递误差进行了仿真计算。
(6)根据齿轮结构的固有频率与激振频率差的绝对值不超过规定值的关系准则,定义齿轮随机结构振动问题的系统可靠度,并结合直齿圆柱齿轮和弧齿锥齿轮提出避免共振的频率可靠性分析方法,讨论了随机参数统计特性的改变对齿轮结构频率可靠性的影响。所得结论为改善齿轮结构,保证齿轮系统的安全运行提供了理论依据。
Gear system plays an indispensable role in the era of industry process changing rapidly, it is widely used in many field such as electric power, energy, transport, national defense and aviation. As the key transmission component in motor, gears always work in the condition of high speed and overload. Besides, it must meet the need of low noise, light weight, high reliability as well as long life span. Therefore, it is significant and meaningful to develop the study of vibration reliability and explore the method of modification to decreases the vibration for gears. Recently, the researches of gear concerning the vibration mainly focus on the aspects of natural characteristics and dynamic responses. However, the stochastic factors of gear parameters such as error, whose stochastic property is accurate to express with probability, are rare to taken into consideration. Therefore, it is possible to adopt reliability theory to design gears in accessory gearbox. Basing on vibration analysis, reliability research can be developed in high speed and overload gears. Still, optimization methods can be use into the analysis of modification and vibration damping. The research is important to enhance the working performance and reliability of gears in accessory gearbox.
The research regards spur gears as the object from the engineering application perspective, and transmission error reliability, nonlinear vibration reliability, structural frequency reliability and profile modification for vibration damping are systematically studied based on integrating some advanced theories and techniques such as the Genetic Algorithm (GA), modification theory, stochastic parameter vibration, nonlinear vibration and dynamic reliability. The main researches are shown as follows:
(1) Basing on nonlinear vibration model of gear sysytem, the probability characteristics of random parameters are reflected in the stochastic response by numerical integration scheme, with the stochastic process introduced into vibration reliability. A method of calculate reliability is formed with failure criteria that random amplitude exceeds the prescribed value in a vibration cycle. For the complexity of gear nonlinear vibration, this method assesses the probability distribution of the vibration response directly by numerical solution, and avoids the error of approximation algorithms. It can be directly used in engineering design.
(2) To determine the parameters of gear profile modification accurately, the meshing process of a pair of gears was simulated via Finite Element Method (FEM). With GA introduced into the process to reduce the fluctuation of transmission error, the profile modification parameters of spur gears were designed optimally and accurately. Obviously, the method is accurate and efficient to decrease the torsion vibration greatly by analyzing the vibration damping effect.
(3) In order to discovery the nature of nonlinear behavior of gear system and avoid the chaotic vibration, nonlinear dynamic model of both single backlash and multiple backlash gear systems are built concerning the transmission error, time-varying mesh stiffness. The model is solved with numerical method, and the nonlinear dynamic characteristic is analyzed. The chaotic character is shown from the phase map and Poincare map, and the bifurcation diagram and largest Lyapunov exponent diagram are plotted while the frequency ratio varies. Lyapunov exponent is applied to identify the chaotic vibration numerically, and to fix the region of chaotic vibration. It provides a basis of nonlinear vibration reliability for gears.
(4) For studying the modification parameters effect on transmission error reliability, the limit state function for the transmission error of gear stochastic parameters is established with Response Surface Method (RSM). And method of Monte-Carlo reliability and reliability sensitivity is applied to acquire the reliability and reliability sensitivity when the transmission error amplitude is not crossing.
(5) Basing on ANSYS Parametric Design Language (APDL) ANSYS, parameter FEM models of involute tooth profile, linear modification profile and parabolic modification profile are established respectively. The model is accurate to simulate the meshing process of gears with stochastic parameters. The time-varying mesh stiffness and transmission error of gear are calculated as well.
(6) The reliability of gear structures with stochastic parameter vibration is defined on the basis of the relation criterion of the difference not beyond special value of the natural frequency and driving frequency. The frequency reliability analysis methods for spur gear and spiral bevel gear are carried out to avoid the resonant. The effect of statistical characteristic change of stochastic parameter is evaluated for frequency reliability of gear structure. The result provides a theoretical basis for improving the gear structure and ensuring the gear system working safely.
本研究从工程应用的角度出发,以直齿圆柱齿轮为研究对象,将遗传算法、修形理论、随机参数振动、非线性振动、动态可靠性理论等先进的理论和技术相结合,对齿轮系统的传递误差可靠性、非线性振动可靠性、结构频率可靠性,齿廓修形减振等多个方面进行了系统的研究,主要研究工作如下：
(1)基于齿轮系统非线性振动模型,通过数值逐步积分法把随机参数的概率特征反映在系统的随机响应中,并将随机过程理论引入振动可靠性的计算中来,以单位啮合周期内振幅超差作为失效准则,形成了一种基于随机过程跨越分析方法的非线性随机参数振动可靠度计算方法。该方法针对齿轮非线性振动的复杂性,采用数值求解并直接对振动响应的概率分布进行评估,避免了近似算法的误差,可直接应用在工程设计中。
(2)为了寻找减振效果最好的修形参数,依靠参数化有限元建模模拟修形齿轮的啮合过程,并引入遗传算法,以减小齿轮啮合传递误差的波动作为目标,对直齿圆柱齿轮齿廓修形参数进行了高精度的优化设计,通过对设计结果进行减振效果分析,说明该方法准确,有效,能大幅度的减小齿轮的扭转振动。
(3)为了揭示齿轮系统振动非线性行为的本质,避免混沌振动状态,在考虑传递误差、时变啮合刚度、齿侧间隙的基础上,建立单间隙以及多间隙的齿轮系统非线性动力学模型,用数值方法对模型进行了求解,分析了系统的非线性动力学特性,从相图和Poincare图上定性判断了混沌性态,并计算了当频率比变化时系统的分岔图和Lyapunov指数图,利用Lyapunov指数定量识别了混沌振动,判断了混沌振动区域,为齿轮非线性振动的可靠性分析奠定了基础。
(4)为了研究修形参数对齿轮啮合传递误差可靠性影响,用响应面法获取齿轮副随机参数啮合传递误差的极限状态函数,并利用Monte-Carlo可靠性及可靠性敏感度分析方法获取传递误差振幅不超差的可靠度及可靠性敏感度。
(5)基于ANSYS软件的参数化设计语言APDL(ANSYS Parametric Design Language)分别建立了渐开线齿廓、齿廓直线修形和齿廓抛物线修形时的齿轮啮合参数化有限元模型。可以准确模拟具有随机参数的齿轮副的啮合过程,并对齿轮时变啮合刚度和齿轮传递误差进行了仿真计算。
(6)根据齿轮结构的固有频率与激振频率差的绝对值不超过规定值的关系准则,定义齿轮随机结构振动问题的系统可靠度,并结合直齿圆柱齿轮和弧齿锥齿轮提出避免共振的频率可靠性分析方法,讨论了随机参数统计特性的改变对齿轮结构频率可靠性的影响。所得结论为改善齿轮结构,保证齿轮系统的安全运行提供了理论依据。
Gear system plays an indispensable role in the era of industry process changing rapidly, it is widely used in many field such as electric power, energy, transport, national defense and aviation. As the key transmission component in motor, gears always work in the condition of high speed and overload. Besides, it must meet the need of low noise, light weight, high reliability as well as long life span. Therefore, it is significant and meaningful to develop the study of vibration reliability and explore the method of modification to decreases the vibration for gears. Recently, the researches of gear concerning the vibration mainly focus on the aspects of natural characteristics and dynamic responses. However, the stochastic factors of gear parameters such as error, whose stochastic property is accurate to express with probability, are rare to taken into consideration. Therefore, it is possible to adopt reliability theory to design gears in accessory gearbox. Basing on vibration analysis, reliability research can be developed in high speed and overload gears. Still, optimization methods can be use into the analysis of modification and vibration damping. The research is important to enhance the working performance and reliability of gears in accessory gearbox.
The research regards spur gears as the object from the engineering application perspective, and transmission error reliability, nonlinear vibration reliability, structural frequency reliability and profile modification for vibration damping are systematically studied based on integrating some advanced theories and techniques such as the Genetic Algorithm (GA), modification theory, stochastic parameter vibration, nonlinear vibration and dynamic reliability. The main researches are shown as follows:
(1) Basing on nonlinear vibration model of gear sysytem, the probability characteristics of random parameters are reflected in the stochastic response by numerical integration scheme, with the stochastic process introduced into vibration reliability. A method of calculate reliability is formed with failure criteria that random amplitude exceeds the prescribed value in a vibration cycle. For the complexity of gear nonlinear vibration, this method assesses the probability distribution of the vibration response directly by numerical solution, and avoids the error of approximation algorithms. It can be directly used in engineering design.
(2) To determine the parameters of gear profile modification accurately, the meshing process of a pair of gears was simulated via Finite Element Method (FEM). With GA introduced into the process to reduce the fluctuation of transmission error, the profile modification parameters of spur gears were designed optimally and accurately. Obviously, the method is accurate and efficient to decrease the torsion vibration greatly by analyzing the vibration damping effect.
(3) In order to discovery the nature of nonlinear behavior of gear system and avoid the chaotic vibration, nonlinear dynamic model of both single backlash and multiple backlash gear systems are built concerning the transmission error, time-varying mesh stiffness. The model is solved with numerical method, and the nonlinear dynamic characteristic is analyzed. The chaotic character is shown from the phase map and Poincare map, and the bifurcation diagram and largest Lyapunov exponent diagram are plotted while the frequency ratio varies. Lyapunov exponent is applied to identify the chaotic vibration numerically, and to fix the region of chaotic vibration. It provides a basis of nonlinear vibration reliability for gears.
(4) For studying the modification parameters effect on transmission error reliability, the limit state function for the transmission error of gear stochastic parameters is established with Response Surface Method (RSM). And method of Monte-Carlo reliability and reliability sensitivity is applied to acquire the reliability and reliability sensitivity when the transmission error amplitude is not crossing.
(5) Basing on ANSYS Parametric Design Language (APDL) ANSYS, parameter FEM models of involute tooth profile, linear modification profile and parabolic modification profile are established respectively. The model is accurate to simulate the meshing process of gears with stochastic parameters. The time-varying mesh stiffness and transmission error of gear are calculated as well.
(6) The reliability of gear structures with stochastic parameter vibration is defined on the basis of the relation criterion of the difference not beyond special value of the natural frequency and driving frequency. The frequency reliability analysis methods for spur gear and spiral bevel gear are carried out to avoid the resonant. The effect of statistical characteristic change of stochastic parameter is evaluated for frequency reliability of gear structure. The result provides a theoretical basis for improving the gear structure and ensuring the gear system working safely.
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