螺旋锥齿轮多体多自由度非线性动力学研究
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摘要
螺旋锥齿轮工作平稳、传动比大、传递动力大、结构紧凑,是各种机器和装备中广泛应用的传动装置,是齿轮传动中最为复杂的一种。其中,包括有偏置距的准双曲面齿轮和无偏置距的弧齿锥齿轮。其传动效率高,传动比稳定,圆弧重叠系数大,承载能力高,传动平稳平顺,工作可靠,结构紧凑,节能省料,节省空间,耐磨损,寿命长,噪音小。随着高速和精密机械传动的发展,对齿轮系统动态特性的要求在不断提高,更为准确的预测其动力学特性就变得越来越重要。由于螺旋锥齿轮啮合的复杂性,如轮齿的时变刚度、啮合时的啮入啮出冲击、齿侧间隙、摩擦阻力和制造及安装误差的存在,整个齿轮系统的振动是轴、齿轮、轴承、箱体及原动机等多种振动耦合的综合作用,并且具有一定的非线性特性,使对螺旋锥齿轮系统的动力学研究变得十分复杂。在现阶段,对齿轮系统的动态特性进行深入的理论和实验研究,寻找出其振动计算的一般规律,具有重要的理论和现实意义。
本文的主要目的就是建立有效的数学模型和分析技术,描述螺旋锥齿轮的啮合和动力学特性,综合考虑各种非线性因素,预测动态响应,以揭示螺旋锥齿轮传动系统的根本物理现象。本文进行了如下研究工作:
首先,通过利用专业螺旋锥齿轮有限元分析软件对齿轮系统进行三维造型建模和有限元建模分析。对螺旋锥齿轮Face-milled和Face-hobbed两种不同的加工方法进行了深入研究。针对两种不同加工方法的加工特点建立螺旋锥齿轮小轮和大轮齿面方程,通过无载齿面接触分析,求解接触路径上的主曲率、主方向和接触椭圆。确定非线性动力学方程中的几何量、力学量和物理量,为今后的动力学分析打下基础。建立基于三维准静态齿面加载接触分析模型,求解不同载荷作用下传动误差,接触区域印迹,时变啮合刚度,啮合点位置和啮合力作用线方向等啮合特征参数。
第二,建立螺旋锥齿轮传动系统多体多自由度非线性动力学分析模型。运用现代的理论和分析方法,通过对螺旋锥齿轮啮合模型和系统动态激励的分析,利用集中参数法建立了螺旋锥齿轮系统的弯-扭-轴-摆多自由度以及包括原动机和负载以及小轮和大轮的多体多自由度非线性动力学模型,该模型中考虑了系统的综合啮合误差、时变啮合刚度、摩擦力、陀螺效应以及齿侧间隙,得到其运动微分方程,运用先进的非线性算法及计算机软件求解动态响应。
第三,研究了混合润滑条件下摩擦力和时变摩擦系数对螺旋锥齿轮系统动力学特性的影响。通过研究弹流润滑理论,区分了齿轮不同的润滑状态。在混合润滑状态下建立基于Mixed-EHL理论的时变摩擦系数模型,利用该模型求解在不同载荷和转速条件下齿面啮合区域每个网格点的瞬时摩擦系数。比较了准双曲面齿轮和弧齿锥齿轮摩擦力行为的特点和各自在时变摩擦力影响下的动态响应。
最后,定性分析安装误差对螺旋锥齿轮系统动力学特性的影响。分别对准双曲面齿轮和弧齿锥齿轮的偏置距误差、轴交角误差、小轮轴向位置误差和大轮轴向位置误差的敏感性进行了对比分析。同时,考察了安装误差对摩擦力行为和动态响应的影响。比较了安装误差和摩擦力的对系统动力学特性的影响程度。
Helical bevel gear including hypoid gear and spiral bevel gear are widely used in power and motion transmission of the machines and equipments. It is one of the most complex types of gear drive system, which has the advantages of smooth working, high ratio of transmission, high efficiency advantages of transmission, compact structure, saving more space, resistant abrasion, long service life and low noise. With the development of high-speed and precision mechanical transmission, the demand of dynamic characteristics and stability of gear transmission system continues to increase. A more accurate prediction of gear dynamic characteristics becomes more and more important. Due to the complexity of gear mesh, such as time-vary mesh stiffness, impaction of tooth mesh in and out, backlash, friction drag, manufacturing and assembly errors and the comprehensive effect of the entire gear transmission system vibration caused by vibration coupling several kinds of vibrations from shaft, gear pair, gear box, engine and nonlinear characteristics, all this factors make the research of gear dynamics quite complicated. Therefore, in-depth theoretical and experimental studying the dynamic characteristics of gearing system and finding the general rule of vibration calculation has current significance at present stage.
The goal of this dissertation research is to establish more effective mathematical model and analytical techniques to characterize the mesh and dynamic behavior, predict vibratory response, and reveal the underlying physics of helical bevel geared rotor system. The major works of this dissertation are listed as follow.
Firstly, the 3-D geometrical model and finite element analysis is conducted by a special and professional finite element analysis software package for helical bevel gear. The different processing methods of face-milled and face-hobbed are investigated. The tooth-surface equation of pinion and gear is established according to these two different types of methods. The principal curvatures, principal direction and contact ellipse of contact path is obtained by unload tooth contact analysis (ULTCA).The method of representation for geometry, mechanics and physics variables in the dynamic equation of spiral bevel gears and hypoid gear drives lay the roots for dynamic analysis.The 3-D quasi-static loaded tooth contact analysis model is established to obtain the transmission error, contact pattern, time-varying mesh stiffness and mesh point and line of action under different load.
Secondly, the multi-body and multiple-degree-of-freedom (MDOF) dynamic model of helical bevel is established. By applying modern analytical theory and method and analyzing mesh model and dynamic excitation, the bend-torsion-axes-swing degree of freedom and load, engine, pinion and gear multi-body model of vibration is obtained by lumped parameter method. This model takes into account synthesized transmission error, time varying stiffness, backlash, and gyroscopic effect. The differential equations of motion of gear system are solved by using advanced nonlinear algorithm and computer software.
Thirdly, the effect of time-varying friction coefficient and friction force on helical bevel gear dynamics is investigated. The elastohydrodynamic lubrication (EHL) theory was used to differentiate the different lubrication situation. Mixed EHL model is also proposed to predict the instantaneous friction coefficient at each contact grid cell along the contact zone under different load and speed condition. Comparison of different frictional behavior and dynamic response for hypoid gear and spiral bevel gear is made.
Finally, the effect of assembly error is qualitative analyzed. The misalignment error including hypoid offset error, shaft angle error, pinion axial position error, gear axial position error of hypoid and spiral bevel gear sensitivities are identified. At the same time, the effect of assembly error on frictional behavior and dynamic response are examined and differentiated. The influence of assembly error and time-varying friction force on gear system dynamic are compared and analyzed.
本文的主要目的就是建立有效的数学模型和分析技术,描述螺旋锥齿轮的啮合和动力学特性,综合考虑各种非线性因素,预测动态响应,以揭示螺旋锥齿轮传动系统的根本物理现象。本文进行了如下研究工作:
首先,通过利用专业螺旋锥齿轮有限元分析软件对齿轮系统进行三维造型建模和有限元建模分析。对螺旋锥齿轮Face-milled和Face-hobbed两种不同的加工方法进行了深入研究。针对两种不同加工方法的加工特点建立螺旋锥齿轮小轮和大轮齿面方程,通过无载齿面接触分析,求解接触路径上的主曲率、主方向和接触椭圆。确定非线性动力学方程中的几何量、力学量和物理量,为今后的动力学分析打下基础。建立基于三维准静态齿面加载接触分析模型,求解不同载荷作用下传动误差,接触区域印迹,时变啮合刚度,啮合点位置和啮合力作用线方向等啮合特征参数。
第二,建立螺旋锥齿轮传动系统多体多自由度非线性动力学分析模型。运用现代的理论和分析方法,通过对螺旋锥齿轮啮合模型和系统动态激励的分析,利用集中参数法建立了螺旋锥齿轮系统的弯-扭-轴-摆多自由度以及包括原动机和负载以及小轮和大轮的多体多自由度非线性动力学模型,该模型中考虑了系统的综合啮合误差、时变啮合刚度、摩擦力、陀螺效应以及齿侧间隙,得到其运动微分方程,运用先进的非线性算法及计算机软件求解动态响应。
第三,研究了混合润滑条件下摩擦力和时变摩擦系数对螺旋锥齿轮系统动力学特性的影响。通过研究弹流润滑理论,区分了齿轮不同的润滑状态。在混合润滑状态下建立基于Mixed-EHL理论的时变摩擦系数模型,利用该模型求解在不同载荷和转速条件下齿面啮合区域每个网格点的瞬时摩擦系数。比较了准双曲面齿轮和弧齿锥齿轮摩擦力行为的特点和各自在时变摩擦力影响下的动态响应。
最后,定性分析安装误差对螺旋锥齿轮系统动力学特性的影响。分别对准双曲面齿轮和弧齿锥齿轮的偏置距误差、轴交角误差、小轮轴向位置误差和大轮轴向位置误差的敏感性进行了对比分析。同时,考察了安装误差对摩擦力行为和动态响应的影响。比较了安装误差和摩擦力的对系统动力学特性的影响程度。
Helical bevel gear including hypoid gear and spiral bevel gear are widely used in power and motion transmission of the machines and equipments. It is one of the most complex types of gear drive system, which has the advantages of smooth working, high ratio of transmission, high efficiency advantages of transmission, compact structure, saving more space, resistant abrasion, long service life and low noise. With the development of high-speed and precision mechanical transmission, the demand of dynamic characteristics and stability of gear transmission system continues to increase. A more accurate prediction of gear dynamic characteristics becomes more and more important. Due to the complexity of gear mesh, such as time-vary mesh stiffness, impaction of tooth mesh in and out, backlash, friction drag, manufacturing and assembly errors and the comprehensive effect of the entire gear transmission system vibration caused by vibration coupling several kinds of vibrations from shaft, gear pair, gear box, engine and nonlinear characteristics, all this factors make the research of gear dynamics quite complicated. Therefore, in-depth theoretical and experimental studying the dynamic characteristics of gearing system and finding the general rule of vibration calculation has current significance at present stage.
The goal of this dissertation research is to establish more effective mathematical model and analytical techniques to characterize the mesh and dynamic behavior, predict vibratory response, and reveal the underlying physics of helical bevel geared rotor system. The major works of this dissertation are listed as follow.
Firstly, the 3-D geometrical model and finite element analysis is conducted by a special and professional finite element analysis software package for helical bevel gear. The different processing methods of face-milled and face-hobbed are investigated. The tooth-surface equation of pinion and gear is established according to these two different types of methods. The principal curvatures, principal direction and contact ellipse of contact path is obtained by unload tooth contact analysis (ULTCA).The method of representation for geometry, mechanics and physics variables in the dynamic equation of spiral bevel gears and hypoid gear drives lay the roots for dynamic analysis.The 3-D quasi-static loaded tooth contact analysis model is established to obtain the transmission error, contact pattern, time-varying mesh stiffness and mesh point and line of action under different load.
Secondly, the multi-body and multiple-degree-of-freedom (MDOF) dynamic model of helical bevel is established. By applying modern analytical theory and method and analyzing mesh model and dynamic excitation, the bend-torsion-axes-swing degree of freedom and load, engine, pinion and gear multi-body model of vibration is obtained by lumped parameter method. This model takes into account synthesized transmission error, time varying stiffness, backlash, and gyroscopic effect. The differential equations of motion of gear system are solved by using advanced nonlinear algorithm and computer software.
Thirdly, the effect of time-varying friction coefficient and friction force on helical bevel gear dynamics is investigated. The elastohydrodynamic lubrication (EHL) theory was used to differentiate the different lubrication situation. Mixed EHL model is also proposed to predict the instantaneous friction coefficient at each contact grid cell along the contact zone under different load and speed condition. Comparison of different frictional behavior and dynamic response for hypoid gear and spiral bevel gear is made.
Finally, the effect of assembly error is qualitative analyzed. The misalignment error including hypoid offset error, shaft angle error, pinion axial position error, gear axial position error of hypoid and spiral bevel gear sensitivities are identified. At the same time, the effect of assembly error on frictional behavior and dynamic response are examined and differentiated. The influence of assembly error and time-varying friction force on gear system dynamic are compared and analyzed.
引文
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