Tooth contact analysis of face-gear drives

Based on the face-gear generation process, the analytical geometry of the face-gear drive and its mathematical model for tooth contact analysis of the face-gear and the spur pinion meshing were derived. The tooth contact paths and the transmission errors due to assembly error along the axis of face-gear, misalignment of crossed and angular displacements between axes of spur pinion and face-gear were analyzed. Furthermore, the conditions of undercutting and pointing in the generation process were studied. Several design charts were then developed. Finally, the conditions of undercutting and pointing for a face-gear drive were all identified in meshing with and without assembly error along the direction of the axis of face-gear and misalignment of angular displacement between axes of spur pinion and face-gear. Several numerical examples were also presented.     

含误差的直齿轮的齿廓修形

依据齿轮传动载荷与轮齿变形的关系,推导出定载荷条件下,修形直齿轮静态传动误差与齿对综合修形参数的关系表达式。提出含制造误差修形齿轮修形参数的确定原则:理论设计修形齿轮的最大综合修形量应能消除含误差齿廓轮齿在啮入和啮出位置产生的几何干涉,理论设计修形齿轮的静态传动误差应保持最小的变化。给出了合制造误差修形直齿轮修形参数的计算公式,阐述了相应的确定方法。系统动态响应计算表明该方法所获得的修形齿轮具有良好的减振效果。     

数控成形砂轮磨齿机在机测量研究

提出了利用数控机床的回转运动（C轴）和径向进给运动（X轴）形成理论渐开线，实现齿形误差的在机测量方法。通过使用Num数控系统的YK73125数控成形磨齿机上的渐开线直齿轮齿形误差的在机测量实验验证了该方法的可行性。     

修形直齿锥形齿轮啮合特性研究

为了避免直齿锥形齿轮发生边缘接触和降低对安装误差的敏感性,对小轮进行齿向弧线修形和齿廓抛物线修形.采用假想齿条的方法,模拟齿条与被加工齿轮的啮合运动关系,推导出齿轮齿面的数学模型,根据两齿面在啮合过程中的连续切触条件,建立了齿轮接触分析模型.仿真表明,修形后的直齿锥齿轮副,对安装误差的敏感性降低,且没有发生边缘接触,改...     

Localised tooth contact analysis of single envelope worm gears with assembly errors

This paper presents the basis of loaded tooth contact analysis and predicts the influence of assembly errors on localised contact stress distribution in single enveloping (cylindrical) worm gearing during a meshing cycle. A method for loaded tooth contact analysis, geometry and kinematics of such gear pairs is developed. The method accounts for the effects of tooth composite deflection caused by bending, shearing, foundation, tooth contact deformation and initial profile separation due to assembly errors. The method includes the determination of contact lines, load and stress distribution due to assembly errors. Because of the complex geometry of worm gear teeth, the tooth bending stiffness is calculated using the slicing technique developed earlier by the authors. Classical Hertz theory is used for calculating contact stress and deformation. A computer program based on the presented method has been developed and used to study the influence of errors on mating teeth contact. It is shown that the governing factors in loaded gears with assembly error are the mesh stiffness and the amount of error which is linked to load sharing between adjacent tooth pairs. A numerical example is presented to further clarify the outlined method.     

Tooth flank, undercutting and tooth pointing of spherical gears

By the developed mathematical model of spherical gears from literature [16], conditions of tooth undercutting of spherical gears with convex and concave teeth and tooth pointing conditions of the proposed concave spherical gear are derived. Besides, tooth flank profile deviations, limit curves of tooth undercutting and the beginning points of tooth undercutting and tooth pointing of the proposed spherical gear can be obtained by applying the spherical gear tooth mathematical model and the developed computer simulation programs, as illustrated by five numerical examples.     

考虑边缘接触直齿面齿轮传动轮齿接触分析

主要分析了直齿面齿轮传动在考虑边缘接触条件下的啮合.从理论上推导了求解边缘接触的数学模型及边缘接触点满足的约束条件,利用相邻接触点主曲率方向相似的方法确定了当发生边缘接触时在接触曲面上的主曲率方向.结果显示,边缘接触更容易在刚进入啮合时产生,并且在边缘接触点处几何传动误差不再为O.     

修形直齿轮的啮合干涉与啮合间隙

修形直齿轮因误差因素在齿对的啮入和啮出位置会出现啮合干涉或啮合间隙。本文利用齿对的啮合变形与齿廓法向修形量、齿廓误差的关系,推导了啮合齿对在啮合线的啮入和啮出位置所存在的几何干涉或啮合间隙,得到了啮合重合度小于2和3的齿轮副啮合齿对的最大干涉量或间隙量的计算表达式,为分析修形直齿轮的啮合状态和修正齿轮修形参数奠定了基础。     

二齿差摆线针轮行星传动中摆线轮等效代换齿廓的研究

二齿差摆线针轮行星传动具有较强的齿面抗胶合能力，但由于二齿差摆线轮齿廓曲线的不连续性，给传动带来了不良的影响，并造成工艺上的复杂性，对此，本文提出了采用完整的短幅外摆线的等距曲线作为摆线轮齿廓，来等效地代换二齿差摆线针轮行星传动中的摆线轮齿廓的理论，并给出了求解等效代换齿廓参数的数学模型，计算方法和求解实例，采用等效代换齿廓既可消除尖点，又能保证传动性能相同。     

摆线传动接触载荷与效率的关联模式及试验验证

为揭示摆线针轮传动的接触载荷与效率的关联模式,基于牛顿法和弹性接触理论提出了一种可实现摆线针轮多齿啮合点精确定位、接触载荷分布确定和传动效率计算的理论分析方法。该方法建立了摆线轮刚体动力学方程,并将传动效率表达为摆线轮所受合外力的函数;通过摆线轮齿廓离散点与针齿圆、输出销/孔的相对几何关系与接触条件来确定摆线轮接触载荷;将摆线轮接触载荷转化为广义力,形成由摆线轮合外力/力矩表达的传动效率函数。以某单级摆线减速器为例,分析设计参数与运行工况对传动效率的影响。通过专用试验台进行对比试验,结果表明：理论计算传动效率不随负载变化;低负载时,传动效率偏低,且随负载增大而快速上升;高负载时,传动效率变化不大,趋于稳定值,且与理论值接近。该方法仅可作为高负载时摆线针轮传动效率评估的简明方法。     

有误差的螺旋锥齿轮传动接触分析

以多体系统误差建模理论和齿轮啮合原理为基础,提出含有机床运动几何误差以及齿轮副安装误差的螺旋锥齿轮齿面接触分析(Error tooth contact analysis, ETCA)方法。以SGM法(大轮展成法加工,小轮变形法加工)加工的弧齿锥齿轮为例,通过ETCA分析,得到机床运动误差和安装误差对螺旋锥齿轮齿面加工质量影响的定量关系,对ETCA和TCA的结果进行对比分析,结果表明机床运动误差和安装误差对螺旋锥齿轮的齿面接触质量有较大的影响,为了通过齿面接触分析达到更准确的反调加工参数的目的,采用ETCA的分析结果指导加工参数反调更为合理。     

Influence of tooth modifications on tooth contact in face-hobbed spiral bevel gears

In this study the influence of tooth modifications induced by machine tool setting and head-cutter profile variations on tooth contact characteristics in face-hobbed spiral bevel gears is investigated. The concept of face-hobbed spiral bevel gear generation by an imaginary generating crown gear is applied. The modifications of tooth surfaces are introduced into the teeth of both members. The lengthwise crowning of teeth is achieved by applying a slightly bigger radius of lengthwise tooth flank curvature of the crown gear generating the concave side of pinion/gear tooth-surfaces, and by the variation of machine tool settings in the generation of pinion/gear teeth. The ease-off in the tooth height direction of meshing tooth surfaces is achieved by applying a head-cutter whose profile consists of two circular arcs, instead of a straight-line. The method of tooth contact analysis applied determines the path of contact, the potential contact lines, the separations along these lines, and the transmission errors. A computer program implements the method. By using this program the influence of the variation of machine tool settings and of head-cutter geometry on tooth contact is investigated and discussed.     

基于啮合特性直齿锥齿轮分支传动静力学均载分析

为了改善直齿锥齿轮分支传动系统性能,提出基于啮合特性的静态均载分析方法。通过建立考虑中心轮安装误差的单级分支系统多体齿轮齿面接触分析(MTCA)方法,获得各齿轮副初始接触间隙;计及该间隙引起的啮合偏差,采用集中参数法主要考虑中心轮的浮动特性,建立考虑其支撑变形、各齿轮扭转变形相互耦合的静力学平衡方程,获得各分支均载系数,分析了载荷、安装误差、支撑刚度等对系统均载系数的影响。结果表明,随支撑刚度、安装误差的增大,均载系数逐渐增大,轴向对正误差和偏置对正误差对均载系数的影响是等效的;随转矩的增加,均载系数逐渐减小;各分支齿轮副的几何传动误差大小反映了均载系数变化规律,即齿面初始间隙越大,承担的载荷越小。研究结果为高精度直齿锥齿轮分支系统的均载分析提供了理论参考。     

Dynamic interaction between contact loads and tooth wear of engaged plastic gear pairs

This study investigates the interaction between the dynamic contact load and the tooth profile wear of POM and Nylon 66 plastic gear pairs. A dynamic model of a plastic gear pair is presented. This model incorporates the effects of position-varying tooth mesh stiffness, damping ratio, load sharing, tooth profile wear and temperature on the dynamic contact load. The tooth wear equation developed by Flodin and Andersson [Simulation of mild wear in spur gears. Wear 1997;207:16–23] is utilized to simulate tooth wear and tooth profile variation. The variation of the contact load generated by the cumulative tooth profile wear is simulated and examined. A computational algorithm is developed to simulate the interaction between a dynamic contact load and tooth profile wear. Numerical results demonstrate that the dynamic load histogram of an engaged plastic gear pair can change markedly due to tooth wear.     

摆线针轮啮合副加工误差的综合分析

本文根据摆线针轮啮合副的工作原理及性能要求,利用误差计算方法对摆线针轮啮合副的误差进行了综合分析与计算,从而推导出摆线针轮啮合副误差的一般计算公式,进而运用这些公式对摆线全液压转向器（马达）的摆线齿轮（转子）的齿形和针轮（定子）的齿形的误差进行了实际计算。     

变位非正交面齿轮副小轮齿向修形轮齿接触分析

建立了变位非正交面齿轮的加工坐标系和啮合坐标系,推导了变位小轮及变位非正交面齿轮的齿面方程,计算得到了面齿轮数值齿面,分析了变位对非正交面齿轮齿宽的影响。在变位的基础上研究了对小轮进行齿向鼓形修形,而面齿轮不修形的修形方式。分别对未变位、变位、变位加小轮齿向修形的三种非正交面齿轮传动形式进行考虑安装误差的轮齿接触分析。研究表明：随着变位系数增大,非正交面齿轮最小内半径、最大外半径及极限齿宽均减小;变位不影响非正交面齿轮副的接触规律;小轮齿向修形能降低接触轨迹对安装误差的敏感性,会引起幅值较小的直线型传动误差。     

Load sharing and stress analysis of single enveloping worm gearing considering transmission errors

A procedure, known as the slicing method, which has been developed earlier by the authors to estimate the load share of single enveloping (cylindrical) worm gear drives and to calculate the instantaneous tooth meshing stiffness and stresses, is extended in this study to investigate the effect of transmission error on loading and stresses. Tooth composite deflection caused by bending, shearing, contact deformation and initial separation due to profile mismatch are determined in the analysis as well as tooth stiffness variation during the course of contact of both the worm and the wheel. The elasticity theory was used to calculate fillet stresses and deformation of both the worm and wheel. Numerical examples are presented to illustrate the response of worm gear set in the presence of transmission error and under given loading conditions. The procedure was proven to be safe and reliable in rating the load capacity and useful in the design of worm gear drives with compact size and high power transmission.     

行星线齿变速器基于装配误差的传动精度分析

针对一种以线齿轮为核心传动元件的行星线齿变速器,研究了装配误差对其传动精度的影响。介绍了该行星线齿变速器的结构组成及不同传动方案的特点,建立了轮系空间啮合坐标系,以及啮合时线齿行星轮、线齿太阳轮以及外线齿圈的参变量之间的定量关系;分析了外线齿圈与线齿行星轮之间以及线齿行星轮与线齿太阳轮之间的角度误差和位移误差的产生原因,并采用齿面接触分析方法(TCA),对行星线齿变速器的装配误差模型进行了数值分析。结果表明,在行星线齿变速器的装配中,绕水平轴旋转的角度误差、外线齿圈纵截面内的位移误差以及线齿行星轮横截面内的位移误差,对传动精度有较大影响。研究对该行星线齿变速器的设计和装配具有指导意义。     

含侧隙齿轮副的动载荷分析

以振动理论为基础，提出一种考虑齿轮拍击振动的齿轮动载荷的数值计算方法。建立计算动载荷的齿轮冲击模型，在模型中考虑了齿轮正、反冲击时实际的啮合刚度，并给出啮合柔度的计算方法。分析在考虑静态传递误差、啮合刚度、侧隙、摩擦力及外部扭矩变化等多种激励时，作用在轮齿上的动态载荷以及整个齿轮上的综合动态载荷的计算公式。最后通过实例分析作用在轮齿上的动态载荷、综合动态载荷变化规律以及相关激励参数对动态载荷的影响。     

摆线减速器轮齿啮合的数学模型及求解

以非赫兹弹性接触问题对齿轮传动轮齿啮合进行研究，建立起了新的摆线针轮单齿啮合的柔度矩阵数学模型，并结合摆线减速器针齿修形计算，重点讨论了通过计算机编程，正确求解这一数学模型的方法。