Tooth-root stress calculation of high transverse contact ratio spur and helical gears

In this paper, a non-uniform model of load distribution along the line of contact of spur and helical gears, obtained from the minimum elastic potential criterion, has been used, combined with the equations of the linear elasticity, to evaluate the tooth-root stress of high transverse contact ratio gears. The values of both critical stress and load conditions have been obtained and a complete analysis of the tooth bending strength has been carried out. As the load per unit of length at any point of the line of contact and any position of the meshing cycle has been described by a very simple equation, a complete study of the location and the value of the tooth-root stress has been carried out. From this study, a recommendation for the calculation of the bending load capacity of high transverse contact ratio spur and helical gears is proposed.     

基于载荷分担理论的渐开线斜齿轮热混合弹流润滑分析

沿接触线把斜齿轮分成许多小薄片,每一薄片看成具有当量角速度的直齿轮,根据欧拉方程得到任一接触点处的曲率半径和表面速度.然后基于载荷分担、弹流润滑和粗糙线接触理论,建立了考虑表面粗糙度的斜齿轮传动混合热弹流润滑模型.研究了斜齿轮传动稳态载荷分布下牛顿流体和Carreau流体时的润滑特性.结果表明:牛顿流体和Carreau非牛顿流体模型下,中心油膜厚度、油膜承载比例、油膜温升随时间和接触线的变化规律相同.牛顿流体下的摩擦系数较工程实际偏大.Carreau非牛顿流体模型下摩擦系数和工程实际相符,其随接触线啮合位置的变化规律与油膜厚度正好相反.     

表面粗糙度特征对齿轮接触区润滑特性的影响

大部分工程实际粗糙表面符合非高斯分布,并对齿轮接触副润滑特性有重要影响.将渐开线齿轮啮合过程中齿面接触等效为三维无限长线接触,建立了一个可分析直齿轮和斜齿轮的混合弹流润滑计算模型;采用基于快速傅里叶变换的数值仿真方法生成给定参数的非高斯粗糙表面;运用该模型对直齿轮和斜齿轮啮合过程进行分析,求得不同表面粗糙度特征齿轮在各个啮合点的油膜厚度、接触区载荷以及接触区比例的情况.结果表明:对于标准差相等的非高斯粗糙表面,偏度值对齿轮润滑状况的影响与工况紧密相关,在润滑良好的条件下,偏度值越小润滑状况越优;润滑恶劣的条件下,偏度值越大润滑状况越优;而在各种工况下,峰度值对齿轮润滑状况的影响都表现出峰度值越大润滑状况越优的特点.     

Modeling and analyzing of torsional dynamics for helical gear pair considered double and three teeth drive-side meshing

Helical gears are generally considered to be more stable than spur gears. But rattling of the helical gear transmission is found in the engineering practice. The torsional dynamics equations of helical gear pair in high-speed railway gearbox are established in order to reveal the rattling mechanism of helical gear transmission. Double and three teeth pair drive-side meshing are considered. The multi-state meshing zone, load distribution rate and time-varying stiffness determined by contact ratio are analyzed and calculated. The dynamic characteristic transition process of the system is analyzed according to the bifurcation diagrams and the corresponding top Lyapunov exponent (TLE) diagrams, phase portraits, Poincaré maps and time history spectrums of dynamic meshing force based on the calculation of these parameters. The tooth disengagement, tooth back-side contact and their parameter range are found. This study can provide theoretical basis for rattling suppression and transmission stability improvement of helical gear pair.

     

Comparison of spur,helical and curvilinear gear drives by means of stress and tooth contact analyses

Involute tooth surfaces are a successful technical solution for both spur and helical gear drives since they provide linear contact and a low-level function of transmission errors under good conditions of meshing. Tip relief is usually required to improve contact conditions during the transfer of meshing between adjacent pairs of teeth. Yet, unfavorable conditions of contact appear when shaft deflections and misalignments are present. Localization of contact through lead crowning is a solution that increases the cost of machining in both spur and helical gear drives. In this sense, the generation process of curvilinear gear drives provides localization of contact with no additional cost. Comparison of stresses and transmission error functions in spur, helical and curvilinear gear drives is investigated to show if the application of curvilinear gear drives yields some advantages respect to spur and helical gear drives. The three mentioned types of cylindrical parallel-axis gear drives are provided, firstly, with linear contact, and, secondly, with localized contact, for the purpose of comparison. Different misalignments conditions are taken into account by means of several numerical examples.     

渐开线斜齿轮非稳态弹流润滑数值模拟研究

建立了渐开线斜齿轮啮合的弹流润滑计算模型,将斜齿圆柱齿轮啮合的齿面接触等效为有限长线接触的弹流润滑问题.考虑斜齿轮啮合的实际因素,将斜齿轮啮合过程中的等效曲率半径和齿面载荷的变化反映到弹流润滑计算模型中,应用统一Reynolds方程方法求得轮齿在1个完整啮合周期内的瞬时弹流润滑数值解.结果表明：斜齿轮啮合线上各点处的膜厚、压力均有较大不同,各接触点处的油膜厚度受综合曲率半径的影响较大;斜齿轮传动非稳态效应相对较弱;小齿轮齿根附近和节点位置处润滑状态较差;适当增大压力角可以改善齿轮的润滑.     

斜齿轮弹流润滑下的接触疲劳寿命计算

经典齿轮接触疲劳强度理论是基于光滑表面赫兹干接触理论,而实际齿面具有粗糙度,且啮合轮齿多数处于混合润滑状态.本文基于齿轮润滑接触分析建立了渐开线斜齿轮的接触疲劳寿命计算模型.模型由齿轮润滑接触分析模型和基于次表面应力分布的疲劳寿命模型组成.首先将斜齿圆柱齿轮一对齿的瞬时啮合等效为两反向圆锥的接触问题,建立了齿轮的有限长弹流润滑计算模型,考虑了齿轮啮合周期内瞬时载荷、接触线长、卷吸速度等因素的影响,基于统一雷诺方程方法求得啮合齿对间的润滑压力和油膜厚度分布;在此基础上,计算轮齿接触区次表面的米歇斯应力分布,根据Zaretsky接触疲劳寿命计算模型,对齿轮组的接触疲劳寿命进行模拟预测.针对不同工况参数下接触疲劳寿命计算表明:润滑油黏度、轮齿表面粗糙度等因素对齿面接触疲劳寿命均有显著的影响.     

Non-Newtonian mixed elastohydrodynamics of differential hypoid gears at high loads

Prediction of friction and transmission efficiency are design objectives in transmission engineering. Unlike spur and helical involute gears, there is a dearth of numerical analysis in the case of hypoid gear pairs. In particular, it is important to take into account the side leakage of the lubricant from the contact as the result of the lubricant entrainment at an angle to the elliptical contact footprint. In the automobile differential hypoid gears, high loads result in non-Newtonian behaviour of the lubricant, which may exceed its limiting shear stress, a fact which has not been taken into account in the open literature. This results in conditions which deviate from observed experimental tractive behaviour. The paper takes into account these salient practical features of hypoid gear pair analysis under high load. It highlights a non-Newtonian shear model, which limits the lubricant shear behaviour. Prediction of friction and transmission efficiency is in line with those reported in the literature.     

渐开线直齿轮弹流润滑条件下的多轴疲劳寿命预估

本文中基于弹流润滑分析和次表面应力建立了渐开线直齿轮多轴疲劳寿命计算模型.相对于传统的单轴疲劳模型,考虑了齿轮固定点的应力历史和材料属性对疲劳寿命的影响,并可以得到齿轮在完整啮合过程中的寿命分布.首先建立齿轮的有限长弹流计算模型,得到齿轮啮合过程中的油膜压力和油膜厚度,再根据油膜压力计算出次表面的应力分布;通过分析齿轮计算区域随啮合过程移动的关系,得到固定点的应力历史,再根据基于应力历史的多轴疲劳寿命模型对齿轮的完整啮合过程进行寿命预估.计算分析了不同粗糙度幅值对轮齿各点寿命大小和分布的影响.研究表明:齿面粗糙度对疲劳寿命的影响显著,随着粗糙度幅值的增大,表层下最大应力向齿面移动,导致低疲劳寿命区向齿面发展且逐步扩展到整个单齿啮合区;而表面粗糙度降低到一定程度则对疲劳寿命的影响变得不明显.     

Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.     

Dynamics of a Gear System with Faults in Meshing Stiffness

Gear box dynamics is characterised by a periodically changing stiffness. In real gear systems, a backlash also exists that can lead to a loss in contact between the teeth. Due to this loss of contact the gear has piecewise linear stiffness characteristics, and the gears can vibrate regularly and chaotically. In this paper we examine the effect of tooth shape imperfections and defects. Using standard methods for nonlinear systems we examine the dynamics of gear systems with various faults in meshing stiffness.     

磨损与动力学耦合的行星传动齿轮动力学研究

行星齿轮磨损会导致齿轮齿侧间隙非线性增大、传动精度下降、齿面冲击力增大, 进而会导致齿轮传动系统振动加剧, 因此需要对行星齿轮的齿面磨损与动力学耦合特性进行研究. 本文构建了齿轮非线性磨损与考虑齿轮齿侧间隙的非线性动力学耦合计算模型, 对行星传动齿轮磨损动力学特性进行了研究. 首先建立齿轮啮合非线性动力学模型, 获得齿轮运行过程中的非线性啮合力; 进一步将非线性啮合力与齿轮齿面磨损模型相结合, 研究齿轮齿面磨损分布规律; 并根据齿轮磨损后的齿侧间隙对齿面重构, 同时对齿轮动力学模型进行更新; 进而得到行星齿轮传动中动态啮合力和磨损特性的变化趋势, 并获得齿轮传动系统齿轮齿向振动响应. 数值计算结果表明, 行星齿轮磨损导致齿轮在单?双齿交替啮合时产生的冲击增大, 同时太阳轮?行星轮啮合齿对对磨损较为敏感, 齿面啮合条件剧烈恶化, 是造成行星齿轮传动性能退化的主要原因, 本文研究结果为行星齿轮传动系统运行状态评估与可靠性预测提供了理论基础.      

双联行星轮角度偏差对系统均载特性的影响

为了研究双联行星齿轮在实际设计参数下,其相对角度偏差对复合行星传动系统动力学特性的影响,采用集中参数法建立了3K-I型行星齿轮动力学模型,模型中将双联行星齿轮的相对角度偏差转化为啮合副齿侧间隙的变化,考虑了双联齿轮角度偏差、轮齿侧隙和时变啮合刚度等非线性因素,采用龙格库塔法求解了系统的时域响应并计算其均载系数。分析了不同工况、偏差下系统的动态特性。结果表明,存在双联行星轮角度偏差时,轻载下更容易发生齿轮的脱齿与冲击,系统的均载系数随着双联行星轮角度偏差差值及系统的负载降低而增大,各组行星轮角度偏差分布越集中,角度偏差对系统均载特性的影响越小;角度偏差分布同号时,对系统中某一对齿轮的承载影响明显;角度偏差分布异号时,对系统均载特性的影响最大。     

基于热弹流润滑的双渐开线齿轮温度场研究

针对现有双渐开线齿轮温度场计算模型不考虑油膜润滑影响的问题,根据双渐开线齿轮啮合特点,提出采用“分段法”建立适合双渐开线齿轮的热弹流润滑模型,综合有限元法和热弹流润滑方法对其本体温度进行研究,并以润滑油膜为热源对其瞬时温升进行研究,最后分析了齿腰分阶参数对双渐开线齿轮温度场影响以及与普通渐开线齿轮温度场差异. 结果表明:双渐开线齿轮本体温度沿齿宽方向呈非对称分布,主动轮最高本体温度偏向齿根啮入端,从动轮偏向齿顶啮出端;啮合齿面间的油膜瞬时温升明显高于两齿轮界面温升,且主动轮界面瞬时温升高于从动轮;齿腰分阶参数变化对双渐开线齿轮温度场影响较小;双渐开线齿轮与普通渐开线齿轮的本体温度及齿面瞬时温升区别不大.      

应用边界元方法的正齿轮接触应力分析

本文应用边界元方法,采用在齿廓上按赫兹分布的载荷,直接计算了在接触区附近齿表面和齿体内的真实接触应力。并通过计算定量地分析了各齿轮参数及齿面摩擦对接触应力的影响。为高精度的齿轮设计、寿命预测和渗碳层厚度选择提供了可靠的依据。     

渐开线直齿圆柱齿轮接触疲劳失效成因再分析

联合采用表面失效分析和有限元应力分析的方法,研究了渐开线直齿圆柱齿轮接触疲劳失效的成因.结果表明:由啮入线至第一次双对轮齿啮合结束部分齿面上密集的表面点蚀与该段啮合齿面相对滑移大,所消耗的摩擦功最大,摩擦应力大于第二次双齿啮合部分,而且最大剪应力更靠近齿面等因素有关.靠近节线的齿根齿面上的片状大块剥落属次表面点蚀,是由于该部位的次表面剪应力最大,位置最深,并且承受了前一对轮齿脱离啮合带来的冲击作用.紧邻啮入线附近齿面的较浅的剥落点蚀是由于承受啮入冲击,最大剪应力较大且出现位置较浅,齿面相对滑移最大所造成的,并与该处的表面点蚀坑有关.     

Direct calculation of AGMA geometry factor J by making use of polynomial equations

The available sources and procedures for determination of AGMA geometry factor J are tables, charts and semi-analytical methods. When computerized gear design is considered, usage of tables requires a number of interpolations; usage of charts requires curve fitting; and usage of semi-analytical methods needs a numerical algorithm and may have convergence problems. As an alternative to these, polynomial equations for direct calculation of AGMA geometry factor J are derived for external spur gears. Thus, it is made possible to evaluate the J factor easily and with minimum process time. J factors are determined being independent of the highest point of single tooth contact (HPSTC). Derived equations can be used to calculate the tooth root stresses corresponding to loads acting on any point on the involute tooth profile. Thus, cases where the center distance is increased for providing backlash or for operating the gears at a desired exact center distance can easily be handled by determining the corresponding new HPSTC. A computer program is developed to demonstrate the usage of the derived equations. The method can also be used for determination of the J factors for gears with non-standard proportions.     

渐开线直齿轮牛顿流体非稳态热弹流润滑分析

齿轮的非稳态弹流润滑问题由于啮合过程中滑滚比、曲率半径、卷吸速度和载荷变化范围较大,因此数值计算稳定性很差。而考虑热效应的齿轮非稳态弹流润滑问题,数值计算就更困难。本文应用多重网格技术,求得了齿轮牛顿流体润滑情况下,非稳态热弹流润滑问题的完全数值解。     

Using Hob Offset to Balance Dynamic Strength in Spur Gears*

An analytical study of the effect of hob offset on dynamic tooth strength of spur gears is presented. The study was limited to equal and opposite offset values applied to the pinion and gear to maintain the standard operating center distance. The analysis presented is performed using a new version of the NASA gear dynamics code DANST.

The operating speed of a transmission has a significant influence on the amount of hob offset required to equalize dynamic stresses in the pinion and gear. In the transmission studied, at low speeds, the optimum hob offset was found to fluctuate within a range. At higher speeds, the optimum value is constrained by the minimum allowed thickness at the tip of the pinion tooth. For gears that must operate over a range of speeds, an average offset value can be used. Spur gears designed with the procedure presented here can have significant improvements in load capacity.     

Simulation of impacts in geartrains using different approaches

Gear hammering in diesel engines is a well-known phenomenon in geared drives, exhibiting not only noise but also influencing the performance and durability of diesel engines. Gear hammering is characterised by flanks in contact that lift off and cause impacts when the contact reestablishes, which induces high, sharp dynamic loads. The knowledge of these contact forces is very important for the design of gears. Since contact forces in meshing gears are extremely difficult and expensive to measure, the simulation of these forces plays an important role. Nowadays, these contact simulations are usually carried out within overall models of entire engines using commercial multibody programs that provide submodels for gear contacts, usually based on rigid-body models. However, to reduce inertia effects, gears in geartrains are often designed with very thin bodies, whose elastic compliance influences the contact behaviour to a large extent. For a closer insight into the dynamic behaviour, and especially the influence of thin gear bodies during impact, a typical gear pairing is selected and impacts between one tooth pair are investigated for different boundary and initial conditions with three different models. Besides a multibody model, similar to those used in commercial multibody programs, a fully nonlinear finite-element model and a modally reduced model in combination with a local force law is used. The results of the different approaches are benchmarked in terms of accuracy and numerical effort.