EFFECT OF INVOLUTE CONTACT RATIO ON THE DYNAMIC PERFORMANCE OF SPUR GEAR WITH NOTOOTH PROFILE MODIFICATION

The effect of involute contact ratio on the torsional vibration behavior ofspur gear-pair is studied analytically through a mass-spring model. The tooth stiffness in model notonly has a relation with time, as many prior studies presented, but, more important, with involutecontact ratio (ICR) as well. The ICR embodies its impact on the spur gear's dynamic performancethrough changing the proportion of tooth stiffness when there are n+1 teeth in contact to toothstiffness when there are n teeth in contact. A couple of curves about impact of ICR on the gear'sdynamic performance are presented, and they clearly demonstrate that the model can accuratelydescribe the effects of ICR on dynamic transmission error. Finally, some conclusions useful toreduce vibration and noise of gear-pair are proposed.

Effects of Comprehensive Eccentricity of Involute Cam on Gear Profile Deviations

The manufacturing accuracy of ultra-precision master gears signifies the technological capability of the ultra-precision gear.Currently,there is little report about the manufacturing technologies of ultra-precision master gears at home and aboard.In order to meet the requirement of grinding ultra precision master gear,the gear grinder with flat-faced wheel Y7125 is chosen as the object machine tool and the geometric model of its precision generating part,the involute cam,is established.According to the structure of the involute cam,the effective working section and its adjustable range of the cam are determined,and the mathematical expressions of the effects of comprehensive eccentricity of the involute cam on gear profile deviations are derived.According to the primary harmonic trends of the deviation curve,it is shown that gear profile form and slope deviations in different work generating sections of the involute cam are different which the latter changes with the cam eccentricity obviously.Then,the issues of extreme values and methods of error compensation are studied and the conclusion that large adjustable range is benefit to search the optimal involute-cam section which is responding to the minimum gear profile deviations is obtained.A group of examples are calculated by choosing master gears with d=120 mm and m=2-6 mm and an involute cam with base diameter djcam =117 mm.And it is found that the maximum gear profile deviation counts for no more than 5% of the cam eccentricity after error compensation.A gear-grinding experiment on the master gear with m=2 mm is conducted by choosing different sections of the involute cam and the differences of gear profile deviations then the existence of the cam eccentricity are verified.The research discloses the rule of gear profile deviations caused by the comprehensive eccentricity of the involute cam and provides the theoretical guidance and the processing methods for grinding profile of the ultra precision master gear.

STUDY ON THE LUBRICATION FACTOR OF INVOLUTE SPUR GEARS

Based on a lot of numerical solutions to the problems of the thermalnon-Newtonian elastohydrodynamic lubrication and some fatigue tests with rollers, the lubricationfactor of involute spur gears (called gear for short) is investigated. The results suggest that gearlubrication effects bear close relations to a dimensionless parameter D which is syntheticallydetermined by gearing rotational speed, load, material, dimension and lubricant viscosity. When D<=8, the gear fatigue life increases as the lubricant viscosity is increased; When D>8, however, thelife decreases with the viscosity addition, which is in marked contrast to the lubrication factorZ_L recommended by the International Standard for Computing Cylindrical Gear Strength. At the end, aset of formulae for calculating gear lubrication factors suitable for different working conditionsare advanced.

Characteristic of involute slope modification of asymmetric spur gear

The meshing characteristic of asymmetric involute spur gear was studied, the equations of the geometric shape of the asymmetric gear for both sides were deduced, and the equations of contact ratio and the key points of contact were also obtained.Meanwhile, an involute slope modification method considering the effects of static transmission errors was proposed based on the meshing properties. The characteristic of the involute slope modification was analyzed by changing different modification parameters.The mesh stiffness and synthetic mesh stiffness of unmodified and modified asymmetric spur gears were investigated. Furthermore,the spectrums of synthetic mesh stiffness under different modification parameters were compared. Research results showed that the modification parameters influence the meshing performance of gear pairs, and the proposed modification method was feasible to improve the transmission performance of gear pairs with appropriate modification parameters.

Oil Film Stiffness of Double Involute Gears Based on Thermal EHL Theory

Lubrication failure is one of the main failure forms of gear failure.Time varying meshing stiffness is an important factor affecting the dynamic behavior of gears.However,the influence of oil film stiffness is usually ignored in the research process.In this paper,according to the meshing characteristics of double involute gears,based on the non-Newtonian thermal EHL theory,a new calculation method of normal and tangential oil film stiffness for double involute gears is established by the idea of subsection method.The oil film stiffness difference between double involute gears and common involute gears is analyzed,and the influence of tooth waist order parameters,working conditions,and thermal effect on the oil film stiffness are studied.The results reveal that there are some differences between normal and tangential oil film stiffness between double involute gears and common involute gears,but there is little difference.Compared with the torque,rotation speed and initial viscosity of the lubricating oil,the tooth waist order parameters have less influence on the oil film stiffness.Thermal effect has a certain influence on normal and tangential oil film stiffness,which indicates that the influence of thermal effect on the oil film can not be ignored.This research proposes a calculation method of normal and tangential oil film stiffness suitable for double involute gears,which provides a theoretical basis for improving the stability of the transmission.

Application of Isogeometric Analysis Method in Three-Dimensional Gear Contact Analysis

Gears are pivotal in mechanical drives,and gear contact analysis is a typically difficult problem to solve.Emerging isogeometric analysis(IGA)methods have developed new ideas to solve this problem.In this paper,a threedimensional body parametric gear model of IGA is established,and a theoretical formula is derived to realize single-tooth contact analysis.Results were benchmarked against those obtained from commercial software utilizing the finite element analysis(FEA)method to validate the accuracy of our approach.Our findings indicate that the IGA-based contact algorithmsuccessfullymet theHertz contact test.When juxtaposed with the FEA approach,the IGAmethod demonstrated fewer node degrees of freedomand reduced computational units,all whilemaintaining comparable accuracy.Notably,the IGA method appeared to exhibit consistency in analysis accuracy irrespective of computational unit density,and also significantlymitigated non-physical oscillations in contact stress across the tooth width.This underscores the prowess of IGA in contact analysis.In conclusion,IGA emerges as a potent tool for addressing contact analysis challenges and holds significant promise for 3D gear modeling,simulation,and optimization of various mechanical components.

EFFICIENCY OF INVOLUTE GEAR DRIVES WITH UNDERCUT PINIONS USED IN WRIST WATCHES

In wrist watches gear drives with undercut pinions are used. Those pinions have got a tooth number as few as 5—7. The efficiency of such gear drives is important because of the limited energy supply. The mean efficiency of gear drives can be expressed as Em=1/（φ2-φ1）（integral from φ1 to φ2（E（φ）dφ）） in which 01, <sub>2 are the approach angle and the recess angle respectively. In the discussion we assume that the friction coefficient between the two tooth flanks is constant because the sliding speed is relatively low in wrist watches and the main purpose is to find out the relation between gear parameters and its efficiency rather than to calculate the exact value of the efficiency. In normal conditions o1 and o2 can be found from the gear geometry without much difficulty. However when the pinion is an undercut one we have to first find the forming diameter where the involute profile starts. In this paper equations for the computation of the efficiency and the contact ratio of those gear drives are discussed.

TRANSMISSION MOMENTARY EFFICIENCY BASED ON THE D'ALEMBERT-LAGRANGE EQUATION FOR INVOLUTES GEARS

The D'Alembert-Lagrange equation is introduced and used to derive theformulas of momentary efficiency for external gearing of standard involutes spur gears. The gearingswith correct and increased center distance are discussed. The momentary efficiency formula iscalculated and analyzed using software Matlab. The derived formula of momentary efficiency is alsocompared with the traditional formula.

Load distribution of involute gears along time-varying contact line

Base on the theory of energy minimization, a numerical algorithm is established to calculate load distribution, and the relationship curve of spur gear load distribution is obtained, and the load distribution ratio changes from 033 to 067 in double contact zone. This theory is adopted to compute the load distribution of helical gear along time-varying contact line, and the load distribution varies with the instantaneous position of the meshing point and the length of contact line, and the maximum value of load appears at the pitch point. Compared with the load distribution results, the helical gear changes more smoothly than spur gear. The load distribution provides a basis for calculate tooth bending deformation and critical stress.

非对称齿轮啮合刚度精确求解及齿廓参数分析

非对称齿轮是一种轮齿两侧采用不同压力角的新型渐开线齿轮,其齿廓参数对齿轮齿廓形状、运动特性、承载能力、传动效率和动力学特性等有至关重要的影响。基于齿轮啮合原理,运用李特文矢量法,根据加工刀具齿廓参数方程推导了非对称齿轮全齿廓方程;应用材料力学理论,建立非对称齿轮时变啮合刚度的数值求解模型,并通过有限元方法验证了该模型的有效性。基于所建立数值分析模型进行算例研究,总结了刀具圆角、压力角和变位系数对齿轮啮合刚度和重合度的影响规律。结果表明,随着轮齿工作侧压力角的增大,齿轮齿根厚度增加,单齿对的啮合刚度加大,齿轮的承载能力得到有效提高。加工工作侧轮齿齿廓刀具圆角的增大,也有利于提高齿轮的啮合刚度;同时,工作侧齿廓压力角的增大会使齿轮的齿顶厚度降低、啮合重合度减小,会导致齿轮的齿顶强度和冲击韧性降低。非对称齿轮齿廓设计时,应根据实际工况合理设置压力角参数。

渐开线变齿厚齿轮啮合冲击分析

基于弹性接触动力学理论和齿轮传动物理模型,对渐开线变齿厚齿轮啮合冲击问题进行了分析,建立了变齿厚齿轮全齿啮合接触冲击模型.通过模拟轮齿啮合接触状态,对轮齿啮合区域展开了讨论,利用LS-DYNA软件进行接触碰撞分析,分别研究了冲击合力和冲击应力随转速和冲击位置的变化规律.结果表明:转速和冲击位置对冲击合力和冲击应力影响显著,且冲击最大应力主要集中在齿顶和轮齿的大小端附近;转速越大,轮齿受载面与非受载面之间的冲击应力相差越大.

渐开线圆柱齿轮接触分析计算与仿真研究

为方便实现渐开线圆柱齿轮接触分析,开发出渐开线圆柱齿轮啮合接触分析和齿廓修形算法程序.采用渐开线直齿轮参数方程精准建模,获得齿廓接触点的曲率半径,在考虑实际载荷的情况下,改进了赫兹接触公式来求解齿轮啮合接触应力.为减缓齿轮啮合冲击,在算法程序中提供了5种齿廓修形方式及其修形后的接触应力分析方法.与行业公认的齿轮设计仿真软件MASTA计算结果进行对比.结果表明,该算法程序接触应力求解精度较高,相对MASTA平均误差约为5.04%,齿廓修形后的接触应力分布一致,验证了该算法程序的合理性.

考虑弹性误差的渐开线圆柱齿轮半解析接触模型与传动精度分析

在弹性和误差作用下,齿轮接触的高效、精准建模是研究齿轮传动精度的重要基础。为准确分析渐开线圆柱齿轮的弹性和误差对传动精度的影响,提出了一种考虑弹性误差的齿轮半解析接触模型。以渐开线斜齿圆柱齿轮为例,首先,结合齿轮的齿面方程与齿廓误差建立误差齿面模型,得到了误差齿面方程。然后,将解析法和有限元法相结合,建立了弹性误差作用下的齿轮半解析接触模型,并推导了其变形协调方程和力平衡方程。最后,建立齿轮的完整和局部有限元模型,通过提取接触齿面的柔度矩阵来计算齿轮的传动误差。应用所构建的半解析模型分析了齿廓误差和弹性变形对齿轮传动精度的影响规律,并与有限元仿真结果进行对比,验证了该半解析模型的高效性和精准性。研究结果可为渐开线圆柱齿轮在工程中的应用提供理论指导。

蜗杆斜齿轮传动副时变啮合刚度计算及非线性动力学特性分析

蜗杆斜齿轮传动副作为一种交错轴点接触传动形式,掌握其时变啮合刚度及啮合侧隙与误差等引起的非线性动力学行为是减振降噪的核心。文中基于势能法修正并求解蜗杆斜齿轮传动副理论时变啮合刚度,建立考虑齿侧间隙-综合误差及时变啮合刚度的动力学模型,采用4~5阶Runge⁃Kutta法对系统动力学方程进行数值求解,得到蜗杆斜齿轮传动副动力学特性曲线。结合时域图、Poincaré图及FFT频谱图,分析齿侧间隙、阻尼比及载荷等因素对传动副振动和冲击状态的影响,结果表明:较小的误差幅值及较大的阻尼比可使系统处于稳定的周期运动状态。研究结果对优化蜗杆斜齿轮传动系统的振动噪声有指导意义。

正交面齿轮传动的两类界线

为了建立系统的正交面齿轮传动啮合界线与曲率干涉界线理论,本文通过求解非线方程组获得了两类界线的数值解。证明了啮合界线在小轮齿面上是渐开线;其共轭线也是渐开线。发现干涉界线一般有2条,近齿面干涉界线存在于面齿轮内端附近,大致沿齿高走向。近齿面干涉界点存在平凡解,是干涉界线和啮合界线共轭线的交点,位于传动副瞬时相对转动轴上。基于近齿面干涉界线,建立了面齿轮无根切内径公式,通过数值迭代可以获得内径准确值;还导出了无根切内径估算公式,便于设计中应用。

基于锥形铣刀的渐开线齿轮两轴仿形倒角计算与加工

圆柱齿轮的加工毛刺与运输磕碰会对齿轮副的传动性能产生重要影响。对圆柱齿轮齿廓倒角加工技术进行研究,构建齿廓倒角加工的数学模型,基于仿真软件构建机床模型,编写加工程序完成齿廓倒角的仿真与分析。以此为基础,在铣齿机上完成了倒角加工实验,验证了齿廓倒角加工技术的可行性,为工业生产中渐开线齿轮齿廓倒角加工提供了新的解决方案。

基于NURBS曲线拟合的渐开线圆柱齿轮齿形偏差计算

为了提高现有齿轮齿廓曲线的拟合精度,针对现有的齿形偏差计算方法,提出了一种基于非均匀有理B样条(NURBS)曲线拟合的渐开线圆柱齿轮齿形偏差计算方法。首先,建立渐开线圆柱齿轮齿形误差的理论数学模型,然后根据NURBS曲线拟合原理和三坐标测量机测量的齿廓坐标点,用Matlab编程实现了NURBS曲线对实际齿廓曲线的拟合,结合渐开线等效拟合法,计算出了齿轮的齿形偏差。拟合结果表明,与最小二乘法(LS)的拟合结果对比,NURBS曲线拟合齿廓曲线精度更高,误差小,具有良好的拟合效果。齿形偏差计算结果表明,采用的基于NURBS曲线拟合的齿形偏差与实测值一致,表明该方法准确、有效。

基于平面共轭对数的弧齿锥齿轮齿面设计研究

为了提高弧齿锥齿轮建模模拟在承载工作的情况下所得出数据的准确性,在球面渐开线弧齿锥齿轮齿面成形设计原理的基础上,引入平面共轭对数对球面渐开线弧齿锥齿轮齿面形成理论进行进一步的研究与改进,建立与法向截面圆弧齿形点位相接触的弧齿锥齿轮数学模型,并利用修改后的齿面设计理论进行弧齿锥齿轮工作齿面、齿顶根曲线及齿根过渡曲面的齿面精确数学建模,使大小轮之间的共轭曲面啮合精度得到提升,从而提高弧齿锥齿轮在循环啮合过程当中的接触强度与稳定性,并对原先球面渐开线理论中所具有的对弧齿锥齿轮传动时的轴角可分性进行分析,进行存在轴角误差时对该设计下齿轮的传动误差的影响。

基于最小势能原理的双渐开线齿轮载荷分布研究

双渐开线齿轮接触线上的载荷非均匀分布,使得载荷计算困难。采用现有的方法或过于简化导致精度不足,或过于复杂使得计算量大、耗时长,并不适用于双渐开线齿轮载荷分布研究。为此,将双渐开线齿轮接触线等分为若干段,建立了双渐开线齿轮载荷分布模型,综合有限元法对轮齿载荷分布进行了研究。首先,根据双渐开线齿轮的啮合特点,求解了时变接触线,将每一啮合时刻下的齿轮接触线“分段”;然后,基于最小势能原理,建立了双渐开线齿轮的载荷分布模型,综合有限元法对齿轮轮齿载荷分布进行了研究;最后,对双渐开线齿轮与普通渐开线齿轮的载荷分布进行了对比分析,并研究了输入扭矩、齿宽对双渐开线齿轮轮齿载荷分布的影响。研究结果表明:载荷分布模型与有限元仿真结果之间的误差在10%之内,载荷分布模型合理可靠;双渐开线齿轮沿接触线方向载荷分布不均匀,节线附近载荷值最大;同参数、同工况下的双渐开线齿轮载荷波动幅度小于普通渐开线齿轮;输入扭矩增加,双渐开线齿轮节线附近载荷分布发生突变;齿宽增加,双渐开线齿轮沿接触线载荷分布不均匀程度增加。

考虑磨损故障的双渐开线齿轮传动系统动态特性研究

针对目前双渐开线齿轮(DIG)研究过程中齿廓磨损对双渐开线齿轮动态特性的影响问题,对考虑磨损故障的双渐开线齿轮传动系统动态特性进行了研究。首先,进行了双渐开线齿轮接触线长度与齿面载荷计算,考虑相对滑移距离,建立了等效接触模型;然后,进行了双渐开线齿轮齿面累积磨损量、齿轮齿面累积磨损深度的计算,建立了双渐开线齿轮啮合刚度计算模型,进行了磨损故障的双渐开线齿轮刚度的计算;最后,以一对双渐开线齿轮为研究对象,采用集中质量法建立了6自由度动力学模型,研究了齿面磨损对双渐开线齿轮动力学特性的影响。研究结果表明:在运行1×104次载荷循环后,混合弹流润滑状态下,双渐开线齿轮与普通渐开线齿轮(CIG)在节线附近齿根位置的最大累计磨损量差值为0.00207μm,且双渐开线齿轮节线靠近齿顶位置的磨损小于普通渐开线齿轮的磨损;双渐开线齿轮系统在磨损后振动加速度幅值小于普通渐开线斜齿轮的幅值。通过对不同磨损情况下的时域响应进行研究,发现磨损导致双渐开线齿轮传动系统的振动幅值增大。