TOOTH CONTACT ANALYSIS OF CONICAL INVOLUTE GEARS

The mathematical model of conical involute gears is developed based on the theory of gearing and the generating mechanism. Tooth contact analysis （TCA） is performed to examine the meshing and bearing contact of the conical involute gear pairs with intersected and crossed axes. In addition, the principal directions and curvatures of the gear surfaces are investigated and the contact ellipses of the mating tooth surfaces are also studied. Finally, the numerical illustrative examples are provided to demonstrate the computational results, test gears are made for tooth-bearing tests, and the conclusion is verified that the theory has the applicability.

Geometry Design and Tooth Contact Analysis of Crossed Beveloid Gears for Marine Transmissions

Beveloid gears,also known as conical gears,gain more and more importance in industry practice due to their abilities for power transmission between parallel,intersected and crossed axis.However,this type of gearing with crossed axes has no common plane of action which results in a point contact and low tooth durability.Therefore,a geometry design approach assuming line contact is developed to analyze the tooth engagement process of crossed beveloid gears with small shaft angle for marine transmission applications.The loaded gear tooth contact behavior is simulated by applying a quasi-static analysis to study the effects of gearing parameters on mesh characteristics.Using the proposed method,a series of sensitivity analyses to examine the effects of critical gearing parameters such as shaft angle,cone angle,helix angle and profile-shift coefficient on the theoretical gear mesh is performed.The parametric analysis of pitch cone design shows that the dominant design parameters represented by the angle between the first principle directions（FPD） and normal angular factor are more sensitive to the shaft and cone angles than they are to the helix angle.The theoretical contact path is highly sensitive to the profile-shift coefficient,which is determined from the theoretical tooth contact analysis.The FPD angle is found to change the distribution of contact pattern,contact pressure and root stress as well as the translational transmission error and the variation of the mesh stiffness significantly.The contact pattern is clearly different between the drive and coast sides due to different designed FPD angles.Finally,a practical experimental setup for marine transmission is performed and tooth bearing test is conducted to demonstrate the proposed design procedure.The experimental result compared well with the simulation.Results of this study yield a better understanding of the geometry design and loaded gear mesh characteristics for crossed beveloid gears used in marine transmission.

Mathematic model and tooth contact analysis of a new spiral non-circular bevel gear

A novel spiral non-circular bevel gear that could be applied to variable-speed driving in intersecting axes was proposed by combining the design principles of non-circular bevel gears and the manufacturing principles of face-milling spiral bevel gears.Unlike straight non-circular bevel gears,spiral non-circular bevel gears have numerous advantages,such as a high contact ratio,high intensity,good dynamic performance,and an adjustable contact region.In addition,while manufacturing straight non-circular bevel gears is difficult,spiral non-circular bevel gears can be efficiently and precisely fabricated with a 6-axis bevel gear cutting machine.First,the generating principles of spiral non-circular bevel gears were introduced.Next,a mathematical model,including a generating tooth profile,tooth spiral,pressure angle,and generated tooth profile for this gear type was established.Then the precision of the model was verified by a tooth contact analysis using FEA,and the contact patterns and stress distributions of the spiral non-circular bevel gears were investigated.

Complex geometric modeling and tooth contact analysis of a helical face gear pair with arc-tooth

A complex geometric modeling method of a helical face gear pair with arc-tooth generated by an arc-profile cutting(APC)disc is proposed,and its tooth contact characteristics are analyzed.Firstly,the spatial coordinate system of an APC face gear pair is established based on meshing theory.Combining the coordinate transformation matrix and the tooth profile of the cutter,the equations of the curve envelope of the APC face gear pair are obtained.Then the surface equations are solved to extract the point clouds data by programming in MATLAB,which contains the work surface and the fillet surface of the APC face gear pair.And the complex geometric model of the APC face gear pair is built by fitting its point clouds.At last,through the analysis of the tooth surface contact,the sensitivity of the APC face gear to the different types of mounting errors is obtained.The results show that the APC face gear pair is the most sensitive to mounting errors in the tooth thickness direction,and it should be strictly controlled in the actual application.

Dynamics of a Dual Power-split Transmission Based on Loaded Tooth Contact Analysis

In order to implement the dynamic characteristic of a dual power-split transmission, a dynamic me- chanics model is built. Firstly, according to the method of theoretical analysis of the tooth contact analysis （TCA） and loaded tooth contact analysis （LTCA）, the actual meshing process of each gear pairs is simulated, and the time-varying mesh stiffness excitations are obtained, which can improve the numerical precision. Second- ly, by using the lumped mass method, the bending-torsional coupling three dimensional dynamic model of the dual power-split transmission is established, and the identical dimensionless equations are deduced by elimina- ting the effect of rigid displacement and the method of dimensional normalization. Finally, by the method of the fourth order Runge-Kutta algorithm with variable step lengths, the responses of this system in a frequency domain and time domain are obtained, and the dynamic load change characteristics of each gear pairs are analyzed. The results show that the establishment, solution and analysis of the system dynamics model could provide a basis for the dynamic design, and have an important significance for the dynamic efficiency analysis and dynamic perform- ance optimization design of the dual power-split transmission.

弧齿锥齿轮TCA接触印痕与传动误差协同求解方法研究

弧齿锥齿轮齿面接触分析(Tooth contact analysis,TCA)可以精确计算得到反映传动性能的重要评价指标接触印痕以及传递误差。然而由于存在强烈的非线性和数值耦合关系,导致求解精度差、效率低和稳定性差。因此,提出了一种同时考虑接触印痕与传动误差的TCA协同求解新方法。首先基于啮合原理进行齿面参数化表达,同时在三维坐标空间中利用旋转矩阵模拟齿轮副的啮合运动进行接触轨迹和传递误差求解;再根据数值齿面的曲率特征,进行接触印痕精确解析求解。算例结果验证了本文所提方法的可行性。

Contact analysis and load distribution of double-envelop hourglass worm gearing affected by errors and load

An approach for the contact analysis and load distribution of double-envelop hourglass worm gearing is presented, which is based on a 3-D elastic contact finite element method (FEM) model that accommodates the influence of errors and load. As compared with existing tooth contact analysis model that assumes rigidity for the contacting surfaces, the proposed model provides a more realistic analysis on the contact patterns, the distribution of contact load and transmission errors. It is also capable of exploring the influence of different errors on meshing performances, the contact deformation, the shift of the contact zone and load share among the meshing tooth-pairs under different load.

Generation Principle of Helix Tooth Profile in Hypocycloid Pinw heel Transmission and Its Contact Characteristics

The transmission characteristics of gear drive can be improved with the use of novel tooth profiles.A theoretical study on tooth profile of the hypocycloid pinwheel transmission and contact analysis of gear pair based on finite element method(FEM) are carried out,respectively.The line contact between mated tooth surfaces becomes point contact according to a plus movement.Through loaded tooth contact analysis(LTCA),the contact stress and load distributions for the proposed hypocycloid pinwheel transmission and the traditional one are discussed.The analysis results show that the developed tooth surfaces have anticipatory point contact characteristics under loads and contact fatigue dangerous area locates around the ultimate contact position.

安装误差对双圆弧齿轮啮合特性影响分析

为深入研究安装误差对双圆弧齿轮副啮合特性的影响,基于有限元法建立齿轮加载分析模型,研究啮合过程中单齿凸、凹齿廓的接触性能,对比分析安装误差对接触性能以及传动误差的影响规律。结果表明:啮合周期内,双圆弧齿轮的凹齿廓先接触且承担较大的法向接触力,但凸齿廓的啮入端接触应力最大,凸、凹齿面接触压力差与转矩成正比;安装误差对齿面接触性能影响较大,其中中心距误差容易引起齿轮分阶处应力集中,轴线平面的安装误差会引起齿面接触偏载,且对安装误差影响最大。

内凹刀盘加工的直齿锥齿轮齿面接触分析

为了在国产数控弧齿锥齿轮机床上采用内凹刀盘加工直齿锥齿轮,对使用内凹刀盘加工直齿锥齿轮的原理和齿面接触分析(Tooth Contact Analysis,TCA)方法进行了研究。基于内凹刀盘的加工原理和结构特点,建立了内凹刀盘加工的直齿锥齿轮的齿面数学模型,给出了相应的齿面接触分析方法;编写齿面接触分析程序,计算得到了齿轮副的接触斑点和传动误差曲线;通过实际的切齿加工和接触斑点检验,验证了齿面数学模型以及TCA方法的正确性。研究为锥齿轮加工参数及刀具参数的确定与啮合质量控制提供了理论依据。

直纹面面齿轮共轭小齿轮的滚齿加工研究

针对直纹面面齿轮与圆柱齿轮不满足共轭关系而限制其在小传动比应用场合的情况,提出与直纹面面齿轮共轭的小齿轮及其滚齿加工方法。首先建立了直纹面面齿轮及其共轭小齿轮的齿面方程,获得共轭小齿轮相比圆柱齿轮的齿面偏差形貌;随后推导了阿基米德齿轮滚刀的齿面方程,并根据滚齿加工运动坐标系和啮合原理建立了圆柱齿轮的数学模型;在滚齿加工圆柱齿轮的运动基础上,通过增加滚刀的径向移动和工件的附加转动使滚刀切削刃逼近共轭齿面,附加运动由泰勒多项式表示,通过建立敏感矩阵方程并采用奇异值分解法迭代求解方程获得附加数控运动多项式系数,最后获得数控加工齿面;为评价直纹面面齿轮副的传动性能,进行了齿面承载接触分析,并与传统面齿轮-圆柱齿轮副进行了对比。研究结果表明:相同参数下,直纹面面齿轮副与传统面齿轮副的承载性能基本相同,完全能够代替传统面齿轮副传动;数值算例和仿真加工对该方法的验证表明,其数值算例误差不超过1μm,仿真结果误差不超过6μm,证明了该方法的可行性和准确性。

Design of Pinion Machine Tool-settings for Spiral Bevel Gears by Controlling Contact Path and Transmission Errors

This paper proposes a new approach to design pinion machine tool-settings for spiral bevel gears by controlling contact path and transmission errors. It is based on the satisfaction of contact condition of three given control points on the tooth surface. The three meshing points are controlled to be on a predesigned straight contact path that meets the pre-designed parabolic function of transmission errors. Designed separately, the magnitude of transmission errors and the orientation of the contact path are subjected to precise control. In addition, in order to meet the manufacturing requirements, we suggest to modify the values of blank offset, one of the pinion machine tool-settings, and redesign pinion ma- chine tool-settings to ensure that the magnitude and the geometry of transmission errors should not be influenced apart from minor effects on the predesigned straight contact path. The proposed approach together with its ideas has been proven by a numerical example and the manufacturing practice of a pair of spiral bevel gears.

齿轮TCA技术及发展

通过对齿轮齿面接触分析TCA技术的研究,介绍了传统齿面接触分析TCA技术、承载齿面接触分析LTCA、真实齿面接触分析RTCA及边缘接触分析ETCA技术的基本原理、发展现状以及发展动态。

基于几何接触算法的摆线针轮传动误差分析

在摆线针轮传动的轮齿接触分析(ToothContactAnalysis,TCA)模型中,通常将误差值设为一个定值,易造成计算结果不合理,且计算过程中常出现计算不收敛等问题。对此,建立了包含修形、加工误差等多因素综合作用下的摆线针轮传动误差分析模型;在对啮合点矢量方程分析后,提出了一种解决数值不收敛等问题的方法,即在求得的摆线轮和针齿上纵横坐标相等的点中,找到法向量相等的啮合点。对回转体误差形式分析后,提出了一种解决计算结果不合理的方法,即令摆线轮周节误差、摆线轮齿廓径向跳动误差和针齿中心圆半径误差在公差带内呈周期性变化。以RV-80E减速器为试验对象,对比试验结果,验证了理论模型的正确性;研究各误差对传动误差的影响,开展了各误差灵敏度分析。结果表明,针齿半径误差和针齿与针齿壳间隙综合作用下的法向间隙是影响传动误差的主要因素。

含有不同裂纹类型的弧齿锥齿轮副啮合特性分析

在重载和交变载荷工况下,弧齿锥齿轮易产生齿根裂纹导致轮齿断裂,需明晰裂纹对啮合特性的影响机理,为故障诊断提供理论基础.利用ANSYS软件对弧齿锥齿轮齿副进行静态接触仿真分析.通过单元节点替换的方法设置了多种裂纹类型的故障模型,探讨了裂纹对弧齿锥齿轮副啮合特性的影响.研究结果表明:当轮齿接触椭圆移动到轮齿裂纹区域,啮合刚度随着裂纹的严重程度增加而逐渐下降,平面裂纹工况、空间裂纹工况和断齿工况的刚度最大下降量分别为27.58%,14.12%,32.82%.轮齿啮合的接触椭圆位置、裂纹位置以及深度会使齿面接触应力以及齿根弯曲应力产生不同的增减趋势.

考虑误差、轴变形和轴承刚度的弧齿锥齿轮LTCA

加工误差、安装误差、轴变形及轴承刚度会对弧齿锥齿轮的接触轨迹及传递误差曲线产生影响,基于加载接触有限元分析原理,给出考虑加工误差、安装误差、轴变形及轴承刚度的弧齿锥齿轮有限元加载接触分析方法,以一对准双曲面齿轮为对象,研究在考虑上述条件下接触轨迹的变化及传递误差曲线的变化情况,为高质量弧齿锥齿轮传动提供了一种有效设计与分析方法。

双重螺旋法准双曲面齿轮啮合性能研究

针对采用双重螺旋法(全工序法)加工的准双曲面齿轮副,运用数值分析的方法研究了安装误差、螺旋运动系数与啮合性能之间的作用规律。利用矢量法建立切齿加工数学模型,求得齿面方程并建立齿轮副装配数学模型,利用齿面接触分析(TCA)研究含多种安装误差的齿面接触特征;建立齿面失配数学模型,分析螺旋运动系数对齿面接触特征、几何形貌的影响规律,揭示双重螺旋法对轮齿工作面、非工作面不同的修形原理;结合刀具与齿坯之间的相对位置关系,阐述了展成线的求解过程,研究了螺旋运动系数对展成线位置的影响规律。对准双曲面齿轮副(7×43)进行磨齿和滚检实验,与数值分析结果进行对比验证了分析模型的正确性。

新型内啮合S型齿轮啮合效率计算与分析

为了准确计算新型内啮合S型齿轮的啮合效率,引入了考虑轮齿表面滑差、润滑油状况及时变载荷等因素的弹流润滑时变摩擦模型;分析啮合齿面润滑机制,通过轮齿接触分析(Tooth ContactAnalysis,TCA)及轮齿承载接触分析,计算了啮合齿面滑动摩擦因数及摩擦损失功率;在此基础上,获得内啮合S型齿轮啮合周期内的瞬时啮合效率和平均啮合效率,并给出了计算实例。研究结果表明,在相同设计参数下,新型内啮合S型齿轮较渐开线齿轮有更高的啮合效率;经螺旋线修形后,新型内啮合S型齿轮的啮合效率有所提高且随修形量的增大而增大。

齿轮齿面接触分析TCA技术及发展动态

通过对齿轮齿面接触分析TCA技术(利用电子计算机进行齿面接触区域的分析和修正)的研究,介绍了齿面接触分析TCA技术的基本原理、发展现状以及发展动态,阐明了TCA技术在现代工业生产中重要意义。

基于时变啮合刚度的修形齿轮承载接触分析

为了解决传统齿轮承载接触分析力学解析方法难以考虑齿廓修形的问题,以齿廓修形渐开线直齿轮为研究对象,基于时变啮合刚度,提出了一种齿轮承载情况下齿间载荷分配系数与齿面接触应力解析算法。通过对比齿轮承载接触时接触点在啮合线方向的变形量与修形量,提出修形齿轮单双齿啮合状态判定方法,并计算不同状态下的啮合刚度;推导修形齿轮的啮合力、刚度与变形的力学关系,建立各啮合齿对的齿间载荷分配系数计算模型,应用赫兹接触理论求解齿面接触应力;分别在不同修形高度、修形曲线与配对齿数情况下,计算分析多种齿廓修形算例的齿间载荷分配系数与齿面接触应力。结果表明,解析算法求解的齿间载荷分布系数、齿面接触应力及相应单双齿啮合区间与有限元结果相一致,相对误差较小,证明了该算法有较高的精度与稳定性,为齿廓修形齿轮承载接触分析提供一种高效便捷的新方法。