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.

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.

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。

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.

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.

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。

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。

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。

计入齿面摩擦的斜齿轮啮合动力学特性研究

针对斜齿轮啮合过程中,建立了考虑齿面摩擦的斜齿轮啮合副动力学模型。首先对斜齿轮副啮合进行齿面接触分析和加载齿面接触分析,得到齿面接触路径、接触位置法向位移,在此基础上计算了时变啮合刚度激励、齿面滑动摩擦系数和摩擦力臂。其次,建立并求解了考虑齿面摩擦的斜齿轮副动力学模型,计算了动态啮合力/速度/位移。由齿面上的滑动摩擦系数、摩擦力臂和动力学方程构成斜齿轮副啮合动力学耦合系统,通过解耦计算,对模型系统方程进行了求解。最后通过实例验证了该模型的有效性。

数控铣齿机上切齿模型建立及TCA分析

基于格里森机床运动学方法和啮合方程推导出被加工齿轮的齿面方程,并利用轮齿接触分析(TCA)技术分析了该齿轮工作时的传动误差,为机床的设计与改造提供理论依据。

基于LTCA的高速齿轮传动系统非线性动态特性分析

为研究高速齿轮传动系统的动态特性问题,建立了考虑时变啮合刚度激励、误差等因素的非线性弯-扭耦合动力学模型。采用承载接触仿真(LTCA)方法,结合当量误差啮合理论,计算了模型中的时变啮合刚度和啮合误差激励条件。利用数值方法求解得到系统的动态响应特性。结果表明:在时变啮合刚度和啮合误差激励因素影响下,系统表现出很强的非线性特性;转速从7000r/min变化到15000r/min时,时间历程一直表现为非周期变化;增大轮齿啮合刚度和提高加工精度等级可改善动态响应特性。为高速齿轮传动系统的动态稳定性设计提供理论基础。

考虑LTCA技术的功率双分支轮系动态特性分析

双路功率分支传动系统较多应用于航空系统传动装置中,为了揭示其动态振动机理,建立该系统的动态力学模型。采用承载接触分析(LTCA)技术计算齿轮副时变啮合刚度激励;利用集中质量法建立弯扭耦合的动力学微分方程,进行消除刚体位移和量纲归一化处理,采用数值解法,得到系统的时域-频域响应,给出动载系数与输入转速的函数关系。结果表明:系统动态特性受到输入转速的影响较大;啮合阻尼的增大,可以降低临界转速附近的动载荷;实验和理论具有较好的一致性。为进一步对该系统的优化设计提供理论基础。

Pinion Tooth Surface Generation Strategy of Spiral Bevel Gears

Aviation spiral bevel gears are often generated by spiral generated modified（SGM） roll method.In this style,pinion tooth surface modified generation strategy has an important influence on the meshing and contact performances.For the optimal contact pattern and transmission error function,local synthesis is applied to obtain the machine-tool settings of pinion.For digitized machine,four tooth surface generation styles of pinion are proposed.For every style,tooth contact analysis（TCA） is applied to obtain contact pattern and transmission error function.For the difference between TCA transmission error function and design objective curve,the degree of symmetry and agreement are defined and the corresponding sub-objective functions are established.Linear weighted combination method is applied to get an equivalent objective function to evaluate the shape of transmission error function.The computer programs for the process above are developed to analyze the meshing performances of the four pinion tooth surface generation styles for a pair of aviation spiral bevel gears with 38/43 teeth numbers.The four analytical results are compared with each other and show that the incomplete modified roll is optimal for this gear pair.This study is an expansion to generation strategy of spiral bevel gears,and offers new alternatives to computer numerical control（CNC） manufacture of spiral bevel gears.

Mesh stiffness calculation of cycloid-pin gear pair with tooth profile modification and eccentricity error

Cycloid speed reducers are widely used in many industrial areas due to the advantages of compact size, high reduction ratio and high stiffness. However, currently, there are not many analytical models for the mesh stiffness calculation, which is a crucial parameter for the high-fidelity gear dynamic model. This is partially due to the difficulty of backlash determination and the complexity of multi-tooth contact deformation during the meshing process. In this paper, a new method to calculate the mesh stiffness is proposed including the effects of tooth profile modification and eccentricity error. The time-varying mesh parameters and load distribution of cycloid-pin gear pair are determined based on the unloaded tooth contact analysis （TCA） and the nonlinear Hertzian contact theory, allowing accurate calculations of the contact stiffness of single tooth pair and the torsional stiffness of multi-tooth pairs. A detailed parametric study is presented to demonstrate the influences of tooth profile modification, applied torque and eccentricity error on the torsional mesh stiffness, loaded transmission error, Hertzian contact stiffness and load sharing factor. This model can be applied to further study the lost motion and dynamic characteristics of cycloid speed reducer and assist the optimization of its precision, vibration and noise levels.

基于ETCA的螺旋锥齿轮研磨技术

以多体系统误差建模理论和齿轮研磨原理为基础,建立研齿时V/H调整与齿面啮合关系的数学模型。利用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)技术来实现研磨控制,并通过在格里森600HTL上进行研齿实验表明:基于ETCA的研磨技术要比基于TCA的研磨技术更为合理,研磨效果也更优。

NUMERICAL APPROACH TO DETERMINING INSTANTANEOUS CONTACT REGION FOR CONJUGATE SURFACES

According to the defect of traditional method of determining instantaneous contact regions for conjugate surfaces, a numerical approach to the determination is proposed. A local coordinate system is built by using the surface unit tangent and unit normal at the contact point. Considering that the gap forming the boundary of instantaneous contact region in the direction of the common normal vectors is given, a system of nonlinear equations is built to find the instantaneous contact boundary in local coordinate system, a modified Powell＇s hybrid algorithm of finite-difference approximation to the Jacobian used to solve the system. The new method simplifies the task of determining instantaneous contact regions without calculating curvatm＇e and relative curvature. The validity of the proposed approach is verified by an example of hypoid gears.