Photoelastic optimization of the profiles of Wildhaber-Novikov gears

Wildhaber-Novikov gears having circular flanks hold out a very promising future in the field of gearing for power transmission in very heavily loaded machineries. In order to optimize the profile parameters of these gears, a photoelastic investigation was carried out. Models of Wildhaber-Novikov gears made of photoelastic material, Araldite-D, were used in the investigation. Included were the models with various profile radii and fillet radii at different pressure angles. The flank and bending stresses of these gears were determined in each case and were compared with those of involute gears. From this investigation, it was concluded that the load-carrying capacity of circular-arc gears is 3.5 times that of involute gears. It was found that with profile radius up to 1.5 module, the bending stresses of these gears are lower than those of corresponding involute gears, under identical conditions of loading. It was observed that high pressure angles, when used for Wildhaber-Novikov gears having comparatively smaller profile radii, produce high flank stresses. Gears with larger profile radii, at smaller pressure angles were seen to have a high value for the ratio of flank stress to the bending stress. On the other hand, gears with smaller profile radii, at higher pressure angles, were observed to have a smaller value for this ratio of stresses.     

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.     

An analytical determination of speed factor in gear designs

In practical gear design, the “speed factor” is usually employed which is defined as a ratio of the allow-able stress at required running speed of the gear system to the stress at zero speed. At present, several expressions for the speed factor are recommended by several authorities; however, those are almost all empirical and not analytical. The present paper aims to contribute to an analytical determination of the speed factor for spur gears as stated below. First, from a dynamic photoelastic test of a pair of spur gears, the following results were obtained:

1. (1)
   At a contant speed and torque and at the same point of engagement, the maximum fillet stress did not have a specific constant value but had somewhat scattered values due to the vibration of gear system under operation.     
   
   

基于分形理论的双渐开线齿轮接触应力研究

综合考虑接触面粗糙度、材料特性等因素对齿轮接触应力的影响,基于分形理论和经典Hertz接触理论建立双渐开线齿轮分形接触模型. 该模型中,影响载荷和实际接触面积的主要因素包括分形维数、粗糙度幅值和材料特性参数. 理论分析表明:分形维数一定时,真实接触面积随着载荷的增大而增大;载荷一定时,接触面积随着粗糙度幅值的增大而减小;随着材料特性参数值的增加,在一定程度上加强了软材料轮齿承载能力,同时会使得微凸体由弹性变形到塑性变形的临界面积减小. 对比分形接触模型和有限元模型两种计算双渐开线齿轮轮齿接触应力方法,结果证明了分形接触模型计算双渐开线齿轮接触应力的有效性.      

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.     

齿轮接触有限元分析

通过接触仿真分析研究了通用接触单元在轮齿变形和接触应力计算中的应用。建立了一对齿轮接触仿真分析的模型，并使用新的接触单元法计算了轮齿变形和接触应力，与赫兹理论比较，同时也计算了摩擦力对接触应力的影响。计算分析了单元离散、几何、边界范围与加载或约束处理方式的误差，建立了一个计算轮齿变形和接触应力的标准，说明了新的接触单元法的精确法、有效性和可靠性。     

Boundary slippage for generating hydrodynamic load-carrying capacity

Boundary slippage is used to generate the load-carrying capacity of the hydrodynamic contact between two parallel plane surfaces. In the fluid inlet zone, the fluidcontact interfacial shear strength on a stationary surface is set at low to generate boundary slippage there, while in the fluid outlet zone the fluid-contact interfacial shear strength on the stationary surface is set at high enough to prevent the occurrence of boundary slippage. The fluid-contact interfacial shear strength on the entire moving surface is set at high enough to prevent boundary slippage on the moving surface. These hydrodynamic contact configurations are analyzed to generate the pronounced load-carrying capacity. The optimum ratio of the outlet zone width to the inlet zone width for the maximum load-carrying capacity of the whole contact is found to be 0.5.     

Photoelastic analysis of I-beams with elliptic-type web cutouts

An investigation of the fully plastic load-carrying capacity of aluminum-alloy I-beams with web cutouts leads to an interest in the elastic stress distribution for these cutouts. The photoelastic procedure was used to evaluate the elastic stress distribution at the surface of the web cutouts. The shear-difference method was used to establish the probable location of the fully plastic hinges which might be expected to form in the aluminum-alloy prototype. The data obtained were in good agreement with experimental results obtained on aluminum-alloy I-beams.     

A new look at the bending strength of gear teeth

This paper reports on an extensive two-dimensional photoelastic study of the basic deformation behavior of gear teeth under load. Limitations of existing bending-strength design procedures are used to formulate a test program which considers the magnitude of the critical root-fillet stresses, the effects of varying the load position on the flank, the effects of friction forces at contact and the relationship between bending and shearing deformations. Particular interest is devoted to a study of the observed movement of the maximum-stress position around the fillet as load moves up the tooth flank and to the associated variation in stress-concentration factors (around the fillets) due to radial, bending and shear loads. Note is also made of the problems associated with observed ‘proximity effects’ and of the procedures carried out in an attempt to qualify the nature of the effect. The results of the program have been used to, explain certain anomalies observed in earlier three-dimensional photoelastic model tests^1.7 and reference is made to the development of a new design procedure for the bending strength of helical gears.     

Dynamic characteristics and load-sharing performance of concentric face gear split-torque transmission systems with time-varying mesh stiffness,flexible supports and deformable shafts

Meccanica - The concentric face gear split-torque transmission system is a new gear mechanism which integrates the advantages of face gears, split-torque transmissions and concentric transmissions....     

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.     

Photoelastic investigation of a shrink-fit problem

Stress analysis of the components of a shrinkfit assembly has been carried out by the three-dimensional photoelastic technique. Experimental results obtained from the analysis of a typical shrink-fit assembly, wherein a cylindrical wheel is shrink fitted onto a hollow cylindrical hub in an asymmetrical position. Photoelastic models made from hot-setting Araldite B were loaded at the critical temperature in a stress-freezing oven and the meridional and the hoop slices were observed in a transmission polariscope. It is concluded from this investigation that the bending induced in the components due to the asymmetrical placement of the wheel and the contact shear at the interface influences the stress distribution quite considerably. Also, the dimensions of the components in the axial direction have an influence on the mode and contribute to the magnitude of the variation of stress components.     

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.

     

Compensation of errors of alignment and contact pattern repositioning in hypoid gears with low crossing shaft angle

The investigation of the influence of different errors of alignment on the mesh behavior of hypoid gears with low crossing shaft angle has been carried out. As a result, a simplified methodology of compensation of errors of alignment and correction of the contact pattern by controlled axial displacements to the pinion and wheel of the hypoid gear set has been proposed. The shaft angle error and the offset error mainly influence the position of the contact pattern in profile direction. The pinion and wheel axial position errors influence the position of contact pattern along the tooth trace direction and change the position slightly along the tooth profile direction. All the assembly errors increase the peak-to-peak value of transmission errors, except positive values of the pinion axial position error that allows decreasing the peak-to-peak transmission error slightly in an appropriate range. Finite element analysis has been performed to verify the compensating ability of the developed methodology.

     

Enhanced model of load distribution along the line of contact for non-standard involute external gears

The presence of undercut at the tooth root, non-equal addendum on pinion and wheel, non-standard tooth height or non-standard center distance may have decisive influence on the load distribution along the line of contact of spur and helical gear teeth. The curve of variation of the meshing stiffness along the path of contact, quite symmetric respect the midpoint of the interval of contact, loses its symmetry for non-standard geometries and operating conditions. As a consequence, the critical contact points for bending and wear calculations may be shifted from their locations for standard gears. In this paper, a non-uniform model of load distribution along the line of contact of standard spur and helical gears, obtained from the minimum elastic potential criterion, has been enhanced to fit with the meshing conditions of the above mentioned non-standard cylindrical gear pairs. The same analytical formulation of the initial model may be used for the non-standard gears by considering appropriate values of a virtual contact ratio, which are also presented in the paper.     

Planetary gearbox with localised bearings and gears faults: simulation and time/frequency analysis

Planets bearings of planetary gear sets exhibit high rate of failure; detection of these faults which may result in catastrophic breakdowns have always been challenging. The objective of this paper is to investigate the planetary gears vibration properties in healthy and faulty conditions. To seek this goal a previously proposed lumped parameter model (LPM) of planetary gear trains is integrated with a more comprehensive bearing model. This modified LPM includes time varying gear mesh and bearing stiffness and also nonlinear bearing stiffness due to the assumption of Hertzian contact between the rollers/balls and races. The proposed model is completely general and accepts any inner/outer race bearing defect location and profile in addition to its original capacity of modelling cracks and spalls of gears; therefore, various combinations of gears and bearing defects are also applicable. The model is exploited to attain the dynamic response of the system in order to identify and analyze localized faults signatures for inner and outer races as well as rolling elements of planets bearings. Moreover, bearing defect frequencies of inner/outer race and ball/roller and also their sidebands are discussed thoroughly. Finally, frequency response of the system for different sizes of planets bearing faults are compared and statistical diagnostic algorithms are tested to investigate faults presence and growth.     

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.     

A Balanced Maximum Fillet Stresses on Normal Contact Ratio Spur Gears to Improve the Load Carrying Capacity Through Nonstandard Gears

This article presents an idea to remove the inequality in maximum fillet stresses developed between pinion and gear of a step up gear drive. This uniform fillet strength of the gear drive can be achieved by using nonstandard pinion and gear with appropriate addendum modifications generated by nonstandard basic racks of respective tooth thickness not equal to 0.5πm at the pitch circle. The influence of gear parameters such as gear ratio, pressure angle, addendum factor, pinion teeth number, and addendum modifications on the maximum fillet stress on the nonstandard pinion and gears of different tooth thickness has been analyzed through finite element method and finally the optimum value of rack tooth thickness coefficients (k _pc and k _gc ) are suggested for the given gear drive (defined by i) that improves the fillet capacity in bending. This study has been extended for various drives like S _std , S _o , S ₊, and S _? drives.     

The use of precision castings in the three-dimensional photoelastic analysis of threaded tubular connections

A new, room-temperature formulated, epoxy-resin system with superior stress-freezing photoelastic properties is presented. The ease of producing relatively large, high-quality, precision castings is discussed. Stress-optical, rheological, and mechanical properties are given. Cylindrical tubes joined by integrally cast Stub Acme and American Standard rolled thread profiles with typical geometric parameters are loaded in tension and analyzed using the stressfreezing method. Results are presented for surface axial stress distributions for the wall surfaces opposite the threaded profiles. Additionally, the fillet stress distribution for the first engaged thread as a function of the distance along the thread helix from the location of initial contact is presented. Comparison with a finite-element analysis is given. Paper was presented at the 1983 SEM Spring Conference on Experimental Mechanics held in Cleveland, OH on May 15–19.