Time-varying mesh stiffness calculation of spur gears with spalling defect

Considering the effects of extended tooth contact (ETC), revised fillet-foundation stiffness under double-tooth engagement region, nonlinear contact stiffness and tooth spalling defect, an analytical model for time-varying mesh stiffness (TVMS) calculation of spur gears is established. In addition, the analytical model is also verified by comparing the TVMS under different spalling widths, lengths and locations with that obtained from finite element method. The results show that gear mesh stiffness decreases sharply with the increase of spalling width, especially during the single-tooth engagement; the spalling length only has an effect on the beginning and ending of gear mesh stiffness reduction; the spalling location can affect the range of gear mesh stiffness reduction, and the range will reduce when the spalling location is close to the addendum. This study can provide a theoretical basis for spalling defect diagnosis.     

Dynamic features of three-dimensional helical gears under sliding friction with tooth breakage

Mass eccentric and mesh stiffness variation induced by tooth breakage will change the dynamic features of helical gears. However, the internal excitation in helical gears under sliding friction with tooth breakage is seldom considered to reveal the dynamic features. In this study, the calculation method of mesh stiffness in helical gears with tooth breakage is proposed based on the time-varying contact line. A three-dimensional analytical helical gear pair model is developed by considering the mass eccentric and mesh stiffness induced by tooth breakage with sliding friction. The effects of tooth breakage on the dynamic responses of helical gears are performed. The results show that the amplitude modulation phenomenon emerged in the dynamic transmission error by considering the mass eccentric caused by tooth breakage. The oscillation of the dynamic response change significantly in the tooth breakage area, especially with the growth of the breakage size. Sliding friction plays a certain role in inhibiting the amplitude of the frequency. Tooth breakage results in the presence of rotational frequency and sidebands around mesh frequency and its harmonics. The rotational frequency increases significantly by considering the mass eccentricity due to tooth breakage defect.     

Review on dynamics of cracked gear systems

Dynamic characteristics of cracked gear systems, also known as cracked-gear rotor systems, have received increasing interests among industry and academy in the past two decades. This paper reviews published papers on the dynamics of cracked gear systems. These studies mainly focused on three topics: crack propagation prediction, time-varying mesh stiffness (TVMS) calculation and vibration response calculation; Study objects involve the spur gear, helical gear and planetary gear; Different modeling methods including analytical method, finite element (FE) method, combined analytical-FE approach were adopted. More specifically, this review is composed of three related parts according to the above three topics. The first part involves the prediction of the crack propagation path based on two-dimensional (2D) or three-dimensional (3D) gear models, which provides a basis for the hypothesis of crack path in the process of TVMS calculation of cracked gear pairs. The second part summarizes the TVMS calculation methods including analytical methods, FE methods, combined analytical-FE approaches and experimental methods. The final part reviews the dynamic models for vibration analysis of cracked gear systems including lumped mass models and FE models, where the crack effects are characterized by introducing TVMS of cracked gear pairs into the system dynamic models. The well known open problems about cracked gear dynamics are finally stated, and some new research interests are also pointed out. The review will provide valuable references for future studies on dynamics of cracked gears.     

Effects of gear center distance variation on time varying mesh stiffness of a spur gear pair

Gear center distance variation is one of the most common defects of gear transmission systems. The changes in the gear center distance as well as other faults (e.g. tooth crack, pitting) have a direct influence on the Time Varying Mesh Stiffness (TVMS) which further modifies gear vibration behaviors. Accurately estimating gear TVMS under fault conditions is crucial in gear vibration dynamic simulation. Common methods used to evaluate TVMS are generally based on the assumption that the gear pair is perfectly mounted and that all mesh points are at their theoretical positions. This assumption prevents these methods from modeling deviations in gear center distance. To address this shortcoming, this paper proposes a new gear mesh kinematic model that can evaluate the actual contact positions of tooth engagement with time varying gear mesh center distance. With the proposed kinematic model, the actual TVMS of both healthy and cracked gear teeth are computed under conditions of perfect mounting, constant gear center distance deviation, and also time-varying gear center distance. Numerical simulations indicate that gear center distance variation has a significant effect on gear TVMS. Comparison between the effect of multiple faults and summed individual effects on TVMS indicates that the TVMS modification due to multiple-faults do not appear to combine in a linear manner. The proposed model for actual TVMS enables gear system dynamic models to be used to study the effects of assembly errors, gear run-out errors, shaft bending, and bearing deformation on the vibration behavior of gear transmission systems.     

Improved time-varying mesh stiffness model of cracked spur gears

Based on our previous work (Ma et al., 2014, Engineering Failure Analysis, 44, 179–194), this paper presents an improved analytical model (IAM) for the time-varying mesh stiffness (TVMS) calculation of cracked spur gears. In the improved analytical model, the calculation error of TVMS under double-tooth engagement due to repeatedly considering the stiffness of the fillet-foundation is revised, and the effects of reduction of fillet-foundation stiffness of cracked gears and extended tooth contact (ETC) are also considered, which have a great influence on TVMS, especially under the condition of large torques and crack levels. Moreover, the comparisons among the IAM, traditional analytical model (TAM) and finite element (FE) model are also carried out under different torques and crack depths. IAM is also verified by comparing TVMS and vibration responses obtained by FE model, which can be considered as a gauge to evaluate the calculation error. The results show that the maximum error of IAM is about 12.04%, however, that of TAM can be up to 32.73%.     

Effect of shaft misalignment and friction force on time varying mesh stiffness of spur gear pair

Shaft misalignment and sliding friction between meshing teeth are considered as primary excitation to generate vibrations and extra dynamic loads on transmitting gear teeth. Time varying mesh stiffness (TVMS) is an important parameter to understand the dynamics of meshing gear pair. Potential energy method is one of the most acceptable methods to calculate TVMS. This paper proposes a computer simulation based approach to study the effect of shaft misalignment and friction on total effective mesh stiffness for spur gear pair. The results showed clearly that misalignment and friction affect TVMS of gear pair. The effect of misalignment and friction has also been studied for cracked gear pair and results are discussed.     

Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects

Spalling is one of the common tooth surface failures of gear teeth and is defined as the formation of deeper cavities that are mainly developed from subsurface defects. The time varying mesh stiffness (TVMS) of gear pairs, gives significant information about the health of the system. The change indirection of time varying friction on both sides of the pitch line causes the change of gear mesh stiffness. This article proposes a computer simulation based approach to study the effect of time varying friction coefficient on the total effective mesh stiffness for the spur gear pair. An analytical method to calculate the TVMS of the spur gear for different spall shapes, size and location considering sliding friction is also proposed in this study. The results show that spall shape, size and location are very important parameters that need to be considered for calculation of TVMS and subsequently to know the dynamic response of the gear pair in the presence of a spall.     

完整约束下齿轮啮合转子系统的弯扭耦合振动稳态响应

在不脱齿等基本假设下，根据齿轮啮合原理和轮齿的齿面方程，推导了齿轮形心的横向位移和齿轮扭转角之间的约束关系式，从Lagrange方程出发，同时考虑齿轮啮合和不平衡效应，建立了直齿齿轮啮合转子-轴承系统的弯扭耦合动力学模型。分别在质量偏心和扭转激励作用下，分析了系统的弯扭耦合振动稳态响应。结果表明：两者均会引起弯曲振动和扭转振动，并且响应的幅值与系统的参数有关。     

Detection and Advancement Monitoring of Distributed Pitting Failure in Gears

Conventional methods (i.e. time, frequency and cepstrum) can routinely be used to reveal fault-indicating information in the vibration signal. In recent years, Wavelet analysis, which can lead to the clear identification of the nature of faults, are widely used to describe machine condition. Capability of this method in the detection of any abnormality can be further improved when its low-order frequency moments are considered. This paper presents the use of vibration-based techniques in the early detection and advancement monitoring of distributed pitting fault. The pits were seeded on all of the gear tooth surfaces in differing degrees of severity, and intended to replicate the pitting damage due to misalignment. With each fault severity, the helical gears were tested and the resulting vibration data were recorded. The application of employed vibration-based methods (i.e. time, frequency, cepstrum, and wavelet transform: scalogram and its mean frequency variation) to each set of experimental data are presented. It has been found that presence of pitting fault cannot be clearly revealed by the conventional unless fault severity is significantly large. In contrast, the scalogram and especially its mean frequency variation provide early indications of presence and progression of pitting faults in gears even when the fault severity is considerably smaller.     

Improved calculation method for load-dependent gear losses

The topic of this study is to investigate the load-dependent losses of practical spur gears, for which the existing calculation method is analyzed in detail and improved.In drive technology, spur gears are frequently used for the transformation of torque and speed. Power transmission always involves power losses. Thereby, particular interest has to be given to the load-dependent gear loss as it often accounts for a large share of the total loss.According to the state of the art, the calculation of load-dependent gear loss is mainly based on (i) gear loss factors derived from simplified load distributions and (ii) a mean coefficient of friction derived from empirical equations based on spur gears without flank modifications. In practice, however, helical gears for improved NVH (Noise-Vibration-Harshness) behavior and flank modifications for uniform contact patterns are used. The modified mesh and contact conditions of modified helical gears affect the load-dependent gear losses significantly. Hence, significant deviations compared to the state of the art can occur.Comprehensive experimental investigations at the FZG efficiency test rig and FZG Bearing power loss test rig have been carried out to investigate the load-dependent gear losses of modified helical gears. The results are used to improve an existing calculation method that distinguishes between geometrical and tribological influence factors. Thereby, the local mesh and contact conditions along the plane of action are considered by the gear loss factor H_VL according to Wimmer, whereas the calculation of the mean coefficient of friction µ_mz has been enhanced by regression analyses. A comparison of the improved calculation method to the state of the art shows improved accuracy for determining the load-dependent gear losses and hence the efficiency of gear boxes.     

圆柱斜齿轮精密成形数值模拟与试验研究

基于浮动凹模的思想对圆柱斜齿轮的成形及脱模进行了计算机仿真和石蜡试验研究,分析了成形过程的变形特点及脱模后斜齿轮齿形的应变情况,验证了方案的可行性.结果表明:对圆柱斜齿轮采用浮动凹模成形及脱模的工艺方案,可以获得符合精度要求的锻件.     

单级人字齿轮减速器振动噪声影响因素研究

本文以船用人字齿轮减速器为研究对象,依据人字齿轮传动结构特点,综合考虑齿轮时变啮合刚度、误差等激励以及人字齿轮轴向定位与滑动轴承支撑等因素,建立了传动系统弯-扭-轴耦合动力学模型,通过求解得到了传动系统轴承动载荷。以轴承动载荷为激励,采用FEM/BEM方法计算了齿轮箱噪声辐射,得到了齿轮箱声场声压分布云图与各场点噪声谱。系统讨论了人字齿轮基本参数(包括齿顶高系数、顶隙系数、齿宽、螺旋角及压力角)以及减速器结构特征（人字齿轮中间连接刚度、轴向定位刚度）对减速器振动噪声的影响,为减速器的减振降噪设计提供了理论基础。     

浮动凹模温精密成形圆柱斜齿轮变形规律分析

目的研究采用浮动凹模工艺温精密成形圆柱斜齿轮时,不同凹模运动速度下齿轮力能参数和各种场量变化规律。方法结合浮动凹模原理和圆柱斜齿轮结构特点,利用Defrom-3D软件建立变形-传热耦合有限元模型,模拟圆柱斜齿轮采用浮动凹模温精密成形过程,分析不同凹模运动速度下的变形规律。结果通过模拟分析,得到了凹模运动速度不同时的温成形斜齿轮成形载荷特点、坯料流动速度场分布、等效应力-应变分布、温度场分布等规律。结论采用浮动凹模工艺成形圆柱斜齿轮,可以减小成形力,当凹模运动速度大于凸模下行速度时,齿轮成形性更好。     

Meshing characteristics of spur gear pair under different crack types

The effects of three different gear crack types such as, for example, the crack along tooth width uniformly and the crack propagating in the depth direction (crack type 1, CT1), the crack along tooth width non-uniformly and the crack propagating in both the depth and the tooth width directions (crack type 2, CT2), and the spatial crack propagating in the depth, the tooth width and the tooth profile directions (crack type 3, CT3) on the time-varying mesh stiffness (TVMS) of spur gear pairs are investigated in this study. Firstly, an analytical model for studying these three types of cracks is established based on potential energy method. A finite element (FE) model of the cracked spur gear pair is also built in the ANSYS software as well. In order to verify the analytical method, the TVMS obtained from analytical method is compared with that obtained from FE method under different crack types. Moreover, the effects of the depth, the length and the height of crack are discussed. The equivalent stress, contact pressure and displacement of tooth are also analyzed under different crack types by using the FE method. The results show that the effect of crack depth on TVMS is the largest, while that of the crack height is the smallest, and the non-penetrating crack for CT2 and CT3 will generate the non-uniform load distribution along tooth width.     

同功重比修形斜齿和直齿面齿轮性能对比研究

为深入了解同功重比修形斜齿与直齿面齿轮的性能差异,选择更适合于高速重载工况下的面齿轮传动.基于啮合原理推导了修形斜齿与直齿面齿轮齿面方程,基于CATIA建立了修形斜齿与直齿面齿轮三维模型,采用有限元接触分析方法,以接触应力、弯曲应力和重合度为面齿轮传动性能指标展开研究.研究结果表明:修形斜齿面齿轮相比修形直齿面齿轮接触应力大幅降低,算例最大接触应力降低16.3%;修形斜齿面齿轮相比修形直齿面齿轮弯曲应力大幅降低,算例最大弯曲应力降低32.4%;修形斜齿面齿轮相比修形直齿面齿轮重合度大幅提高,算例重合度提高10.3%.所以同功重比情况下,修形斜齿面齿轮传动性能优于修形直齿面齿轮,前者更适合于高速重载工况下的轻量化设计.     

Investigations on surface quality of WEDM-manufactured meso bevel and helical gears

This article presents investigations on and analysis of surface finish of meso bevel and helical gears made of stainless steel (SS 304) manufactured by wire electric discharge machining (WEDM) process using thin soft plain brass wire of 0.25?mm diameter. Effects of eight WEDM process parameters, namely, peak current, pulse-on time, pulse-off time, wire feed rate, wire tension, servo-gap voltage, dielectric pressure, and cutting speed on average and maximum surface roughness of the meso bevel and helical gears have been studied by conducting 31 experiments using one-factor-at-a-time approach to identify their optimum ranges/values for further experiments. Tooth profile, microstructure, microhardness, and topography of tooth flank surface have been studied for the best quality meso gears. Average and maximum surface roughness of tooth flank surfaces of meso bevel and helical gears increase with increase in peak current, servo-gap voltage, pulse-on time, wire feed rate, wire tension and cutting speed, and decrease with increase in pulse-off time while dielectric pressure does not significantly influence surface roughness. This work establishes that WEDM process can be an economic and sustainable manufacturing alternative for net-shaped meso-sized bevel and helical gears having better surface finish which will eliminate need of any subsequent finishing processes.     

圆柱斜齿轮成形及脱模工艺研究

针对斜齿轮的特点,提出了旋转挤压成形和旋转脱模的工艺方案,并对该工艺方案速度参数进行了精确的计算。利用有限元软件模拟分析了斜齿轮的挤压及脱模过程的等效应变、成形载荷等。结果表明:对圆柱斜齿轮采用旋转成形及脱模的工艺方案,一定程度上可以满足零件的精度要求。     

齿轮振动的边带分布特征与故障诊断实例

从理论上讨论了齿轮集中缺陷和分布缺陷的频域特征,特别是边带分布特点。集中故障的边带数目多,强度起伏小;而对于分布故障,若调制波为单频,则理论上只在啮合频率ｆ ｍ 和它的 ２ 倍频２ｆｍ 两侧各形成 １ 个边带;若调制波含有２ 次谐波,则在啮合频率ｆｍ 和它的２ 倍频２ｆｍ 两侧各形成２ 个边带。还结合测试实例,提出了运用振动信号诊断齿轮故障所应注意的问题     

基于EEMD的混合陶瓷球轴承故障双冲击特征提取

疲劳剥落是导致滚动轴承失效的主要原因,当滚道出现剥落故障时滚动体在进入和退出剥落区时的加速度振动信号表现出不同特征：进入故障区时产生以较低频率成分为主的阶跃响应;退出剥落区则引起频带较宽的脉冲响应。有效分离这两类信号特征,对实现对混合陶瓷球轴承剥落区长度的测量有重要意义。本文提出了一种基于总体经验模态分解（EEMD)的混合陶瓷球轴承剥落故障双冲击特征提取方法,该方法首先用AR模型对原始振动信号进行预白化处理,然后利用EEMD对白化后的振动信号进行去噪,并结合Hilbert包络提取算法实现对剥落故障混合陶瓷球轴承振动信号双冲击特征的有效分离提取。仿真及试验研究表明该方法能够有效地分离出混合陶瓷球轴承故障双冲击特征。     

Identifying process parameters influencing gear runout

The hardening distortions with respect to base body, clutch teeth and helical gears are investigated for a serial‐produced main shaft gear of a 20NiCrMoS6‐4 steel. The influences of casting geometry, annealing heat treatment and stress relief annealing of blanks, as well as vertical and horizontal loading arrangements during case hardening, are studied. The concentricity, roundness and runout of clutch teeth and helical gears are measured in the soft machined, hardened and hard‐machined conditions. The Brinell hardness is measured on blanks obtained from different manufacturing routes showing differences in hardness and scatter. Stress relief annealing lowers the hardness and the scatter for all groups, but has no significant effect on distortions. The case depth, core hardness and surface hardness are measured after hardening. The study shows that the surface hardness correlates with the oil flow measured in the quench tank. The effect of casting geometry is stronger for the clutch teeth compared to the helical gears. For the clutch‐teeth roundness and runout, significantly lower values are found for square geometry compared to rectangular. It is also seen that the major part of the runout comes from roundness errors which are mainly induced by the hardening. Horizontal loading reduces roundness errors and runout but produces conical base‐bodies with worse backplane flatness.