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.     

Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth

Gear tooth crack will cause changes in vibration characteristics of gear system, based on which, operating condition of the gear system is always monitored to prevent a presence of serious damage. However, it is also a unsolved puzzle to establish the relationship between tooth crack propagation and vibration features during gear operating process. In this study, an analytical model is proposed to investigate the effect of gear tooth crack on the gear mesh stiffness. Both the tooth crack propagations along tooth width and crack depth are incorporated in this model to simulate gear tooth root crack, especially when it is at very early stage. With this analytical formulation, the mesh stiffness of a spur gear pair with different crack length and depth can be obtained. Afterwards, the effects of gear tooth root crack size on the gear dynamics are simulated and the corresponding changes in statistical indicators – RMS and kurtosis are investigated. The results show that both RMS and kurtosis increase with the growth of tooth crack size for propagation whatever along tooth width and crack length. Frequency spectrum analysis is also carried out to examine the effects of tooth crack. The results show that sidebands caused by the tooth crack are more sensitive than the mesh frequency and its harmonics. The developed analytical model can predict the change of gear mesh stiffness with presence of a gear tooth crack and the corresponding dynamic responses could supply some guidance to the gear condition monitoring and fault diagnosis, especially for the gear tooth crack at early stage.     

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.     

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%.     

The effects of spur gear tooth spatial crack propagation on gear mesh stiffness

Due to the presence of non-uniform load distribution, local non-homogeneity of material quality and potential misalignment of gear shafts and bearings, etc., spatial cracks may occur in the fillet region of spur gear teeth. These cracks will eventually propagate in three distinct directions either individually or simultaneously. These directions are the crack depth direction, the tooth width direction and the tooth profile direction. In this paper, an analytical investigation of the influence of spatial crack propagation on the time-varying Gear Mesh Stiffness (GMS) and also the Load Sharing Ratio (LSR) is presented. In order to quantitatively define the spatial crack propagation scenario, the involute spur gear tooth geometry cut with a typical double rounded rack is first determined using two parametric equations. The effects of some gear design parameters and initial crack locations on GMS and LSR are determined and compared with the results from previous papers that used Finite Element Analysis (FEA) in order to verify the proposed analytical model. Finally, a quasi-parabolic crack propagation scenario is assumed, in which 7 propagation cases and 3 typical crack growth paths on the tooth surface are investigated to determine their effect on the maximum reduction of GMS when compared to normal conditions. The results are important for the dynamic simulation of gear transmission behavior, and consequently helpful for the monitoring of gearbox working condition and detection of early crack damage that may exist in gear sets.     

Analytical model for mesh stiffness calculation of spur gear pair with non-uniformly distributed tooth root crack

Gear tooth crack is likely to happen when a gear transmission train is working under excessive and/or long-term dynamic loads. Its appearance will reduce the effective tooth thickness for load carrying, and thus cause a reduction in mesh stiffness and influence the dynamic responses of the gear transmission system, which enables the possibility for gear fault detection from variations of the dynamic features. Accurate mesh stiffness calculation is required for improving the prediction accuracy of the dynamic features with respect to the tooth crack fault. In this paper, an analytical mesh stiffness calculation model for non-uniformly distributed tooth root crack along tooth width is proposed based on previous studies. It enables a good prediction on the mesh stiffness for a spur gear pair with both incipient and larger tooth cracks. This method is verified by comparisons with other analytical models and finite element model (FEM) in previous papers. Finally, a dynamic model of a gear transmission train is developed to simulate the dynamic responses when cracks with different dimensions are seeded in a gear tooth, which could reveal the effect of the tooth root crack on the dynamic responses of the gear transmission system. The results indicate that both the mesh stiffness and the dynamic response results show that the proposed analytical model is an alternative method for mesh stiffness calculation of cracked spur gear pairs with a good accuracy for both small and large cracks.     

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.     

Experimental investigation of spur gear tooth mesh stiffness in the presence of crack using photoelasticity technique

Gear mesh stiffness plays a very important role in gear dynamics and it varies in the presence of gear fault such as crack. The measurement of stress intensity factor can lead to the determination of gear tooth mesh stiffness variation in the presence of crack in a spur gear system. In this paper, the technique of conventional photoelasticity has been revisited to explore the possibility of using it as a supplementary technique to experimentally measure the variation of gear mesh stiffness. An attempt has been made to calculate the variation of mesh stiffness for a pinion having a cracked tooth and a gear tooth with no crack of a spur gear pair. An analytical methodology based on elastic strain energy method in conjunction with total potential energy model has been adopted and implemented within the mesh stiffness calculations. To visualize the state of stress in a structure using finite element and other currently available methods, photoelasticity is considered to be one of the oldest and most developed experimental technique. An experimental methodology based on conventional photo-elasticity technique for computing stress intensity factor (SIF) for cracked spur gear tooth is presented for different single tooth contact position and crack length. The relation between contact position, crack length, crack configuration, SIF and the variation of total effective mesh stiffness have been quantified. Finally, a comparison has been made and the results obtained from finite element method (FEM) based on linear elastic fracture mechanics (LEFM), analytical method and proposed experimental method has been outlined.     

Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness

Most of the gear dynamic model relies on the analytical measurement of time varying gear mesh stiffness in the presence of a tooth fault. The variation in gear mesh stiffness reflects the severity of tooth damage. This paper proposes a cumulative reduction index (CRI) which uses a variable crack intersection angle to study the effect of different gear parameters on total time varying mesh stiffness. A linear elastic fracture mechanics based two dimensional FRANC (FRacture ANalysis Code) finite element computer program is used to simulate the crack propagation in a single tooth of spur gear at root level. A total potential energy model and variable crack intersection angle approach is adopted to calculate the percentage change in total mesh stiffness using simulated straight line and predicted crack trajectory information. A low contact ratio spur gear pair has been simulated and the effect of crack path on mesh stiffness has been studied under different gear parameters like pressure angle, fillet radius and backup ratio. The percentage reduction of total mesh stiffness for the simulated straight line and predicted crack path is quantified by CRI. The CRI helps in comparing the percentage variation in mesh stiffness for consecutive crack. From the result obtained, it is observed that the proposed method is able to reflect the effect of different gear parameters with increased deterioration level on total gear mesh stiffness values.     

Influences of tooth spalling or local breakage on time-varying mesh stiffness of helical gears

Time-varying mesh stiffness (TVMS) provides important information about the health condition of geared systems. Tooth faults, like crack, pitting, spalling and breakage, will change the TVMS. This work aims to reveal the influences of tooth spalling and local breakage on the TVMS of helical gears. Firstly, an analytical model is developed to incorporate the faults by combining slicing method, discrete integral method and potential energy method. Then, parametric study is conducted. Finally, conclusions are arrived and the results indicate that length, width and location of these defects are very important parameters that need to be considered for calculating TVMS of helical gears. This study provides a theoretical basis for fault diagnosis of helical geared transmissions.     

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.     

Fault mode analysis and detection for gear tooth crack during its propagating process based on dynamic simulation method

Gearbox is one of the most important parts of rotating machinery, therefore, it is vital to carry out health monitoring for gearboxes. However, it is still an unsolved problem to disclose the impact of gear tooth crack fault on gear system vibration features during the crack propagating process, besides effective crack fault mode detection methods are lacked. In this study, an analytical model is proposed to calculate the time varying mesh stiffness of the meshing gear pair, and in this model the tooth bending stiffness, shear stiffness, axial compressive stiffness, Hertzian contact stiffness and fillet-foundation stiffness are taken into consideration. Afterwards, the vibration mechanism and effects of different levels of gear tooth crack on the gear system dynamics are investigated based on a 6 DOF dynamic model. Then, the crack fault vibration mode is studied, and a parametrical-Laplace wavelet method is presented to describe the crack fault mode. Furthermore, based on the maximum correlation coefficient (MCC) criterion, the optimized Laplace wavelet base is determined, which is then designed as a health indicator to detect the crack fault. The results show that the proposed method is effective in fault diagnosis of severe tooth crack as well as the early stage tooth crack.     

基于齿轮传动系统横-扭-摆耦合非线性动力学模型的齿廓修形优化设计

基于齿轮传动系统动力学模型的齿廓修形优化设计可真实地反映修形参数对齿轮动态特性的影响。考虑几何偏心、陀螺力矩和齿向偏载力矩,建立了单级齿轮传动系统10自由度横-扭-摆耦合非线性动力学模型。提出了考虑齿轮实际运动状态并可适用于齿廓修形齿轮的啮合刚度模型,并采用解析法计算啮合刚度。为了降低齿轮传动系统的振动和噪声,以减小齿轮传动系统的动载系数为目标,建立了基于齿轮传动系统横-扭-摆耦合非线性动力学模型的齿廓修形优化模型。对某重载车辆齿轮传动系统进行了齿廓修形优化设计,优化结果有效的降低了齿轮的动载荷,可为设计低振动和低噪声的齿轮传动系统提供依据。     

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.     

Following Spur Gear Crack Propagation in the Tooth Foot by Finite Element Method

The objective of this study was to follow the crack propagation in the tooth foot of a spur gear by using Linear Elastic Fracture Mechanics (LEFM) and the Finite Element Method (FEM). The tooth foot crack propagation is a function of Stress Intensity Factors (SIF) that play a very crucial role in the life span of the gear. A two-dimensional quasi-static analysis is carried out using a program that determines the gear geometry, coupled with the Finite Element Code (ANSYS). The study estimates the stress intensity factors and monitors their variations on the tooth foot according to crack depth, crack propagation angle, and the crack position. An appropriate methodology for predicting the crack propagation path is applied by considering gear tooth behavior in bending fatigue. The results are used to predict/prevent catastrophic rim fracture failure modes from occurring in critical components.     

变位直齿轮副齿顶修形参数设计

该文以变位直齿轮副的齿顶修形为研究对象,考虑齿轮啮合的非线性接触、修正基体刚度以及延长啮合的影响,建立了考虑齿顶修形的变位直齿轮副时变啮合刚度解析模型,并通过有限元方法验证了该模型的正确性;以对刚度进行快速傅里叶变换（FFT）得到的前五阶幅值之和最小为设计目标,获得了齿顶修形的最优参数范围,并通过有限元模型进行应力分析,反证了该范围的正确性。研究结果表明:基于啮合刚度FFT前5阶幅值之和最小的设计方法可以更为高效地计算并锁定最优修形参数的范围,通过有限元进行应力分析可进一步验证齿顶修形的最优参数;齿轮齿顶修形后,刚度谐波量和应力均明显减小,有助于降低齿轮系统的振动和噪声。研究结果可为变位直齿轮副齿顶修形设计提供理论方法与依据。     

APPLYING NUMERICAL METHODS FOR DETERMINING THE SERVICE LIFE OF GEARS

Abstract The service life of gears with a crack in a tooth root can only be determined by numerical methods. An initial fatigue crack is assumed on the tensile side of a tooth root at the site of maximum reference stress and is assumed to commence growth perpendicularly to the surface. An initial assessment can be made using a two dimensional analysis, which is relatively quick and cheap to perform. However, if we wish to take into account the influence of the contact area of load transfer, which can be distributed along the tooth width in different ways, the gear has to be treated by a three dimensional finite element analysis. Crack profile advance is made in stages, each using the strain energy release rate criteria and giving ultimately the stress intensity factor as a function of average crack depth. With known fracture mechanics material characteristics of different gear layers, through which the crack propagates, the service life of a gear is then determined by numerical integration of Paris' equation. A one-sided contact area causes the crack to propagate several times faster than the preferred load distribution across the middle of the tooth.     

保角映射法求解渐开线直齿轮齿根应力

目前国际上确定渐开线齿轮齿廓保角映射函数的方法均为数值回归法,计算复杂,映射精度的提高受到限制．为此,本文给出了渐开线齿轮齿廓保角映射函数的解析解,计算方便,使映射精度大幅度提高。以上述工作为基础,笔者给出了渐开线直齿轮齿根应力的解析公式,并由算例证实了计算结果和边界元法计算值之间的一致性。     

An elastic–plastic asperity contact model and its application for micro-contact analysis of gear tooth profiles

This paper presents a continuous elastic–plastic asperity contact model with or without the consideration of friction to investigate the micro-contact properties of gear tooth profiles. The model for normal or side contact analysis is established according to Hertz contact theory and the asperity morphology feature, which yields to similar results as obtained from the model proposed by Chang W.R., Etsion I., and Bogy D.B. (CEB model) and the model proposed by Kogut L. and Etsion I. (KE model). More importantly, this model avoids the constant average contact stress as predicted by the CEB model, and the noncontinuous contact stress and deformation within the ultimate strength as given by the KE model. As a application of the present theoretical model in micro-contact analysis of rough tooth profiles, a finite element model (FE model) for elastic–plastic asperity in normal or side contact is established according to the measured surface parameters of a spur gear pair. It is shown that the extreme point of Von Mise stress of the asperities along the normal vector is ascertained by FE model, and that the extreme point is relative to the initial occurrence of the asperities plastic deformation. Compared with the present theoretical model, the similar normal contact stress along the contact radius is attained by FE model. Though the contact stress isogram in the specific plane in normal or side contact of the asperities is a circle or ellipse respectively when the plastic deformation is expanded from the inside of the asperities to their surfaces, it is in line with the distribution of elastic and plastic region of the theoretical model. Compared with CEB model, KE model, and FE model, the consistent results are attained by the present theoretical model in elastic–plastic asperity contact analysis. The results indicate that the theoretical model is applicable to the elastic–plastic asperity contact analysis on the rough surface of a spur gear drive.     

基于动力学分析的大重合度直齿圆柱齿轮强度计算

为了探索大重合度直齿圆柱齿轮的承载能力,以2排交错布置、具有纵向重合度的直齿圆柱齿轮为研究对象,基于由动力学分析得到的轮齿间动态法向力的变化规律和齿轮的啮合点位置,采用有限元分析法计算其齿根弯曲应力,利用Hertz公式计算其齿面接触应力。通过计算可知：与静态工况相比,考虑齿轮传动系统动力学效应时,大重合度直齿圆柱齿轮的最大齿根弯曲应力增大了22.49%,最大齿面接触应力增大了10.94%;与渐开线标准直齿圆柱齿轮相比,大重合度直齿圆柱齿轮的最大齿根弯曲应力减小了17.08%,最大齿面接触应力减小了43.79%。结果表明,大重合度直齿圆柱齿轮的强度大幅提升,应用潜力巨大。