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.     

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.     

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.     

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.     

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.     

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.     

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.     

Differential diagnosis of spall versus cracks in the gear tooth fillet region

This paper presents a technique to differentially diagnose two localized gear tooth faults: a spall and a crack in the gear tooth fillet region. These faults could have very different prognoses, but existing diagnostic techniques only indicate the presence of local tooth faults without being able to differentiate between a spall and a crack. The effects of spalls and cracks on the behavior/response of gear assemblies were studied using static and dynamic simulation models. Changes in the kinematics of a pair of meshing gears due to a gear tooth root crack and a tooth flank spall were compared using a static analysis model. The difference in the variation of the transmission error caused by the two faults reveals their characteristics. The effect of a tooth crack depends on the change in stiffness of the tooth, while the effect of a spall is predominantly determined by the geometry of the fault. The effect of the faults on the gear dynamics was studied by simulating the transmission error in a lumped parameter dynamic model. A technique had previously been proposed to detect spalls, using the cepstrum to detect a negative echo in the signal (from entry into and exit from the spall). In the authors’ simulations, echoes were detected with both types of fault, but their different characteristics should allow differential diagnosis. These concepts are presented prior to experimental validation in hopes that the diagnostic techniques will be useful in the failure analysis community prior to the validation by ongoing experimental testing of the concepts and the evaluation of how metallurgical defects may influence fault development and detection.     

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.     

Influence of different load models on gear crack path shapes and fatigue lives

A computational model for determination of the service life of gears with regard to bending fatigue at gear tooth root is presented. In conventional fatigue models of the gear tooth root, it is usual to approximate actual gear load with a pulsating force acting at the highest point of the single tooth contact. However, in actual gear operation, the magnitude as well as the position of the force changes as the gear rotates. A study to determine the effect of moving gear tooth load on the gear service life is performed. The fatigue process leading to tooth breakage is divided into crack‐initiation and crack‐propagation period. The critical plane damage model has been used to determine the number of stress cycles required for the fatigue crack initiation. The finite‐element method and linear elastic fracture mechanics theories are then used for the further simulation of the fatigue crack growth.     

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。     

Wavelet analysis for gear crack identification

There are many typical damages and faults that can cause problems in relation to gear unit operation, a crack in the tooth root being probably the least desirable among them. It often results in failure of gear unit operation. Fault analyses presented in this article are based on gear units with real damages or faults, produced on the basis of real operating conditions. A test plant has been used. A possible damage can be identified by monitoring vibrations. The influences of a crack in a single-stage gear unit on produced vibrations are presented. A fatigue crack in the tooth root causes significant changes in tooth stiffness, whereas, in relation to other faults, changes of other dynamic parameters are more expressed. Different methods are used to analyse signals acquired by experiments. Signal analysis has been carried out in relation to a non-stationary signal, using the family of Time Frequency Analysis tools, such as Wavelets Analyses. Typical spectrogram and scalogram patterns resulting from reactions to faults or damages indicate the presence of damages in a very reliable way.     

Molecular dynamics simulation of crack initiation and propagation in bcc iron under load within spur gear tooth root

Spur gears are widely used in practice, and one of their typical failures is tooth breakage. In general, the tooth breakage occurs at tooth root, and the amount of crack growth during a meshing cycle is in atomistic scale. This work aims at identifying the mechanisms of crack initiation and propagation at tooth root by using molecular dynamics simulation. The results prove that there are phase transition regions and edge dislocations at crack tips. According to the distribution characteristic of the atomic potential, its concentration can be observed obviously by visualization software. In these concentration regions, microvoids come into being and expand gradually, which results in the subcrack initiation. Additionally, the microvoids and subcracks propagate along the high potential direction and then come together to accelerate the crack growth. Through carrying out a comparative simulation, the effects of heavy load at single meshing area on crack initiation and propagation are addressed.     

Crack behavior in a high contact ratio spur gear tooth and its effect on mesh stiffness

The efficiency of high contact ratio (HCR) gearing can be achieved by proper selection of gear geometry for increased load capacity and smoother operation despite of their high sliding velocities. The prediction of variation in mesh stiffness of HCR gearing is critical as the average number of teeth being in contact is high at a given time as compared to conventional low contact ratio (LCR) gearing. In this paper, linear elastic fracture mechanics (LEFM) based finite element method is used to perform the crack propagation path studies of HCR spur gear having tooth root crack for two gear parameters viz. backup ratio and pressure angle. A total potential energy model has been adopted to analytically estimate the mesh stiffness variation. The results depict the mesh stiffness reduction in the presence of the crack. The percentage change in mesh stiffness with increasing crack length is an important parameter in fault diagnosis of geared transmission. Higher the percentage change in mesh stiffness, easier to detect the fault. Two gear parameters viz. back-up ratio and pressure angle has been studied and the effect of crack length on mesh stiffness have been outlined. With the increase of deterioration level gears having lower back-up ratio fault can be detected at an early stage, similarly, chances for early fault detection is more for gears having higher pressure angle.     

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。     

The spur planetary gear torsional stiffness and its crack sensitivity under quasi-static conditions

The sun–planet and ring–planet tooth mesh stiffness variations and the resulting transmission errors are the main internal vibration generation mechanisms for planetary gear systems. This paper presents the results of torsional stiffness analysis of involute spur planetary gear systems in mesh using finite element methods. A planetary gear model with three planet gears and fixed ring gear and its subsystem models have been developed to study the subsystem and overall torsional stiffnesses. Based on the analysis of torsional mesh stiffness, predictive models for single branch sun–planet–ring and overall planetary gear torsional stiffnesses have been proposed. A crack coefficient was introduced to the sun–planet and ring–planet meshes to predict the effect and sensitivity of changes to the overall torsional mesh stiffness. The resulting mesh stiffness crack sensitivity of the overall gear system was analysed under quasi-static conditions. It was found that the carrier arm stiffness has great influence on the crack sensitivity while the overall stiffness was most sensitive to the crack on the sun–planet mesh.     

Analysis of crack propagation during tooth interior fatigue fracture

Tooth interior fatigue fracture is a failure mode that is initiated as a fatigue crack in the interior of the tooth of a gear. TIFF cracks have been observed in case hardened idler gears. A fracture mechanical analysis of a TIFF crack is performed utilising FEA. A 3D TIFF crack is modelled at a position in the tooth that corresponds with an observed crack surface. The different material properties in the case and the core, determined by mechanical testing, are considered, as well as the residual state of stress due to case hardening. Various crack lengths are analysed to estimate crack propagation both into the core and into the case. The following results have been found:

• A TIFF crack initiated slightly under the case layer will propagate into the case layer where it stops.

• The main crack propagation will take place in the core.

• The crack propagation is only a small portion of the total life (order of 10⁵ cycles).

• After reaching the case layer the TIFF crack eventually deflects toward the tooth root and the upper part of the tooth falls off. The crack deflection is due to redistribution of contact loading. Several gear teeth pairs are simultaneously in contact and the cracked tooth is loaded less than the uncracked during this stage of life.

Fatigue failure of a star–ratchet gear

We present an analysis of the fatigue failure of an 18 tooth star–ratchet gear (SRG). The subject gear was implemented in the freewheel assembly of a mountain bicycle. After 6 years of service, the gear failed unexpectedly during a typical off-road ride. The unique geometry of SRGs precluded a simple comparison to existing gear lifetimes. Scanning Electron Microscopy (SEM) analysis of the failed gears showed crack initiation at the root of the gear teeth, followed by fatigue crack propagation and eventual chip-out. A biomechanical analysis of pedaling forces, coupled with explicit power data obtained from instrumented rides over the same trails, in conjunction with a Finite Element Analysis (FEA) of the gear, were used to determine stress amplitudes for fatigue calculations. Energy dispersive spectroscopy (EDS) determined the alloy composition of the gear and thus set the strength and fatigue properties of the gear. Basquin’s law, Goodman’s mean stress correction, and Miner’s rule were used to estimate the lifetime, in bike rides, of the gear. Our analysis led to an estimate of 2288 rides, while failure was reported after roughly 312 rides. Given the uncertainties in fatigue life estimation and service use, we find this estimate acceptable.     

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.     

Investigation of crack propagation scatter in a gear tooth’s root

This paper describes the problem of determining crack initiation location and its influence on crack propagation in a gear tooth’s root. Three different load positions on the gear tooth’s flank were considered for this investigation of crack initiation and propagation. A special test device was used for the single tooth test. It can be concluded from the measurements that a crack can be initiated at very different locations in a tooth’s root and then propagate along its own paths. A numerical investigation into a crack initiation’s position and its influences on its propagation were carried out within the framework of linear fracture mechanics. The influence of a tooth’s load position, the geometry of the tooth’s root, and the influence of non-parallel load distribution on the tooth’s flank were considered when investigating the crack initiation’s position. Results show that linear fracture mechanics can be used for determining crack propagation, if better initial conditions for crack initiation are considered.