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.     

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%.     

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.     

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.     

Time-varying mesh stiffness calculation of cracked spur gears

Considering the misalignment of gear root circle and base circle and accurate transition curve, an improved mesh stiffness model for a healthy gear pair is proposed and validated by the finite element method (FEM). Based on the improved method, three mesh stiffness calculation methods (method 1: straight lines for crack path and limiting line; method 2: straight line for crack path and parabolic curve for limiting line proposed in Ref. [1]; method 3: parabolic curves for crack path and limiting line) for cracked gear pair are presented and compared with FEM. The results show that there is a significant difference between method 1 and FEM under large crack condition and the results of methods 2 and 3 are quite close to FEM result, which also shows that the parabolic curve as a limiting line is appropriate. Mesh stiffness of method 2 is very close to that of method 3, which also shows that it is acceptable to assume the crack path to be a straight line.     

含裂纹平板的振动及裂纹扩展分析

针对含裂纹平板的振动与疲劳问题,研究含裂纹平板耦合动力学建模方法。首先在变形相似性原则下通过力学平衡原理推导出含裂纹项的平板振动方程,进而基于应力关系式形成裂纹项的表达式。在此基础上,利用Galerkin法将含裂纹板简化成单自由度振动系统,根据Berger经验方法产生方程的非线性项构建含裂纹板的非线性振动模型。最后通过算例探讨含裂纹板的动力学特性,协同Paris方程研究含裂纹平板动力学与裂纹扩展的耦合行为。研究结论表明,阻尼大小和激励力的变化对含裂纹板的振动特性及裂纹扩展规律具有显著影响。所提出的含裂纹平板耦合动力学建模方法,考虑了振动与裂纹扩展的耦合效应,为飞行器板结构抗振动疲劳设计提供理论依据。     

考虑裂纹表面摩擦阻尼的振动疲劳裂纹扩展分析

以含表面裂纹悬臂梁为研究对象,研究了裂纹面摩擦效应对裂纹疲劳扩展的影响。分析时,用双线性弹簧描述裂纹呼吸行为,用Galerkin方法把呼吸裂纹梁简化为单自由度系统,基于Coulomb摩擦模型和能量耗散理论推导了摩擦阻尼损耗因子,运用广义的Forman方程模拟疲劳裂纹扩展,通过振动分析与裂纹扩展计算同步进行的方法考虑振动与疲劳的耦合效应,探讨了摩擦阻尼对裂纹梁疲劳裂纹扩展寿命的影响。结论表明,摩擦阻尼损耗因子随裂纹扩展呈单调递增趋势,摩擦阻尼对振动疲劳裂纹扩展的影响不容忽视     

J-Q characterization of propagating cracks

An investigation is performed to determine to what extent the state at a growing crack tip vicinity can be characterised by J and Q calculated from FE analyses of successively stationary crack tip positions. FE models in two-dimensionals of single edge notch bend and double edge cracked panel specimens with several different crack lengths are used to cover a range of load and constraint levels. The stress and strain fields are compared between different specimens keeping J- and Q-values equal. A remeshing technique in the commercial FE-code ABAQUS is used to enhance the efficiency of the analysis. The results show that the J-Q-theory provides reasonably accurate crack tip characterization also for growing cracks. This leads to the conclusion that FE analyses of successive stationary cracks rather than full FE propagation analyses are sufficient. The limit of validity for propagation is similar to the validation limit for the stationary case, although somewhat more restrictive.     

含裂纹悬臂梁的振动与疲劳耦合分析

基于Paris方程和同步分析方法考虑振动与疲劳裂纹扩展耦合之影响,提出一种含裂纹梁的振动疲劳寿命分析思路.振动分析过程中,利用线性弹簧等效裂纹段,复弹性模量引入阻尼损耗因子,得到考虑裂纹扩展、激励频率和阻尼等因素影响的动应力响应.结果表明:裂纹扩展、激励频率和阻尼等因素对疲劳寿命具有重要的影响.通过振动分析与疲劳裂纹扩展寿命估算同步进行,可进一步提高疲劳寿命估算精度.     

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.     

Longitudinal vibration of a continuous cracked bar

A continuous cracked bar vibration theory is developed for longitudinal vibration of rods with an edge crack. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked bar as a one-dimensional continuum. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack found with fracture mechanics methods.

The results of three independent evaluations of the lowest natural frequency of longitudinal vibrations of a bar with a single edge crack are presented: the continuous cracked bar vibration theory, the lumped crack bar vibration analysis, and experimental results obtained on aluminum bars with fatigue cracks. Experimental results fall between the values predicted by the two analytical methods. Moreover, the continuous bar theory agrees better with the experimental results than the lumped crack flexibility theory for small cracks. For larger cracks, a/h>0.4, experimentation was difficult due to the co-existence of several coupled modes and no reliable results could be obtained.     

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.     

Influence of Crack on Structure Vibration of Gear Tooth

This paper presents a theoretical model for analyzing the dynamic characteristic such as natural frequencies and vibration mode shapes of cracked gears. The influence of crack position and crack size on the dynamic characteristic of gears is thoroughly investigated using the proposed theoretical model as well as the finite element method (FEM) for the sake of model validation. The theoretical analysis and numerical simulation show that the influence of crack size and crack position on the dynamic responses of cracked gears is significant and that the influence of crack position is larger than that of crack size. The natural frequencies drop with the increase of crack size and the low order natural frequencies drop more notably. The natural frequencies drop more significantly when crack is located at tooth root than at tooth tip. The vibration mode shapes of cracked gear tooth are very different from those of gear tooth without crack, and the vibration amplitude increases significantly in crack neighborhood. These observations are very valuable for damage detection and fault diagnosis of gear system.     

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.     

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.     

Vibration signal modeling of a planetary gear set for tooth crack detection

In a planetary gearbox, there are multiple vibration sources, and the transmission path of vibration signals changes due to the rotation of the carrier. This study aims to model the vibration signals of a planetary gearbox and investigate the vibration properties in the healthy condition and in the cracked tooth condition. A dynamic model is developed to simulate the vibration source signals. A modified Hamming function is proposed to represent the effect of the transmission path. By incorporating the effect of multiple vibration sources and the effect of transmission path, resultant vibration signals of a planetary gearbox are obtained. Through analyzing the resultant vibration signals, some vibration properties of a planetary gearbox are revealed and the fault symptoms of sun gear tooth crack are identified and located. Finally, the proposed approach is experimentally verified.     

FATIGUE CRACK GROWTH PREDICTION IN SPECIMENS SIMILAR TO SPUR GEAR TEETH

Abstract— The problem of fatigue crack propagation in surface treated specimens similar to gear teeth is analysed. Experimental fatigue tests were carried out on carburized, and carburized and shot peened, specimens. In order to predict crack propagation and to consider the effect of the treatments, different models of the cracked specimens were realized. Two numerical approaches were followed: the finite element method and the weight function technique. Two- and three-dimensional finite element models were constructed, and the stress-intensity factors were evaluated by considering the effect of both the load and the residual stresses due to the treatments. The agreement between these and those obtained through the weight function technique is good. The weight function approach was used directly in the computer software package (which allows crack propagation predictions) and considered the effects of hardness and residual stresses. The comparison between theoretical predictions and experimental results validated the approach followed.     

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.