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 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.     

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.     

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.     

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.     

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.     

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.     

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.     

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。     

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.     

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.     

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.     

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。     

内激励作用下的单对齿轮振动噪声分析

综合考虑齿轮时变啮合刚度及齿轮误差等内部激励的影响,建立了单对齿轮扭转振动物理模型,采用龙哥库塔法进行求解,得到了齿轮节点位移。分析齿轮几何参数对齿轮啮合的振动噪声的影响,并采用修正Kato公式法对振动产生的噪声进行了定量计算。结果表明该方法可以较好地对齿轮振动所产生的噪声进行定量计算,为齿轮系统的降噪设计奠定了基础。     

一种改进的齿轮非线性动力学模型

在考虑齿面摩擦、齿轮时变啮合刚度和齿侧间隙的情况下,推导出了改正的齿轮副系统的非线性动力学模型,应用符号运算软件,编写符号运算程序,得到了齿轮副非线性振动微分方程。该模型在计算摩擦力时,考虑了载荷在啮合区的动态分配,并根据啮合区单双齿交替的特点提出用周期扩大法建立摩擦力、齿轮时变刚度的模型,改正的齿轮非线性动力学模型是一个周期系数分段线性的非自治系统,与以前所建立的模型相比,该模型的参变系数是具有相同周期的周期函数,新的齿轮非线性动力学模型的建立为求解时变的齿轮动力学方程近似解析解带来方便。     

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%.     

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.     

Three-Dimensional Analyses by Finite Element Method of a Spur Gear: Effect of Cracks in the Teeth Foot on the Mesh Stiffness

In this paper, a finite element method with a three-dimensional survey is presented. The effect of crack dimension and the direction of crack propagation, in the teeth foot, on the mesh stiffness is studied. For spur gears, the mesh stiffness is affected in a meaningful manner by the presence of a foot crack of one or more teeth. This study is an attempt to estimate the effect of crack size, position, and direction on the spectrum of the gear mesh stiffness.     

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.     

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.