裂纹故障对行星齿轮传动系统动力学特性的影响

针对行星传动装置动态特性复杂、故障率高的问题,拟从动力学角度探索行星传动系统的故障机理。采用改进能量法,仿真分析正常与含裂纹齿轮时变啮合刚度,考虑时变啮合参数影响,运用集中参数法建立了行星齿轮传动系统动力学模型;求解得到了正常与含故障齿轮传动系统动态响应,并对比分析了裂纹故障对动力学特性的影响;通过台架实验,分析了裂纹故障对齿轮动态响应的影响,结合小波分析与EEMD方法对齿轮振动信号进行频谱分析,并对比分析了正常与故障齿轮的频域特性差异,揭示了行星齿轮传动系统的故障机理。研究表明:所建立的动力学模型精度较高,能够很好地描述含故障齿轮传动系统的动力学特性;由于裂纹故障引起传动系统振动的调制效应,导致在齿轮啮合频率附近出现明显边频带,故障齿轮箱的振动能量主要集中在高频段。     

An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack

When a tooth crack failure occurs, the vibration response characteristics caused by the change of time-varying mesh stiffness play an important role in crack fault diagnosis. In this paper, an improved time-varying mesh stiffness algorithm is presented. A coupled lateral and torsional vibration dynamic model is used to simulate the vibration response of gear-rotor system with tooth crack. The effects of geometric transmission error (GTE), bearing stiffness, and gear mesh stiffness on the dynamic model are analyzed. The simulation results show that the gear dynamic response is periodic impulses due to the periodic sudden change of time varying mesh stiffness. When the cracked tooth comes in contact, the impulse amplitude will increase as a result of reductions of mesh stiffness. Amplitude modulation phenomenon caused by GTE can be found in the simulation signal. The lateral–torsional coupling frequency increases greatly within certain limits and thereafter reaches a constant while the lateral natural frequency nearly remains constant as the gear mesh stiffness increases. Finally, an experiment was conducted on a test bench with 2 mm root crack fault. The results of experiment agree well with those obtained by simulation. The proposed method improves the accuracy of using potential energy method to calculate the time-varying mesh stiffness and expounds the vibration mechanism of gear-rotor system with tooth crack failure.     

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.     

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.     

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.     

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.     

裂纹梁动态响应有限元分析中的线弹簧模型

本文提出了一种以线弹簧模型为基础来分析裂纹梁动态响应的新数值方法．应用能量原理和断裂力学理论首次建立了线弹簧模型的刚度矩阵，从而确立了一种能使二维裂纹问题转化为一维分析的梁的有限元模型．使用这个模型，研究了不同裂纹长度和裂纹位置对悬臂梁固有频率和振型的影响，并把计算所得到的数值结果与现有的实验数据作了比较．结果表明，当无量纲裂纹长度小于０．６时，两者吻合得非常好；反之，两者之间存在较大的误差．对这种误差产生的原因，本文也作了解释．     

Natural frequencies of cracked functionally graded material plates by the extended finite element method

In this paper, the linear free flexural vibration of cracked functionally graded material plates is studied using the extended finite element method. A 4-noded quadrilateral plate bending element based on field and edge consistency requirement with 20 degrees of freedom per element is used for this study. The natural frequencies and mode shapes of simply supported and clamped square and rectangular plates are computed as a function of gradient index, crack length, crack orientation and crack location. The effect of thickness and influence of multiple cracks is also studied.     

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.     

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.     

Modal parameter identification and vibration based damage detection of a multiple cracked cantilever beam

This paper presents a detailed investigation on the modal parameter identification and vibration based damage detection of a multiple cracked cantilever beam with hollow circular cross-section. To consider multiple crack effects, a cantilever beam including cracks is considered for six damage scenarios. Finite element models are constituted in ANSYS software for numerical solutions. The results are validated by experimental measurements. Ambient vibration tests are performed to extract the dynamic characteristics using Enhanced Frequency Domain Decomposition (EFDD) and Stochastic Subspace Identification (SSI) methods. Calculated and measured natural frequencies and mode shapes for undamaged and damaged beams are compared with each other. Automated model updating is carried out using the modal sensitivity method based on Bayesian parameter estimation to minimize the differences for damage detection. In addition, modal assurance criterion (MAC) and coordinated modal assurance criterion (COMAC) factors are obtained from the mode shapes and two set of measurements to establish the correlation between the measured and calculated values for damage location identification.     

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.     

基于弹性力学的裂缝梁自由振动分析

基于弹性力学平面应力理论,采用Chebyshev-Ritz法分析裂缝梁的自由振动特性。将梁分成三个子梁,取边界函数与Chebyshev多项式的乘积作为每个子梁的位移试函数,保证解的快速收敛性,并使该方法适用于不同的几何边界条件。用里兹法列出每个子梁的振动特征方程,并根据各子梁在界面上的位移连续性条件得到整个裂缝梁的振动特征方程。计算结果与文献数据和有限元分析吻合很好。最后分析了裂缝深度和梁的高跨比对动力特性的影响。     

Natural Frequencies of Stable Griffith Cracks

A quasi-statically growing stable crack, if perturbed from its equilibrium position, will accelerate back towards it. Within quasi-static, ideal, Griffith fracture theory, vibrations of the crack and the structure have characteristic natural frequencies. We explore this feature of Griffith fracture theory in two simple geometries: a crack between a bar and a substrate, and a crack in a double-cantilever beam (DCB) specimen. For small perturbations about the stable quasi-static configuration, the dynamic equations of motion reduce to simple eigenvalue problems, leading to exact expressions for natural frequencies and mode shapes. An interesting feature of the mode shapes is that they correspond to force-free or moment-free conditions at the crack tip. Using an extended form of Hamilton's principle, we have developed a variable-length finite element technique to calculate natural frequencies and mode shapes of deformations perturbed from the stable equilibrium state. Its accuracy is demonstrated by application to the two problems analyzed previously. The possibility of crack tip oscillations in real brittle materials with irreversibility in crack tip decohesion is discussed in light of Rice's generalization of the Griffith theory.     

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.

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.     

Inertial effects on fatigue crack growth

In this paper, a cohesive zone model is used to study the influence of inertial effects on crack growth considering cyclic loading in homogenous rate‐independent materials. Quasi‐static and dynamic solutions are compared in order to establish the conditions in which the inertial effects become important in the analysis. It is discussed how speed and frequency of the loading and specimen sizes modify crack growth characteristics. In general, an increase in the loading frequency leads to a higher propagation velocity. Very high loading frequencies may lead to the formation of microcracks ahead of the crack tip and may change the failure mode of the cracked structure from crack propagation to uniform debonding. This work shows that inertial effects are specially noticeable for frequencies in the kHz range. However, applied frequencies close to natural frequencies of the cracked specimen can give rise to strong inertial effects and then a substantial reduction of fatigue life for much lower frequencies. This work also shows that critical frequencies depend on the specimen size.     

多轴系耦合作用下齿轮系统转子裂纹故障的振动特性研究

为了探究多轴系耦合齿轮系统中的转子裂纹故障与单轴系转子裂纹故障振动响应特性的异同点,基于Jones轴承建模理论,建立滚动轴承的拟静力学模型;利用Timoshenko梁单元建立传动轴的有限元模型;考虑时变啮合刚度、齿轮传递误差、陀螺效应等因素,利用集中参数法建立齿轮副的动力学模型。将轴承、传动轴与齿轮副模型进行集成,建立齿轮系统非线性动力学模型;利用能量释放率理论与应力强度因子为零法分析裂纹转子单元的呼吸效应,利用Newmark-?数值积分法对转子裂纹故障进行动力学仿真,研究转子裂纹故障的振动响应特征。结果表明：与单轴系转子裂纹故障不同,当齿轮系统发生转子裂纹故障时,由于齿轮啮合的引起的耦合效应及转子裂纹引起的呼吸效应,时域响应表现出明显的幅值调制现象,频域中转频及其2倍频幅值增加明显,在啮合频率处伴有明显的边频带。研究结果为齿轮系统转子裂纹故障的监测与诊断提供了理论基础。     

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.     

基于角域平均和连续小波变换的齿轮故障诊断研究

针对齿轮箱升降速过程中振动信号非平稳的特点,将阶次跟踪、角域平均和连续小波变换相结合,提出了基于角域平均和连续小波变换的齿轮箱故障诊断方法。首先对齿轮箱升降速瞬态信号进行时域同步采样,再对时域信号进行等角度重采样,转化为角域平稳信号,然后对角域信号进行角域平均,以消除干扰噪声的影响,最后对角域平均信号进行连续小波变换,根据小波幅值图和相位图,就可提取齿轮的故障特征。通过对齿轮齿根裂纹故障实验信号的分析,表明该方法能有效地诊断齿轮的故障状态。