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Analytical modelling is an effective approach to obtaining a gear dynamic response or vibration pattern for health monitoring and useful life prediction. Many researchers have modelled this response with various fault conditions commonly observed in gears. The outcome of such models provides a good idea about the changes in the dynamic response available between different gear health states. Hence, a catalogue of the responses is currently available, which ought to aid predictions of the health of actual gears by their vibration patterns. However, these analytical models are limited in providing solutions to useful life prediction. This may be because a majority of these models used single fault conditions for modelling and are not valid to predict the remaining life of gears undergoing more than one fault condition. Existing reviews related to gear faults and dynamic modelling can provide an overview of fault modes, methods for modelling and health prediction. However, these reviews are unable to provide the critical similarities and differences in the single-fault dynamic models to ascertain the possibility of developing models under combined fault modes. In this paper, existing analytical models of spur gears are reviewed with their associated challenges to predict the gear health state. Recommendations for establishing more realistic models are made especially in the context of modelling combined faults and their possible impact on gear dynamic response and health prediction. [ABSTRACT FROM AUTHOR]

Analytical modelling is an effective approach to obtaining a gear dynamic response or vibration pattern for health monitoring and useful life prediction. Many researchers have modelled this response with various fault conditions commonly observed in gears. The outcome of such models provides a good idea about the changes in the dynamic response available between different gear health states. Hence, a catalogue of the responses is currently available, which ought to aid predictions of the health of actual gears by their vibration patterns. However, these analytical models are limited in providing solutions to useful life prediction. This may be because a majority of these models used single fault conditions for modelling and are not valid to predict the remaining life of gears undergoing more than one fault condition. Existing reviews related to gear faults and dynamic modelling can provide an overview of fault modes, methods for modelling and health prediction. However, these reviews are unable to provide the critical similarities and differences in the single-fault dynamic models to ascertain the possibility of developing models under combined fault modes. In this paper, existing analytical models of spur gears are reviewed with their associated challenges to predict the gear health state. Recommendations for establishing more realistic models are made especially in the context of modelling combined faults and their possible impact on gear dynamic response and health prediction.

This paper established the system dynamics model for two kinds of tooth cracks of different depths of the sun gear and inner gear ring to study the influence mechanism of crack failure on the tooth root strain of the planetary gear transmission system. Combined with the finite element model of the inner gear ring, the tooth root strain of the ring was solved. Experiments verified the correctness of the solution method. The root strain under the crack fault of the sun gear and the tooth crack fault of the inner gear ring is analyzed, and the following conclusions are drawn: Periodic fault impact occurs in the strain signal of the tooth root of the inner gear ring during the crack fault of the sun gear root. The fault can be extracted by the fast spectral kurtosis method (FSK), and the fault components are used to determine whether the sun gear cracks. The Lempel–Ziv index showed a tendency to increase gradually during the process of solar wheel crack deepening, which could be used as the damage index of crack depth. The results can provide a basis and reference for fault diagnosis. [ABSTRACT FROM AUTHOR]

The current research on wind turbine gearboxes (WTG) considering multiple-factors influence mainly focuses on dynamic response. However, load sharing performance (LSP) is also extremely important for gear transmission system reliability. In this paper, considering the comprehensive effects of manufacturing errors, assembly errors, clearance floating between stages, mesh stiffness and time-varying load, a coupled translation-torsion dynamic model for WTG was developed to study the LSP. Based on the amplitude variations of time-varying mesh stiffness caused by tooth crack, an approach is provided to introduce the tooth crack into the dynamic model. The results indicate the LSP is mostly sensitive to multiple-errors and severe tooth crack, followed by the time-varying load. The LSP of a system is periodically affected by tooth crack, and the fluctuation ranges increase with the crack growth. Furthermore, the crack of sun gear has greater influence on LSP than that of planet and ring gear. [ABSTRACT FROM AUTHOR]

Dynamic modelling of wind turbine drive-train system (WTDS) is important to analyse the behaviour of the vibration and load sharing characteristics of the gearbox. The WTDS mostly contains one planetary gear stage and two parallel gear stages. Sun gear is a significant component in the planetary gear stage and is prone to fail under fatigue loading due to the bending and shear effect. Crack is most likely to occur in the gear tooth root and it decreases the tooth thickness and load-carrying capacity of the gear. The presence of crack decreases the tooth contact number and tooth contact position during the time of meshing. This leads to a variation and quantified reduction in the time varying mesh stiffness (TVMS) of a meshing gear pair. Hence, a dynamic model is developed by including different crack depth levels in the sun gear. The governing equations for the model are derived using Lagrange's formulation and dynamic responses are obtained numerically by step-by-step direct integration method. The dynamic responses like contact forces and angular velocities are estimated at 0%, 10%, and 30% of crack depth in the sun gear tooth and analysed to study the dynamic behaviour of the drive-train for un-cracked and cracked gears beforehand. [ABSTRACT FROM AUTHOR]

This study aimed to evaluate the stress distribution that results from temperature changes in teeth restored with gold inlay and to predict the fatigue life of the teeth. A thermal load regimen for finite element analysis (FEA) was established in an in vivo experiment, in which participants with gold inlays drank hot and cold water. An extracted human maxillary molar was converted into a 3D FEA model. The tooth models with Class I and II preparations were restored with gold inlay in the experimental groups. The stress distribution was analyzed under the thermal loads. Also, lifetime prediction was conducted using Basquin's formulation. The gold inlay groups showed a higher stress distribution than the preparation-only groups regardless of the type of preparation. The Class II gold inlay group showed the shortest fatigue life of 2.48 x 1011 cycles, whereas the Class I preparation group showed the longest fatigue life of 4.49 x 1016 cycles. [ABSTRACT FROM AUTHOR]

Abstract An improved variable cross‐section cantilever beam model for evaluating the time‐varying mesh stiffness (TVMS) of the perfect gear tooth is developed in which the tooth number of driving gear is less than 42 and that of driven is more than 42. The TVMS obtained by the proposed method is compared with the result without considering the misalignment between the base circle and gear root. Four types of root crack models and changes in TVMS of 13‐crack levels are presented. The fault vibration characteristic of a single‐stage spur gear reducer with root crack is analyzed and the correctness is qualitatively verified by the vibration signals of an experimental gearbox with crack or missing failure. The results presented in this paper are of great significance for a deep understanding of the possible causes of vibration and noise of gears and provide a theoretical foundation for the fault diagnosis of the gearbox.

In the current work, the nonlinear dynamic behavior of an eight degrees of freedom gear system is investigated. Tooth crack is introduced into the model. The main sources of excitation are the time varying mesh stiffness (TVMS), time varying mesh damping, backlash and friction inter teeth. By using the potential energy method applied into the cantilevered beam, the TVMS is calculated. The backlash is considered as a dead zone. The effect of backlash, friction and tooth crack on the vibration frequencies is highlighted. The crack detection is based on the time-frequency analysis. [ABSTRACT FROM AUTHOR]

Optical coherence tomography (OCT) is an imaging technique that can visualize the internal biological structure without X-ray exposure. Swept-source OCT (SS-OCT) is one of the latest version of OCT, wherein the light source is a tunable laser that sweeps near-infrared wavelength light to achieve real-time imaging. The imaging depth of OCT is highly influenced by the translucency of the medium. The medium that does not transmit light and the deeper structure beyond the range of light penetration depth are not relevant for OCT imaging. In OCT, sound enamel is almost transparent at the OCT wavelength range, and enamel and dentin can be distinguished from each other as the dentin–enamel junction (DEJ) appears as a dark border. Demineralized enamel and dentin are imaged as bright zones because of the formation of numerous micro-porosities where the backscatter of OCT signal is increased. In cavitated caries at interproximal or occlusal hidden zone, the upper margin of the cavity reflects the signal showing a distinct bright border in the SS-OCT image. SS-OCT is capable of determining crack penetration depth even when the cracks extended beyond the DEJ. SS-OCT has a high degree of sensitivity and specificity for the detection of dental caries and tooth cracks. SS-OCT is also capable of detecting non-carious cervical lesions and occlusal tooth wear in cross-sectional views to estimate the amount of tooth structure loss.

This paper established the system dynamics model for two kinds of tooth cracks of different depths of the sun gear and inner gear ring to study the influence mechanism of crack failure on the tooth root strain of the planetary gear transmission system. Combined with the finite element model of the inner gear ring, the tooth root strain of the ring was solved. Experiments verified the correctness of the solution method. The root strain under the crack fault of the sun gear and the tooth crack fault of the inner gear ring is analyzed, and the following conclusions are drawn: Periodic fault impact occurs in the strain signal of the tooth root of the inner gear ring during the crack fault of the sun gear root. The fault can be extracted by the fast spectral kurtosis method (FSK), and the fault components are used to determine whether the sun gear cracks. The Lempel–Ziv index showed a tendency to increase gradually during the process of solar wheel crack deepening, which could be used as the damage index of crack depth. The results can provide a basis and reference for fault diagnosis.

Featured Application: The proposed TVMS calculation method can be used for building accurate dynamic model of the gear system. The time-varying mesh stiffness (TVMS) is the crucial parameter of the dynamic model of the gear system. Accurate calculation of TVMS is essential for effective fault diagnosis of the gear system. A mesh stiffness improved method considering both the gear tooth transition curve and deformation of the gear body is presented in this paper, and the stiffness calculation expressions under healthy and crack states are given, respectively. Based on the lumped parameter method, the dynamic model of planetary gear with crack in sun gear is constructed, and the vibration response is solved. The simulation results show when the tooth root cracks appear, the vibration response of the tooth has obvious shock response characteristics. The characteristic frequency and frequency multiplication of sun gear fault can be found obviously by envelope analysis. The simulation signal and the test signal obtained by the drivetrain dynamics simulator gearbox test rig are compared and verified. The comparison and verification results show that the proposed TVMS calculation method and the dynamic model are accurate, which can provide a certain theoretical basis for the fault diagnosis of planetary gear with crack fault. The test results are consistent with the numerical analysis results, which provides a theoretical basis for the fault diagnosis of tooth crack. [ABSTRACT FROM AUTHOR]

Optical coherence tomography (OCT) is an imaging technique that can visualize the internal biological structure without X-ray exposure. Swept-source OCT (SS-OCT) is one of the latest version of OCT, wherein the light source is a tunable laser that sweeps near-infrared wavelength light to achieve real-time imaging. The imaging depth of OCT is highly influenced by the translucency of the medium. The medium that does not transmit light and the deeper structure beyond the range of light penetration depth are not relevant for OCT imaging. In OCT, sound enamel is almost transparent at the OCT wavelength range, and enamel and dentin can be distinguished from each other as the dentin–enamel junction (DEJ) appears as a dark border. Demineralized enamel and dentin are imaged as bright zones because of the formation of numerous micro-porosities where the backscatter of OCT signal is increased. In cavitated caries at interproximal or occlusal hidden zone, the upper margin of the cavity reflects the signal showing a distinct bright border in the SS-OCT image. SS-OCT is capable of determining crack penetration depth even when the cracks extended beyond the DEJ. SS-OCT has a high degree of sensitivity and specificity for the detection of dental caries and tooth cracks. SS-OCT is also capable of detecting non-carious cervical lesions and occlusal tooth wear in cross-sectional views to estimate the amount of tooth structure loss. [ABSTRACT FROM AUTHOR]

Accurate mesh stiffness models for cracked spur gears are critical to obtain gearbox failure characteristics. However, the influence of crack on gear body deflection is often ignored by most researchers. In this study, analytical models of mesh stiffness for cracked spur gears considering gear body deflection are proposed. Both the cracked tooth and the adjacent tooth are included to simulate the mesh stiffness with tooth crack. Effects of crack length and inclination angle on mesh stiffness are investigated by these analytical models. Dynamic simulation is conducted by a six-degrees-of-freedom dynamic model incorporating the developed analytical mesh stiffness models. Frequency spectrum analysis and statistical indicators are adopted in order to evaluate the effects of tooth crack. The simulated results reveal that tooth crack produces mesh stiffness reduction in a complete rotation period for both the cracked tooth and the adjacent tooth. The total mesh stiffness is not only affected by the cracked tooth but also by the adjacent tooth due to tooth crack. The proposed analytical models can provide relatively accurate mesh stiffness for cracked spur gears by considering gear body deflection. The dynamic response obtained in this simulation by the proposed model has more obvious impulsive feature than the one without considering the effect of crack on the adjacent tooth and gear body, which are more convenient for tooth crack fault diagnosis. [ABSTRACT FROM AUTHOR]

• Tooth spalls and cracks change transmission error in different ways. • Vibration features of spalls and cracks are simulated and measured. • Timing of fault-related events can be used to distinguish different fault types. • Mesh phasing can be used to identify which planet is faulty. • A comprehensive diagnostic procedure for planet gears is presented. Planetary gearboxes have a wide range of applications, but due to their complex layout, the diagnostics of planetary gearboxes is more challenging than that of fixed-axis gearboxes. This is especially so for planet gears, as multiple planets mesh simultaneously, share the same rotating speed and are in mesh with both the sun and ring gears. To enable the development of reliable diagnostic and, even more critically, prognostic algorithms, the detection of a fault needs to be complemented with information about its location and the dominant failure mode. In this study, we present a comprehensive procedure to diagnose localised planet gear faults, adding to fault detection the differential diagnosis of cracks and spalls and the identification of the faulty planet. To do this, first the transmission error (TE) of planet cracks and spalls is investigated using a finite element model. This is aimed at providing a thorough understanding of how these two types of faults result in different excitations of the system. A lumped parameter model (LPM) is then used to link the changes in TE to specific vibration patterns, which are identified as characteristic of either tooth cracks or spalls. These observations are then combined with a recently proposed concept that uses mesh phasing to identify which planet gear carries a fault, resulting in a comprehensive framework that can both diagnose the type of fault and determine which planet gear it is on. The methodology is verified using experimental data obtained from a planetary gearbox test rig. [ABSTRACT FROM AUTHOR]

Spur gear systems are widely used in power transmission systems in the industry. One of the common defects of the gears is tooth crack. Tooth crack increases the vibration and also generates noise. Previous studies have shown that tooth stiffness will decrease due to any crack and it is important to estimate the magnitude of reduction of tooth stiffness. This research suggests a new analytical approach for crack modeling and determining the reduction of time-varying gear mesh stiffness by Elastic Spring Method (ESM). Based on this approach, two or more cracks can be considered in one tooth. However, previous studies have primarily concentrated on one crack. In addition, it should be voted that each crack is replaced by one linear and one torsional spring in the present study. The results that were obtained from this method are validated through a comparison with Limit Line Method (LLM) and Finite Element Method (FEM). [ABSTRACT FROM AUTHOR]

As the prompt development of modern railway transportation towards high-speed and high load-capacity, the high-power locomotive is urgently required. Under this situation, the wheel–rail dynamic interaction is becoming more and more intensified which will deteriorate the vibration condition of the key elements of locomotive, such as the gear transmissions. Once gear failures are present, such as gear tooth crack or breakage, it is likely to threaten the operation safety of the locomotive. Thus, deep insight into the fault features of the locomotive gear transmission is urgently necessary for prevention of the induced disastrous consequences. This paper is concentrated on the fault vibration feature extraction of a locomotive in presence of gear tooth root crack under the complicated dynamic excitations from both the gear transmissions and the nonlinear wheel–rail interactions. The locomotive–track coupled dynamics model considering the dynamic effect of the mechanical power transmission path is employed, and the time-varying mesh stiffness of the gear pair with tooth root crack fault and the rail geometric irregularities are then incorporated into the dynamics model to obtain the vibration responses. Then, angular synchronous average technique is proposed to enhance the fault vibration features, and the statistical indicators extracted in frequency domain are developed to reveal the evolution law for the crack propagation scenarios along crack depth or tooth width. The analyzed results indicate that the angular synchronous average technique could effectively reveal the fault vibration feature, and the M8A in the selected statistical indicators is most sensitive to the tooth crack propagation in frequency domain. [ABSTRACT FROM AUTHOR]

• The slice method is used to derive the meshing relationship of spiral bevel gears. • TVMS can be evaluated effectively through the slice model. • The influence of the tooth crack on the mesh stiffness is investigated. • The dynamic response with crack fault is solved and verified. Time-varying mesh stiffness (TVMS) is one of the main dynamic excitation sources of gear transmission systems. The accurate calculation of TVMS is the foundation to comprehensively investigate the vibration of powertrain. Compared with the spur gear, the meshing process of spiral bevel gear is spatially time-varying, which makes the analytical calculation of TVMS more difficult. In this paper, the meshing relationship of spiral bevel gears is first established by extending the slice method of spur and helical gears to spiral bevel gear. Secondly, the analytical models of perfect and cracked spiral bevel gears are presented to evaluate TVMS. Meanwhile, the effects of crack length and depth on mesh stiffness are investigated. Thirdly, the influence of crack length and depth on the dynamic response are simulated based on the 8-DOF dynamic model of spiral bevel gear. Finally, the experiments are conducted to verify the effectiveness of the analytical model. Experimental results demonstrate that the existence of tooth crack fault makes the response characterized by periodic impact with the rotating frequency of cracked pinion. [ABSTRACT FROM AUTHOR]

The time-varying mesh stiffness (TVMS) is the crucial parameter of the dynamic model of the gear system. Accurate calculation of TVMS is essential for effective fault diagnosis of the gear system. A mesh stiffness improved method considering both the gear tooth transition curve and deformation of the gear body is presented in this paper, and the stiffness calculation expressions under healthy and crack states are given, respectively. Based on the lumped parameter method, the dynamic model of planetary gear with crack in sun gear is constructed, and the vibration response is solved. The simulation results show when the tooth root cracks appear, the vibration response of the tooth has obvious shock response characteristics. The characteristic frequency and frequency multiplication of sun gear fault can be found obviously by envelope analysis. The simulation signal and the test signal obtained by the drivetrain dynamics simulator gearbox test rig are compared and verified. The comparison and verification results show that the proposed TVMS calculation method and the dynamic model are accurate, which can provide a certain theoretical basis for the fault diagnosis of planetary gear with crack fault. The test results are consistent with the numerical analysis results, which provides a theoretical basis for the fault diagnosis of tooth crack.

Abstract The manuscript is devoted to determine the time variable mesh stiffness (TVMS) and crack propagation behaviour in spur gear tooth using extended finite element method (XFEM). The TVMS of gear pairs provides significant information about the condition of the gear system. Thus, the accurate estimation of TVMS is essential to get the information of the primary source of vibration and noise. Extended finite element method is very effective for discontinuity due to faults like cracks etc. The capability of XFEM to enrich approximate spaces locally makes it appropriate method to compute the values of TVMS in spur gear. A gear tooth crack problem has been solved using XFEM and the obtained results are compared with the analytical results for validation. Further, a problem of crack at the root resign in thin rim gear is simulated to investigate the effect of rim thickness on gear tooth crack propagation path. The estimated crack propagation path shows good agreement with the results available in literature. Highlights • The value of time variable mesh stiffness is used to identify crack of spur gear. • Extended finite element method estimates crack propagation path without remeshing. • Crack propagation path for spur gear depends on backup ratio and crack initial point. [ABSTRACT FROM AUTHOR]

Planetary gearbox (PG), especially multi-stage PG, is widely used in various commercial and military applications. A dynamic model of PG is a valuable way to investigate its vibration behaviors in good and faulty statuses for designing a highly reliable PG and corresponding robust monitoring techniques. In this paper, a modified dynamic model is firstly constructed for a two-stage PG with a varying crack. The modifications include the crack propagation path settings as well as the coupled structure of flexible and rigid gear components, which makes a balance between the model accuracy and computational burden. Secondly, using this model, its vibration responses from a healthy PG and faulty PGs with different cracks are collected for the analyses of its frequency components and statistic features. Experiments are also conducted on a two-stage PG in similar conditions. Both results in simulations and experiments demonstrate the usefulness of the presented dynamic model. Moreover, sidebands and some statistic features are verified to be useful tools to monitor and evaluate the healthy status of PG. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Time-varying gearmesh stiffness (TVGS) is the main cause of gear vibration, and its accuracy affects the responses of dynamic models. An exponential curve model based on the Saint Venant's Principle is proposed to calculate the gearmesh stiffness of cracked spur gears in this paper. With the proposed model, the TVGS under the circumstances of healthy condition and four crack cases are computed, whose results have a good agreement with those of finite element method (FEM). Therefore, the exponential curve model can be used to estimate the TVGS and an alternative to FEM in gearmesh stiffness calculation is provided. [ABSTRACT FROM AUTHOR]

Hilbert-Huang Transform (HHT) has been renowned for its capacity to reveal fault indicating information issue from vibration signals. It uses Empirical Mode Decomposition (EMD) to decompose a signal accordingly to its contained information into a set of Intrinsic Mode Functions (IMFs). Then, the instantaneous frequencies are performed of each IMF using Hilbert Transform (HT). However, the HHT has some disadvantages which are caused by the EMD technique. The EMD has the mode mixing problem that may occur between IMFs, it causes the End Effect phenomenon, which leads to a wrong instantaneous values at both sides of the signal. Furthermore, its lack of mathematical basis. To overcome the HHT inherent problems, we propose the use of the Empirical Wavelet Transform (EWT) which designs an appropriate wavelet filter bank fully depends on the processed signal with HT in the early detection and condition monitoring of tooth crack fault. In this paper, we develop a dynamic model describing a single stage spur gear in normal and abnormal functioning. Results of analyzing the pinion’s vibration displacement show that the proposed approach denoted (HEWT) successfully detect the tooth crack at a much earlier stage of damage development even though in noisy environment. Performance evaluation and comparison between HEWT and HHT methods show that the HEWT is better sensitive to tooth crack fault detection in gearbox systems. [ABSTRACT FROM AUTHOR]

Abstract: The early fault of tooth crack in a planetary gearbox is difficult to be detected duo to its weak yet complex effects on the system's vibration signals. In order to identify fault signals aroused from small tooth crack of the sun gear in a planetary gear train, a lateral-rotational-coupled nonlinear dynamic model that incorporates the tooth-crack-induced time-varying mesh stiffness is proposed. A three-dimensional finite element model of gear pair is established to formulate the mesh stiffness of the gear pair with and without tooth crack. Based on the proposed nonlinear dynamic model, the vibration signals of a planetary gear system in both healthy and tooth-cracked conditions are predicted. Though a distinct reduction in the time-varying mesh stiffness of the sun-planet gear pair can be observed after introducing the effects of tooth crack, the time-domain vibration signals for both healthy and fault conditions are almost the same. However, the vibration signals in frequency-domain under two conditions are quite different in that multiplicities of sidebands spaced by characteristic frequency of the sun gear around the harmonics of mesh frequencies can be observed in the fault condition. Hence, from the perspective of fault diagnosis, the dynamic-model-based frequency spectra can be used to detect the early fault of tooth crack in a planetary gear train. [ABSTRACT FROM AUTHOR]

• An improved TVMS method considering the crack position is proposed. • The proposed method agrees well with FEM for both root and surface crack types. • Effects of the crack position and depth on the vibration behaviors are analyzed. • The effectiveness of several health condition indicators is assessed. The occurrence of a crack could change the gear mesh stiffness and consequently influence gear vibration properties. Cracks may occur on the tooth root and tooth surface, but the surface crack type has been seldomly concerned. This study proposes a comprehensive time-varying mesh stiffness (TVMS) calculation method in which the crack position is considered and emphasized. Two tooth crack types (Root crack type: the crack occurs on the root fillet; Surface crack type: the crack occurs on the tooth surface) are modeled and analyzed, and corresponding analytical calculation expressions of TVMS are derived. The finite element method is applied to validate the applicability and accuracy of the proposed analytical method for both root and surface crack types. Besides, by establishing a nonlinear dynamic model for a spur gear system, the influences of the crack depth and crack position on the gear dynamic characteristics are investigated. Several statistic parameters are also applied to evaluate the health conditions for the spur gear system. The method and analysis results reported here may be of interest to researchers and engineers attempting to handle the gear fault analysis analytically and effectively. [Display omitted] [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• A novel model of pitting-crack coupling and its evolution process are studied. • The meshing stiffness equations of several types of fault combinations are deduced. • The dynamic response of coupling faults in time and frequency domains are obtained. • Some statistical indexes are adopted to evaluate the gear fault severity. As an important part of power transmission in mechanical system, the health status of gear needs to be judged accurately. However, most of researchers focus on single fault gear, and very few papers are reported on the performance of multiple faults. So the gear mathematical model with pitting-crack coupling faults is derived in this study, and it is validated by a three-dimensional finite element model. Furthermore, several types of fault combinations from health to severity are presented to investigate the effect of coupling fault evolution on gear time-varying meshing stiffness. The stiffness excitation is applied into an 8 degrees of freedom model to obtain the dynamic characteristics of the gear-rotor system in terms of displacement and acceleration, and a reasonable diagnosis of the gear status can be made by analyzing the dynamic characteristics in time domain and frequency domain. In the next step of research, the statistical indicators are applied to further reveal the degree of coupling faults, which provides effective reference on the diagnosis of coupling fault gears. [Display omitted] [ABSTRACT FROM AUTHOR]

Considering the effects of profile shift and tooth crack, Time-varying mesh stiffness (TVMS) is determined based on an analytical model and the result accuracy is also verified by Finite element (FE) model. Then, by introducing TVMS into the FE model of a gear rotor system, the combined effects of profile shifted gear pair under positive, negative and zero gear transmission conditions and tooth crack on the system vibration responses are analyzed. The time-domain waveform, frequency spectrum and instantaneous energy at a constant rotational speed are used to indicate the crack feature, and the amplitude frequency response is adopted to display the crack-induced response change. The results show that different modification conditions have great influences on the TVMS due to the change of contact ratio and tooth thickness. The sideband frequency and instantaneous energy both can be adopted to diagnose the tooth crack and evaluate crack levels, and the latter is more sensitive than the former. The amplitude frequency responses of profile shifted gear under small crack condition show slight change compared with those of normal gear, and the change of acceleration is more obvious than that of displacement. [ABSTRACT FROM AUTHOR]

Planetary gear transmission has a wide application in different areas due to its advantages such as compactness, large torque-to-weight ratio, reduced noise and vibrations. However, its dynamic responses are much more complex due to the complicated structures and relative motions, which make it difficult in the fault feature extractions at the view point of fault detection. Better understanding on the dynamic features of a planetary gear transmission and the corresponding internal excitation sources will benefit the fault feature extractions. In this paper, an analytical model for mesh stiffness calculation is developed based on the potential energy principle and uniformly curved Timoshenko beam theory, which enables exploring the effects of the tooth root crack fault and the flexible ring gear rim on the dynamic responses. Based on the developed model, the frequency spectrum structures of the planetary gear transmission can be revealed and analyzed theoretically in the presence of tooth crack and flexible ring gear. A case study is performed to demonstrate the effectiveness of the developed model, where the tooth root cracks are seeded in a tooth of the sun, planet, and ring gears. The simulated results indicate that the complicated modulation phenomenon can be observed where the causes of different frequency components can be revealed. This study is expected to be able to give some theoretical guidance on the identification of vibration sources for planetary gear transmissions. [ABSTRACT FROM AUTHOR]

For the purpose of simulation and vibration-based condition monitoring of a geared system, it is important to model the system with an appropriate number of degrees of freedom (DOF). In earlier papers several models were suggested and it is therefore of interest to evaluate their limitations. In the present study a 12 DOF gear dynamic model including a gyroscopic effect was developed and the equations of motions were derived. A one-stage reduction gear was modelled using three different dynamic models (with 6, 8 and 8 reduced to 6 DOF), as well as the developed model (with 12 DOF), which is referred as the fourth model in this paper. The time-varying mesh stiffness was calculated, and dynamic simulation was then performed for different crack sizes. Time domain scalar indicators (the RMS, kurtosis and the crest factor) were applied for fault detection analysis. The results of the first model show a clearly visible difference from those of the other studied models, which were made more realistic by including two more DOF to describe the motor and load. Both the symmetric and the asymmetric disc cases were studied using the fourth model. In the case of disc symmetry, the results of the obtained response are close to those obtained from both the second and third models. Furthermore, the second model showed a slight influence from inter-tooth friction, and therefore the third model is adequate for simulating the pinion׳s y -displacement in the case of the symmetric disc. In the case of the asymmetric disc, the results deviate from those obtained in the symmetric case. Therefore, for simulating the pinion׳s y -displacement, the fourth model can be considered for more accurate modelling in the case of the asymmetric disc. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• An improved method for mesh stiffness of planetary gear set with crack is proposed. • Influences of crack size and inclination angle on mesh stiffness are investigated. • Effect of crack level on vibration response in time and frequency domains is studied. • Statistical indicators are used to characterize the evolution of crack propagation. The crack fault often appears in the complicated gear system, and even causes tooth broken. To solve this problem, an improved model is proposed in this paper, which considers the coupling effect of cracks on both gear tooth and gear foundation, and a 2D finite element model is established to verify the proposed method. The influences of crack size and inclination angle on meshing stiffness are investigated. The results are used to simulate the situation of the crack occurring at the sun gear in an 18 degrees of freedom dynamic model of the planetary gear set, so as to obtain the dynamic responses of the crack gear system both in time domain and frequency domain. Finally, two different methods are used to process the original vibration signals to obtain the residual signals. Combining the residual signals, several statistical indicators are employed to characterize the evolution of crack propagation. [ABSTRACT FROM AUTHOR]

This paper established the system dynamics model for two kinds of tooth cracks of different depths of the sun gear and inner gear ring to study the influence mechanism of crack failure on the tooth root strain of the planetary gear transmission system. Combined with the finite element model of the inner gear ring, the tooth root strain of the ring was solved. Experiments verified the correctness of the solution method. The root strain under the crack fault of the sun gear and the tooth crack fault of the inner gear ring is analyzed, and the following conclusions are drawn: Periodic fault impact occurs in the strain signal of the tooth root of the inner gear ring during the crack fault of the sun gear root. The fault can be extracted by the fast spectral kurtosis method (FSK), and the fault components are used to determine whether the sun gear cracks. The Lempel-Ziv index showed a tendency to increase gradually during the process of solar wheel crack deepening, which could be used as the damage index of crack depth. The results can provide a basis and reference for fault diagnosis.

Highlights: [•] FE modeling and analysis of HCR spur gear having tooth crack for different gear parameters. [•] To obtain crack propagation path for different backup ratio and pressure angle. [•] Calculations of gear mesh stiffness for teeth pair in mesh by variable crack intersection angle. [•] Comparison of total effective mesh stiffness for different crack length and backup ratio. [•] Comparison of total effective mesh stiffness for different crack length and pressure angle. [Copyright &y& Elsevier]

Gear tooth root crack, as one of the popular gear tooth failures, is always caused by the dynamic load or excessive load applied to the tooth. It will devastate the working performance of the gear system, by problems such as vibration and noise, or even lead to a broken tooth, which will stop the normal working process of the gear system. It has attracted wide attention from researchers. However, the previous studies focused their concentration only on the mesh stiffness reduction due to tooth root crack, while the tooth plastic inclination due to tooth bending damages like gear tooth root crack is seldom considered. In this paper, a tooth plastic inclination model for spur gear with tooth root crack is developed by regarding the cracked tooth as a cantilever beam. It influences not only the displacement excitation but also the mesh stiffness and load-sharing factor among tooth pairs in mesh. The simulation results obtained by incorporating the tooth plastic inclination deformation model together with the tooth root crack model into a 21-Degree-of-Freedom planetary gear dynamic model indicate that the tooth plastic inclination has a significant effect on the performance of the gear system rather than the mesh stiffness reduction due to tooth root crack. [ABSTRACT FROM AUTHOR]

Abstract: Planetary gear is widely used in different areas due to its advantages such as compactness, large torque-to-weight ratio, large transmission ratios, reduced noise and vibrations. However, the tooth faults like cracks are seldom concentrated. In this paper, a mesh stiffness model of internal gear pair with a tooth root crack in the ring gear is derived based on the potential energy principle. The mesh stiffness model is incorporated into the dynamic model of a one-stage planetary gear set with 21-degree-of-freedom (DOF) to investigate the effect of the internal gear tooth root crack. The crack cases with different dimensions are designed in this paper to demonstrate their influences on the mesh stiffness and the dynamic performance of the planetary gear set. The simulated results show that bigger reduction in mesh stiffness is caused by the growth in the crack size. And the impulsive vibrations and sidebands can be observed in the dynamic response of the planetary gear set in time and frequency domains, respectively. Both their amplitudes increase as the crack propagation which supply the possibility for them to be the indicators in the condition monitoring and fault diagnosis of planetary gear system. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: Gear tooth deviations are the main excitation sources for gear dynamic responses. However, it is seldom considered in the analytical mesh stiffness model except for some computational analysis like finite element method. To make up this gap, a general analytical mesh stiffness model is proposed in this paper to include the effect of the gear tooth errors. This proposed model establishes the relationship between the gear tooth errors and the total mesh stiffness, load sharing among different tooth pairs in mesh and loaded static transmission errors (LSTE). It is suitable for not only the gear pairs with low contact ratio (LCR), but also the gear pairs with high contact ratio (HCR). Two spur gear pair models, namely one with LCR between 1 and 2 and the other one with HCR between 2 and 3, are used to demonstrate the effectiveness of the proposed mesh stiffness model. Influences of the TPM, applied torque and gear tooth root crack on the mesh stiffness, load sharing and loaded static transmission errors are also investigated. [Copyright &y& Elsevier]

Abstract: This study focuses on the dynamic and vibration characteristics of a four-degree of freedom gear system with local defects to explore the failure mechanism. The dynamic models of the gear system with tooth crack and spalling failure are respectively established to investigate the effect of the failures on mesh stiffness and dynamic response by numerical simulation. The differences and comparisons between the vibration signals with the tooth crack and spalling failure are discussed by using time history, phase contrail, Poincaré section, spectrum analysis and fractal dimension, which shows the different dynamic characteristics between the tooth crack and spalling failure. An experimental gearbox with tooth crack and spalling failure was designed and experiments was carried out to determine the dynamic characteristics of the faulty vibration signals. The results obtained herein show good agreement qualitatively with the theoretical analysis, which provides a theoretical basis for the fault diagnosis of gearboxes. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]