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[Display omitted] • A machine-vision-inspection method is proposed to detect the gear surface defects. • The EHL model of defective gear is established, considering roughness and defects. • The tribological behavior is investigated for defective spur gear contact. • The mesh stiffness calculation model is proposed for spur gear with surface defects. As typical failure mode of gear system, the tooth surface pitting, spalling can be detected in most long-running gear system, especially under heavy-load and high-speed condition, or under lubricant-starvation condition, the tooth pitting is characterized by irregular contour and random distribution. Most previous study on the defective gear system mainly based on manually detection of defective region, or just rely on geometric simplification of defects, leading to inaccurate results with low-efficient method, therefore, the machine-vision-based defect inspection method is proposed in the study of defective gear system. First, the pitting defects on gear tooth surface is detected and segmented based on involutional neural network U-net, then the tooth surface with segmented defective region is mapped to the elastohydrodynamic lubrication model of spur gear system, finally, the tribological behavior in addition to the mesh stiffness under lubrication condition of defective spur gear system are investigated and discussed. The results reveal that the machine-vision-based defects inspection could improve the accuracy and efficiency of the failure study for gear system. [ABSTRACT FROM AUTHOR]

Tooth root crack generally causes large deformation of meshing gear pairs, which results in delayed meshing out of the contact point. In this paper, an improved dynamic model of spur gears considering delayed meshing out of the contact point due to root crack is proposed, which takes several factors of mesh stiffness, mesh damping coefficient and delayed meshing out force into account. The effects of different crack depths and angles on the dynamic characteristics of the gear system are investigated through various simulation and experimental tests. Further, the improved gear dynamic model is validated by comparison with acceleration responses obtained from the experimental test rig under different working conditions. The findings indicate that the acceleration responses of the improved dynamic model are consistent with experimental results, in both time and frequency domains under different crack depths and angles. [ABSTRACT FROM AUTHOR]

• An analytical-finite-element method for calculating mesh stiffness of spur gears is proposed. • A local rigid area technique is used to avoid the inappropriate local distortion in the contact position. • The effects of different gear foundation types on mesh stiffness of spur gears are analyzed. • The effects of different crack propagation paths on the mesh stiffness of spur gears are studied. Based on the finite element theory and the loaded tooth contact analysis, an analytical-finite-element model considering the complex foundation types and the crack propagation paths is proposed to calculate the mesh stiffness of spur gears. The complex gear foundation types (including web foundation and slots foundation) and the crack propagation paths (including tooth fracture and rim fracture) are considered in the proposed method, and the effects of various foundation types, crack propagation paths and crack lengths on the mesh stiffness are analyzed. In order to verify the proposed method, the simulated crack paths are confirmed by those derived from the experimental and the extended finite element methods, and the mesh stiffness obtained from the proposed method is validated by that of the potential energy method and the three-dimensional finite element method. Compared with the potential energy method, the proposed model can be used to calculate the mesh stiffness of spur gears with different gear foundation types and real crack propagation paths. Meanwhile, the proposed method has higher computational efficiency compared with the finite element method (e.g., the finite element method takes about 3.5 h for a mesh cycle and the proposed method costs 25 s). The results indicate that the web thickness has greater influence on the mesh stiffness of spur gears than the rim thickness. The results show that two crack types, namely the tooth fracture and rim fracture, have more obvious influence on the mesh stiffness of single-tooth contact zone relative to that of the double-tooth contact zone. [ABSTRACT FROM AUTHOR]

Highlights • A shape-independent method was proposed to model tooth spalls. • The proposed method utilizes defect ratios to model the features of tooth spalls. • The proposed method is applicable to model different types of gear tooth spalls. • The proposed method is convenient and straightforward. Abstract Accurately evaluating the gear mesh stiffness with tooth fault(s) plays an important role in gear dynamic simulations. Common approaches to evaluating the gear TVMS with spall assume a specific geometry (rectangular, round, elliptical, V-shape etc.) to approximate the features of the spall. The accuracy of geometry-based methods on modeling gear tooth spall depends on how close the utilized geometry is to the actual shape of the spall. In practice, gear tooth spalls often occur in very irregular shapes and may also have randomly distributed features under multi-spall conditions. Due to shape irregularities and uncertainties in their distribution, it is often difficult to apply existing methods to accurately evaluate gear TVMS with spalls as observed in practice. To address these shortcomings, this paper proposes a shape-independent method to model tooth spall(s) based on defect ratios instead of a specific geometry. The proposed method is applicable to tooth spalls of various sizes, shapes, numbers and distribution conditions, and thus is capable of modeling any kind of tooth spall. A powerful advantage of the proposed method is that to change the type of spall being modelled, only the functions of the defect ratios need to be modified rather than the entire algorithm. Therefore, the proposed method is general in nature and much easier to implement in practice. The effectiveness of the proposed shape-independent approach to modelling different kinds of tooth spalls is validated by finite element results. [ABSTRACT FROM AUTHOR]

Due to the lack of efficient formulas, fillet foundation stiffness under double tooth engagement has long been a challenging problem for mesh stiffness calculation of spur gears by analytical method. Direct summation of the mesh stiffness of each meshing tooth pair overestimates the total mesh stiffness, which may lead to a large calculation error. An improved fillet foundation stiffness calculation method is proposed considering the structure coupling effect, namely, one gear body is shared by two meshing teeth simultaneously. On the basis of the proposed method, a mesh stiffness model and a load sharing model are analytically established. In the mesh stiffness model, fillet foundation stiffness correction factor is introduced to improve the calculation accuracy. Based on the minimum potential energy principle (MPEP), the load sharing ratio is determined by the proposed load sharing model. The accuracy of the proposed models is validated by comparing with finite element method (FEM). [ABSTRACT FROM AUTHOR]

Time-varying mesh stiffness is one of the main internal excitation sources of gear dynamics. Accurate evaluation of gear mesh stiffness is crucial for gear dynamic analysis. This study is devoted to developing new models for spur gear mesh stiffness evaluation. Three models are proposed. The proposed model 1 can give very accurate mesh stiffness result but the gear bore surface must be assumed to be rigid. Enlighted by the proposed model 1, our research discovers that the angular deflection pattern of the gear bore surface of a pair of meshing gears under a constant torque basically follows a cosine curve. Based on this finding, two other models are proposed. The proposed model 2 evaluates gear mesh stiffness by using angular deflections at different circumferential angles of an end surface circle of the gear bore. The proposed model 3 requires using only the angular deflection at an arbitrary circumferential angle of an end surface circle of the gear bore but this model can only be used for a gear with the same tooth profile among all teeth. The proposed models are accurate in gear mesh stiffness evaluation and easy to use. Finite element analysis is used to validate the accuracy of the proposed models. [ABSTRACT FROM AUTHOR]

Gearbox is widely used in industrial and military applications. Due to high service load, harsh operating conditions or inevitable fatigue, faults may develop in gears. If the gear faults cannot be detected early, the health will continue to degrade, perhaps causing heavy economic loss or even catastrophe. Early fault detection and diagnosis allows properly scheduled shutdowns to prevent catastrophic failure and consequently result in a safer operation and higher cost reduction. Recently, many studies have been done to develop gearbox dynamic models with faults aiming to understand gear fault generation mechanism and then develop effective fault detection and diagnosis methods. This paper focuses on dynamics based gearbox fault modeling, detection and diagnosis. State-of-art and challenges are reviewed and discussed. This detailed literature review limits research results to the following fundamental yet key aspects: gear mesh stiffness evaluation, gearbox damage modeling and fault diagnosis techniques, gearbox transmission path modeling and method validation. In the end, a summary and some research prospects are presented. [ABSTRACT FROM AUTHOR]

• A new calculation model of the tooth torsional deformation caused by the NDL along TWD is presented. • A double-layer iterative model for TVMS and load distribution of gears is proposed considering the effects of NDL along TWD. • The proposed model can achieve the decoupling calculation of TVMS and the load along TWD caused by the early-stage crack. • FE method proves the accuracy and high efficiency of the proposed model. • The effects of crack parameters and torque on the tooth torsional deformation, TVMS and load distribution are revealed. To reveal the coupling relationship between the time-varying mesh stiffness (TVMS) and the load distribution along the tooth width direction (TWD) of gears with early-stage crack (ESC), a double-layer iterative analytical model for the TVMS and load distributions of gears is proposed considering the effects of the non-uniformly distributed load (NDL) along TWD caused by the ESC. In the proposed model, an analytical model of tooth torsional deformation and a parallel slice stiffness model of the tooth pair with ESC are separately developed based on the slicing method. On this basis, a double-layer iterative calculation method for the TVMS and load distributions is proposed, in which the coupling relationships between the slice stiffness and load distribution along TWD as well as the TVMS and load distribution between the meshing tooth pairs are respectively presented with the inner- and outer- layer iterations. Finite element (FE) models are established to verify the proposed double-layer iterative model. The effects of crack parameters and applied torque on the tooth torsional deformation, TVMS, and load distributions of the gear with ESC are finally investigated based on the proposed model. The results show that the proposed model can realize the accurate and fast decoupling calculation of the TVMS and load distributions of gears with ESC. This study can provide a basis for the establishment of the refined ESC fault diagnosis method and the rapid evaluation of the load distributions of gears with asymmetric errors or faults along TWD. [ABSTRACT FROM AUTHOR]

Locomotive is used to drag trailers to move or supply the braking forces to slow the running speed of a train. The electromagnetic torque of the motor is always transmitted by the gear transmission system to the wheelset for generation of the tractive or braking forces at the wheel–rail contact interface. Consequently, gear transmission system is significant for power delivery of a locomotive. This paper develops a comprehensive locomotive–track vertical-longitudinal coupled dynamics model with dynamic effect of gear transmissions. This dynamics model enables considering the coupling interactions between the gear transmission motion, the vertical and the longitudinal motions of the vehicle, and the vertical vibration of the track structure. In this study, some complicated dynamic excitations, such as the gear time-varying mesh stiffness, nonlinear gear tooth backlash, the nonlinear wheel–rail normal contact force and creep force, and the rail vertical geometrical irregularity, are considered. Then, the dynamic responses of the locomotive under the tractive conditions are demonstrated by numerical simulations based on the established dynamics model and by experimental test. The developed dynamics model is validated by the good agreement between the experimental and the theoretical results. The calculated results reveal that the gear transmission system has strong dynamic interactions with the wheel–rail contact interface including both the vertical and the longitudinal motions, and it has negligible effect on the vibrations of the bogie frame and carbody. [ABSTRACT FROM AUTHOR]

Considering the factors such as the nonlinearity backlash, static transmission error and time-varying meshing stiffness, a three-degree-of-freedom torsional vibration model of spur gear transmission system for a typical locomotive is developed, in which the wheel/rail adhesion torque is considered as uncertain but bounded parameter. Meantime, the Ishikawa method is used for analysis and calculation of the time-varying mesh stiffness of the gear pair in meshing process. With the help of bifurcation diagrams, phase plane diagrams, Poincaré maps, time domain response diagrams and amplitude-frequency spectrums, the effects of the pinion speed and stiffness on the dynamic behavior of gear transmission system for locomotive are investigated in detail by using the numerical integration method. Numerical examples reveal various types of nonlinear phenomena and dynamic evolution mechanism involving one-period responses, multi-periodic responses, bifurcation and chaotic responses. Some research results present useful information to dynamic design and vibration control of the gear transmission system for railway locomotive. [ABSTRACT FROM AUTHOR]

• A polygonal approximation method for holes is developed to calculate the mesh stiffness with weight reduction holes. • The deflection decomposition method for the fast calculation of the mesh stiffness of lightweight gears with modifications. • Tooth modification reduces the vibration fluctuation caused by holes. Though the special gear foundation structure and tooth modification are effectively applied to weight optimization and vibration control, the lack of an effective mathematical model limits the investigation of their effects. The present work develops a general polygonal approximation method for weight reduction holes, which is introduced into an analytical-finite element (AFE) method along with the decomposition method to calculate the time-varying mesh stiffness (TVMS) of modified spur gear pairs with complex foundation (thin webs and weight reduction holes). The comparisons with the traditional finite element (FE) method show the superiority of the model in calculation accuracy and efficiency. Furthermore, the actual TVMS is introduced into the dynamics model of a high-speed gear-rotor-bearing system to analyze the dynamic responses, and the effectiveness of the model is verified by experiments. The results indicate that the thin web has a greater influence on the reduction of average amplitude in stiffness than holes. The thin web significantly controls the system vibration, while the tooth modification reduces the vibration fluctuation caused by holes at medium and high speeds. The analysis and experimental results provide an important reference for gear weight optimization and vibration reduction. [ABSTRACT FROM AUTHOR]

This paper presents a new gear dynamic model for flexibly supported gear sets aiming to improve the accuracy of gear fault diagnostic methods. In the model, the operating gear centre distance, which can affect the gear design parameters, like the gear mesh stiffness, has been selected as the iteration criteria because it will significantly deviate from its nominal value for a flexible supported gearset when it is operating. The FEA method was developed for calculation of the gear mesh stiffnesses with varying gear centre distance, which can then be incorporated by iteration into the gear dynamic model. The dynamic simulation results from previous models that neglect the operating gear centre distance change and those from the new model that incorporate the operating gear centre distance change were obtained by numerical integration of the differential equations of motion using the Newmark method. Some common diagnostic tools were utilized to investigate the difference and comparison of the fault diagnostic results between the two models. The results of this paper indicate that the major difference between the two diagnostic results for the cracked tooth exists in the extended duration of the crack event and in changes to the phase modulation of the coherent time synchronous averaged signal even though other notable differences from other diagnostic results can also be observed. [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]

• Challenges of dynamic transmission error in gear crack severity assessment. • Removal of transfer function, and separation of profile errors and deflections. • Estimation of meshing compliance and geometric transmission error. • Automated and accurate quantification of root-crack depth. • Validation using encoder measurements from a spur gear rig. Gear transmission error (TE) has been acknowledged as a prominent diagnostic tool in condition monitoring. Its advantages vs vibration include being closer to the source and less affected by transmission path effects. TE has recently demonstrated its enormous capabilities in gear wear severity assessment. For crack diagnostics, despite some promising preliminary results, a methodology for an accurate and reliable TE-based estimation of crack size was previously not available. In order to achieve this result two main obstacles needed to be overcome: (i) the removal of dynamic TE effects due to speed and (ii) the separation of profile error and compliance-related tooth deflections. This paper proposes a new crack-severity-estimation methodology that solves these problems. The proposed technique exploits different speeds and loads, often experienced by geared transmissions, and includes an automated identification of the cracked tooth and a crack-depth estimation based on a comparison with a theoretical tooth stiffness model. The proposed methodology is validated using experimental encoder measurements collected from a spur gear test rig with artificially seeded root cracks. [ABSTRACT FROM AUTHOR]

• The traditional method for mesh stiffness calculation of spur gear pair is improved. • A new method for mesh stiffness calculation of helical gear pair is proposed. • Time-varying backlash effect is taken into consideration. • Sensitivity analysis of parameters affecting the mesh stiffness of helical gear pair is carried out. The mesh stiffness characteristics of the gear pair have important influence on its transmission error (TE), and then affect the vibration and noise level of the gears. It is of great significance to develop a calculation method that can take both accuracy and efficiency into account. First, considering the difference between base circle radius and root circle radius, the potential energy method is improved to calculate the mesh stiffness of involute spur gear pair. Then, on the basis of mesh stiffness theory of spur gear and time-varying backlash effect, a new method for calculating the mesh stiffness of involute helical gear pair is proposed by improving the traditional slicing method. The analysis results of time-varying mesh stiffness curves show that, compared with traditional loaded tooth contact analysis (LTCA) method, the analytical method proposed in this paper is closer to finite element (FE) method, and the calculation cost is relatively low. Finally, the sensitivity analysis of the parameters affecting the mesh stiffness characteristics of helical gear pair is carried out, and the analysis results are discussed. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a 26 degree of freedom gear dynamic model of three shafts and two pairs of spur gears in mesh for comparison of localised tooth spalling and damage. This paper details how tooth spalling and cracks can be included in the model by using the combined torsional mesh stiffness of the gears. The FEA models developed for calculation of the torsional stiffness and tooth load sharing ratio of the gears in mesh with the spalling and crack damage are also described. The dynamic simulation results of vibration from the gearbox were obtained by using Matlab and Simulink models, which were developed from the equations of motion. The simulation results were found to be consistent with results from previously published mathematical analysis and experimental investigations. The difference and comparison between the vibration signals with the tooth crack and spalling damage are discussed by investigating some of the common diagnostic functions and changes to the frequency spectra results. The result of this paper indicates that the amplitude and phase modulation of the coherent time synchronous vibration signal average can be effective in indicating the difference between localised tooth spalling and crack damage. [Copyright &y& Elsevier]