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Gear is one of the most crucial components of the transmission system, and the performance of gear directly affects the efficiency and reliability of the transmission system. Conventional methods for designing gear parameters involve several time-consuming and complex steps, which may not guarantee optimal performance. Therefore, we propose a new method for designing gear parameters that aims to improve efficiency and accuracy. First, the tooth surface equations of spur and helical involute gears suitable for symmetric and asymmetric teeth are deduced based on the gear-forming machining principle. Second, the performance evaluation models for load capacity, dynamic performance, efficiency, and power density of the gears are established based on the precise gear surface. The design objectives are standardized and evaluated comprehensively using a linear weighting method. Finally, a forward performance-driven design method of gear parameters is established. The proposed method is applied to a helical gear pair design case, and the results show that 90.7% of the individuals in the Pareto optimal front are asymmetric gears, with 9.3% being symmetric gears. This suggests that asymmetric gears have more opportunities to be optimal than symmetric gears. The highest-ranked gear designed using the proposed method is superior to the gear designed using conventional methods. [ABSTRACT FROM AUTHOR]

Gear tooth spalling is a typical tooth surface defects in gear transmission. The accurate calculation of the time-varying mesh stiffness of the gear pair with spalling defect under mixed elastohydrodynamic lubrication (EHL) plays an important role in tooth damage prediction and fault feature analysis. In this work, an improved time-varying mesh stiffness (TVMS) model of spalling faulty gear in mixed EHL line contact considering the coupled effect of rough surface topography is established. The proposed model considers the combined effect of spalling defect, surface roughness topography, and lubrication on the contact characteristics of the meshing interface to obtain a revised contact stiffness, which is further used to determine the comprehensive TVMS. The cylindrical contact coefficient for curved meshing interface with spalling fault is derived and incorporated into the statistical micro-contact Greenwood and Williamson model (GW model) to obtain the asperity contact stiffness. The revised contact stiffness at the rough faulty gear interface is obtained from the parallel action of the asperity contact stiffness and oil film stiffness under load-sharing concept. The dynamic responses of the gear system are analyzed using the established mesh stiffness model through the six degrees of freedom translational-torsional dynamic model, in which the dynamic meshing force is calculated iteratively using the transit surface topography and geometrical parameters along the line of contact instead of a quasi-static meshing force. Effects of multi-parameters as surface roughness, spalling fault size, and lubricant on the mesh characteristics and dynamic responses of the gear system are further investigated and discussed. [ABSTRACT FROM AUTHOR]

This paper discusses the possible early crack detectability performances of low contact ratio (LCR) and high contact ratio (HCR) spur gears. CAD models of the LCR and HCR gears are created using the rack cutter type tool. Static bending stress analyses are performed to define the starting point of the cracks and crack propagation analyses are conducted to define the realistic crack paths. Using the healthy and cracked gear geometries, the single tooth stiffness (STS) and the time-varying mesh stiffness (TVMS) of the LCR and HCR gears are calculated numerically. A six degrees of freedom dynamic model of the single-stage gear system is proposed. Dynamic force variation of the LCR and HCR gears are calculated by using the dynamic model. Statistical failure indicators, which are kurtosis, RMS, crest factor, mean, standard deviation, and variance, are used for the determination of the crack in the early stage. The results show that all statistical indicators increase with the increase in the crack length. The statistical indicators increase more for LCR gears than for HCR gears. Although HCR gears have many advantages over LCR gears, it has been determined that their crack detection capacity is lower than that of LCR gears. [ABSTRACT FROM AUTHOR]

A design method of internal meshing gear based on variable center meshing line (VCML) is presented in this paper. The meshing line is an arc line connecting the intersection of the indexing circle connecting the inner and outer gears and the intersection of the tooth top circle, and the center of the line of engagement freely slides on the vertical bisection line of the line connecting the two intersection points. The top tooth profile equation of internal and external gears is derived by rack equation. According to the normal method of gear profile, the equation of gear root meshing line is derived by using the tooth top profile equation of the gear. According to the principle of plane meshing, the root profile equation of gear is derived by using the root meshing line equation of gear. The tooth profile of VCML gear can be deduced quickly by adjusting the center position of the meshing line. The stress, meshing stiffness, transmission error, contact ratio and sliding ratio of the VCML gears are analyzed by finite element method, and the area of significant performance improvement of the meshing line center of the VCML gears is found. The research results show that compared with involute gears, the maximum average stress of VCML gears is reduced by 64.99%, the meshing stiffness is increased by 193.67%, and the transmission error is reduced by 65.95%. This method improves the efficiency of new tooth profile design and transmission performance. By changing the center coordinates of the meshing line, the tooth profile of the VCML gear can be quickly redesigned, which greatly improves the efficiency of the new tooth profile design and transmission performance. This design method provides a new idea for the tooth profile design of internal meshing gear. [ABSTRACT FROM AUTHOR]

Time-varying meshing stiffness has an important contribution to the vibration control in gear systems. However, the design of gears in the electric vehicles and aerospace field is more focused on lightweight. The accurate and efficient evaluation of the time-varying meshing stiffness of lightweight spur gears has become a challenging problem, such as considering the thin-walled or oblique web. Therefore, a new approach is proposed to calculate the mesh stiffness of the lightweight spur gear by combining an analytical model and the dynamic model of the gear pair. The gear model is constructed by adopting the shell element to save computer resources and model preparation time. The validity of the presented model is investigated by comparing it with the traditional analytical and finite element models. The analytical results indicate that the proposed model has the same level of computational efficiency as the analytical model while maintaining the precision of the finite element model. The advantages of the proposed method are the ability to consider the effects of thin-walled web and oblique spoke plate structures on the mesh stiffness. [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. [ABSTRACT FROM AUTHOR]

Gear transmission systems are widely used to transfer energy and motion and to guarantee the accuracy of the entire machine system. The modification technique is a common method that improves the gear profile and reduces the transmission error. Based on the parametric model, a modified gear can be established for the evaluation of static and dynamic characteristics. The influences of profile modification parameters and gear parameters are investigated while changing the rules of different kinds of factors. Based on sensitive parameters, a two-stage profile modification curve is proposed to improve the performance of gear pairs. Thus, considering the time-varying mesh stiffness and backlash, a novel, dynamic modified gear model is established to analyze the dynamic performance, such as the dynamic transmission error. Based on the proposed curve, the range and amplitude of the transmission error can be decreased. Additionally, the vibration displacement and noise can be reduced to improve the running characteristics. [ABSTRACT FROM AUTHOR]

Plastic gears have been widely used in fields such as instruments, meters, and automotive electronics. In particular, metal–plastic gear pairs are commonly used in the low-speed stages of multi-stage gear transmissions to enhance load capacity and achieve lightweight design. However, due to the small elastic modulus and significant temperature sensitivity of plastic gears, off-line meshing phenomenon occurs in metal–plastic gear pairs during operation, which affects the overall system stiffness and dynamic behavior. Therefore, this study takes 45# steel-POM gear pair as the object of study, and considers the effect of temperature on the modulus of elasticity of the gear and the effect of out-of-line meshing on the time-varying stiffness of the gear pair , in which the 45# steel-POM gear pair refers to the combination of the steel and plastic gear pairs, with the pinion gear consisting of 45# steel gears, and the large gear made of POM gears. A lumped parameter method is used to establish a single-degree-of-freedom model of the gear transmission system, and the effect of load on the system's dynamic behavior is investigated. Experimental validation shows that the stiffness varies linearly with the load, and the dynamic performance of the system deteriorates with increasing load. In the experimental study described in this article, as the load increased from 5 to 10 N m, the maximum dynamic transmission error increased by 62%. Under the same load, the system with consideration of off-line meshing exhibits better transmission accuracy compared to the one without consideration. Disregarding the dominant frequency of the system when there is no external meshing, the dominant frequency of the system increases by 1.4 times when considering external meshing. This study provides a theoretical basis for the design and transmission of plastic gears in the future. [ABSTRACT FROM AUTHOR]

This paper discusses the possible early crack detectability performances of low contact ratio (LCR) and high contact ratio (HCR) spur gears. CAD models of the LCR and HCR gears are created using the rack cutter type tool. Static bending stress analyses are performed to define the starting point of the cracks and crack propagation analyses are conducted to define the realistic crack paths. Using the healthy and cracked gear geometries, the single tooth stiffness (STS) and the time-varying mesh stiffness (TVMS) of the LCR and HCR gears are calculated numerically. A six degrees of freedom dynamic model of the single-stage gear system is proposed. Dynamic force variation of the LCR and HCR gears are calculated by using the dynamic model. Statistical failure indicators, which are kurtosis, RMS, crest factor, mean, standard deviation, and variance, are used for the determination of the crack in the early stage. The results show that all statistical indicators increase with the increase in the crack length. The statistical indicators increase more for LCR gears than for HCR gears. Although HCR gears have many advantages over LCR gears, it has been determined that their crack detection capacity is lower than that of LCR gears.

Gear surface wear will inevitably occur under the complicated excitations in a locomotive. The continuous accumulated wear would induce the deterioration of the meshing state of the gear pair and it affects the stability of operation and shorten the service life of components. Therefore, it is the premise of fault maintenance to accurately grasp the evolution law of wear and its effect on the components' vibrations. However, there is little work on the wear calculation of the gear pain in a locomotive considering the complicated excitations. To fill this gap, a locomotive-track spatially coupled dynamics model with the discrete gear model is employed in this paper, which combines with the typical Archard's wear calculation equation. The calculation method of load distribution of discrete tooth slices under misalignment is proposed. The time-varying mesh stiffness, track irregularity, and time-varying wear coefficient are comprehensively taken into account. In addition, the developed wear depth, deemed as a kind of profile error, is taken into account in the dynamic transmission error (DTE). The results indicate that the accumulated wear depth of the driving gear is more severe than that of the driven gear. There is an apparent difference in wear distribution for different discrete slices along the tooth width under misalignment. In addition, the developed cumulative wear depth could exert a vital influence on the dynamic performance of components in the vehicle system. This study can provide a theoretical guideline for service life prediction and structural design. [ABSTRACT FROM AUTHOR]

Due to the complex structure of the planetary gear train, the influence of early fault on the dynamic characteristics of the planetary gear train is complex and changeable. This paper proposes a time-varying meshing stiffness (TVMS) calculation model of internal meshing considering the coupling effect between teeth; and a multi-tooth coupling stiffness calculation model of the planetary gear train. The Archard equation is used to calculate the wear depth; moreover, the potential energy method calculated The TVMS of the planetary gear train under the wear state. The wear fault will cause the TVMS of the gear to decrease. In the two-stage planetary gear train, the TVMS of the first-stage sun-planet gear decreases more obviously. In addition, there is a specific coupling effect between the sun, ring, and planetary gear. The influence of the wear fault on the two-stage planetary gear train is analyzed by calculating the synthetic stiffness considering the meshing phase difference. The results show that under the same wear fault condition, the more teeth involved in meshing, the more significant the decrease in coupling stiffness. [ABSTRACT FROM AUTHOR]

Taking the involute spur gear pairs with profile shift as the research object, the mesh stiffness calculation models including nonlinear Hertzian contact stiffness, tooth profile curve, fillet foundation stiffness with structure coupling effect were established based on the potential energy method. Considering the influences of gear profile shifts, the calculation formulas for tooth profile curve, half tooth angle of base circle and half tooth angle of dedendum circle in mesh stiffness calculation were revised. The fillet foundation stiffness models were revised by considering tooth surface frictions. The pressure angle models of gear mesh point were derived by considering geometric eccentricity errors. The effects of profile shifted, tooth surface frictions and geometric eccentricity errors on mesh stiffness were studied. The results show that the center distance, stiffness amplitude and contact ratio of the profile shifted gear pairs have great changes. Owing to the profile shift, the moment when the direction of frictional force changes on a pair of meshing gears no longer occurs within the single tooth meshing interval, thereby affecting the characteristics of the mesh stiffness. Geometric eccentricity increases the peak-to-peak values of the mesh stiffness, and at the same time, sideband frequencies with intervals corresponding to the rotational frequencies appear in the frequency domain of the meshing stiffness. In the presence of eccentricity errors in both the driving and driven gears, a new sideband frequency arises with an interval equal to the difference in rotation frequency. Considering the combined effects of three factors, the mesh stiffness characteristics may be significantly influenced. The research results provide the references for further researches of the dynamics of profile shifted gears. [ABSTRACT FROM AUTHOR]

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This work studies on analytical model of the single–stage spur planetary gear in form of lumped–parameter model. It includes key nonlinear factors of planetary gear vibration such as mesh stiffness and backlash of meshing gears. The planetary gear set is modeled as a set of lumped masses and springs and dynamic model of the planetary gear set is presented. Nonlinear equations of motion are presented for each component and each component has three degrees of freedom in planar motion. We have numerically evaluated a set of linear and nonlinear equations to obtain natural frequencies and analysis of vibration behavior of the system under nonlinear factors and fixed output. In previous researches, natural frequencies of planetary gears were evaluated from two points of view: the first one considered fixed output for planetary gears and the second one consists of free rotational output. In this research, the influence of the output flexibility is evaluated on natural frequencies of the planetary gear. Meanwhile, by considering the fixed output, vibration behavior of the nonlinear system is investigated. Results show that the most important effect of the output flexibility is on the principal natural frequency. Because of chaotic phenomenon, vibration behavior of components in translational directions is different.

Gear transmission systems are widely used to transfer energy and motion and to guarantee the accuracy of the entire machine system. The modification technique is a common method that improves the gear profile and reduces the transmission error. Based on the parametric model, a modified gear can be established for the evaluation of static and dynamic characteristics. The influences of profile modification parameters and gear parameters are investigated while changing the rules of different kinds of factors. Based on sensitive parameters, a two-stage profile modification curve is proposed to improve the performance of gear pairs. Thus, considering the time-varying mesh stiffness and backlash, a novel, dynamic modified gear model is established to analyze the dynamic performance, such as the dynamic transmission error. Based on the proposed curve, the range and amplitude of the transmission error can be decreased. Additionally, the vibration displacement and noise can be reduced to improve the running characteristics.

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.

Gear mesh excitations are widely concerned in the dynamic studies of the gear transmission system. Meanwhile, intentional and unintentional tooth profile deviations often occur in gears. At present, the established calculation models of gear mesh excitations consider tooth profile deviations as displacement excitation. However, gear mesh excitations calculated by such models have reduced stability compared with the actual situation. Therefore, in this study, an improved analytical model of gear mesh excitations with tooth profile deviations is established. This established model considers tooth profile deviations, extended tooth contact, and the structure coupling effect of the gear body simultaneously. More importantly, the model considers the strong correlation among tooth contact parameters, contact force, and tooth profile deviations to better reflect the actual gear mesh. A calculation flowchart with a simple calculation method of contact forces is also proposed to calculate the gear mesh excitations. Finally, the effects of tooth profile deviations on gear mesh excitations are studied. The results show that the effects of tooth profile deviations on tooth contact position, the direction of contact force, and equivalent basic circle radii should be considered in the calculation of gear mesh excitations because of smaller system transmission errors, larger double-teeth meshing area, and slighter extended tooth contact. Tooth profile deviations also cause jumps in tooth contact position and time-varying mesh stiffness. Thus, our findings show that the proposed model can be used to calculate the gear mesh excitations more accurately when the tooth profile deviates greatly. [ABSTRACT FROM AUTHOR]

Parametric instabilities alongside severe vibration within gear systems are as due to variation in mesh stiffness, the meshing tooth stiffness changes with changes in the number of contact teeth. Two-stage gear trains are used in the investigation of the operation conditions resulting in parametric instability, among them idler gear alongside configurations of the countershaft. Two gear pairs with half the breadth as the one of the primary gear pair are used in replacing each of the traditional two-mesh gear pairs. An investigation is done into the primary, secondary, as well as combination instabilities. An analysis is conducted on the impacts of the parameters of the mesh stiffness, among the frequency of mesh, amplitudes of stiffness variation, and contact ratio alongside mesh phasing on such instability. Design formulas have been developed for the case of mesh stiffness variations having rectangular waveforms for regulation of the instability regions through adjustments to the mesh phasing as well as contact ratios. Numerical solutions are applied in the validation of the analytical findings where it is noted that the substituted four gear pairs having proper mesh stiffness parameters remove the instability regions of a traditional two-stage gear train. [ABSTRACT FROM AUTHOR]

Gear transmission systems are widely used in various fields. The occurrence of gear cracks, tooth pitting, and other faults will lead to the dynamic characteristics deterioration of the transmission system. In order to calculate the meshing stiffness of faulty gear pairs more effectively and precisely, this article improves the loaded tooth contact analysis (LTCA) method by analyzing the influence of different fault types on gear deformation, including bending-shearing deformation and contact deformation, which combines the accuracy of the finite element method (FEM) and the rapidity of the analytical method (AM). The improved LTCA method can model the fault areas accurately and optimize the deformation coordination equation under the actual meshing situation of the faulty gear tooth, making it suitable for calculating the meshing stiffness of faulty gears. Based on the calculation results of the finite element method, the accuracy of the improved meshing stiffness calculation method has been verified, and the sensitivity of different fault type parameters on meshing stiffness has been studied. [ABSTRACT FROM AUTHOR]

The mesh stiffness calculation and load distribution among teeth of helical gear are the basis of its dynamics analysis and strength calculation. Aiming at the accuracy of approximate replacement method of mesh stiffness of helical gear, the quasi-static elastic model of helical gear was established based on the change rule of contact length, quasi-parabolic model of single tooth mesh stiffness per unit of contact length and ISO6336-1 standard, and a comparative study was conducted with the approximate replacement method based on quasi-static rigid model and ISO6336-1 standard. The results showed that the results calculated by quasi-static elastic model were in good agreement with the ISO standard results when the ratio of the minimum stiffness to the maximum stiffness of single tooth αk is 0.55, and the errors of the quasi-static rigid model and ISO standard were large. Based on the quasi-static elastic model, the effects of geometric parameters and broken teeth failure on meshing stiffness and load distribution coefficient of helical gear were analyzed and discussed, which provided a basis for the study of fault dynamics.

This work studies on analytical model of the single-stage spur planetary gear in form of lumped-parameter model. It includes key nonlinear factors of planetary gear vibration such as mesh stiffness and backlash of meshing gears. The planetary gear set is modeled as a set of lumped masses and springs and dynamic model of the planetary gear set is presented. Nonlinear equations of motion are presented for each component and each component has three degrees of freedom in planar motion. We have numerically evaluated a set of linear and nonlinear equations to obtain natural frequencies and analysis of vibration behavior of the system under nonlinear factors and fixed output. In previous researches, natural frequencies of planetary gears were evaluated from two points of view: the first one considered fixed output for planetary gears and the second one consists of free rotational output. In this research, the influence of the output flexibility is evaluated on natural frequencies of the planetary gear. Meanwhile, by considering the fixed output, vibration behavior of the nonlinear system is investigated. Results show that the most important effect of the output flexibility is on the principal natural frequency. Because of chaotic phenomenon, vibration behavior of components in translational directions is different. [ABSTRACT FROM AUTHOR]

A classical Lagrangian staggered‐grid hydrodynamic (SGH) method on triangular grids is prone to cell‐to‐cell oscillations. The causes of these oscillations can be divided into two categories. One is dissipation and the other is fluid deformation. For dissipation, when the kinematic energy stored on nodes are dissipated into internal energy in cells (shock wave is the most typical case), the dissipated energy may be unevenly assigned among cells, thus causes oscillations. This kind of oscillations can be controlled to a small level through a carefully designed viscous stress tensor or artificial heat flux, as studied widely in references. For fluid deformation, as the grid composed of straight lines cannot deform as flexibly as a continuous fluid, the volume of a grid cell would not accurately represent that of a fluid element, and this volume error will lead to oscillations too. In this article, we study in the SGH framework a matter‐flow method, whose design principle is representing the under‐grid bending motion of fluid elements by matter transport through grid edges. We first derive governing equations of the matter flow velocity defined at each edge of a grid, then derive formulas for the mass, energy, and momentum fluxes on the edge with the matter flow velocity. We test the matter‐flow method in largely deforming fluids problems on triangular meshes. It is shown that serious oscillations arise in the regular SGH simulations, while they are well controlled after implementing the matter‐flow method. In order to get a knowledge about the adaptability of the matter‐flow method, we also test it in shock problems that do not fall within its design goal. For this purpose, an artificial heat flow is introduced in the SGH scheme to play a major role in stabilizing the shock capture. The test results show that the matter‐flow method can help improve the shock simulations. The source code can be downloaded from the GitHub repository: https://github.com/zhaoli0321/Matterflow. [ABSTRACT FROM AUTHOR]

Due to ignoring the effects of the change of the tooth attachment position caused by the cracks, traditional time vary mesh stiffness (TVMS) calculation models and dynamic simulations for cracked gears will lose their precision in the body crack case. To address this shortcoming, a new analytical TVMS calculation model of cracked gear considering tip relief (TR) is developed based on a proposed variable-angle deformation energy integration method. On this basis, a dynamic model of the gear system for the analysis of fault vibration characteristics is established. The effectiveness and accuracy of the proposed TVMS calculation model are verified by the finite element method. A comprehensive investigation is finally carried out to reveal the effects of the parameters of TR, load and crack on the TVMS and dynamic characteristics of the cracked gears. The study results indicate that the proposed models can meet the accurate TVMS calculation and dynamic simulation for both the tooth- and body-cracked gears, and the influences of the tooth attachment position change caused by the crack cannot be ignored. This study could provide a systematic methodology and meaningful reference for the dynamic modelling, simulation and fault diagnosis of gear systems with crack failures. [ABSTRACT FROM AUTHOR]

Gear time-varying mesh stiffness (TVMS) plays an important role in the analysis accuracy of a gear transmission mechanism. In previous research, the TVMS of a modified straight bevel gear (SBG) was mainly obtained by the finite element method (FEM), whose computation time is usually long to ensure computational accuracy. By contrast, this work proposes a fast calculation method with high accuracy for the TVMS of a modified SBG. First, B-spline surfaces are introduced to reconstruct a modified SBG using discrete tooth surface points. Next, based on the tooth contact analysis and loaded tooth contact analysis, combined with energy conservation principle and infinitesimal method, a semi-analytical instantaneous single mesh stiffness (SMS) model for the modified SBG is derived. Then, using the additional deformation compatibility condition and principle of force balance, the calculation method of the TVMS of the modified SBG is obtained. Finally, a comparison of the mesh stiffness calculation results between the method of this study and that of FEM is completed. The results show that the semi-analytical TVMS model used in this study can be accurately and quickly solved. [ABSTRACT FROM AUTHOR]

As one of the most typical fault forms of the helical gear, the crack will change the dynamic excitation and further affect the dynamic behaviors of the transmission systems. Due to the complicated structure of the helical gears, the coupling effect between the neighboring loaded teeth is usually ignored in the mesh stiffness calculation, making it considerably overestimated especially in the case of the crack fault. An improved mesh stiffness calculation method of helical gear with spatial crack is proposed to make up this gap. The interactions between the loaded neighboring teeth induced by the gear body flexibility were considered to improve the calculation accuracy and applicability. Besides, the load distribution law for the engaged cracked tooth along the tooth width and profile can be obtained. The results indicated that the mesh stiffness of the multi-tooth engagement calculation using this model could be further improved compared with the traditional methods. Finally, the effects of the helix angle, crack depth, and crack propagation length on the mesh stiffness and load distribution were investigated using the proposed method. [ABSTRACT FROM AUTHOR]

Transmission error (TE) in polymer gears is significantly affected by the elastic deflection of meshing teeth. The present work describes the deflection characteristics of the asymmetric Nylon 66 spur gears under similar and dissimilar engagements. The load sharing and deflection behavior were estimated numerically using a linear viscoelastic material model. Experimental determination of mesh deflection was carried out through static load tests. The peak mesh deflection in a cycle for the 20°/34° configuration was found to be 8% and 12% smaller as compared to 34°/20° and 20°/20° configurations, respectively. The peak tooth deflection was found to be highest for the 34°/20° configuration and lowest for the 20°/34° configuration, with the deviation between their peak deflections being 15% in metal-polymer pairs and 17% in polymer-polymer pairs. In the 34°/20° configuration, the higher drive side pressure angle increased the single tooth contact period, causing greater peak-to-valley amplitude in a cycle. The negligible tooth compliance of steel gear caused unsymmetrical load sharing in the double tooth contact zones of metal-polymer engagements. The greater compliance of driver gear in polymer-polymer engagements increased the mesh deflections of 20°/20° and 20°/34° configurations by 45% and that of 34°/20° configuration by 37%. Also, gear pairs with similar material stiffness balanced the load sharing in the addendum and dedendum double tooth contact zones. The large tooth deflection extended the contact duration of all configurations for both the similar as well as for the dissimilar engagements, respectively. This was signified by the contact path extension at the peripheries of the mesh cycle. [ABSTRACT FROM AUTHOR]

The internal gear is part of the planetary and epicyclic gear pairs in the transmission system of the helicopter. Gear tooth wear is one of the most usual gear failures. This paper establishes an analytical model to evaluate the influence of tooth wear on the mesh relationship. A new mesh relationship can be derived for internal spur gears with tooth wear by the proposed analytical model. Consequently, using the new mesh relationship, the two most important meshing excitations, mesh stiffness and unloaded static transmission error (USTE), are quantitatively evaluated for the internal gear with tooth wear. The results indicate that tooth wear mainly affects the meshing ranges of single-tooth and double-teeth in mesh stiffness, rather than its amplitude. Additionally, the amplitudes of USTE increase with the increasing wear depth. Finally, this study can offer a foundation for the dynamic modeling and fault diagnosis of internal spur gears with wear faults. [ABSTRACT FROM AUTHOR]

Gears are critical machine elements that transmit power and motion in diverse implementation fields. Over time, gears may produce a series of faults due to harsh operating conditions, fatigue, manufacturing errors, etc., leading to severe performance degradation. During the meshing process, the stiffness of a single tooth controls the load sharing, vibration, and noise characteristics of a geared system. An undetected fault could decrease the gear stiffness and thus may lead to a fatal breakdown, substantial economic losses, or even human casualties in safety-critical applications such as helicopters, high-speed trains, and wind turbines. Hence, the accurate quantification of the gear stiffness emerges as an important research area to obtain reliable gear designs. With this in mind, the asymmetric tooth concept offers superior bending strength, fatigue propagation life, and the ability to lessen vibration over the standard (symmetric) designs in applications where unidirectional loadings are predominant. This study investigates the influence of tooth root cracks on the single-tooth and meshing stiffness characteristics of the standard and asymmetric involute spur gears. To this end, the numerical crack propagation paths obtained in our previous works were introduced to the created 3D CAD geometries. Subsequently, the single tooth stiffness of both healthy and cracked (2 5%-50%-75%-100%) gears was calculated through the ANSYS® Workbench, and the time-varying mesh stiffness was obtained. The present study evaluated the effects of backup ratio and the tooth asymmetry on the spur gears' meshing stiffness characteristics simultaneously and further expanded the scope of the research work. The results indicated that the single tooth stiffness and mesh stiffness could be improved by 35% and 22%, respectively, as the drive side pressure angle increased from 20° to 35°. It has been noted that the gear stiffness decreased as the crack level increased, while the increment of the backup ratio further increased the reduction in the stiffness. The findings could provide significant outputs for a better understanding of the influence of tooth asymmetry on the gear dynamics characteristics, life prediction, and early fault diagnosis. [ABSTRACT FROM AUTHOR]

In this study, dynamic response analysis of functionally graded gears (FGGs) has been performed using a 6-degree of freedom dynamic model. The pinion and gear are divided into homogeneous sub-domains of uniform thickness, conforming to the gear tooth profile. The material composition varies radially according to power-law gradation with metal at the innermost and ceramic at the outermost surface. Mesh stiffness and transmission error of FGGs have been evaluated using a finite-element-based numerical method employing contact analysis. Results show that for considered values of gradient index (GI), FGGs show a 10% to 50% reduction in mesh stiffness with a 30% to 60% reduction in weight compared to steel gears of exact specifications. Also, FGGs show a 4% to 12% reduction in dynamic factor and a 9% to 17% reduction in peak-to-peak displacement amplitude than steel gears over the selected values of GI. [ABSTRACT FROM AUTHOR]

Tooth root fatigue crack is one of the most common failure forms in the service process of gear transmission system， and it will threaten the safety of gear transmission systems or even the whole equipment. It is of great significance to carry out the fault diagnosis of the gear tooth in gear transmissions， where the dynamic action mechanism and vibration feature of the tooth root crack fault are the core. However， the influence of tooth root rack on tooth coupling has not be considered in the traditional calculation model of gear mesh stiffness. With respect to this problem， a finite element model of a spur gear is established to analyze the coupling mechanism between the adjacent teeth caused by the structural deformation of the gear body. And the effect of the force， the hub hole radius and the depth of tooth root crack on gear body stiffness are investigated， especially for the coupling stiffness between the adjacent teeth. The results show that the tooth root crack has a significant effect on both the gear body stiffness and the coupling stiffness between the adjacent teeth， which should be considered in the excitation calculation of the time-varying mesh stiffness with the presence of tooth root crack， so as to improve the accuracy of gear excitation model with tooth root crack.

• A general definition of mesh stiffness function for spur, helical gears and its limitations are presented. • A novel multi-foundation model of mesh interface for spur and helical gears is proposed. • Extensive 3D finite elements results are given for several spur and helical gears. • In depth comparisons between classic thin-slice, finite element and multi-foundation results are presented. • The influence of gear body deflections and elastic couplings between adjacent teeth is highlighted. • The role of axial displacements on helical gear mesh stiffness is emphasised. A definition of mesh stiffness as a scalar function connecting spur and helical three-dimensional pinions and gears is presented along with its conceptual limitations. Since it relies on global parameters namely, variations in torque and transmission errors, it can therefore be applied to the vast majority of the models found in the literature. In parallel, a multi-foundation (MF) model of mesh interface is introduced with the objective of being as accurate as three-dimensional finite element (FE) simulations for highly reduced computational costs. Numerous results on mesh stiffness and tooth contact conditions are presented for unmodified spur and helical gears. It is shown that the proposed mesh stiffness functions based on global parameters are sound and that the MF and FE results compare well, particularly when gear body deflections are limited. The contributions of off-line of action contacts, elastic couplings when several tooth pairs are loaded and axial deflections in helical gears are analysed. Finally, the notion of mesh stiffness function representative of the elasticity of a pinion-gear pair for static and dynamic conditions is examined. [ABSTRACT FROM AUTHOR]

Tooth surface spalls fault is the irregularly shaped partial pits on the gear surface. In the calculation of gear mesh stiffness. Common method is to simplify the tooth surface spalls to a specific regular shape for research, which has great limitations. Based on image processing, this paper extracted the edge features of the spalls fault, dissociated spalls area according to the pixels and fit the spalls width function. Based on the traditional potential energy method, mesh stiffness of the faulty gear is calculated more accurately and dynamics simulation are performed. Compared with the rectangular spalls fault, the mesh stiffness of the faulty gear after image processing can more accurately reflect the contour characteristics. Eliminating the stiffness and acceleration response mutation points caused by simplification. The proposed method has high versatility and is applicable to tooth spalls of arbitrary shapes.

Rail corrugation is usually generated in modern railway transportations, such as high-speed railway, urban railway, and heavy-haul railway. It is one of the major excitations to the wheel--rail dynamic interaction, which will cause extra vibration and noise, failures, or even risk of derailment to the vehicle and its components. A dynamics model of a heavy-haul locomotive considering the traction power from the electric motor to the wheelset through gear transmission is employed to investigate the nonlinear dynamic responses of the locomotive. This dynamics model couples the motions of the vehicle, the track, and the gear transmission together. In this dynamics model, excitations from the rail corrugation, the nonlinear wheel--rail contact, the time-varying mesh stiffness, and the nonlinear gear backlash are considered. Then, numerical simulations are performed to reveal the dynamic responses of the locomotive. The calculated results indicate that different nonlinear phenomenon can be observed under the excitation of the rail corrugation with different amplitude and wavelength. The high frequency vibrations excited by the time-varying mesh stiffness are usually modulated by the low frequency vibrations caused by the rail corrugation. However, this is likely to vanish under the chaotic conditions with some corrugation wavelength. The vibration level of the vehicle and the gear transmission increases generally with the corrugation amplitude. However, some corrugation lengths have been found to be more responsible for the vibration of the dynamics system, which should be concerned greatly during the locomotive operation. Meanwhile, involvement of gear transmission systems will cause different dynamic responses between the wheelsets under rail corrugation and gear mesh excitations. [ABSTRACT FROM AUTHOR]

There have been many researches focusing on the dynamic characteristics of gear faults, such as spalling, crack and broken, to improve the accuracy of fault diagnosis methods. However, the tooth pitch error caused by actual gear manufacture precision is usually neglected, which may have a direct influence on the gear vibration response. In order to settle the problem of the underlying vibration mechanism with the combined effect of pitch error and spalling fault, a dynamic modeling method of the spalling gear coupling with pitch error is proposed, in which the relationship between the gear precision level and the pitch error is considered. The changes in both meshing stiffness phase and displacement excitation on the contact line caused by pitch error are analyzed, where the additional effect of spalling fault on stiffness excitation is also taken into consideration. Base on the proposed dynamic model, the dynamic responses and vibration characteristics of spur gear with coupling excitation of pitch error and spalling fault are investigated. The simulation results show that vibration characteristics caused by the early spalling fault are easy to be affected by the pitch error, especially when the gear precision is poor. [ABSTRACT FROM AUTHOR]

This paper develops a mesh stiffness model for flexible internal meshing with favorable accuracy and computational efficiency, where the interactions between multiple ring-planet branches are introduced. Based on the curved beam element submodel and the minimum potential energy principle (MPEP), the estimation of mesh stiffness is transformed into an optimization problem. Mesh forces are iteratively generated to better satisfy the MPEP until the predetermined thresholds are reached. The mesh stiffness is then calculated from the final group of mesh forces. Simulations of the finite element method validate the merits of this model. By changing the ring thickness, the number of fixed supports, and especially the number of planets, the variation rules of mesh stiffness as well as mesh force are summarized. It is demonstrated that the fluctuations of mesh stiffness and mesh force caused by the alternation of single and double tooth engagement become more distinct and tend to present within certain ranges around their mean value due to the increment of the thickness of ring gear or the number of fixed supports. In addition, the low-stiffness area of the mesh stiffness curve for flexible internal meshing is influenced by the number of planets, especially when this number is greater than the number of fixed supports. [ABSTRACT FROM AUTHOR]

As one of the most important dynamic excitation sources of the gear system, time-varying mesh stiffness (TVMS) is a key parameter of the gear system dynamics model. To calculate the mesh stiffness of straight bevel gears, a semi-analytical model for calculating the single mesh stiffness (SMS) was proposed based on the infinitesimal method and energy equivalence theory, considering the real transition surface of the tooth root, elastic coupling effects between tooth segments, and nonlinear contact between tooth pairs. Based on the principle of deformation compatibility and force balance and considering the extended tooth contact, which is the phenomenon of the incoming tooth pair coming in contact ahead of the theoretical start of contact and the outgoing tooth pair coming out of contact later than the theoretical end of contact, a semi-analytical calculation model of TVMS was established. Based on the analytical model, the TVMS with different torques and gear parameters was compared with that obtained from the finite element method (FEM). The causes of errors were then analyzed to verify the validity of the semi-analytical method. The results show that the mesh stiffness calculated by the semi-analytical model is in good agreement with that calculated by the FEM, and the calculation efficiency is considerably higher than that of the FEM. Finally, the influence of the gear parameters on the mesh stiffness is analyzed. This study is expected to be helpful in further enriching and developing the calculation theory of gear stiffness and providing a theoretical tool for gear dynamics research. [ABSTRACT FROM AUTHOR]

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Bevel gear pairs are employed extensively in transmission systems, such as vehicle transmissions (rear axle drive), aircraft engines/turbines and helicopter gears, to transfer power between non-parallel shafts at high speed or high torque. The most complex form of bevel gear is the spiral bevel gear (SBG). SBG pairs are commonly used in applications that require high load capacity at higher operating speeds than are typically possible with other types of bevel gear. When manufactured in a metal-cutting process, spiral bevel gears can either be produced using single indexing (a face-milling method, which is considered in this study) or continuous indexing (a face-hobbing method). Due to manufacturing imperfections and the flexibility of components, the system might experience misalignments that intensify or exert a destructive effect on the gear vibration, which causes disruption in the stress distribution, thereby decreasing the lifetime of the gearbox. The main purpose of this study is to carry out loaded tooth contact analysis (LTCA) and unloaded tooth contact analysis (UTCA) for an SBG pair in the presence of two types of misalignment, axial and radial misalignment, and represent their effects on the mesh stiffness (MS). To calculate the MS, it is essential to determine the geometrical mismatch between two mating tooth profiles by means of UTCA. To conduct LTCA, three main approaches can be utilised: the finite element method (FEM) and experimental and analytical approaches. Due to the development of software packages during the last decade, Transmission3D-Calyx, an FEM-based software, is used in this study to carry out LTCA and UTCA. Finally, the MS for different misalignment cases is compared to represent the effect of misalignment on the SBG pair. [ABSTRACT FROM AUTHOR]

The positioning errors of a gear pair will directly affect the mesh position and the corner contact (via the tooth pair separation distance) which should be considered for correctly modeling the gear time-varying mesh stiffness and dynamic responses. This phenomenon is more complex for a planetary gear set as it includes both the external gear pairs (sun-planets mesh) and internal gear pairs (ring gear-planets mesh). However, in the literature, these effects have not been studied in the dynamic modeling of planetary gear sets possibly due to their complexities. To fill this gap, the actual mesh position and the separation distance considering the gear positioning errors for both the external and internal gear pair are derived via geometric analysis. The mesh stiffness models for both the internal and external gear pairs are established through the load coordination equation considering the corner contact induced by the tooth pair separation distance and the position-dependent Hertzian contact deformation. Finally, the dynamic model for a planetary gear set with position errors is built based on the proposed mesh stiffness models. The load sharing characteristics simulated by the proposed model are compared with the traditional model which only considers the effect of positioning errors on the contact ratio and pressure angle and ignores their effects on the mesh position and the separation distance for both the external and internal gear pairs. The results indicate that the difference of the load sharing characteristics between the proposed model and the traditional models is significant under heavy external load. The simulation results also show that suitable positioning errors can improve the load sharing condition by 10% if the amount of positioning errors is changed from 0.1 to 0.2 mm. [ABSTRACT FROM AUTHOR]

The effect of gear contact state change due to shaft misalignment on meshing stiffness is usually neglected in the traditional stiffness calculation model with misalignment error, the further influence mechanism of shaft misalignment on gear dynamic characteristics is also unclear. To address these shortcomings, a new mesh stiffness calculation model with misaligned gear considering the effects of tooth contact state is proposed by combining the improved loaded tooth contact analysis (LTCA) model. Then the effects of tooth contact state changes aroused by shaft misalignment on the meshing stiffness excitation are investigated. Moreover, a dynamic model of the misaligned gear system with 8 degrees of freedom (DOF) is established, and based on which the dynamic characteristics of the gear system are investigated and verified by experiment. The study results indicate that the proposed model can be used to evaluate the stiffness excitation and dynamic characteristics of the misaligned gear system with the tooth contact state taken into consideration. This study can provide a theoretical method for evaluating and identifying shaft misalignment errors. [ABSTRACT FROM AUTHOR]