Gear Mesh Research Articles

Dynamic response analysis of high-speed train gearboxes excited by wheel out-of-round: experiment and simulation

Due to the special characteristics of rail vehicles, wheel–rail excitation has always had a huge impact on the stable operation of the vehicle system and the dynamic performance of each component structure. This paper takes high-speed train gearbox as the research object and establishes a rigid-flexible coupling dynamics model and analyses the dynamic response and fatigue damage of the gearbox under the excitation of wheel flat and wheel polygon. Combined with the analysis of test data from the gear transmission testrig and actual line operation of high-speed trains, it shows that high-speed train gearboxes are different from those applied in other fields, where the gear meshing excitation is no longer its main excitation source, but the wheel–rail excitation instead. The comparison analysis between the wheel flat and wheel polygon conditions indicates that the response of the gearbox to the wheel flat excitation is similar to the operational modal test and is capable of causing multi-order modal vibration of the gearbox. The effect of the wheel polygon on the gearbox is similar to that of single-frequency excitation, which lead to certain order modal resonance in the gearbox under frequency coupling situations. The impact of both conditions on gear box fatigue damage increases with the vehicle operating speed.

Meshing stiffness characteristics of modified variable hyperbolic circular-arc-tooth-trace cylindrical gears

Abstract. Stiffness excitation is one of the important excitations for the variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear system. Accurate calculation of the gear meshing stiffness is of great significance to investigating dynamic characteristics of the VH-CATT cylindrical gear system. Firstly, based on the forming theory of the modified tooth surface, the modified tooth surface equation of the VH-CATT cylindrical gear was deduced, and the 3D reconstruction was realised. Next, the load tooth contact analysis (LTCA) model of the VH-CATT cylindrical gear was developed to calculate the meshing stiffness of the VH-CATT cylindrical gear, and it was verified by the finite-element calculation. Finally, the influence of the load and modification parameters on the VH-CATT cylindrical gear stiffness was investigated. Research shows that the stiffness calculation method of the VH-CATT cylindrical gear based on LTCA is accurate. The meshing stiffness of the VH-CATT cylindrical gear in the double-tooth meshing area is large, and the meshing stiffness of the VH-CATT cylindrical gear in the single-tooth meshing area is small. The stiffness of the VH-CATT cylindrical gear increases with an increase in the load and cutter inclination angle, the stiffness of the VH-CATT cylindrical gear only in the double-tooth meshing area decreases with an increase in the parabolic coefficient, and the stiffness of the VH-CATT cylindrical gear increases with a decrease in the blade parabolic vertex position value. The research results provide a basis for improving the bearing capacity of the VH-CATT cylindrical gear and optimising design.

Evaluation of vibration behavior at different sensing positions on gearboxes

Monitoring systems for rolling contacts in machinery are widespread. There are multiple solutions in various applications like Health Usage Monitoring Systems (HUMS) for Helicopters and smart bearings from different manufacturers. The number of solutions and possibilities is rising. This paper outlines how the vibrational behavior depends on the measuring position outside and inside the gearbox. The effect of the sensor location is evaluated by examining different locations close to the main excitation of gearboxes—the loaded tooth contact—as well as positions on the gearbox housing. Based on the results, recommendations for an optimal sensor position for gear vibration monitoring are given by considering the efforts of sensor integration. The results can be used to select sensors with the lowest cost to support the widespread gear vibration monitoring.Important locations for health monitoring are the gear meshes inside the gearbox. Due to the requirement of high power density, a compact design is mandatory. This does not allow for sensor application close to the gear meshes but only on the outside of the gearbox. An impediment of signal quality and less chance of early damage detection is the consequence. Hence, efforts have been made to directly integrate multiple sensors inside the gear body without negatively impacting the package.For the investigation of the vibration transmission behavior, an FZG gear test rig is selected. These standardized test rigs are approved and established all around the world for the creation and evaluation of various gear failure data. The presented test rig has been modified to be able to sense vibration in the gear wheel, at the gear wheel, and on the gearbox housing to evaluate different sensor positions and vibration sensors for their suitability on vibration measurement.

Numerical Analysis of Tooth Contact and Wear Characteristics of Internal Cylindrical Gears with Curved Meshing Line

In order to improve the contact strength and reduce the sliding friction of the gear pair, an internal cylindrical gear pair with a curved meshing line is studied in this paper. Firstly, a curved meshing line is designed. The tooth profiles of the internal gear pair with the designed meshing line are calculated by using differential geometry and the gear meshing principle. Secondly, a wear model is established by combining the finite element method and the Archard wear formula. Then, a numerical simulation is conducted; the relative curvature, sliding coefficient, sliding distance, maximum contact pressure, transmission error, and wear depth are calculated. Ultimately, the variation law of tooth surface wear of new gear with and without installation errors is observed under different stress cycles. On this basis, the influence of tooth modification on tooth surface wear is further researched. Through the results, the advantages of the introduced novel internal cylindrical gears in wear resistance are further demonstrated. The study in this paper provides new research ideas and methods for gear wear research and gear design.

An investigation into the effects of profile shifts on helical gear mesh stiffness

Time-varying meshing stiffness (TVMS) is a significant nonlinear periodic excitation in helical gears, it poses significant impacts on gear system’s dynamic characteristics and fatigue life. However, the specific impact of profile shift on it remains unclear. This article establishes an analytical model for the TVMS of helical gears by utilizing the slicing method to address this gap based on the potential energy method, and quantitatively analyze the TVMS with different modification coefficients. The research found that the helical gear’s TVMS decreases with the increase of modification coefficient for individual profile shifts. For the equal compound profile shifts, it rises with the increase of modification coefficient of pinion. For the unequal compound profile shifts, the positive drive increases the TVMS, but the negative drive to the contrary. The research achievements provide valuable guidelines for dynamic design and structure optimization for helical gears.

Research on Calculation and Optimization Methods for Tooth Flash Temperature and Meshing Power Loss of the Gear System in Drum Shearer

The operating conditions of the drum shearer are very complex, and its ranging arm gear system often suffers from gear scuffing and wear. Gear scuffing is caused by the adhesive wear, which is due to the instantaneous friction and flash temperature of the tooth surface, and the gear meshing power loss is also caused by tooth surface friction. In order to resist tooth scuffing and improve meshing efficiency of the transmission system, an improved semi-analytical tooth surface flash temperature calculation method was used. The tooth flash temperature status under various working conditions were analyzed in detail. Based on the mechanical model of the shearer drum picks, the load condition of the drum was analyzed. Under these load and boundary conditions, the misalignments of each gear pair in the ranging arm were calculated. The tooth surface load distribution was calculated under the gear misalignments, and then the theoretical tooth surface flash temperature and meshing power loss were determined. Next, the tooth micro-geometry was modified to reduce flash temperature and meshing power loss. The flash temperature distribution pattern of the optimized tooth surface was studied under various working conditions, and the meshing power loss was also obtained. Finally, experiments were conducted to verify the effects of the optimized tooth surface on the friction temperature rise and the effectiveness of the modification method. Tooth surface optimization aimed at reducing tooth surface flash temperature can also effectively reduce meshing power loss, which has a significant effect on gear anti-scuffing and energy saving.

Comparative study on the causes of rail corrugations in long steep-grade sections under traction and braking conditions

There are significant differences in rail corrugations between the uphill and downhill sections in long steep-grade sections. Comprehensive field investigations and numerical simulations are conducted to investigate rail corrugations under traction and braking conditions. Firstly, field investigations of rail corrugations in long steep-grade sections are carried out. Secondly, the dynamic models of the vehicle-track system under uphill traction and downhill braking conditions are constructed, and the dynamic characteristics of the vehicle-track system are explored. Then, the finite element models of wheel-rail-traction/brake systems are constructed, and the friction self-excited vibration characteristics of the wheel-rail system are studied by the complex eigenvalue method. Finally, the influences of key parameters of traction and brake friction pairs on rail corrugation are studied, and suppression methods of rail corrugations are proposed. The results show that (1) rail corrugation more easily occurs on the transition positions of the easement curve and circular curve, with slope changing. (2) Under the traction and braking conditions, rail corrugations are induced by the friction self-excited vibration of the wheel-rail-traction/brake system. (3) Reducing the traction speed or increasing the gear mesh stiffness, reducing the brake friction coefficient or the brake force can suppress the rail corrugation.

An analytical approach for predicting the fillet and contact stresses in symmetric and asymmetric spur gears under frictional mesh assumptions

A precise estimation of the bending and contact stresses, as well as the true friction value during the gear mesh, are critical aspects of a cost-effective and safe gear design. Previous studies have usually neglected friction by assuming a frictionless contact. Additionally, the impact of friction on the bending stress of spur gears has rarely been addressed. Moreover, to the authors' best knowledge, the impact of friction on the weakest section, stress concentration factor, and Lewis form factor remain unexplored for both symmetric and asymmetric gears. This study presents an analytical model incorporating friction at different contact locations to calculate the variables related to the tooth's weakest section. This model facilitates the evaluation of the stress concentration factor, Lewis form factor, and maximum tensile stress for symmetric and asymmetric gears. The proposed model is especially beneficial for assessing the load-carrying capacity of spur gears operating in challenging conditions. The results showed that the asymmetric spur gear is superior to its equivalent symmetric counterpart in terms of tooth form factor; however, this superiority decreases as the coefficient of friction increases. In addition, friction has been found to have a greater effect on fillet stress than contact stress. For μ = 0.2, at the HPSTC, the contact load and tooth form factor decreased by 7 % and 12 %, respectively, and the fillet stress worsened by 13 % compared to the frictionless assumption. Meanwhile, at the LPSTC, the stress concentration factor and contact stress increased by 9 % and 3 %, respectively.

Development of a comprehensive car safety gears testing procedure for low- to medium-speed lifts in Hong Kong

Operation of the safety gears is the last means of protecting passengers in a high-speed or even freely falling lift car. EN81-20 stipulates that the average retardation when the safety gears are engaged should lie between 0.2 g n and 1.0 g n while g n is the gravitational acceleration. Decades ago, in Hong Kong, to conduct a formal test on the safety gears, the car was fully loaded and the machine brake was manually released to allow the car fall freely until the overspeed governor triggered the engagement of the safety gears. This test was rather vigorous to both the gear jaws and the guiderails. Recently, tests have been conducted with 125% in-car load while the car travels under an inspection (or levelling) speed and the governor is manually triggered. Such a revised test, though much less vigorous, may not be able to verify the average retardation as required. Even a test with full load and free-falling may not reveal the whole truth by merely measuring the resulting contact scratches on the guiderails because a majority of the retardation could have been assisted by the counterweight via the suspension ropes. A method that involves an empty car traveling downward at 1 m/s with the governor manually triggered was first developed in Germany. This method has been improved to form a comprehensive procedure as detailed in this article so that a no-load rated speed, or slightly reduced speed, test is conducted, which is almost non-invasive or much less vigorous. By calculation, the average retardation when all suspension ropes are broken can be estimated. Cross-checking in our experiments revealed the validity of this procedure. The procedure also involves a test to verify the validity of parameters used to perform the estimation and takes care of some modern overspeed governors which cannot be manually triggered. It is hoped that this procedure could encourage maintenance professionals carry out safety gears tests more often to ensure a high safety standard. This comprehensive procedure could encourage more frequent tests on the car safety gears, which is almost non-invasive for manually triggerable overspeed governors or less vigorous for non-triggerable ones. An empty car traveling downward at rated speed, or at slightly reduced speed, is tested by manually triggering the governor while a partially loaded free-falling car is tested if the governor is not manually triggerable. Results can be used to estimate the average retardation of the lift car when all suspension ropes are broken. A pre-test is included to validate critical parameters necessary for the estimation. Experiments conducted so far were on the safety gears of low- to medium-speed lifts. For high-speed lifts beyond 4 m/s, that may be considered in the next stage of research. Although the title of this article includes the city where the procedure was developed, there is no reason why this testing procedure could not be extensively applicable to any traction lift systems around the world at such speed levels.

Computerised design, simulation of meshing and stress analysis of bionic logarithmic spiral spur gear drives

Inspired by the inherent properties of logarithmic spiral in nature, a new cylindrical spur gear drives using logarithmic spiral as a gear tooth profile is proposed. The logarithmic spiral property is explored, and then the equation of meshing, conjugate tooth profile and contact path of the proposed gear pairs are derived. The meshing characteristics of the new pair are investigated, such as pressure angle, contact ratio and sliding ratio. Simulation analysis of the strength, transmission error and transmission efficiency are used to validate the performance of the gear pairs before production of the proposed gear, followed by gluing load capacity and bending fatigue strength testing of the fabricated gear pairs. The results show that the pressure angle at any point of the tooth profile on the proposed gear is constant and equal to the logarithmic spiral strike angle. The contact ratio of the gear decreases with the increase of the strike angle. Compared with that of involute gears, logarithmic spiral gears have better performance in terms of strength, transmission error, transmission efficiency and lifetime.

Analytical method for time-varying meshing stiffness and dynamic responses of modified spur gears considering pitch deviation and geometric eccentricity

Pitch deviation and geometric eccentricity are almost inevitable in gear transmission systems. Tooth profile modification (TPM) is a common method used to improve gear contact properties. However, the coupling interaction between the tooth profile deviation introduced by TPM and manufacturing errors is complex. There is a lack of an analytical model that can clarify the dynamic behaviors of the modified gear pair affected by multiple manufacturing errors, specifically pitch deviation and geometric eccentricity. To fill this gap, a comprehensive analytical gear mesh model is proposed considering TPM, pitch deviation, and geometric eccentricity. The calculation formulas for the time-varying meshing stiffness (TVMS) and load sharing factor (LSF) are derived considering the iterative change of gear pair meshing parameters with phase under the influence of multiple manufacturing errors. Then, the translation-torsion coupling dynamic model of the spur gear pair based on the lumped mass method is established to investigate the dynamic transmission error (DTE) and vibration characteristics. Results show that TPM is an effective method for improving the comprehensive stiffness of a gear pair within the meshing range containing pitch deviation, but it reduces the meshing phase that includes only geometric eccentricity. Compared with the unmodified gear pair, the suppression effect of changing the modification length on the maximum DTE and vibration acceleration is more significant than that of the modification amount. TPM increases the overall fluctuation threshold range of DTE for gear pairs with only geometric eccentricity. It is expected that the proposed method can provide a theoretical basis for exploring vibration suppression in gear pairs with geometric eccentricity and pitch deviation errors.

Study on the mathematical model of the novel drum worm pair

Abstract A lot of work is needed to investigate the design method of planar internal gear enveloping drum worm (PIGEDW). Firstly, based on the gear meshing theory, the meshing equation is deduced. Secondly, the PIGEDW mathematical model is established by MATLAB. Finally, the influence of the inclination angle of the generating plane on the meshing performance of PIGEDW is analyzed. The results show that appropriate changes in geometric parameters are beneficial in enhancing the performance of PIGEDW, which provides a way to optimize its design.

Study on multi-state of meshing-impacting dynamic characteristics of spur gear systems considering friction under localized tooth breakage

Localized tooth breakage is one of the common failures of spur gears, which affects the smooth and safe operation of spur gear transmission systems. The tooth collision cannot be neglected, and it is especially important to reveal the meshing-impacting dynamic characteristics of the gear system under localized tooth breakage to improve the safe and stable operation of the gear system. Based on the gear meshing principle and the dissipative collision contact force model, the drive-side tooth meshing model and back-side tooth impacting model are established with considering the transient nature of tooth back-side contact. The tooth surface meshing model and tooth back collision model under partial breakage are established. According to the contact state and force environment of the gear pair, the multi-state meshing-impacting behaviour under local breakage is classified, and the discrete meshing-impact dynamics model of an involute spur gear system under local breakage is established to explore the influence of local breakage on the meshing stiffness and load distribution. The mechanism of contact force under partial tooth breakage is revealed, and the influence of load coefficient and meshing frequency on the nonlinear dynamics is studied by defining two Poincaré maps. It is found that local tooth breakage affects the contact force of single-and double-tooth meshes and reduces gear load carrying capacity. Larger loads inhibit back-side impact, and smaller loads induce the coexistence behaviour and back-side impact. Larger or smaller meshing frequency induce back-side impact behaviour. The coexistence of chaotic and periodic motions induces back-side impact behaviour, and localized tooth breakage affects the coexistence phenomenon and aggravates the complexity of the dynamic behaviour. The dynamic model of gear system considering energy dissipation and nonlinear vibration under the presence of local tooth breakage of the pinion is explored, and the conditions of back-side impact are studied. This research provides new methods and ideas for nonlinear dynamic modelling and analysis of faulty gear systems.

The Influence of the external meshing gears’ contact ratio on the contact stress in the transmission system

Abstract Using the Hertz contact theory, the calculation method of tooth contact stresses at different boundary points within the double tooth meshing zone for high contact ratio external meshing gear pair is investigated. The radius of curvature and pressure angle at each boundary point of the external meshing gear with a high contact ratio is given. The boundary point meshing coefficients are introduced to convert the comprehensive radius of curvature at the boundary point into the radius of curvature at the node, to make it in line with the form of calculation of the national standard. The analysis examines how the parameters affect the contact stress at different boundary points of the gear pair with a high contact ratio. When the contact ratio of the gears is 2, there is a sudden change in the contact stress on the tooth surface, and HCR gears have smaller contact stress than LCR gears.

Identification of the Asymmetric Transmission Error and Gear Mesh Dynamic Parameters using Full-Spectrum Responses in a Geared-Rotor System

A dominant source of vibration in geared-rotor systems are the gear mesh fault parameters. They include the asymmetric transmission error, phases of transmission error, the gear mesh stiffness, the gear mesh damping and the gear runouts. The present work deals with the experimental identification of aforementioned parameters. A mathematical model of geared-rotor system has been developed using Lagrangian dynamics. Equations of motion are transformed into frequency domain using the full-spectrum response analysis. These transformed equations are used to develop an identification algorithm based on least-squares fit to estimate the transmission error and gear mesh dynamic parameters. The system identification algorithm is initially verified using numerical simulations. The robustness of the algorithm is checked by introducing white Gaussian noise in the simulated responses. A geared-rotor experimental rig was developed and used to measure responses at gear locations in two orthogonal directions. Measured responses are transformed in frequency domain using the full-spectrum analysis and used in the present novel identification algorithm to identify the gear parameters. The identified parameters are validated by comparing the numerically generated full-spectrum response using experimentally estimated parameters and that from the experimental rig. Conflict of Interest Statement The authors declare no conflicts of interest.

Hydrostatic bearing groove multi-objective optimization of the gear ring housing interface in a straight-line conjugate internal meshing gear pump

The lubrication performance of a straight-line conjugate internal meshing gear pump is poor under the low-speed, high-pressure operating conditions of the volumetric servo speed control system, and it is difficult to establish a full fluid lubricating oil film between the gear ring and the housing. This leads to significant wear and severe heating between the gear ring and the housing. The lubrication performance of the interface moving pair of the electro-hydraulic actuator pump gear ring housing can be improved by designing a reasonable lubrication bearing structure for the gear ring housing. In this study, a multi-field coupling multi-objective optimization model was established to improve lubrication performance and volumetric efficiency. The whole model consists of the dynamic model of the gear ring components, the fluid lubrication model of the gear ring housing interface, the oil film formation and sealing model considering the influence of temperature, and the multi-objective optimization model. The comprehensive performance of the straight-line conjugate internal meshing gear pump was verified experimentally using a test bench. The results show that the lubrication performance is improved, the mechanical loss is reduced by 31.52%, and the volumetric efficiency is increased by 4.91%.

A novel semi-analytical method for fillet foundation deflection calculation in presence of a tooth crack propagating into the rim of a spur gear

Existing analytical approaches for calculating fillet foundation deflections in spur gear struggle to adequately address the complexities introduced by tooth cracks, particularly as cracks propagate into the gear body. The finite element methods are accurate but computationally expensive. In response to these challenges, the paper introduces a novel semi-analytical method based on elastic circular ring theory that is capable of calculating the fillet foundation deflection for both healthy and cracked conditions. To establish the usefulness of the method, the fillet foundation deflections obtained are further used to obtain the time-varying mesh stiffness. A methodology centered on compatibility conditions is employed to achieve load distribution during double tooth engagement. The results are verified using two three-dimensional finite element models. In addition, the effect of different hub hole radii, crack lengths and orientations on the fillet foundation deflections and TVMS are investigated. The outcomes suggest that the proposed semi-analytical approach demonstrates superior accuracy in the calculation of spur gear mesh stiffness compared to alternative methods.

Cut-off point (COP) for early gear fault diagnosis via Meshing impact modulation (MIM) analysis

Comparing the number and magnitude differences and variations of the fault-induced features in frequency domain, such as the sidebands of meshing frequency in the raw spectrum, or the rotation frequency of faulty gear in the envelope spectrum between healthy and faulty conditions are the underlying principle of most of classical fault diagnostic methods for gearbox. However, several inevitable factors, for instance, circumferential stiffness anisotropy of gear shaft (including coupling mechanisms like keyways) and gear meshing backlash, can also lead to similar features, which may challenge these classical methods, especially when the fault is in its early stage. In specific, the initial early gear fault may neither lead to the significant increment of sideband or rotation frequency nor stimulate the high amplitude resonance, more than the inevitable factors mentioned above. As such, a meshing impact modulation frequency band is constructed to create a unique feature which does not exist in healthy condition. Considering both the impact nature and the modulation mechanism of gear meshing process, an MIM (Meshing Impact Modulation) index is designed, with which can effectively identify the optimal meshing impact modulation response frequency band with high stability and robustness across entire lifespan of gears. Within the demodulation spectrum of this band, the presence or absence of distinct gear rotation frequency spectral lines and their mirage as sidebands around the meshing frequency can serve as the effective and robust indicators to discern whether the gear is in a healthy "before cut-off point (pre-COP) state " or in an early faulty "after cut-off point (post-COP) state ". The effectiveness and the superiority of the proposed algorithm is testified by the life-test data from a test rig using industrial gearbox.

Cylindrical gear transmission errors influenced by temperature based on thermal network

Cylindrical gear transmission errors influenced by temperature based on thermal network

Linear complementarity formulations for contact analysis and wear prediction in gears

This work presents linear complementarity based formulations for contact analysis and wear prediction in gear trains. Assuming steady motion of gears transmitting constant input torque without transmission errors, three sets of linear complementarity formulations are proposed, first for rigid gear teeth, second for gear teeth with skin compliance, and third for gear teeth with full compliance, the last two being new in linear complementarity literature. The developed formulations are used for contact analysis of two examples of gear trains, a pair of spur gears in mesh, and a planetary gear train. The accuracy of the method is tested by considering various time step sizes and stability of the formulations demonstrated for wide range of time steps. Contact forces between meshing teeth are calculated at a pre-defined set of points on the path of contact, and then wear is calculated on gear tooth flank using Archard’s wear equation. These wear calculations are compared with existing results in literature which demonstrate the applicability of the developed LC formulations. Future work is projected to include more state-of-art friction models and extension to 3D contact analysis in helical gear trains.