Loaded Tooth Contact Analysis Research Articles

Study on the contact performance of the variable hyperbolic circular arc tooth trace cylindrical gear with installation errors

Abstract. Installation errors are important factor for the contact performance of the variable hyperbolic circular arc tooth trace (VH-CATT) cylindrical gear; the study of the relationship between installation errors and contact performances is beneficial for gear vibration and noise reduction. Firstly, based on the meshing theory, the tooth surface equation and contact ellipse of the VH-CATT cylindrical gear were deduced, and the tooth surface imprinting experiment was realized. Next, the tooth contact analysis model of the VH-CATT cylindrical gear was developed to investigate the influence of the installation errors on the elliptical contact area. Then, the calculation formulas of the tooth surface gap and contact point flexibility matrix were carried out to develop the load tooth contact analysis model of the VH-CATT cylindrical gear. Finally, the influence of the installation errors on the load distribution was investigated. Research shows that the elliptical contact area of the VH-CATT cylindrical gear is a line contact within a certain range and is located in the middle section of the tooth width. The elliptical contact area of the VH-CATT cylindrical gear increases or decreases rapidly when meshing in and out, and all the installation errors have a certain influence on the elliptical contact area, except for the center distance error. The single-tooth meshing area with the installation errors is greater than that of the ideal gear pair. The load distribution area with the installation errors deviates from the middle section, except for the center distance error. The installation rotation errors around the x and y axis have a certain influence on the maximum load of the single-tooth meshing area. The research results provide a basis for improving the bearing capacity of the VH-CATT cylindrical gear and optimizing the design.

A stiffness model for EHL contact on smooth/rough surfaces and its application in mesh stiffness calculation of the planetary gear set

A method for calculating time-varying mesh stiffness (TVMS) is proposed by developing a stiffness model for elastohydrodynamic lubrication (EHL) contact and integrating it into three-dimensional loaded tooth contact analysis (3D-LTCA). To enhance the accuracy and efficiency of the calculation, three projects have been implemented: Firstly, the contact deformation is modified, and the average slope is employed for calculating contact stiffness. Secondly, the lubricant film and EHL contact deformation are introduced into the deformation compatibility equations. Finally, the polynomial chaos expansion (PCE) surrogate model is employed to enhance computational efficiency. The proposed method is used to calculate TVMS of the planetary gear set, and the influence of speed, torque, and rough surface are analyzed. The results show that the previous method underestimating the TVMS of planet-ring gear pair in double-teeth meshing area under fluid lubrication, while the proposed method effectively addresses the issue. The different effects of the torque and speed on TVMS under mixed and fluid lubrication also have been revealed, emphasizing the significance of surface topography in gear meshing characteristics.

Mesh Characteristic Analysis of Spiral Bevel Gear Pairs Considering Assembly Errors and Tooth Tip Chipping Faults

Due to machining errors, location inaccuracies, human error, and various other factors, it is challenging to avoid assembly errors during the production of spiral bevel gears (SBGs). When SBG assembly errors occur, it can cause the appearance of edge contact and may even lead to severe tooth tip chipping. In this study, we propose an improved method based on loaded tooth contact analysis (LTCA) to examine mesh characteristics, including time-varying mesh stiffness (TVMS), unloaded transmission error, and contact stress. Furthermore, we explore the effects of assembly errors and tooth tip chipping. Moreover, it is observed that assembly errors can alter the contact area of SBGs and potentially reduce the peak-to-peak value of TVMS. Additionally, the occurrence of tooth tip chipping decreases TVMS within the chipping region, lowers transmission error, and increases maximum contact stress. Notably, when assembly errors are present, the reduction in TVMS due to tooth tip chipping exceeds that of a properly assembled SBG pair.

Forecasting model for hypoid gear elastohydrodynamic lubrication considering entrainment effect

In full consideration of time-varying meshing performance and oil film characteristics, entrainment effect on elastohydrodynamic lubrication (EHL) is developed for establishing accurate forecasting model of the complex hypoid gears. Firstly, dynamic loaded tooth contact analysis (DLTCA) is employed to determine the meshing interface kinematics, load distribution and entrainment speed under dynamic load condition. Then, entrainment speed is introduced into the EHL governing equation and accurate forecasting model is established. With accurate discretization of EHL governing equation, discrete convolution and fast Fourier transformation (DC-FFT) method is employed to compute the flank elastic deformation. Finally, a hypoid gear set in automotive industrial application is exercised to verify entrainment effect on EHL assessment in the proposed model.

Modeling and analysis of elastohydrodynamic lubrication of orthogonal face gear with installation errors

Installation error is one of the most prominent factors affecting the lubrication of face gear. To analyze the lubrication performance of the face gear with installation errors, the installation error model of the orthogonal face gear transmission system is established. Then, the contact path, equivalent curvature radius, load distribution and entrainment velocity are studied using the face gear loaded tooth contact analysis (LTCA). On this basis, the lubrication governing equations of the face gear are established according to the elastohydrodynamic lubrication (EHL) theory. In addition, the oil film thickness, oil film pressure and coefficient of friction in the process of face gear drives are calculated using the method of progressive mesh densification (PMD). Finally, the effects of axis intersection angle error, axis intersection error and axial displacement error on lubrication characteristics are analyzed. The results reveal that when the axial intersection angle error and axial displacement error are negative and the axial intersection error is positive, the minimum oil film thickness of the face gear tooth surface is less than that without error. When the error values are the same, the minimum oil film thickness and coefficient of friction have larger changes with the axial displacement error than with the axis intersection error, and the contact path has a bigger movement range.

Data-Driven Multi-Objective Optimization Approach to Loaded Meshing Transmission Performances for Aerospace Spiral Bevel Gears.

Loaded meshing transmission performance optimization has been an increasingly significant target for the design and manufacturing of aerospace spiral bevel gears with low noise and high strength. An innovative data-driven multi-objective optimization (MOO) method is proposed for the loaded meshing transmission performances of aerospace spiral bevel gears. Data-driven tooth surface modeling is first used to obtain a curvature analysis of loaded contact points. An innovative numerical loaded tooth contact analysis (NLTCA) is applied to develop the data-driven relationships of machine tool settings with respect to loaded meshing transmission performance evaluations. Moreover, the MOO function is solved by using an achievement function approach to accurate machine tool settings output, satisfying the prescribed requirements. Finally, numerical examples are given to verify the proposed methodology. The presented approach can serve as a powerful tool to optimize the loaded meshing transmission performances with higher computational accuracy and efficiency than the conventional methods.

Efficient semi-analytic method for single tooth contact analysis of loaded spiral bevel gears

Accurately and fast calculation of single tooth load contact state of a gear pair is the basis for accurate calculation of multi-tooth load contact analysis. Therefore, this study proposed a semi-analytical quick single tooth load tooth contact analysis (Q-SLTCA) method. First, the ease-off tooth contact analysis (ease-off TCA) method was established, and a numerically stable initial contact stress distribution and contact ellipse analytical model was derived. Then, considering the influence of edge contact, the real-time division method of gear tooth slice was derived. Based on this, the equivalent principle of contact elliptical node and the load correction strategy were innovatively proposed. Then, the calculation method of the meshing characteristics of the tooth slice was established by coupling the boundary element method (BEM) with the equivalent cylindrical Hertz contact theory. The bending deformation, tooth root stress distribution, contact stress, contact deformation and meshing stiffness of the tooth slice were analyzed and calculated, and the instantaneous meshing stiffness of the single tooth was determined by combining the parallel method of the gear tooth stiffness. Finally, a single-tooth finite element method (FEM) meshing model was created for comparison with the method in this study, which verified the calculation accuracy and efficiency of the proposed method. This method can be easily developed into a quick multi-tooth load contact analysis algorithm, which can provide a new method for gear tooth crack and failure analysis.

Numerical modeling and experimental validation of power loss and efficiency in an electric-drive axle system

Power loss and efficiency are pivotal performance metrics significantly impacting the quality of electric-drive axles. This study outlines a numerical method for determining the electric-drive axle system power loss, which includes gear meshing, gear churning, and bearing power losses. Power loss from gear meshing was calculated by analyzing load distribution and friction coefficient. The frictional loaded tooth contact analysis method was employed to ascertain the load distribution during gear meshing. A formula based on a weighting function was utilized to compute the friction coefficient. The finite element method was used to verify the gear load distribution and meshing power loss. Subsequently, empirical formulas were used to calculate the gear churning and bearing power losses. To substantiate the proposed numerical method, an experimental apparatus was employed to measure the power loss and efficiency of an electric-drive axle under various operating conditions. The experimental study demonstrated a strong correlation between the numerical and calculated results. These findings suggest that the proposed method can be an effective tool in predicting the power loss and efficiency of electric-drive axles.

Theoretical and experimental research on tooth root bending stress of face gear

A tooth root bending stress analysis method is proposed to study the tooth surface load distribution and tooth root bending stress during the meshing process of orthogonal face gear. Firstly, the tooth surface equation, transition surface equation and tooth width constraints are derived. Then, Through the tooth contact analysis (TCA), the contact path and spots of the face gear pair are obtained. What’s more, the proposed method based on Load tooth contact analysis (LTCA) is utilized to calculate the tooth surface load distribution and root bending stress of the face gear along the mesh cycle and compared with the finite element method. Meanwhile, root bending stress tests were carried out to verify the proposed method. The comparisons indicate that the proposed method has higher computational efficiency compared with the finite element method (e.g. the finite element method takes about 5 h for a mesh cycle and the proposed method costs 40 s). The results show that the tooth surface load distribution and root bending stress of the proposed method are consistent with those obtained by the finite element method and gear root stress experiment.

Research on noise reduction of drive axle hypoid gear based on tooth surface mismatch modification

In order to reduce the transmission noise of drive axle hypoid gear, a method of tooth surface mismatch modification was proposed. Firstly, on the basis of establishing the grinding mathematical model of hypoid gear with cutter tilt, the theoretical tooth surface equation was derived, and the numerical tooth surface was calculated by dividing the grid points on tooth surface. Secondly, the pinion tooth surface which was full conjugated with gear tooth surface was constructed, and the tooth surface mismatch topography was established and decomposed into the first order and second order mismatch coefficients. According to the modification requirements, the pinion target surface was constructed by modifying the tooth surface mismatch coefficients, and the pinion modified machine setting parameters were acquired by establishing the pinion processing parameters corrected mathematical model. Finally, the loaded tooth contact analysis with finite element simulation and drive axle NVH (Noise, Vibration, Harshness) simulation were carried out for a pair of hypoid gear, therefore the loaded tooth contact areas, transmission errors curves, and noise simulation curves were obtained, and the simulation results show that after modification the tooth contact stress, loaded transmission errors amplitude, and NVH simulation curves value are all reduced. The road test results indicate that the drive axle hypoid gear howling phenomenon under the high speed operation can be eliminated, so the effectiveness of tooth surface modification method and NVH simulation method were verified.

Multi-Point Control for Face-Milled Spiral Bevel Gears with a Predesigned Fourth-Order Motion Curve

This paper presents an ultimate motion methodology of a face-milling spiral bevel gear pair to synthesize the mating tooth surfaces with a predesigned fourth-order motion curve. The methodology is to control some contact points along the contact path in the process of tooth contact analysis via application of an extended local synthesis which permits some transmission errors rather than zero at the concerned contact point. The modified offset motion correction is selected to demonstrate the proposed methodology. Applied torque corresponding to an elastic approach of 0.00635 mm at the mean contact point is calculated and the loaded tooth contact analysis is performed. Numerical results show that the extended local synthesis can effectively control the transmission errors on the predesigned fourth-order motion curve at arbitrarily predesigned contact points along the contact path of the spiral bevel gear pair. The tooth contact pattern for the actual tooth pair is scattered into three segments since the rotational motion of the driven gear at any instant angular position is dependent on the tooth pair with the least transmission error among the three adjacent tooth pairs. The actual tooth contact patterns of the spiral bevel gear pair become continuous when meshing tooth surfaces are elastically deformed.

A Novel Tooth Modification Methodology for Improving the Load-Bearing Capacity of Non-Orthogonal Helical Face Gears

This study proposes a double-crown tooth surface modification technology that improves the load-carrying capacity of non-orthogonal helical tooth surface gears. The tooth modification is determined by a modified rack-cutter, and its feed motion is related to an intentionally designed transmission error. The novelty of the tooth modification design is that the transmission error can be pre-designed. First, changing the tooth profile of the tool enables preliminary modification along the tooth profile direction; second, by modifying the interaction between the tool and the machined gear, subsequent fine adjustments are made to the contact path. This two-stage tooth modification strategy not only retains the advantages of the traditional method but also significantly improves the balance of the load distribution on the tooth surface through an original contact path modification strategy. Through systematic tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA), it was verified that the new method reduces contact stress and tooth root bending stress and improves the gear’s resistance to misalignment errors. This research provides the basis and motivation for further exploring and improving this tooth profile modification technology to solve the challenges faced by more complex gear systems.

Vibration optimization of spur gear based on GSA-SA algorithm.

To determine the optimal design parameters of spur gear under a specific condition, based on the basic theories of gear dynamics theory, gear meshing principle, tooth contact analysis and load tooth contact analysis, a six-degree-of-freedom vibration analysis model of spur gear pair is established, and a gravitational search-simulated annealing hybrid algorithm (GSA-SA) is used to optimize the gear addendum modification coefficient and profile modification parameters. The vibration response of the spur gear pair is evaluated through the optimization objective function established by the combination of the G1 method and variation coefficient method. The study shows that the optimized design parameters effectively reduce the level of the vibration, which proves the effectiveness of the optimization method, and the simultaneous optimization of the addendum modification coefficient and profile modification parameters of the gear has a better result than only optimizing the addendum modification coefficient or profile modification parameters. This method can be used for gear transmission system vibration optimization design in the automotive industry and shipbuilding.

Analytical calculation of mesh stiffness for spiral bevel gears with an improved global tooth deformation model

The paper proposes an efficient and accurate analytical calculation model of time-varying mesh stiffness (TVMS) for the spiral bevel gear. Firstly, an enhanced numerical loaded tooth contact analysis (NLTCA) model that considers both the global tooth and local contact deformations is developed. Based on Tredgold's approximation, spiral bevel gears are sliced into a series of spur gears. An improved global tooth deformation model is developed by accounting for axial tooth deformation and the effect of non-contacting tooth slices. The influence coefficient method is utilized to determine the local contact deformation. Subsequently, based on the results of the NLTCA model, a detailed calculation procedure for determining the TVMS of spiral bevel gears is described. Finally, some numerical examples are provided to illustrate the efficiency and accuracy of the proposed model through comparisons to finite element analysis (FEA) results. The proposed model can achieve satisfactory accuracy without FEA modification. Furthermore, ignoring the effects of the non-contacting tooth slices will lead to excessive elastic deformation when the length of the contact area across the tooth width is significantly shorter than the tooth width.

Vibration characteristics analysis of flexible helical gear system with multi-tooth spalling fault: Simulation and experimental study

Gear systems operating at high speeds and under heavy loads are susceptible to continuous multiple tooth surface spalling. The time-varying meshing stiffness (TVMS) model of helical gear pair with multi-tooth spalling fault is established by combining image recognition method and loaded tooth contact analysis (LTCA) method to target the actual form of the spalling fault, and its effectiveness was verified through finite element method (FEM). The flexible helical gear dynamic model is established by utilizing 8-node shell model and Timoshenko beam model to simulate the gear foundations and shafts, the TVMS in multi-tooth spalling fault condition is introduced as the excitation. The mapping relationship between the meshing process, TVMS and vibration response under the influence of rotational speed, spalling surface morphology scale, spalling depth and location of spalling occurrence is analyzed by combining simulation and experimental results. As the rotational speed increases, the specific meshing frequency coincides with the natural frequency of the gear system, leading to the phenomenon of superharmonic resonance. Continuous multi-tooth spalling fault results in continuous time-domain shocks and the meshing frequency modulation of the rotational frequency of the gear with spalling fault. In the absence of bottoming out, the increase in spalling surface morphology scale exerts a more pronounced influence on the vibration characteristics of the helical gear system compared to the increase in spalling depth. The research results can provide theoretical basis for gear system fault diagnosis.

A novel design method of low noise spiral bevel gear tooth surface for polynomial ease-off topological modification

In order to reduce the vibration noise of the spiral bevel gear transmission system (SBGTS), a polynomial ease-off topological modification (PETM) method, which combines the load tooth contact analysis (LTCA) and vibration analysis of the spiral bevel gear, is proposed in this paper to minimize the normal dynamic meshing force and normal relative displacement to reduce the vibration noise. Firstly, a new PETM equation is established based on the complete conjugation principle, which is applied to the initial clearance calculation of LTCA and an explicit LTCA calculation method with PETM without requiring inverse processing parameters is constructed. The calculation accuracy is better than that of a general algorithm. Secondly, the dynamic response of an eight-degree-of-freedom dynamic system of spiral bevel gear is calculated according to the loaded transmission error and time-varying meshing stiffness calculated by the explicit LTCA of PETM. Finally, a PETM low-noise tooth surface design method is constructed with the goal of minimizing the normal meshing force and the normal relative displacement and the PETM equation coefficient as the optimization variable. The numerical results show that the new PETM low-noise tooth surface design method can improve the meshing performance and dynamic performance of spiral bevel gears to a greater extent than those before modification and after the second-order topological modification, thus greatly reducing the vibration noise of SBGTS.

Exploration of trade-offs between NVH and efficiency in bevel gear design

Minimizing NVH and friction-induced power losses is becoming paramount in the design of geared transmissions. The aim of this paper is to present an automatic methodology to explore Pareto-optimal designs of bevel gears when minimization of noise and frictional losses is essential. In the first part, a semi-empirical model to estimate frictional power losses under elasto-hydrodynamic lubrication is described. The model has been validated against experimental data available in the literature in previous works by the authors. The efficiency calculation is coupled with a state-of-the-art loaded tooth contact analysis (LTCA) tool to obtain accurate predictions of the instantaneous load shared by the mating tooth pairs during the meshing cycle. In the second part, an automatic framework based on multi-objective optimization (MOO) is presented where the tooth micro-geometry is systematically designed. The design variables are represented by few coefficients of a polynomial basis that embodies the tooth flank ease-off topography. To ensure manufacturability, the polynomial modifications are projected onto the feasible set of the machine-tool envelopes. This step is achieved through a state-of-the-art identification algorithm that the authors have developed in previous work. Frictional losses are estimated with the aforementioned model, whereas the NVH level is measured by the loaded transmission error (LTE), directly available from the simulation tool. The maximum contact pressures are limited by the material properties, thus proper nonlinear constraints are prescribed. Application to a test case involving the design of a spiral bevel gearset reveals that the methodology presented allows the designer to obtain Pareto-optimal solutions in a systematic and automatic manner.

Investigation on Tooth Surface Wear of Cycloid Drives Considering Tooth Profile Modifications

Cycloid drives are widely used in various mechanical systems due to their high reduction ratio, compact size, and high efficiency. Tooth surface wear is a major problem that affects the reliability and durability of cycloid drives. However, compared to the research on the wear of involute gears, the prediction of tooth surface wear in cycloid drives is relatively limited and less extensive. To fill this gap, the theoretical wear model of the tooth surface of cycloid-pin gear pairs is proposed based on the Hertz contact theory and Archard’s formula, with consideration of tooth profile modifications. Firstly, the loaded tooth contact analysis model is established to determine the relative sliding velocity and tooth contact stress. Secondly, the calculation steps of single tooth surface wear are presented within one gear mesh cycle. With this, the effects of the tooth profile modifications, the operating conditions such as output torque, input speed, and the assembly eccentricity on the wear depth within one gear mesh cycle are investigated. This study gives a deeper understanding of the tooth surface wear mechanisms of cycloid drives and could be employed to assist gear design and to improve the wear resistance.

Research on Active Precontrol Strategy for Shape and Performance of Helicopter Spiral Bevel Gears

Gear vibration becomes unmanageable under extreme conditions, which shortens the time between overhauls of the helicopter reducer and affects the service efficiency of helicopters. For this purpose, an active precontrol strategy for shape and performance was developed for spiral bevel gear (SBG) systems, thus prolonging the reliable service period of helicopters. Firstly, the perfect pinion surface was built based on the preset contact path (CP) and transmission error (TE). Secondly, through a combination of tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA), the loaded transmission error (LTE), meshing stiffness (MS), and meshing impact (MI) of the SBG transmission were obtained. Thirdly, an eight-degrees-of-freedom (8-DOF) dynamic model was built, and a simplified dynamic model was generated based on this. As regards the dynamic response, the root mean square of normal vibration acceleration (RMA) of the SBG transmission was obtained by solving the differential equation of motion. The RMA of the SBG transmission was also analysed based on the basic theory of vector structural mechanics. Finally, an optimization model aimed at reducing the RMA of the SBG transmission was established using ease-off, and a redesigned SBG transmission with good dynamic performance was obtained through optimization. Simulation analysis showed that the LTE and MS amplitudes of the optimized gear were smaller than those of the original gear. The impact force and impact velocity of the optimized gear were 27.02% and 25.81%, respectively, lower than those of the original gear. The RMA value of the optimized gear was much lower than that of the original gear. The normal vibration velocity and displacement of the optimized gear transmission were also cut down. Therefore, the design approach can effectively increase the performance of gear transmission.

Sideband analysis of cracked planetary gear train considering output shaft radial assembly error

In a planetary gear train (PGT), the accuracy of gear tooth crack fault monitoring is affected by many external factors, such as the radial assembly error of the output shaft. In this paper, a three-dimensional loaded tooth contact analysis (3DLTCA) model is proposed considering the variation of the center distances and tooth root crack fault of the planet gear. The time-varying meshing stiffness (TVMS) caused by the crack fault of the planet gear and the radial assembly error of the output shaft has been revealed. By introducing the TVMS as the vibration excitation source into a coupling PGT-shaft dynamic model, the dynamic response under the joint influence of assembly error and the crack fault is calculated. By comparing with the experimental results, the distribution law of sidebands caused by the radial assembly error of the output shaft and the crack fault of the planet gear is obtained. Based on the distribution law of sidebands, a signal decomposition method is proposed. Sidebands caused by the assembly error and the crack fault are separated. Through the validation of the condition indicators (CIs), it is found that the proposed signal decomposition method based on the distribution law of sidebands can improve the accuracy of diagnosing the crack fault of the planet gear under the influence of the radial assembly error of the output shaft.