Hypoid Gears Research Articles

Study on Building Digital-Twin of Face-Milled Hypoid Gear From Measured Tooth Surface Topographical Data

AbstractA highly accurate digital-twin spiral bevel gear or hypoid gear is often required for dynamic analysis or stress analysis for gear transmission. However, a highly accurate digital-twin solid model is not always available because the final hypoid gear is completed by the gear manufacturer. This study constructs a digital-twin from a sample hypoid gear. The tooth surface of sample gear is digitized as topographical grids using a coordinate measuring machine (CMM) or a gear measurement center. The geometric parameters (the surface position vector, the normal vector, the principal curvatures, and the corresponding principal directions) for the measured tooth surface (MTS) are then extracted using numerical differential geometry (NDG). The machine settings and the cutter parameters for the sample hypoid gear are obtained by minimizing the topographical error between the replicated digital-twin and the sample gear using optimization software. The initial estimation for the machine settings and the cutter parameters is calculated using an explicit form of the modified-roll motion (MRM), which decreases numerical divergence and time that is required for calculation. The machine settings, the cutting tool parameters, and the auxiliary flank modification (AFM) motion are used as the design variables. A numerical example is presented to verify the proposed methodology. The numerical results show that the replicated digital-twin that is developed using the proposed method is sufficiently accurate for industrial applications.

Computational Study on Machine Settings for Face-Milled Hypoid Gears With Low Shaft Angles

Abstract Hypoid gear with small cone angle and large pitch cone distance can be directed at the transmission with low shaft angle (LSA). The manufacturing process has more freedoms of motion to control the tooth surface and ensure higher mesh performance. However, it is difficult to adjust the machine settings due to the extreme geometry. This paper focused on the manufacturing process and machine settings calculation of hypoid gear with low shaft angle (LSA hypoid gear). Based on the generating process, nongenerated gear, and generated pinion manufactured by circular cutter blade, the mathematic model of tooth surface of LSA hypoid gear was developed, and the expressions of principal directions and curvatures of LSA hypoid gear were derived. The relationship of curvatures between pinion and gear was also proposed. Then based on the basic relationships of two mating surfaces, an approach to determinate machine settings for LSA hypoid gear was proposed. Finally, the tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) were directed at the validation of machine settings’ derivation. TCA contact pattern results highly coincide with the preset values. And the LTCA contact pattern also highly coincides with TCA results, it can be considered that the determination approach of machine settings is valid. The TCA transmission error result also shows that the ratio of contact is quite large, which is a little bigger than 2. Thus, the load bearing ability and stability of LSA hypoid gear may be superior.

Vibration and Dynamic Analysis of Right-Angle Geared Drives

<div class="section abstract"><div class="htmlview paragraph">The dynamics of hypoid or spiral bevel gears like most high-speed precision gears employed in the powertrains of automobiles, commercial trucks, and off highway vehicles are significantly influenced by the design of the shafts. The finite element modeling approach is one of the useful methodologies applied to perform gear dynamic analysis. One of the major advantages of the finite element modeling approach is that it can account for the gear-shaft-bearing assembly design more accurately than other modeling approaches, for example, the lumped parameter modeling approach. In this paper, the finite element formulation, which can generally represent more complete characteristics of the gear-shaft-bearing assembly design, is employed to investigate how the temperature change, shaft material type and shaft size influence the dynamics of spiral bevel gear system.</div></div>

Collaborative machine tool settings compensation considering both tooth flank geometrical and physical performances for spiral bevel and hypoid gears

With the increasing demands for low noise and high strength in spiral bevel and hypoid gears transmission, tooth flank optimization considering both geometric and physical performances has been an significant stage in recent design and manufacture. In this paper, a novel collaborative machine tool settings compensation considering both the tooth flank geometric and physical performances is proposed. Firstly, the mathematical tooth flank model is established by simulating the actual generation process. Then, in the tooth flank collaborative optimization design, the geometric constraint on the tooth flank error is developed, as well as the physical constraint on tooth contact performance. Finally, a novel machine settings modification model considering both tooth flank geometric and physical performances is established and solved for the accurate machine settings having the compensation variations. Moreover, the optimal collaborative compensation strategy is provided by using the sensitivity analysis for tooth flank performances. A numerical instance is given to verify the proposed collaborative machine tool settings compensation.

Investigation on contact and bending stress of face-hobbed and face-milled hypoid gear

The tooth strength performance of face-hobbed and face-milled hypoid gears under the same work condition is a controversial research hotspot. In this paper, the load capacity of face-hobbed and face-milled hypoid gears has been studied based on finite element analysis method and conventional international standards. The stress comparison for contact and bending stress between two types of gears have been conducted. The accurate mathematical model of face-hobbed and face-milled hypoid gear were established based on the generating and forming process. The statistical method was used to assess the contact and bending stress. Results show that the values of bending stress of face-milled gear are about 20–30% larger than those of face-hobbed gear. For the contact area with high stress (800–1300 MPa), the proportion of face-milled hypoid gear is much more than that of face-hobbed gear. The standard deviation values of contact point stress for face-milled hypoid gear are 7.1% higher than those of face-hobbed gear. The strength results by standards and finite element analysis results can verify with each other consistently. Both results contribute that face-hobbed hypoid gear has better tooth strength under the same condition.

Parametric relationship between hypoid gear teeth and accurate face-milling cutter

The cutter systems of hypoid gear cutting machines contain groups of inside and outside blades. In these cutter systems, the side cutting edges of the blades machine the convex and concave gear teeth while rotating about the cutter rotation axis. The side cutting edges lay on the rake face formed through the blade, rake, and relief angles; hence, the normal cross-section of the cutter swept surface forms hyperboloid gear teeth. Using the accurate geometry of the cutter system, a relationship between the pressure and spiral angles of the gear tooth and the parameters of the cutter system is developed for the FORMAT machining of a hypoid gear. A new parameterization of the gear tooth surfaces is introduced to determine these angles for the accurate gear tooth by the accurate cutter system. A numerical example with different cutter systems and blade parameters is presented, demonstrating the effects of rake and relief angles over the pressure and spiral angles on mean point projections and gear tooth surface. Finally, the change in pressure and spiral angles with respect to the rake and relief angles are plotted, and the results are analyzed. Finally, it is concluded that the pressure and spiral angles are changed up to a few seconds of a degree in the operating area of the tooth with the change in the back and side rake angles. The side relief angle exhibited little or no effect over the geometry of the gear tooth.

Effect of Shaft Misalignment on Hypoid Gear Pair Driven Through a Universal Joint

An enhanced hypoid geared rotor system model incorporating a universal joint that connects driveline propeller shaft and pinion shaft is introduced in this study. The effect of shaft misalignment and pinion mass unbalance on hypoid gear dynamic response is evaluated under different assembly conditions. A nonlinear 14-DOF lumped parameter model based on coupled multi-body dynamics is applied by considering large rotational displacement in torsional direction. Results demonstrate three main influences from shaft misalignment and mass unbalance including speed and torque fluctuation as well as external bending moment on pinion shaft. It is also observed that bending moment generated can exert additional load on pinion shaft bearings which is found to be more sensitive to misalignment angle compared to dynamic response along line-of-action. Besides, shaft misalignment and mass unbalance excitation will interact with gear internal excitation from transmission error (TE) which can affect mesh parameters.

A global synthesis approach for optimizing the meshing performance of hypoid gears based on a swarm intelligence algorithm

To control the meshing performance of hypoid gear drive, current local conjugate theory and local synthesis method establish a relationship between the relative motion and local geometrical properties of the mating surfaces at one reference point. Theoretically, both local conjugate theory and local synthesis method cannot ensure the contact performance on the entire tooth surface resulting in uncontrolled contact pattern and undesirable transmission error. A global synthesis approach consisting of a machine-tool settings calculation model, a tooth contact analysis model, a meshing quality assessment function, and a swarm intelligence algorithm are proposed. The direction, area, and shape of the contact pattern and the amplitude of the intersection of two adjacent transmission error curves are taken as the evaluation indexes. Three curvatures of the design pitch cone of the pinion are taken as the control variables. A global selection space is then established within the reasonable range of the curvatures. An improved multiple population genetic algorithm is employed to find the optimal set of the curvatures to achieve the target values of the evaluation indexes. To avoid the edge contact and corner contact condition, a feasible region is defined on tooth surface and the contact patterns on the gear and pinion surfaces are all confined within the area. The optimized contact patterns obtained by the loaded tooth contact analysis method are similar to those obtained by the proposed approach, thus demonstrating the effectiveness of this methodology. The main novelty of this approach is to translate the derivation of the mathematical relation between the curvatures of the mating surfaces and the contact properties to solving a multi-objective optimization problem of the meshing quality indexes by the intelligence algorithm.

Investigation of meshing phenomenon of hypoid gears with different offsets based on thermal image using high speed video infrared thermography

Investigation of meshing phenomenon of hypoid gears with different offsets based on thermal image using high speed video infrared thermography

Investigation of the effects with linear, circular and polynomial blades on contact characteristics for face-hobbed hypoid gears

On this research, the effects of the three different types of blade sections including linear, circular and polynomial on mesh characteristics for face-hobbed hypoid gears are investigated. Firstly, with the equations of three blade sections and the transformation matrixes based on the processing method derived, the mathematical model of gear flank and loaded finite element mesh model are proposed. Then, the impacts of different blade sections on tooth surface deviation and mesh characteristics are analyzed. Results show that the effects are almost identical when the parameters choose reasonable for the three blades. However, with the circular radius reducing, the peak-to-peak value of transmission error decreases but the contact pattern area shrinks and the location of maximum bending stress will change. For the polynomial blade, the effects the design parameters on mesh characteristics are obvious.

Multi-objective optimization of hypoid gears to improve operating characteristics

In this paper a multi-objective optimization method of hypoid gears correlating to the operating characteristics is presented. Optimal design of hypoid gears demands that multiple objectives be simultaneously achieved. Four objectives considered in this study are the minimization of the maximum tooth contact pressure, transmission error and the average temperature in the gear mesh, and the maximization of the mechanical efficiency of the gear pair. The goals of the optimization are achieved by the optimal modification of meshing teeth surfaces. In practice, these modifications are introduced by applying the appropriate machine tool setting for the manufacture of the pinion and the gear, and/or by using a tool with an optimized profile. The proposed optimization procedure relies heavily on the loaded tooth contact analysis for the prediction of tooth contact pressure distribution and transmission errors, and on the mixed elastohydrodynamic analysis of lubrication to determine temperature and efficiency. A fast elitist nondominated sorting genetic algorithm (NSGA-II) is applied to solve the model. The effectiveness of the method is demonstrated by using hypoid gear examples. The obtained results have shown that by the optimization considerable improvements in the operating characteristics of the gear pair are achieved.

Contact fatigue life prediction of a bevel gear under spectrum loading

Rolling contact fatigue (RCF) issues, such as pitting, might occur on bevel gears because load fluctuation induces considerable subsurface stress amplitudes. Such issues can dramatically affect the service life of associated machines. An accurate geometry model of a hypoid gear utilized in the main reducer of a heavy-duty vehicle is developed in this study with the commercial gear design software MASTA. Multiaxial stress—strain states are simulated with the finite element method, and the RCF life is predicted using the Brown–Miller–Morrow fatigue criterion. The patterns of fatigue life on the tooth surface are simulated under various loading levels, and the RCF S–N curve is numerically generated. Moreover, a typical torque—time history on the driven axle is described, followed by the construction of program load spectrum with the rain flow method and the Goodman mean stress equation. The effects of various fatigue damage accumulation rules on fatigue life are compared and discussed in detail. Predicted results reveal that the Miner linear rule provides the most optimistic result among the three selected rules, and the Manson bilinear rule produces the most conservative result.

Improving Hypoid Gear Oil Pitting Performance through Friction Reduction

Earlier studies have shown that the load-dependent friction behavior of various gear oils can affect their pitting performance; that is, low friction resulted in a long pitting life. These studies were limited, however, to test methods and running conditions quite different from those occurring in actual gear transmissions. In the present study, a more gear-like twin-disc machine with test specimens and running conditions relevant for gear contacts was used to investigate whether the same trends could be found. To analyze this possible correlation, the first step was to prepare a set of hypoid gear oils and to test their friction performance to compare various ways of improving friction behavior but also to form an understanding of why their friction performance varied. The second step was to test the pitting performance of the oils. The pitting results could then be compared to the friction properties of the oils to analyze the correlation. Other possible mechanisms behind the formation of pits are also discussed. The results show that for the oils included, the antiwear and extreme-pressure additive package and the base oil type affect friction. The results further show that additive combinations and/or base oils that result in low friction lead to enhanced pitting performance.

A Mathematical Model for Grinding a Stick Blade Profile to Cut Hypoid Gears

Abstract Cutter head with multiple stick blades is widely used in the mass production of hypoid and spiral bevel gears because they allow more blades per revolution of the head cutter. However, the stick blade geometry for a head cutter that is used in face-milling and face-hobbing methods to produce hypoid and spiral bevel gears is complicated and difficult to describe. The geometry of a stick blade is defined in terms of cutting parameters such as the rake angle, the hook angle, and the side relief angle that are required to perform cutting and the theoretical cutter profile in the offset plane or the neutral cutter profile in the normal plane of an imaginary generating crown gear. This study uses a 5-axis profile grinder to grind the stick blade. The machine settings for the profile grinder and the corresponding grinding wheel geometry are derived for the grinding of each face of the stick blade. A sensitivity analysis for the machine settings is conducted to determine the accuracy of the profile grinder. The numerical example shows that the proposed mathematical model is sufficiently accurate for industrial applications.

Improved mixed elastohydrodynamic lubrication of hypoid gears by the optimization of manufacture parameters

Extensive wear appears in the case of dry contacts, or when the lubrication of the contacting surfaces is not appropriate. The aim of this research is to improve the mixed elastohydrodynamic lubrication in hypoid gears by the optimization of manufacture parameters for tooth surface processing.A full numerical analysis of the thermal mixed EHL in hypoid gears is applied. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed, and boundary lubrication. In the hydrodynamically lubricated areas the calculation method employed is based on the simultaneous solution of the Reynolds, elasticity, energy, and Laplace's equations. In the asperity contact areas the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. The real geometry and kinematics of the gear pair based on the manufacturing procedure is applied, thus the exact geometrical separation of the mating tooth surfaces is included in the oil film shape, and the real velocities of these surfaces are used in the Reynolds and energy equations. The transient nature of gear tooth mesh is included. The oil viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included. The non-Newtonian behaviour of the lubricant is considered. Using this model, the pressures, film thickness, temperatures, and power losses in the mixed lubrication regime are predicted. By using the developed method, the influence of the manufacturing parameters on the conditions of mixed elastohydrodynamic lubrication is investigated. On the basis of the obtained results recommendations are formulated to improve the mixed EHL and the efficiency of face-milled hypoid gears.

A CNC tooth grinding method for formate face hobbed hypoid gears

Face hobbing is an efficient process for hypoid gear cutting. However, the continuous indexing process renders it inapplicable for tooth grinding, which limits the further improvement of gear accuracy and transmission performance. In this paper, a tooth grinding method using large diameter conical grinding wheel for Formate® face hobbed hypoid gears (FFHHG) was proposed. The generator of conical grinding wheel was used to replace the cutter blade of the face hobbing process. By controlling the conical grinding wheel, the generator of the grinding wheel can be tangentially in contact with the tooth profile curve of the FFHHG. The sweeping of the generator of the grinding wheel along the extended epicycloidal tooth curve can grind one tooth surface accurately without theoretical deviation. Firstly, the tooth surface was formulated and the geometric parameters of the working side of the grinding wheel were determined to avoid the curvature interference between the grinding wheel and the ground tooth surface. Then, a five-axis tooth grinding tool location model was established to position the grinding wheel and the tooth surface correctly. The interference of the non-working side of grinding wheel was checked and if necessary the grinding wheel axis can be tilted to avoid interference. Finally, the above processes were verified by case analysis and grinding simulations. This study can be employed to improve the pitch accuracy of FFHHG and reduce the tooth profile deviation.

Investigation of Tool Errors and Their Influences on Tooth Surface Topography for Face-Hobbed Hypoid Gears

Abstract Tool errors are inevitable in an actual gear-manufacturing environment and may directly affect the accuracy of machined tooth surfaces. In this paper, tool errors including spheric radius, pressure angle, rake angle, regrind angle, and cutting side relief angle errors for three-face blade are defined and considered to establish the accurate tooth surface mathematical model for face-hobbed hypoid gears based on the manufacturing process and the meshing theory. The simulation flowchart for tooth surface modeling and tooth surface topography deviation analysis are proposed and performed. Results show that the tooth surface deviation is positive with positive spheric radius and rake angle errors and contrary results can be found for other three tool errors. In addition, the impact of the pressure angle error is the strongest. In addition, the rake angle error has the weakest effect and the influence of spheric radius error on the tooth surface deviation is unsubstantial. For location of tooth surface deviation, the maximum deviation is at the top on the heel and the minimum deviation is at the middle on the toe for spheric radius error. The maximum and minimum deviations are at the top and the middle tooth on the heel for other factors, respectively.

Modeling of Impact Damping Effect on Hypoid Gear Dynamic Response

Abstract An enhanced hypoid gear mesh model that incorporates the Hertzian impact damping function is developed in this study to reveal the mechanisms of damping during gear mesh. Two types of impact damping models, namely, viscous and nonviscous type based on the first principle of mechanics, are compared to previous empirical damping models with a constant damping coefficient. Parametric studies are performed for both steady-state and transient analyses to investigate the impact damping effect under numerous load conditions or physical parameters of a meshing gear pair. Comparative study between viscous and nonviscous damping models is also performed to examine their effects on dynamic response for various load levels. It is shown that impact damping can significantly reduce the amplitude of the dynamic mesh force. Nonviscous damping has more significant effect on the dynamic response under heavy torque load due to the influence of greater elastic deformation. Furthermore, it is observed that impact damping can turn double-sided impact into single-sided impact and suppress response peaks in a certain mesh frequency range during speed ramp up.

Hypoid gear integrated wear model and experimental verification design and test

Hypoid gear abrasive wear significantly influences transmission error which may worsen gearing performance. Wear and transmission error are best studied jointly. This paper reports on a proposed integrated hypoid gear wear model which combines Archard's wear equation with a conjugate approach, tooth contact analysis. Transmission error calculation is included in this integrated wear model. The model is also tested and verified by an experimental design in which transmission error measures wear rate. This allows hypoid gear wear prediction simulation to be controlled by transmission error. It can be examined, calibrated, and synchronized with solid hypoid gear wear progress. Identical no-wear and used some-wear vehicle hypoid gears with different wear times are applied experimentally. Statistical analysis standard deviations and geometric means are used to quantify transmission error magnitude and fluctuation. Magnitude controls wear prediction geometry updates. Fluctuation examines integrated wear model accuracy by comparing transmission errors. The test suggests that the accuracy of this integrated hypoid gear wear model is acceptable as there is minimal difference between predicted gear wear and actual gear wear.

Torsional Vibration and Bifurcation Characteristics of Automotive Main Reducer Based on Nonlinear Hybrid Model

In the view of integrating dynamics of systems with dynamics of structures, the pinion shaft of automobile main reducer is divided into several torsion rod elements to establish its dynamic model, while the torsional vibration dynamic model of hypoid gears is still built with lumped mass method, thus the hybrid dynamic model of automotive main reducer is obtained by the combination of two kinds of dynamic models. Considering the nonlinear factors such as time-varying mesh stiffness, backlash and static transmission error of gear pairs, Runge-Kutta method is applied to obtain the time domain spectrum, spectrum graph, phase-plane diagram, Poincaré maps, and bifurcation diagrams of the system. The dynamic response characteristics of system with different parameters such as input rotational speed and mesh stiffness are systematically investigated by numerical simulations.