Face-milled Hypoid Gears Research Articles

An exact system of generation for face-milled hypoid gears with uniform depth taper: Application to hypoid gear drives with high gear ratio

An exact system of generation for face-milled hypoid gears with uniform depth taper is proposed. This system defines the location of pinion and wheel, their respective cutting (or grinding) cutters, and the relations of motion in their respective reference coordinate systems attached to the cutting machine. As a result, the grinding cutter geometric parameters and basic machine-tool settings are analytically determined without the need of any further optimization process. The appropriate selection of the mean cutter radius is addressed following the recommendations of the Information Sheet AGMA ISO 22849-A12. The tooth contact analysis of the gear drive shows a localized bearing contact with no unloaded transmission errors in the absence of misalignments. A lower sensitivity of the contact pattern to errors of alignment is observed as the mean cutter radius is decreased. Stress analysis shows that only the application of a tip relief is needed to avoid edge contacts at the tip of the gear tooth surfaces, being the point width of the cutting blades a key parameter to balance bending stresses in the pinion and the wheel.

Computerized determination of the qualified region of main design parameters of face-milled hypoid gears with low shaft angle

The influence of main design parameters of hypoid gears with low shaft angle, namely, the offset, shaft angle, cutter radius, gear ratio, and the spiral angle of wheel, on the geometrical morphology the gear tooth surfaces has been investigated. Checks for obtaining appropriate top-land widths and spiral angles at the toe and heel sides have been developed and applied. The limits for the main design parameters that yield non-adequate tooth surfaces because- of top-land pointing, undercutting, or having a negative value of the spiral angle of wheel at the toe side have been investigated based on the results obtained by the simulation of a large number of cases of design. A specific computerized flow chart has been designed to iterate among the main design parameters to obtain a qualified region for them. With those results, a limiting graph has been proposed to get the list of recommended values of main design parameters and provide a reference for their selection. Finally, the results of loaded tooth contact analysis of two cases of design arbitrarily chosen from the proposed limiting graph of recommended values are shown to verify its applicability to yield to successful designs with favorable conditions of meshing and contact.

Multi-objective optimization of the manufacture of face-milled hypoid gears on numerical controlled machine tool

A new method is presented for advanced manufacture of hypoid gears on numerical controlled machine tool. The tool geometry and machine tool settings are determined to introduce the optimal tooth modifications into the teeth of hypoid gears. The goal is to reduce the maximum tooth contact stresses, angular displacement error of the driven gear, and energy losses in the oil film existing between tooth surfaces. The calculation is based on the optimal variation of machine tool settings on the classical machine tool for hypoid gear manufacture. The novelty of the method is that during the machining process of teeth surfaces, the variation of machine tool settings on the cradle-type hypoid generator is conducted by polynomial functions of fifth-order. By an algorithm, this variation of machine tool settings is transferred to the numerical controlled machine tool for hypoid gear manufacture (hypoid generator). The obtained results have shown that by applying the optimal manufacture process, considerable reductions in tooth contact stresses and angular displacement errors of the driven gear, and a moderate reduction in energy losses were obtained. Therefore, by applying this new method in practice, advanced manufacture of hypoid gears on CNC hypoid generator is made possible, resulting improved operating characteristics of the hypoid gear pair.

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.

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.

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.

Improvements in the mixed elastohydrodynamic lubrication and in the efficiency of hypoid gears

In this paper, 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 full numerical analysis of the 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 are 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. The effectiveness of the presented method is demonstrated by using hypoid gear examples.

Micro tooth surface topography of face-milled hypoid gears

An accurate kinematic model of the face-milling process of hypoid gears has been developed in order to compute the micro tooth surface topography, generated by successive cuts of the head-cutter cutting edges, as they come into contact with the gear blank. The instantaneous gear tooth surface point has been determined at each point on each active cutting edge and from this it has been possible to predict the tooth surface roughness experienced during the machining process. The paper presents a description of the theoretical gear cutting model and the method to calculate the generated roughness of gear tooth flank. The effectiveness of the presented method is demonstrated by using a hypoid gear example.

Pinion development of face-milled spiral bevel and hypoid gears based on contact attributes

This paper proposes a pinion development approach in order to obtain excellent transmission performance of the face-milled spiral bevel and hypoid gears. First, the flank form of the gear pair is determined based on the original designed ease-off topography by adjusting the contact attributes. After the gear pair is finished and measured, the deviation sum can be established by converting the wheel flank deviations into the equivalent pinion flank deviations. Then, the real ease-off topography is obtained, which is an important indicator to evaluate the transmission performance. Next, the real ease-off topography is redesigned with the target of the original designed ease-off topography to compensate the deviation sum by readjusting the contact attributes. Thus, the pinion flank form is redesigned and the corresponding machine-tool settings can be calculated by the active design method. According to which, the pinion is recut/reground and the wheel remains unchanged. Finally, the approach is conducted on a face-milled hypoid gear pair to demonstrate its effectiveness.

An Approach for Determination of Basic Machine-Tool Settings From Blank Data in Face-Hobbed and Face-Milled Hypoid Gears

The conditions of meshing and contact in hypoid gear drives depend substantially on the machine-tool settings to be applied. Determination of gear geometry is the first step in the design process of a hypoid gear drive. An approach for determination of basic machine-tool settings for face-hobbed and face-milled hypoid gears is proposed, covering the cases when the gear is generated and nongenerated. Gear basic machine-tool settings are determined from the blank data that can be obtained from application of Standard ANSI/AGMA 2005-C96. Some machine-tool settings are determined analytically considering the imaginary generation of the gear by a crown gear. Some other machine-tool settings are obtained numerically in order to provide some given blank data as the normal chordal tooth thickness and the normal pressure angles of the gear teeth. The developed theory is illustrated with numerical examples.

Estimation of Bending Fatigue Life of Hypoid Gears Using a Multiaxial Fatigue Criterion

In this study, a crack initiation life prediction methodology for the tooth bending fatigue of hypoid gears is proposed. This methodology employs a previously developed finite-element based hypoid gear root stress model (Hotait et al. 2011, “An Investigation of Root Stresses of Hypoid Gears with Misalignments,” ASME J. Mech. Des., 133, p. 071006) of face-milled and face-hobbed hypoid gears to establish the multiaxial stress time histories within the root fillet regions. These stress time histories are combined with a multiaxial crack initiation fatigue criterion to predict life distributions along roots of the pinion and the gear. The predictions of the multiaxial fatigue model are compared to those from a conventional uniaxial fatigue model to establish the necessity for a multiaxial approach. The model is exercised with an example face-milled hypoid gear set from an automotive application to demonstrate the impact of various misalignments well as the key cutting tool parameters on the resultant tooth bending lives.

Generation of Hypoid Gears on CNC Hypoid Generator

In this study, polynomial functions of orders up to five are applied to induce variations in the cradle radial setting and the velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surfaces corresponding to reduced transmission error amplitudes of a hypoid gear pair. The new CNC hypoid generators have made it possible to perform this nonlinear correction motions for the cutting of the face-milled hypoid gears. An algorithm is developed for the execution of motions on the CNC hypoid generator for the generation of face-milled hypoid gear tooth surface, based on the machine tool setting variation on the cradle-type hypoid generator induced by the optimal polynomial functions up to fifth-order. By using the corresponding computer program, the motion graphs of the CNC hypoid generator are determined for the generation of hypoid gear tooth surface, based on the optimal variation in the velocity ratio in the kinematic scheme and on the variation in the cradle radial setting on a cradle-type generator. The results presented indicate that the variation of the velocity ratio in the kinematic scheme of the hypoid generator induced by a fifth-order polynomial function resulted in a 62% reduction of the maximum transmission error of the gear pair.

On the estimation of continuous mappings from cradle-style to 6-axis machines for face-milled hypoid gear generation

Abstract Modern Computer Numerically Controlled (CNC) machines for grinding face-milled hypoid and spiral bevel gears have six axes and are able to perform complex motions with almost free-form ease-off topography correction capabilities. However, in the gear literature and in the common practice machine-settings, corrections are usually performed within the classical cradle-style framework. We believe that, working in the 6-axis framework, may have advantages with respect to correction performances, and numerical conditioning of the resulting optimization problem. However, since for historical and practical reasons basic kinematic settings are usually given within the cradle-style framework, an imperative intermediate step is the translation of the cradle-style kinematics into the 6-axis (minimal) layout. Therefore, in this paper the estimation of the continuous mappings from the cradle-style to the 6-axis kinematics is investigated. In particular, the performances of two methods are assessed: (i) an existing central difference method (CDM) approach, and (ii) an original Least Squares Method (LSM) approach. Numerical results show that the LSM approach presents some advantages over the CDM approach in terms of accuracy of the reconstructed surface, numerical stability, and ease of implementation.

An Ease-Off Based Method for Loaded Tooth Contact Analysis of Hypoid Gears Having Local and Global Surface Deviations

Actual hypoid gear tooth surfaces do deviate from the theoretical ones either globally due to manufacturing errors or locally due to reasons such as tooth surface wear. A practical methodology based on ease-off topography is proposed here for loaded tooth contact analysis of hypoid gears having both local and global deviations. This methodology defines the theoretical pinion and gear tooth surfaces from the machine settings and cutter parameters, and constructs the surfaces of the theoretical ease-off and roll angle to compute for the unloaded contact analysis. This theoretical ease-off topography is modified based on tooth surface deviations and is used to perform a loaded tooth contact analysis according to a semi-analytical method proposed earlier. At the end, two examples, a face-milled hypoid gear set having local deviations and a face-hobbed one having global deviations, are analyzed to demonstrate the effectiveness of the proposed methodology in quantifying the effect of such deviations on the load distribution and the loaded motion transmission error.

Robust Optimization of the Loaded Contact Pattern in Hypoid Gears With Uncertain Misalignments

This paper presents an automatic procedure to optimize the loaded tooth contact pattern of face-milled hypoid gears with misalignments varying within prescribed ranges. A two-step approach is proposed to solve the problem: in the first step, the pinion tooth microtopography is automatically modified to bring the perturbed contact patterns (as the assembly errors are varied within the tolerance limits) match a target area of the tooth while keeping them off the edges; in the second step, a subset of the machine-tool settings is identified to obtain the required topography modifications. Both steps are formulated and solved as unconstrained nonlinear optimization problems. While the general methodology is similar to the one recently proposed by the same authors for the optimization at nominal conditions, here, the robustness issues with respect to misalignment variations are considered and directly included in the optimization procedure: no a posteriori check for robustness is therefore required. Numerical tests show that nominally satisfactory and globally robust hypoid pairs can be designed by a direct process and within a unified framework, thus avoiding tiresome trial-and-error loops.

1304 創成歯切り勾配歯ハイポイドギヤの三次元歯面の平面展開法(GS-11 ベルト・歯車,研究発表講演)

1304 創成歯切り勾配歯ハイポイドギヤの三次元歯面の平面展開法(GS-11 ベルト・歯車,研究発表講演)

創成歯切り勾配歯ハイポイドギヤの運転性能解析法(S44-5 設計I,S44 伝動装置の基礎と応用)

創成歯切り勾配歯ハイポイドギヤの運転性能解析法(S44-5 設計I,S44 伝動装置の基礎と応用)

Computerized Simulation of Transmission Errors and Shift of Bearing Contact for Face-Milled Hypoid Gear Drive

A computerized investigation of the influence of alignment errors on the transmission errors and the shift of the bearing contact is performed. The investigation is performed for an imaginary hypoid gear drive with conjugate tooth surfaces. The transmission error functions caused by misalignment are shown to be periodic discontinuous almost linear functions with the frequency of the cycle of meshing. The above functions can be totally absorbed by a predesigned parabolic function. The shift of the bearing contact caused by misalignment has been determined as well. The investigation is based on computerized simulation of meshing and contact of gear tooth surfaces. The machine-tool settings for the generation of the designed gear drive have been determined. A numerical example that illustrates the developed theory is given.