Ease-off Topography Research Articles

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.

On the modelling of gear alignment errors in the tooth contact analysis of spiral bevel gears

A penetration-based gear contact model for accurate and numerically efficient tooth contact analysis of spiral bevel gears is used to investigate the simulation of gear alignment errors. The developed gear contact model with surfaces of roll angles, computed for the gear pair’s tooth flanks in the absence of misalignments, is shown to be well capable of predicting a misaligned gear pair’s contact performance. In this approach the contact curves on the three-dimensional tooth flanks are efficiently computed from the surfaces of roll angles, while the flank mismatch is directly determined by the instantaneous position and orientation of the gear tooth surfaces without the need for an ease-off topography. The latter provides a higher robustness to configuration variations. Finally, a novel strategy to parametrically redefine the gear contact element’s surfaces of roll angles in function of a gear pair’s instantaneous misaligned state, is developed to further increase the accuracy of the contact algorithm.

On the determination of fully conjugate hypoid tooth flanks

Presented is a procedure to individually specify fully conjugate hypoid gear tooth flanks. Cartesian coordinates for the hypoid tooth flanks are derived in terms of a planar rack with a specified constant pressure angle. These coordinates define a “coordinate curve” on the tooth flank. Each coordinate curve depends on an axial coordinate and the entire tooth flank is established as a family of these coordinate curves. This process results in global interference between locally conjugate hypoid tooth flanks if the pressure angle is constant. A differential equation is established for the axial variation in the rack’s pressure angle to avoid global interference. The resulting axial variation in normal pressure angle is combined with the determination of coordinate curves to obtain fully conjugate tooth flanks. An automotive hypoid gear pair is presented to showcase the process. Unloaded Ease-off (UEO) topography is used to quantify conjugate action for the example hypoid gear pair. Subsequently, axial and transverse profile modifications are applied individually to the ideal tooth flanks to accommodate generalized loading conditions. Loaded Transmission Error (TE) and maximum contact pressure are reported for a set of input torques.

An Active Ease-Off Topography Modification Approach for Hypoid Pinions Based on a Modified Error Sensitivity Analysis Method

A new active ease-off topography modification approach is proposed to improve the meshing performance of hypoid gears based on a fourth-order predesigned transmission error (PTE) model and a modified error sensitivity analysis method. Ease-off topography modifications that describe local deviations of pinion tooth surfaces can be conducted by converting the fourth-order PTE into equivalent deviations of pinion tooth surfaces. The modified error sensitivity analysis method is developed to investigate the effects of misalignments on the moving velocity of a contact point of a hypoid gear pair. The moving velocity of the contact point can describe transmission error (TE) curve shapes of ease-off tooth surfaces. The ease-off topography modification approach can achieve TE precontrol and modification curvature adjustment of the pinion for stable meshing performance of the hypoid gear pair. Moreover, pinion ease-off tooth surfaces are finished by a five-axis computer numerical control swarf-cutting machine tool. Swarf-cutting tests and TE measurement tests are conducted on hypoid gear pair specimens to demonstrate the feasibility and effectiveness of the proposed methodology.

On the third law of gearing: A study on hypoid gear tooth contact

Presented is a procedure to determine the extreme relative curvatures between conjugate gear flanks. This procedure is a modification of the third law of gearing which establishes the extreme relative curvatures between conjugate tooth flanks in terms of normal pressure angle, spiral angle, and hyperboloidal pitch surfaces. The original formulation for the third law of gearing was based on parabolic contact along a line of contact for the limiting condition of relative curvature between two gear flanks. Penetration or existence of negative Gaussian curvature for the relative flank curvature can occur for skew axis gearing with “conjugate” teeth. In such cases, reciprocity between the tooth normal and the twist defined by the gear rotation axes and instantaneous gear ratio is a necessary condition but not a sufficient condition to ensure suitable gear flanks in skew axis gearing. For cases with penetration, a transverse profile amendment is introduced to achieve parabolic contact (zero Gaussian curvature) for the relative curvature between meshing gear tooth flanks. First and second order reciprocity are applied in the transverse direction to determine profile amendment. Ease-off topography and unloaded transmission error (UTE) are used to assess conjugate action between mating gear flanks in direct contact.

A novel collaborative manufacturing model requiring both geometric and physical evaluations of spiral bevel gears by design for six sigma

Collaborative manufacturing requiring both geometric and physical evaluations has been an increasingly important trend in the industrial applications of spiral bevel gears. It is an innovative attempt that design for six sigma (DFSS) is introduced to establish a model of above collaborative manufacturing system. In the DFSS-based modeling, in addition to establishing the basic framework, the related data-driven decision and optimization are proposed, respectively. The house of quality (HOQ) is firstly established to convert the voice of customer (VOC) into the critical to qualities (CTQs) and determine the geometric and physical evaluations. Then, the conventional machine setting modification is extended to the multi-objective optimization (MOO) modification considering both geometric and physical evaluations. The whole system is divided into the three data-driven sub-problems for their respective decision and optimization. First, the accurate measurement and compensation is used to operate the noise factors in the actual manufacturing. Second, a prescribed ease-off topography is identified as the target flank by MOO of the physical evaluations. An improved high-order modification of the optimal machine setting is performed to optimize the geometric performance. A case of study on measurement-modification-machining (3 M) collaborative manufacturing system is provided to verify the proposed model.

A methodology for simulation-based, multiobjective gear design optimization

Design optimization of geared transmissions has become more of a necessity than ever before. Typically, conflicting design goals must be concurrently achieved. The difficulty of such a multiobjective design optimization problem is exacerbated by the fact that modern design practices rely on increasingly sophisticated, computationally-expensive simulation tools for tooth contact analysis. Their intrinsic nonlinearities add complexity to the problem, hampering gear designers’ efforts to obtain globally optimal solutions. Practical optimization problems of this class have often been solved by evolutionary algorithms, but their computational burden may well be inappropriate for CPU-intensive simulation models. The present work details an algorithmic framework inspired by deterministic multiobjective optimization methods, specially combined with a direct-search global optimization algorithm to obtain globally Pareto-optimal solutions. Nonlinear constraints are enforced through an exact penalty formulation. A comprehensive description of all theoretical and algorithmic details is provided, with the intention of enabling gear designers to implement or adapt the proposed methodology to their design optimization purposes. Two tests on a challenging gear design problem, namely ease-off topography optimization of a hypoid gear set for maximum efficiency and minimum contact stress, demonstrate that the proposed method can efficiently obtain solutions belonging to the global Pareto front.Declarations of interest: none.

A data-driven optimization model to collaborative manufacturing system considering geometric and physical performances for hypoid gear product

Abstract A data-driven optimization model to collaborative manufacturing system considering geometric and physical performances is proposed to improve competitiveness of hypoid gear product development facing with economic globalization. Firstly, to deal with the vagueness or impreciseness of the voice of customer (VOC), the fuzzy quality function deployment (fuzzy-QFD) using fuzzy weighted average method in the fuzzy expected value operator is introduced into hypoid gear manufacturing. It can convert them into the critical to qualities (CTQs), and the technical geometric and physical performance requirements. And then, the multi-objective optimization (MOO) modification of machine settings is proposed to establish a basic data-driven model for collaborative system. Different with the conventional modification only considering geometric performance, it provides an improved modification also considering the physical performances. Finally, a double-curved shell model of hypoid gear finite element is used to perform the numerical loaded tooth contact analysis (NLTCA), as well as to establish the data-driven relationships between machine settings and physical performance evaluations. Immediately, whole development is divided into three sub-problems: i) optimal operations of the noise factors by measurement and numerical control (NC) compensation, ii) identification of the prescribed ease-off topography by multi-objective optimization using iterative reference point approach and iii) modification considering geometric performance by a trust region algorithm with step strategy. The numerical instance in practical applications is given to verify the proposed methodology.

A method for grinding face gear of double crowned tooth geometry on a multi-axis CNC machine

The stability requirement of bearing contact and a favorable function of transmission errors motivates the investigation of design and manufacture technology of the face-gear drives. Application of a grinding disc wheel with axial profile modification and a longitudinal ease-off topography design of face gear tooth flanks could be proper utilized to obtain a double crowned tooth geometry, which results in the pre-designed bearing contact and desired transmission errors. With the profile crowned tooth geometry of the gear generated by the modified wheel cutter, finding of the execution motions to generate the tooth geometry with the longitudinal ease-off topography design on a multi-axis Computer Numerical Control (CNC) machine with the same tool is a major issue. Therefore, a universal method on the Design Optimization Techniques is proposed to obtain the execution motions for such problem. The proposed optimization methodology is also applicable for the tooth flank error correction by a worm hob or a grinding cutter on the CNC generator. With the term of additional movement of execution motion is defined, a detailed description of additional movements for the ease-offs of the real crowned tooth geometry is also investigated. The feasibility of proposed method is validated by numerical examples.

A function-oriented active form-grinding method for cylindrical gears based on error sensitivity

This paper proposes a function-oriented form-grinding approach to obtain excellent and stable contact performance of cylindrical gears by designing modification forms based on a predesigned controllable fourth-order transmission error (TE) function and error sensitivity evaluation. First of all, a predesigned fourth-order TE polynomial function is assigned to the gear drive. Mathematical models of modified tooth surfaces that can describe their local deviation and ease-off topography are then obtained with the predesigned fourth-order TE function. The corresponding error sensitivity analysis is applied for investigation that reflects inherent relationships between contact attributes of modified tooth surfaces and misalignments. Moreover, the form-grinding wheel’s profile equation, the coordinate transformation matrix during form-grinding, and settings of computer numerical control (CNC) form-grinding programs for this active design method can be determined. This approach is ultimately conducted on three involute cylindrical gear pairs to demonstrate its feasibility and effectiveness.

Design of double-crowned tooth geometry for spiroid gear produced by precision casting process

A novel double-crowned tooth geometry is proposed by the application of ease-off topography for spiroid gear manufactured by precision casting process, with the goals of localizing the bearing contact and obtaining a perfect function of transmission errors. The modified tooth surface is applied as the reference geometry to machine the die cavity geometry that will produce such geometry of the gears. The tooth geometry of crowned gear was achieved first from a pre-designed controllable function of transmission errors along the desired contact path. Then, the desired ease-off topography along the contact line is designed and calculated computationally from the given mathematic model of surface modification. The geometry of double-crowned spiroid gear could be reconstructed by superimposing the ease off of contact line direction on the profile-crowned tooth surface. The article provides numerical examples to validate the feasibility of ease-off modification methodology that was used to produce the double-crowned tooth geometry for the gears, while tooth contact analysis is performed computationally to investigate the stability of bearing contact and function of transmission errors to alignment.

Deproximating Tredgold's Approximation

Presented is Tredgold's Approximation for using an equivalent cylindrical gear with spur teeth to “approximate” a bevel gear with straight teeth. This relation is extended to spiral bevel and hypoid gears by utilizing pitch surface curvature in the direction perpendicular to the gear tooth spiral to establish an equivalent gear. Subsequently, the envelope of a planar gear tooth profile in this perpendicular direction is presented. The envelope of the gear tooth profile is used to determine fully conjugate gear teeth profiles for spatial gear elements. This procedure is valid for any tooth profile along with circular and non-circular gears. To validate the methodology, a virtual model of a bevel gear pair (“presented model”) is created and an unloaded tooth contact analysis is performed. The procedure used to perform the unloaded tooth contact analysis and determine the corresponding unloaded transmission error is based on the concept of ease-off topography. An example of a face-milled bevel gear pair (“literature model”) serves as a reference of correctness in determining the ease-off for unloaded contact. To conclude the paper, the obtained ease-off topography and unloaded transmission error of the presented model are calculated and displayed, demonstrating the specification of fully conjugate teeth.

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.

Multi-Objective Ease-Off Optimization of Hypoid Gears for Their Efficiency, Noise, and Durability Performances

Microgeometry optimization has become an important phase of gear design that can remarkably enhance gear performance. For spiral bevel and hypoid gears, microgeometry is typically represented by ease-off topography. The optimal ease-off shape can be defined as the outcome of a process where generally conflicting objective functions are simultaneously minimized (or maximized), in the presence of constraints. This matter naturally lends itself to be framed as a multi-objective optimization problem. This paper proposes a general algorithmic framework for ease-off multi-objective optimization, with special attention given to computational efficiency. Its implementation is fully detailed. A simulation model for loaded tooth contact analysis is assumed to be available. The proposed method is demonstrated on a face-hobbed hypoid gear set. Three objectives are defined: maximization of gear mesh mechanical efficiency, minimization of loaded transmission error, minimization of maximum contact pressure. Bound constraints on the design variables are imposed, as well as a nonlinear constraint aimed at keeping the loaded contact pattern inside a predefined allowable contact region. The results show that the proposed method can obtain optimal ease-off topographies that significantly improve the basic design performances. It is also evident that the method is general enough to handle geometry optimization of any gear type.

Optimization of Face Cone Element for Spiral Bevel and Hypoid Gears

In the blank design of spiral bevel and hypoid gears, the face cone is defined as an imaginary cone tangent to the tops of the teeth. Traditionally, the face cone element or generatrix is a straight line. On the other hand, the tooth root lines, which are traced by the blade tips, are normally not straight lines. As a result, the tooth top geometry generally does not fit the mating member’s real root shape, providing an uneven tooth root-tip clearance; additionally, in some cases root-tip interference between the tooth tip and the root tooth surfaces of the mating gear members may be observed. To address this issue, this paper describes a method of determining an optimized face cone element for spiral bevel and hypoid gears. The method is based on the incorporation of calculation of tooth surface and root geometries, the conjugate relationship of the mating gear members, the ease-off topography, and the tooth contact analysis. The resulting face cone element may not be a straight line but generally an optimized curve that, in addition, to avoidance of the interference, offers maximized contact ratio and even tooth root-tip clearance. Manufacturing of bevel gear blanks with a curved face cone element can be implemented by using computer numerically controlled machines.

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.

A novel ease-off flank modification methodology for spiral bevel and hypoid gears

Gear flank modification is an important technique for reducing gear running noise and avoiding edge contact for spiral bevel and hypoid gears. This work proposes a novel ease-off flank modification methodology for spiral bevel and hypoid gears made by a modern Cartesian-type hypoid gear generator. The first step is to obtain the desired ease-off topography by calculating and then synthesizing the ease-off along the contact path from the predesigned transmission error and along the contact line from the predesigned bearing ratio. The second step establishes the ease-off sensitivity matrix that corresponds to the pinion head cutter and the six-axis motion parameters of a hypoid generator based on a mathematical model of a modern Cartesian hypoid generator. Finally, linear regression identifies the corrective machine settings for modifying the pinion tooth flanks so that they approximate the desired ease-off topography. We demonstrate the validity of this flank modification method using a numerical example of the Gleason Triac face-hobbed hypoid gear made by a Cartesian-type CNC machine. This proposed flank modification methodology can serve as a basis for developing a general technique of flank modification for other gear types.

An Ease-Off Based Optimization of the Loaded Transmission Error of Hypoid Gears

Loaded transmission error (LTE) is one of the primary sources of gear noise and vibration. While ease-off topography has been shown to be powerful in improving the contact properties of a gear drive, its optimization to minimize LTEs has been an open problem in the gear literature. Through the formulation of an appropriate nonlinear optimization problem, this study proposes a novel methodology to systematically define optimal ease-off topography to simultaneously minimize LTEs and contact pressures, while concurrently confining the loaded contact pattern within a prescribed allowable region on the tooth surface to avoid any edge- or corner-contact condition. Effectiveness of this optimization is presented using a face-milled and a face-hobbed hypoid gear examples. These example analyses reveal particularly promising results that feature both a drastic reduction in LTE and an appreciable decrease in the maximum contact stress. Although the method is employed here for hypoid gears, its intrinsically systematic formulation enables straightforward applicability to any kind of gears. The methodology presented in this work can be a useful aid for gear engineers to determine optimal ease-off topographies without having to rely on time-consuming trial-and-error approaches or on a priori subjective judgments.

A load distribution model for hypoid gears using ease-off topography and shell theory

A computationally efficient load distribution model is proposed for both face-milled and face-hobbed hypoid gears produced by Formate and generate processes. Tooth surfaces are defined directly from the cutter parameters and machine settings. A novel methodology based on the ease-off topography is used to determine the unloaded contact patterns. The proposed ease-off methodology finds the instantaneous contact curve through a surface of roll angles, allowing an accurate unloaded tooth contact analysis in a robust and accurate manner. Rayleigh-Ritz based shell models of teeth of the gear and pinion are developed to define the tooth compliances due to bending and shear effects efficiently in a semi-analytical manner. Base rotation and contact deformation effects are also included in the compliance formulations. With this, loaded contact patterns and transmission error of both face-milled and face-hobbed spiral bevel and hypoid gears are computed by enforcing the compatibility and equilibrium conditions of the gear mesh. The proposed model requires significantly less computational effort than finite elements (FE) based models, making its use possible for extensive parameter sensitivity and design optimization studies. At the end, comparisons to the predictions of a FE hypoid gear contact model are also provided to demonstrate the accuracy of the model under various load and misalignment conditions.