Face-milled Spiral Bevel Gears Research Articles

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.

Contact pattern expansion regulation model for spiral bevel gear transmission

Focusing on digital assessment and optimization for time-varying contact pattern, an innovative contact pattern expansion (CPE) regulation model is established for spiral bevel gear transmission. By building a parametric optimization design bridge between loaded contact pattern and geometric boundary, the proposed method can get an accurate and effective forecasting, control and decision-marking of geometric topography and contact performances. Firstly, in order to reflect the influence of misalignment on CPE, an improved TCA solution method is developed for loaded contact pattern determination. Considering influence of tooth surface geometric boundary, accurate parametric CPE definition and assessment are proposed. Then, CPE regulation model is established by providing tooth flank deflection method. Where, by optimizing the regulation amounts of each assessment items, it can get an accurate and effective CPE regulation. In order to establish the parametric relations between regulation targets and deflection surface, an innovative TCA-oriented prescribed surface expression is developed. Levenberg-Marquardt method is used to solve the established model for the machining setting parameters. Finally, the proposed model is exercised with an example face-milled spiral bevel gear set from an aerospace engine application to demonstrate the impact of the key machine settings on the result CPE regulation.

Reverse engineering of spiral bevel gear drives reconstructed from point clouds

The replacement of spiral bevel gears represents a challenge when the design data is missing and only the physical gears are available. In this case, application of reverse engineering is required to determine the design data and machine-tool settings that would allow the default gears to be replaced and/or improved. This paper proposes a methodology of reverse engineering of spiral bevel gears reconstructed from point clouds. Point clouds of the gear tooth surfaces are obtained from non-contact metrology machines using a 3D laser scanner. Non-uniform rational B-splines surfaces are obtained to fit a set of predefined points of the point cloud. Once the reconstructed tooth surfaces are available, a bound-constrained optimization algorithm is used to derive the finishing machine-tool settings of the gears to produce the closer tooth surfaces to the reconstructed ones. The resulting reconstructed gear drive can be analyzed throughout tooth contact and finite element analyses to evaluate its mechanical performance and improve it if necessary. A numerical example, comprising a given design of a face-milled spiral bevel gear drive and the corresponding manufactured gears, shows the advantages and disadvantages of the proposed procedure to capture the designer’s intent and analyze the existing gear drive.

A Novel Taper Design Method for Face-Milled Spiral Bevel and Hypoid Gears by Completing Process Method

A Novel Taper Design Method for Face-Milled Spiral Bevel and Hypoid Gears by Completing Process Method

Active design and manufacture of face-milled spiral bevel gear by completing process method with fixed workpiece axis based on a free-form machine tool

In the design and manufacture of face-milled spiral bevel gear, the completing process method has significant advantages, such as increased efficiency, cost reduction, machining accuracy improvement, and tooth strength enhancement. Since the intrinsic limitations of the cradle-type machine tool, we develop a mathematical model for the completing process method directly based on the free-form machine tool. It is applied directly to the free-form machine tool, eliminating the need for the equivalent conversion of machine settings from the cradle-type machine tool to the free-form machine tool, and taking advantage of the flexibility and freedom offered by the free-form machine tool. Thus, except the first-order and second-order parameters of the mean contact point of each pinion flank controlled by the existing methods, each point of the contact path on the frequently used drive-side flank of pinion can be guaranteed additionally simultaneously by this method. Furthermore, the workpiece axis linkage is dispensed with during the machining process, avoiding the error caused by poor motion accuracy of the workpiece shaft and the difficulty in its manufacture. The target tooth surface and tooth depth can be guaranteed simultaneously. Finally, the method is verified by tooth contact analysis (TCA), grinding test, and rolling test.

A semi-analytical model for cutting force prediction in face-milling of spiral bevel gears

In the machining process, prediction of the cutting force is the key to cutting simulation and optimization. Specially, for the face milling of the spiral bevel gear, the un-deformed chip varies during the process due to the complicated generating kinematics, bringing great challenge to the prediction of cutting force.Regarding this, with a thorough investigation carried out, it is found that the un-deformed chips can be summarized as six cases and described by quadratic polygon interpolation. On such basis, a highly efficient and accurate analytical cutting force prediction model for generated face-milling spiral bevel gear is proposed. Compared to the Boolean operation based method, the proposed method is faster; compared to the chip-simplify method, the method is more precise; and compared to the semi-analytical method, the method is more effective as the chip forming rule is fully considered, which in turn simplifies the operation.Further, the proposed method is validated through both FEM simulation and cutting force experiment with a tolerance within 10%.

Adaptive data-driven collaborative optimization of both geometric and loaded contact mechanical performances of non-orthogonal duplex helical face-milling spiral bevel and hypoid gears

Considering the duplex helical face-milling characteristics, an innovative adaptive data-driven collaborative optimization of both tooth flank geometric accuracy and loaded contact mechanical performance evaluations is developed for non-orthogonal spiral bevel and hypoid gears. Firstly, an advanced duplex helical face-milling is simulated for tooth flank modeling and an improved tooth contact analysis (TCA) is proposed for the sensitive assembly problem. Numerical loaded tooth contact analysis (NLTCA) is used to determine data-driven relations between the loaded contact mechanical performance evaluations and assembly error. Then, a new adaptive data-driven collaborative optimization model is established by modifying assembly error evaluations. In addition to tooth flank geometric accuracy, the loaded contact mechanical evaluations including loaded contact pattern, loaded transmission error, loaded contact pressure and loaded contact stress are used as main targets. Here, to get high accuracy and efficiency, the decision-making of collaborative optimization is divided into two sub-systems: i) Loaded contact mechanical performance multi-objective optimization (MOO) for target flank determination. Here, an achievement function approach is used to get Pareto optimal solution. ii) Tooth flank geometry optimization by assembly error modification. Where, sensitivity analysis strategy is applied to select the optimal design variables. The given numerical instance can verify the proposed method.

Tooth surface modification method of face-milling spiral bevel gears with high contact ratio based on cutter blade profile correction

To avoid tooth edge contact of high contact ratio spiral bevel gear under misalignment or heavy load, a tooth surface modification method based on cutter blade profile correction is proposed. Generally, an arc blade is used to substitute the straight blade of cutter-head to get the ideal tooth surface. Firstly of all, the mathematical model of the cutter blade profile is established. Secondly, the computational method of the unknown parameters used to determine the cutter blade profile is presented in details. Thirdly, the modified tooth surfaces of pinion and gear are deduced based on the principle of gear mesh and the theory of differential geometry. Finally, a numerical example is carried out to verify the effectiveness of the tooth surface modification method with the method of finite element analysis (FEA) and tooth contact analysis (TCA). The TCA result shows that the tooth surface modification method can effectively reduce tooth edge contact without affecting the location of contact path. It can also ensure the symmetry of geometric transmission error curve. The results of the FEA show that the tooth surface modification method can not only effectively avoid tooth edge contact but can also reduce the maximum contact stress on the tooth surface. It can also reduce the adverse effects of misalignment.

Curvature synthesis for face-milling spiral bevel gears with high-order transmission errors

A modification methodology based on curvature synthesis for face-milling spiral bevel gears is proposed in this paper. This scheme includes the calculation of machine-tool settings and determination of the instantaneous modified roll ratio based on different predesigned high-order transmission error (TE) curves. Firstly, the principal curvatures and directions of ease-off pinion surface at mean contact point are carried out based on the fully conjugate pinion surface. The machine-tool settings of the pinion can be determined to meet with the meshing conditions between the cradle and the pinion. Then, compared with the parabolic TE, the amplitudes of loaded TE under a series of pinion torque for fourth-order and sixth-order TE are decreased based on the results of loaded tooth contact analysis. Finally, an example is provided to verify the proposed method. The parabolic, fourth-order and sixth-order transmission error curves are used to illustrate how to obtain the modified roll curve.

The fundamental roughness model for face-milling spiral bevel gears considering run-outs

Surface roughness is important for the machined product, especially the gear. With its influence on wear, friction and lubrication properties in driving, researches have been made on surface roughness in different machining processes.However, due to the complicated kinematics for tooth generation and tooth contact correctness, there still lacks the roughness model for face-milling of spiral bevel gears, hindering the prediction of the driving performance.For a highly accurate and efficient calculation, through fully investigating the generating process, tooth surface is found to be formed by the intersection of cutter edges in specific cutter cycles with a specific order. On such basis, the paper proposes an algorithm, which takes the insert run-out errors into consideration, for the roughness of the face-milling spiral bevel gear. Further, through both numerical study and experiment, the influence of the design and process parameters on roughness distribution is revealed.

An ease-off based approach to designing a high-order transmission motion for face-milled spiral bevel gears

An ease-off based approach to designing a high-order transmission motion for face-milled spiral bevel gears

An ease-off based approach to designing a high-order transmission motion for face-milled spiral bevel gears

This work proposes an approach for designing a fourth-order transmission motion based on ease-off modification for spiral bevel gears. Four control parameters derived from three feature points on a predetermined fourth-order motion graph are proposed to obtain an ease-off modification. This approach differs from other methods in that the polynomial coefficients of the transmission motion are not directly used to obtain the ease-off. Instead, parameters of the predetermined transmission motion with actual physical meanings are used. A spiral bevel gear pair is designed with a fourth-order transmission motion to show the effectiveness. The simulated loaded contact pattern (LCP) shows good agreement with the loaded tooth contact experiment. The results show that the loaded transmission error (LTE) can be reduced by 29.6%, the maximum contact pressure can be reduced by 10.6%, and the maximum root bending stress can be reduced by 17.4% and 23.8% for the pinion and wheel, respectively.

An analytical model for accurate and numerically efficient tooth contact analysis under load, applied to face-milled spiral bevel gears

An analytical model that allows for a computationally efficient analysis of face-milled spiral bevel gears, is presented. The methodology builds on the consideration that the mating tooth flanks are designed to transmit motion in a nearly conjugate manner. A multibody approach to tooth contact analysis is proposed that assumes contact between rigid surfaces. By taking advantage of the action surfaces for each flank pair, contact is detected in a computationally efficient and accurate way. An analytical load distribution model is used to translate the detected penetration into resulting contact forces, under the assumption that the flank penetration matches the deformation of the teeth if they were flexible. To account for the global tooth deformation Tredgold’s approximation in combination with a set of expressions based on beam theory are utilized, while the local contact deformation is modeled based on Hertz theory. The methodology is validated against nonlinear finite element simulations. A comparison in terms of transmission error, contact patterns and contact pressure is provided. Contrary to FE simulations the proposed methodology requires significantly less computational effort, allowing further extension to optimization or system analysis problems.

An innovative determination approach to tooth compliance for spiral bevel and hypoid gears by using double-curved shell model and Rayleigh–Ritz approach

In high-performance sophisticated flank design for the spiral bevel and hypoid gears, tooth compliance is always an important input for identifying the tooth mechanical properties. Distinguished with the conventional modeling, a new accurate finite element structure model is established to determine tooth compliance by using elasticity-based deformation solution. In full consideration of the flank flexural behavior characteristic, a double-curved shell model with varying thickness is established to get a more accurate representation of geometric shape for face-milling or face-hobbing gear than a beam or plate model. In determination of tooth compliance, the high order shear theory of Bhimaraddi shell is used to set the displacement assumption, and the Rayleigh–Ritz approach having algebraic polynomial trail functions is performed to obtain an expression for the transverse deflection and shear rotation by considering the geometric boundary conditions. A given numerical result for the face-milling spiral bevel and hypoid gear is provided to verify the proposed methodology by comparing with the loaded tooth contact analysis (LTCA) using well-known finite element method (FEM).

A novel method for gear tooth contact analysis and experimental validation

This paper proposes a new method for gear tooth contact analysis, which concentrates on solving two considerable disadvantages of the generalized algorithm: 1) numerical instability due to adopting a system of five nonlinear equations to search for each contact point; 2) complexity of the computation process caused by differences in the mathematical models for the tooth surface contact and edge contact. To address the first issue, the instantaneous conjugate contact curve, and its modified curve, were defined and utilized to effectively separate the transmission error; thus, the contact point and instantaneous contact curve can be effortlessly located and the number of required nonlinear equations is reduced from 5 to 2. The second disadvantage was addressed using a universal mathematical model for both the tooth surface contact and edge contact. Finally, the proposed algorithm was used to simulate the meshing of a face-milled spiral bevel gear set and similar results to using the Gleason software were attained. Real contact patterns were verified with the gear-rolling test, and comparing these results to our simulations, the effectiveness of the proposed methodology was demonstrated.

Computational approach to design face-milled spiral bevel gear drives with favorable conditions of meshing and contact

The micro-geometry of the tooth surfaces of spiral bevel and hypoid pinions has to be fine-adjusted to obtain enhanced meshing and contact characteristics during the meshing process with their corresponding mating gears. In this paper, a new methodology is proposed to design face-milled spiral bevel gear drives to, firstly, derive favorable orientation and dimensions of the contact pattern between the mating surfaces of the gears and, secondly, obtain a predesigned parabolic function of negative transmission errors with limited magnitude of maximum transmission errors. The proposed approach is based on the definition of the desired topography for the active surfaces of the pinion followed by a numerical derivation of their finishing machine-tool settings through a bound-constrained optimization algorithm. Increasing mechanical strength and reducing the levels of noise and vibration of face-milled spiral bevel gear drives constitute the main objectives of the proposed design process. A numerical example is provided to illustrate the applicability of the developed theory .

Numerical approach for determination of rough-cutting machine-tool settings for fixed-setting face-milled spiral bevel gears

Spiral bevel gear drives manufactured by the so-called five-cut method are designed to achieve the best conditions of meshing and contact since convex and concave pinion tooth surfaces are generated independently, allowing the achievement of optimized contact patterns and favorable functions of transmission errors for both flanks of gear tooth surfaces, and reducing with it noise and vibration and increasing the endurance of the gear drive. As part of the process of manufacturing, roughing of both flanks of the pinion tooth surfaces is needed. In this paper, a novel numerical approach for determination of machine-tool settings for roughing of the pinion by using a spread-blade face-milling cutter is proposed. The main purpose of the proposed procedure is the reduction of the manufacturing cycle time and maximizing the material cut during the rough-cutting operation, without damaging the to-be-finished surfaces. The procedure is based on the resolution of a constrained optimization problem. A numerical example shows the applicability of the developed numerical approach. The obtained numerical results disclose that acceptable rough-cut geometries are possible to be managed during the first iterations as trade-off solutions.

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.

Computerised Design and Simulation of Meshing and Contact of Formate Hypoid Gears Generated with a Duplex Helical Method

The duplex helical method has higher machining efficiency for face-milling spiral bevel and hypoid gears. An accurate and practical approach for calculating the basic machine-tool settings of spiral bevel and hypoid gears manufactured with the duplex helical method is proposed in the present work. The gear tooth surface vector functions and curvature parameters based on basic machine, and head-cutter settings are calculated. Two types of curvature parameters for pinion tooth surfaces are obtained by utilizing the conjugation of gear and pinion tooth surfaces, and the conjugation of pinion tooth surfaces and pinion head-cutter surfaces at the reference point. Next, the basic machine settings for generating pinions in accordance with the two types of curvature parameters for pinion tooth surfaces are determined. Finally, a numerical examples using the duplex helical method are performed and validated by comparing with experimental results. New hypoid gear software has been developed using the new approach.

A new geometric meshing theory for a closed-form vector representation of the face-milled generated gear tooth surface and its curvature analysis

The tooth surface and its curvature are fundamental inputs to evaluate the contact and transmission of spiral bevel gears or hypoid gears. Currently, the tooth surface of the face-milled generated gear is represented as an implicit form of three parameters, and one of which can be eliminated according to the well-known equation of meshing. Unfortunately, it is not easy to solve the equation of meshing due to the complex process of gear generation. Moreover, it is complicated to calculate the derivatives of the equation of meshing, and this makes it inefficient to use the results of well-established differential geometry to conduct curvature analysis. To address this problem, a new geometric meshing theory is proposed to obtain a closed-form (explicit) vector representation of the tooth surface, and curvature analysis is directly implemented with differential geometry equations. As a consequence, the calculation is straightforward and efficient. The example of a face-milled spiral bevel gear is presented as the application of the proposed method.