Involute Pinion Research Articles

Face gear drives: Nominal contact stress calculation for flank load carrying capacity evaluation

Face gear drives are angular gear units in which an involute pinion meshes with a face gear wheel. Spur and helical face gear drives are possible with or without center offset. Face gear drives can be manufactured on conventional gear cutting machines. Furthermore, the pinion is free to move axially and such displacements have no effect on the contact pattern. This makes them a promising gearing for various applications. Due to the lack of established calculation standards, the design of face gear drives is a challenge and often results in oversizing or early failures. This may also be why face gear drives have not yet established themselves in power transmissions. Instead, the main application can be found in kinematic/auxiliary drives. In this paper, two methods for contact stress calculation for face gear drives with any configuration between helix angle and center offset are developed. Calculation examples compare the two methods and confirm the applicability. Overall, this study provides a new standard-capable easy-to-program calculation method for the contact stress based on existing standards for cylindrical gears.

Face gear drive manufacturing: Pinion/cutter design for a crowned contact pattern

Face gear drives are special angular gears in which an involute pinion and face gear wheel mesh. They enable high transmission ratios in small construction spaces. The face gear is typically manufactured by gear hobbing or gear shaping using an involute pinion-type cutter. Face gear drives with identical pinion and cutter parameters form a conjugated pairing. Conjugate pairings tend to have unfavourable edge contact, which is usually counteracted by geometric flank modifications during manufacture. A common modification technique is the crowning of the flanks. This usually requires changes to the machine kinematics. Due to their geometry and manufacturing process, face gear drives offer an easy way to achieve a crowned contact pattern in the facewidth direction without changing the machine kinematics, making them promising for multiple applications. Using a pinion with fewer teeth than the cutter results in a crowned contact pattern (lead crowning). The position of the contact point along the facewidth can be adjusted by varying the profile shift coefficient of the pinion or the cutter. This paper presents a straightforward method for selecting proper design parameters depending on the contact distances. The contact distances can be estimated analytically to evaluate the chosen design parameters. Example calculations are given as proof of the developed method presented in this paper. Overall, this study offers a novel evaluation strategy for an ideal cutter/pinion design for manufacturing durable face gear drives with optimised contact patterns.

A study on design of tooth plane for Face Gear

The purpose is to simplify the tooth flank design of the face gear and the setting of optimum specifications. In this case, the target was the face gear that meshes with the helical involute pinion gear in offset position. The curved surfaces of the tooth of the face gear can be generated by the action of constant velocity meshing with the tooth surfaces of the pinion gear. The meshing between the pinion and the face gear proceeds at a constant speed along the straight line of action. And the line of action of the straight line is continuous and exists as a curved surface. By quantifying and comparing the size of the effective action curved surface, it can be used as an index for determining the parameters of the specifications of the face gear.

A grinding method of face gear mating with a conical spur involute pinion

A grinding method of face gear manufactured by a disk wheel is proposed in this paper. The computerized generation of the face gear by application of a conical spur involute shaper is presented. The tooth surfaces of the face gear are ground point by point with the disk wheel. In addition, a numerical model for determination of envelope residuals of the ground face gear surfaces is established. This model is used to determine the grinding tracks of the disk wheel according to the prescribed processing precision. The manufacture of four different cases of face gears is simulated by applying a simulation platform to illustrate the effectiveness of the proposed grinding method. The results show that deviations of excess greater than the prescribed processing precision only exist on the fillet surfaces and the root of the ground tooth, and increase from the fillet surfaces to the mean section of the root of the face gear. However, this has no influence on the meshing of the face gear drive. The processing efficiency of the face gear grinding is sufficiently affected by the prescribed processing precision.

Computerized generation and meshing simulation of a backlash-adjustable face-gear drive with a tapered involute pinion

This paper focuses on the face-gear drive with a tapered involute pinion. Two types of pinion geometries are derived: 1) the profile crowned pinion implemented by a crowned rack-cutter, and 2) the double-crowned pinion by a generating worm. The tooth surfaces and the limiting conditions of the pinion and the face-gear are obtained based on the meshing and geometry theories. The tooth contact analysis (TCA) and stress analysis are performed to evaluate the performance of the presented face-gear drive. The results indicate that the backlash adjusting of the face-gear drive can be achieved by axially moving the pinion. The application of double-crowned pinion modifies the contact patterns on the gear surfaces, providing lower contact and bending stresses compared with the face-gear drive using a profile crowned pinion. Both the profile crowned and double-crowned pinions can relieve the edge contact effects. Numerical examples are given to illustrate the results.

The mathematical model and a case study of the cosine face gear drive that has a predesigned fourth order function of transmission errors and a localized bearing contact

Traditionally, a face gear drive uses an involute pinion. If the number of teeth of pinion is smaller than the minimum number of teeth without undercutting, the pinion will be undercut. To avoid undercutting, it is necessary to quit using involute pinion. This paper proposes a new face gear drive called cosine face gear drive. The cosine face gear drive has a predesigned fourth order function of transmission errors and a localized bearing contact. Therefore, it has a smoother motion curve and no edge contact problem. According to coordinate transformation theory, the theory of gearing, and differential geometry, the mathematical model of the cosine face gear drive is created, including position vectors, unit normal vectors, principal curvatures, principal directions, the condition of contact, and the condition for producing the predesigned fourth order function of transmission errors. A numerical case is studied to verify the mathematical model. The geometric solid model of the case is created. The actual form of the function of transmission error is analyzed. Contact paths and contact ellipses are analyzed. The effects of assembly errors on contact paths and on the function of transmission errors are analyzed. Contact and fillet stresses are analyzed using finite element analysis.

A Comparative Study between a Pair of Cosine Gears and a Pair of Involute Gears: Contact Stress, Root Fillet Stress, and Contact Ratio

Contact stresses, root fillet stresses, and contact ratios of a pair of cosine gears and a pair of involute gears are analyzed and compared in this paper. The mathematical models of pinions and gears are created based on the coordinate transformation theory and the theory of gearing. Static structural finite element analysis is performed by using ANSYS. The assumption of contact of one tooth is not adopted in order to comply with the real situation of meshing. Numerical results have shown that maximum contact stress and maximum root fillet stress of cosine pinion are smaller than those of involute pinion, maximum contact stress of cosine gear is approximately equal to that of involute gear, and maximum fillet stress of cosine gear is smaller than that of involute gear. The pair of cosine gears has a higher strength but a lower contact ratio in comparison with the pair of involute gears. Maximum contact and root fillet stresses can be reduced largely when the number of teeth in contact is increased from 1 to 2.

Loaded multi-tooth contact analysis and calculation for contact stress of face-gear drive with spur involute pinion

The analytical method based on “Hertz theory on normal contact of elastic solids” and the numerical method based on finite element method (FEM) calculating the contact stress of face-gear drive with spur involute pinion were introduced, and their relative errors are below 10%, except edge contact, which turns out that these two methods can compute contact stress of face-gear drive correctly and effectively. An agreement of the localized bearing contact stress is gotten for these two methods, making sure that the calculation results of FEM are reliable. The loaded meshing simulations of multi-tooth FEM model were developed, and the determination of the transmission error and the maximal load distribution factor of face-gear drive under torques were given. A formula for the maximal load distribution factor was proposed. By introducing the maximal load distribution factor in multi-tooth contact zone, a method for calculating the maximal contact stress in multi-tooth contact can be given. Compared to FEM, the results of these formulae are proved to be reliable, and the relative errors are below 10%.

The Correction of Face Gear Tooth Flank Deviation

A orthogonal-shaft face gear drive with spur involute pinion is considered. The mathematical model of the face gear tooth flank was studied and the correction parameters was taken into account. The measurement grid schema was accomplished on the rotary projection of theoretical tooth flank, with the real tooth flank data from three-dimensional measuring machine or gear measuring center, the tooth flank deviation could be calculated. The effect of correction parameters on tooth flank deviation was investigated and the correction method for the single index generating grinding was proposed. Finally a numerical example was presented and the results show that the method proposed can decrease the tooth flank deviation.

Effect of Tooth Surface Modification on the Load Sharing and Strength of Offset Face Gear Drive with Spur Involute Pinion

Load sharing is one of the main factors that determine gear strength. In this paper, Tooth Contact Analysis (TCA) and Loaded Tooth Contact Analysis (LTCA) have been performed to investigate the effect of tooth surface modification on the contact ratio, load sharing and strength of an orthogonal offset face gear drive with spur involute pinion. The results indicate that the contact ratio of 2.0 or higher could be achieved. The maximum load carried by single tooth and bending stress are significantly reduced when appropriate tooth surface modification is applied to the orthogonal offset face gear drive.

Diagonal shaving of an involute pinion: optimization of the geometric and kinematic parameters for the pinion finishing operation

Investigation of diagonal shaving of an involute pinion and optimization of parameters for the pinion finishing operation and of the shaving cutter design are the objectives of the current research. The paper covers the following topics: (1) analysis of the local topology of the pinion tooth surface and shaving cutter; (2) exploration of the parameters of geometry of contact between the involute tooth surfaces of the pinion and the shaving cutter; (3) determination of the optimum parameters of the shaving cutter design and of the kinematics of the diagonal shaving operation. An approach for the computation of the optimum parameters for the diagonal shaving operation of an involute pinion is developed. The approach is based on fundamental results in the theory of gearing obtained by Buckingham (Analytical mechanics of gears, Dover Publications, New York). The developed approach enables the computation of the optimum parameters for the gear finishing operation. Its application can be easily extended to and employed for the optimization of parameters for other methods of rotary shaving operations of involute and non-involute pinions, i.e., axial (transverse or conventional) shaving, tangential (or underpass) shaving, and plunge shaving operations. Computer simulation of the diagonal shaving operation proves that application of the developed approach enables shaving time reduction or reduction of the actual pinion tooth surface deviation from the true involute tooth surface.

Computation of Parameters of a Form Grinding Wheel for Grinding of Shaving Cutter for Plunge Shaving of Topologically Modified Involute Pinion

Abstract The paper is targeting on the finishing of precision gears for low-noise/noiseless transmission for cars and light trucks. Transmission error is the predominant cause of gear noise. The application of a topologically modified pinion results in reduction of transmission error up to two times. The required modification of the pinion tooth surface is provided on a plunge shaving operation with application of a shaving cutter of an appropriate design. A novel approach for computation of parameters of a form grinding wheel for grinding of the shaving cutter for plunge shaving of a precision involute pinion with topologically modified tooth surface is reported in the paper. The developed approach for computation of parameters of the form grinding wheel is focused on application of the shaving cutter grinder with a lack of CNC articulation. The problem under consideration is solved using the DG/K-based approach of part surface machining earlier developed by the author. (The DG/K-approach is based on fundamental results obtained in differential geometry of surfaces, and in kinematics of multi-parametric motion of a rigid body in E3 space (See Radzevich, S.P., Sculptured Surface Machining on Multi-Axis CNC Machine. Monograph, 1991, Vishcha Shkola Publishers, Kiev (in Russian). See also Radzevich, S.P., 2001, Fundamentals of Surface Machining. Monograph, Rastan, Kiev (in Russian).) An analytical solution to the problem is discussed in the paper. The solution has been used for developing software for the Mitsubishi ZA30CNC shaving cutter grinder for the needs of the automotive industry. Computer simulation reveals high accuracy of the ground shaving cutter.

Design of Shaving Cutter for Plunge Shaving a Topologically Modified Involute Pinion

The purpose of this paper is to solve the problem of designing a shaving cutter for plunge shaving a topologically modified involute pinion. Application of topologically modified pinion allows noise reduction. Due to the desired topologically modified pinion tooth surface is determined not analytically, and discretely by a grid of points, the problem under consideration cannot be solved by direct application of methods developed in theory of enveloping surfaces. A modified kinematical approach is developed to establish one-to-one correspondence between points that determine tooth surface of a pinion to be machined, and points that determine tooth surface of a shaving cutter to be applied. The developed approach is based on ideas, which could be traced back to publications by E. Buckingham. The following three considerations are essential: a) tooth surface of an initial nonmodified pinion, b) desired deviations of the topologically modified pinion tooth surface relatively to the nonmodified pinion tooth surface, and c) relative motion that shaving cutter performs in gear finishing operation. Consequently, the algorithm, and software for computing coordinates of points, which determine the modified gear-shaving cutter tooth surface is developed.

Face-gear drive with spur involute pinion: geometry, generation by a worm, stress analysis

A face-gear drive with a spur involute pinion is considered. The generation of the face gear is based on application of a grinding or cutting worm whereas the conventional method of generation is based on application of an involute shaper. The authors have developed an analytical approach for determination of: (i) the worm thread surface, (ii) avoidance of singularities of the worm thread surface, (iii) dressing of the worm, and (iv) determination of stresses of the face-gear drive. A computer program is developed for simulation of meshing and contact of the pinion and face gear. Correction of machine-tool settings is proposed for reduction of the shift of the bearing contact caused by misalignment. An automatic development of the model of five contacting teeth has been proposed for stress analysis. Numerical examples for illustration of the developed theory are provided.

Stress analysis of a helical gear set with localized bearing contact

This study investigates the contact stress and bending stress of a helical gear set with localized bearing contact, by means of finite element analysis (FEA). The proposed helical gear set comprises an involute pinion and a double crowned gear. Mathematical models of the complete tooth geometry of the pinion and the gear have been derived based on the theory of gearing. Accordingly, a mesh-generation program was also developed for finite element stress analysis. The gear stress distribution is investigated using the commercial FEA package, ABAQUS/Standard. Furthermore, several examples are presented to demonstrate the influences of the gear's design parameters and the contact positions on the stress distribution.

Bearing contact of a helical gear pair with involute teeth pinion and modified circular-arc teeth gear

AbstractThis paper investigates the contact ellipse of a helical gear set, composed of an involute pinion and a modified circular-arc gear based on curvature analysis. This gear drive exhibits point contact and parabolic transmission error due to the double crowning effect of the gear, i.e. crowning in both profile and longitudinal directions. The principal directions and curvatures of the generating tool surfaces were derived by means of differential geometry and Rodrigues’ equation. The principal directions and curvatures of the pinion and gear surfaces were obtained directly from the generating surfaces. Finally, the determination of the contact ellipses of the mating tooth surfaces was achieved. Numerical examples are also provided to demonstrate the computational results.

Bearing contact of a helical gear pair with involute teeth pinion and modified circular-arc teeth gear

This paper investigates the contact ellipse of a helical gear set, composed of an involute pinion and a modified circular-arc gear based on curvature analysis. This gear drive exhibits point contact and parabolic transmission error due to the double crowning effect of the gear, i.e. crowning in both profile and longitudinal directions. The principal directions and curvatures of the generating tool surfaces were derived by means of differential geometry and Rodrigues’ equation. The principal directions and curvatures of the pinion and gear surfaces were obtained directly from the generating surfaces. Finally, the determination of the contact ellipses of the mating tooth surfaces was achieved. Numerical examples are also provided to demonstrate the computational results.

Study on Finishing of Pinion Cutters by Form Grinding. 2nd Report. Finishing of Tooth Flanks by Multiple Motions.

This report describes a new method to finish tooth flanks of pinion cutters by form grinding with multiple relieving motions, which are the approaching motion between the grinding wheel and the pinion cutter, the rotational motion of the pinion cutter and the shifting motion of the grinding wheel. The method can successfully finish the whole depth of the cutting edge of the pinion cutter with an arbitrary profile by one-pass grinding for each tooth flank. The motions were decided as to minimize the cutting edge errors at two regrinding positions. As an example of profile calculation and the finishing test, the involute helical pinion cutter with a protuberance was selected. The profile of grinding wheel was calculated by applying the element removal method. The finishing test was conducted on the NC form gear-grinding machine and the pinion cutter with almost no profile error and a small regrinding error was obtained.

Contact Ratios and Transmission Errors of a Helical Gear Set with Involute-Teeth Pinion and Modified-Circular-Arc-Teeth Gear.

Contact ratio (CR) and transmission error (TE) are two significant indices for gear tooth strength and dynamics. This work investigated the CR, contact teeth (CT) and TE of a helical gear pair, composed of an involute pinion and a modified circular-arc gear. Point contact and built-in parabolic TE are obtained due to the modification of the gear’s tooth profile. Tooth contact analysis(TCA) is applied to determine the TE as well as the CR of the proposed helical gear set under various assembly condition. The effect of gear design parameters on the CRs and TEs are also investigated in numerical examples.

Study on Finishing of Pinion Cutters by Form Grinding. 1st Report. Finishing of Tooth Flanks by Excessive Helical Motions.

A new form relief grinding method for an involute pinion cutter with almost no regrinding error is proposed. In the method, the tooth flanks of pinion cutter are finished as those of involute helical gears with excessive(lager and smaller)helix angles than that of the pinion cutter. The obtained results are as follows : (1)In the conventional generating method, it is impossible to finish an involute pinion cutter with no profile error. (2)The profiles of grinding wheel to finish a pinion cutter under excessive helical motions are calculated. (3)Profile errors of a pinion cutter being finished by the proposed method are smaller than those finished by the conventional method. (4)The finishing test of pinion cutter(mn=2.5mm, N=39, β=15°)was conducted on the NC form grinding machine and the pinion cutter with the accurate cutting edges was obtained.