Chung-Biau Tsay

Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears Manufactured With Shaper Cutters

Pitch curves of a conjugate noncircular gear pair are derived based on kinematic considerations. A method for considering the inverse mechanism relationship and the equation of meshing, is proposed here to derive a complete mathematical model of noncircular gears manufactured with involute-shaped shaper-cutters. The proposed method is similar to the contact line method and the envelope method, but is easier to apply to the determination of tooth profiles. A computer program is developed for generation the tooth profile with backlashes. Undercutting analysis is also investigated by considering the relative velocity and equation of meshing. Finally, modified elliptical gears are presented to illustrate the tooth profile generation when the proposed mathematical model is applied, and to investigate the phenomenon of tooth undercutting.

Mathematical model and undercutting analysis of elliptical gears generated by rack cutters

The theory of gearing and geometry of the straight-sided rack cutter are employed to develop a mathematical model of elliptical gears that takes backlash into consideration. The mathematical model also includes the bottom land of the elliptical gear, which is equidistant from the pitch ellipse. A computer program is developed to generate the tooth profile of elliptical gears. The profile of elliptical gears obtained by applying the proposed method is compared with that obtained by the evolute method. Undercutting of elliptical gears is also investigated. The results of this research should be helpful in the design, manufacture, and measurement of elliptical gears. Copyright 1996 Elsevier Science Ltd

Computer-aided manufacturing of spiral bevel and hypoid gears with minimum surface-deviation

The linear regression method to minimize the deviations of a real cut gear-tooth-surface is investigated in this paper. Based on the Gleason hypoid gear generator, a mathematical model of the tooth surface is proposed. Applying the proposed mathematical model, the sensitivities of tooth surface due to the variations of machine-tool setting are also investigated. The corrective machine-tool settings, calculated by using the sensitivity matrix and the linear regression method, are used to minimize the tooth-surface deviations. The minimization problem was solved by using the singular value decomposition (SVD) method. The result of this paper can improve the conventional development process and be also applied to different manufacturing machines and methods. Two examples are presented to demonstrate the proposed methodology.

Computer-aided manufacturing of spiral bevel and hypoid gears by applying optimization techniques

Abstract A mathematical model of an ideal spiral bevel and hypoid gear-tooth surfaces based on the Gleason hypoid gear generator mechanism is proposed. Using the proposed mathematical model, the tooth surface sensitivity matrix to the variations in machine–tool settings is investigated. Surface deviations of a real cut pinion and gear with respect to the theoretical tooth surfaces are also investigated. An optimization procedure for finding corrective machine–tool settings is then proposed to minimize surface deviations of real cut pinion and gear-tooth surfaces. The results reveal that surface deviations of real cut gear-tooth surfaces with respect to the ideal ones can be reduced to only a few microns. Therefore, the proposed method for obtaining corrective machine–tool settings can improve the conventional development process and can also be applied to different manufacturing machines and methods for spiral bevel and hypoid gear generation. An example is presented to demonstrate the application of the proposed optimization model.

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 gears design parameters and the contact positions on the stress distribution.

MATHEMATICAL MODEL OF SPIRAL BEVEL AND HYPOID GEARS MANUFACTURED BY THE MODIFIED ROLL METHOD

Based on grinding mechanism and machine-tool settings of the Gleason modified roll hypoid grinder, a mathematical model for the tooth geometry of spiral bevel and hypoid gears is developed. Since all the machine-tool settings and machine constants are involved in the proposed mathematical model, excellent correlation between the mathematical model and actual manufacturing machines can be obtained. An example is given to illustrate the application of the proposed mathematical model. Surface deviation between a real cut spiral bevel gear surface and the model surface is investigated. Bearing contacts and kinematic errors in the spiral bevel gear set are also studied.

Mathematical model and bearing contacts of the ZN-type worm gear set cut by oversize hob cutters

The ZN-type oversize hob cutter for cutting of worm gears is investigated. The mathematical model of the ZN-type worm gear set is developed based on the cutting mechanism and hob cutter parameters. Influences of oversize hob cutter regrinding, worm-thread number and assembly errors on the bearing contacts and kinematical errors of the ZN-type worm gear set are investigated. A numerical example is given to demonstrate the bearing contacts and kinematical errors of the worm gear set.

Mathematical model and undercutting of cylindrical gears with curvilinear shaped teeth

A rack cutter with a curved-tooth is considered the generating tool for the generation of the proposed gear, and a mathematical model of cylindrical gears with curvilinear shaped teeth is developed according to the gearing theory. The developed computer program can develop profiles of the curvilinear-tooth gear with point contacts. The developed gear tooth mathematical model is used to investigate the tooth undercutting of curvilinear-tooth gears via differential geometry and the numerical method.

A general mathematical model for gears cut by CNC hobbing machines

A hobbing machines cutting mechanism is a mechanism with multi-degree offreedom during the cutting process. In this paper, we propose a general gear mathematical model simulating the generation process of a 6-axis CNC hobbing machine based on the cutting mechanism of CNC hobbing machine and worm-type hob cutter. The proposed gear mathematical model can be applied to simulate different types of gear cutting. Some examples are included to verify the mathematical model. Also, a novel type of gear named Helipoid which can be used in crossed axes transmission is proposed. The proposed general gear mathematical model can facilitate a more thorough understanding of generation processes and toward the development of novel types of gears.

Contact characteristics of beveloid gears

This study investigates the contact characteristics of beveloid gear pairs with intersected, crossed and parallel axes. Tooth contact analysis (TCA) is performed to examine the meshing and bearing contact of the beveloid gear pairs composed of a pinion and a gear. In addition, the principal directions and curvatures of the pinion and gear surfaces are investigated and the contact ellipses of the mating tooth surfaces are also studied. Numerical illustrative examples are provided to demonstrate the computational results.