scholarly journals Parametric surface and properties defined on parallelogrammic domain

2014 ◽  
Vol 1 (1) ◽  
pp. 27-36 ◽  
Author(s):  
Shuqian Fan ◽  
Jinsong Zou ◽  
Mingquan Shi
Keyword(s):  
Mathematical Model ◽  
Logarithmic Spiral ◽  
Tooth Surface ◽  
Parametric Surface ◽  
Surface Geometry ◽  
Spiral Bevel Gears ◽  
Geometrical Properties ◽  
Bevel Gears ◽  
Geometrical Characteristics ◽  
Spiral Bevel

Abstract Similar to the essential components of many mechanical systems, the geometrical properties of the teeth of spiral bevel gears greatly influence the kinematic and dynamic behaviors of mechanical systems. Logarithmic spiral bevel gears show a unique advantage in transmission due to their constant spiral angle property. However, a mathematical model suitable for accurate digital modeling, differential geometrical characteristics, and related contact analysis methods for tooth surfaces have not been deeply investigated, since such gears are not convenient in traditional cutting manufacturing in the gear industry. Accurate mathematical modeling of the tooth surface geometry for logarithmic spiral bevel gears is developed in this study, based on the basic gearing kinematics and spherical involute geometry along with the tangent planes geometry; actually, the tooth surface is a parametric surface defined on a parallelogrammic domain. Equivalence proof of the tooth surface geometry is then given in order to greatly simplify the mathematical model. As major factors affecting the lubrication, surface fatigue, contact stress, wear, and manufacturability of gear teeth, the differential geometrical characteristics of the tooth surface are summarized using classical fundamental forms. By using the geometrical properties mentioned, manufactura-bility (and its limitation in logarithmic spiral bevel gears) is analyzed using precision forging and multi-axis freeform milling, rather than classical cradle-type machine tool based milling or hobbing. Geometry and manufacturability analysis results show that logarithmic spiral gears have many application advantages, but many urgent issues such as contact tooth analysis for precision plastic forming and multi-axis freeform milling also need to be solved in a further study.

scholarly journals Surface Geometry of Circular Cut Spiral Bevel Gears

1982 ◽  
Vol 104 (4) ◽  
pp. 743-748 ◽  
Author(s):  
R. L. Huston ◽  
J. J. Coy
Keyword(s):  
Logarithmic Spiral ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Surface Geometry ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Spiral Bevel

An analysis of the surface geometry of spiral bevel gears formed by a circular cutter is presented. The emphasis is upon determining the tooth surface principal radii of curvature of crown (flat) gears. Specific results are presented for involute, straight, and hyperbolic cutter profiles. It is shown that the geometry of circular cut spiral bevel gears is somewhat simpler than a theoretical logarithmic spiral bevel gear.

Surface Geometry of Straight and Spiral Bevel Gears

1987 ◽  
Vol 109 (4) ◽  
pp. 443-449 ◽  
Author(s):  
Y. C. Tsai ◽  
P. C. Chin
Keyword(s):  
Mathematical Modeling ◽  
Tooth Surface ◽  
Surface Geometry ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Geometrical Characteristics ◽  
Spiral Bevel

The fundamental geometrical characteristics of bevel gears have been discussed in this study. The mathematical modeling of the tooth surface geometry of bevel gears can be developed based on the basic gearing kinematics and involute geometry along with the tangent planes geometry. The parametric representations of the spherical involute and the involute spiraloid, which are the tooth surface geometry of straight bevels and spiral bevels, respectively, have been derived in this paper. This study may provide some fundamentals for computer numerical controlled manufacturing of bevel gears.

Kinematical Optimization of Spiral Bevel Gears

1992 ◽  
Author(s):  
Zhang-Hua Fong ◽  
Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Tooth Contact ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Contact Patterns ◽  
Spiral Bevel ◽  
Kinematic Errors

Abstract Kinematical optimization and sensitivity analysis of circular-cut spiral bevel gears are investigated in this paper. Based on the Gleason spiral bevel gear generator and EPG test machine, a mathematical model is proposed to simulate the tooth contact conditions of the spiral bevel gear set. All the machine settings and assembly data are simulated by simplified parameters. The tooth contact patterns and kinematic errors are obtained by the proposed mathematical model and the tooth contact analysis techniques. Loaded tooth contact patterns are obtained by the differential geometry and the Hertz contact formulas. Tooth surface sensitivity due to the variation of machine settings is studied. The corrective machine settings can be calculated by the sensitive matrix and the linear regression method. An optimization algorithm is also developed to minimize the kinematic errors and the discontinuity of tooth meshing. According to the proposed studies, an improved procedure for development of spiral bevel gears is suggested. The results of this paper can be applied to determine the sensitivity and precision requirements in manufacturing, and improve the running quality of the spiral bevel gears. Two examples are presented to demonstrate the applications of the optimization model.

Theoretical and experimental analyses of spiral bevel gears with two contact paths

2017 ◽  
Vol 232 (4) ◽  
pp. 665-676 ◽  
Author(s):  
Rulong Tan ◽  
Bingkui Chen ◽  
Dong Liang ◽  
Changyan Peng
Keyword(s):  
Differential Geometry ◽  
Contact Stress ◽  
Logarithmic Spiral ◽  
Transmission Efficiency ◽  
Tooth Surface ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Generating Method ◽  
Spiral Bevel ◽  
Maximum Contact

This paper investigates the geometrical design principal of the spiral bevel gears with two contact paths from spatial conjugate curve theory. Differential geometry and gearing kinematics are introduced to derive this model. In this process, the calculation method of contact paths and tooth surface generating method are presented. According to the arguments in this paper, a process of designing the tooth surface of logarithmic spiral bevel gears with two contact paths is investigated. Then, through this process, the design of a pair of logarithmic spiral bevel gears with two contact paths is completed. Besides, the prototype is manufactured and the performance experiment is completed. Results show the maximum contact stress of spiral bevel gears with two contact paths is reduced compared to those with one contact path. Besides, the transmission efficiency of the spiral bevel gears with two contact paths can reach 98.2%.

Kinematical Optimization of Spiral Bevel Gears

1992 ◽  
Vol 114 (3) ◽  
pp. 498-506 ◽  
Author(s):  
Zhang-Hua Fong ◽  
Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Tooth Contact ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Contact Patterns ◽  
Spiral Bevel ◽  
Kinematic Errors

Kinematical optimization and sensitivity analysis of circular-cut spiral bevel gears are investigated in this paper. Based on the Gleason spiral bevel gear generator and EPG test machine, a mathematical model is proposed to simulate the tooth contact conditions of the spiral bevel gear set. All the machine settings and assembly data are simulated by simplified parameters. The tooth contact patterns and kinematic errors are obtained by the proposed mathematical model and the tooth contact analysis techniques. Loaded tooth contact patterns are obtained by the differential geometry and the Hertz contact formulas. Tooth surface sensitivity due to the variation of machine settings is studied. The corrective machine settings can be calculated by the sensitive matrix and the linear regression method. An optimization algorithm is also developed to minimize the kinematic errors and the discontinuity of tooth meshing. According to the proposed studies, an improved procedure for development of spiral bevel gears is suggested. The results of this paper can be applied to determine the sensitivity and precision requirements in manufacturing, and improve the running quality of the spiral bevel gears. Two examples are presented to demonstrate the applications of the optimization model.

Mathematical Model and 3D Modeling of Involute Spiral Bevel Gears for Nutation Drive

2013 ◽  
Vol 694-697 ◽  
pp. 503-506 ◽  
Author(s):  
Zheng Lin ◽  
Li Gang Yao
Keyword(s):  
Mathematical Model ◽  
3D Modeling ◽  
Bevel Gear ◽  
Transition Curve ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Spiral Bevel ◽  
The Mathematical Model

The mathematical model and 3D modeling of involute spiral bevel gears for nutation drive are considered. The basic tooth profile of involute is composed of involute curve and dedendum transition curve, and the equations have been established. The mathematical model of crown gear with involute profile is obtained, and then the mathematical models of the involute spiral bevel gears are developed. The tooth surface modeling of involute spiral bevel gear is proposed, and the 3D modeling of the involute spiral bevel gear for nutation drive is illustrated.

scholarly journals Computerized Design of Low-Noise Face-Milled Spiral Bevel Gears

1994 ◽  
Author(s):  
F. L. Litvin ◽  
Yi Zhang ◽  
R. F. Handschuh
Keyword(s):  
Low Noise ◽  
Tooth Surface ◽  
Principal Curvatures ◽  
Surface Geometry ◽  
Local Synthesis ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Gear Drive ◽  
Geometry Design ◽  
Spiral Bevel

Abstract An advanced design methodology is proposed for the face-milled spiral bevel gears with modified tooth surface geometry that provides a reduced level of noise and has a stabilized bearing contact. The approach is based on the local synthesis of the gear drive that provides the “best” machine-tool settings. The theoretical aspects of the local synthesis approach are based on the application of a predesigned parabolic function for absorption of undesirable transmission errors caused by misalignment and the direct relations between principal curvatures and directions for mating surfaces. The meshing and contact of the gear drive is synthesized and analyzed by a computer program. The generation of gears with the proposed geometry design can be accomplished by application of existing equipment. A numerical example that illustrates the proposed theory is presented.

The geometric principles of general spiral bevel gears of local bearing contact from spatial conjugate curves

2015 ◽  
Vol 231 (9) ◽  
pp. 1651-1663 ◽  
Author(s):  
Rulong Tan ◽  
Bingkui Chen ◽  
Changyan Peng ◽  
Dong Liang ◽  
Dongyun Xiang
Keyword(s):  
Mathematical Model ◽  
Tooth Surface ◽  
Element Analysis ◽  
Theoretical Derivation ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Contact Points ◽  
Bearing Contact ◽  
Spiral Bevel ◽  
The Mathematical Model

This paper aims at obtaining the mathematical model of the general spiral bevel gears of local bearing contact from spatial conjugate curve theory. Differential geometry and gearing kinematics are introduced to derive this model. Meshing-correctly conditions are set in the theoretical derivation process. The final model is represented in the form of equations and inequalities. According to the arguments in this paper, a process of designing the tooth surface of spiral bevel gears of local bearing is proposed. Based on this process, the numerical example of a pair of these gears with specific profiles is represented by applying the finite element analysis. Results show that the magnitudes of the deviations between theoretical contact points and real contact points are small. Therefore, the results agree with the mathematical model of the spiral bevel gears of local bearing contact in this paper.

Manufacture of Optimized Face-Hobbed Spiral Bevel Gears on Computer Numerical Control Hypoid Generator

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Vilmos V. Simon
Keyword(s):  
Machine Tool ◽  
Angular Displacement ◽  
Transmission Error ◽  
Numerical Control ◽  
Tooth Surface ◽  
Surface Geometry ◽  
Tooth Contact ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Spiral Bevel

In this study, a method is proposed for the advanced manufacture of face-hobbed spiral bevel gears on CNC hypoid generators with optimized tooth surface geometry. An optimization methodology is applied to systematically define optimal head-cutter geometry and machine tool settings to introduce optimal tooth modifications. The goal of the optimization is to simultaneously minimize tooth contact pressures and angular displacement error of the driven gear (the transmission error). The optimization is based on machine tool setting variation on the cradle-type generator conducted by optimal polynomial functions. An algorithm is developed for the execution of motions on the CNC hypoid generator using the relations on the cradle-type machine. Effectiveness of the method was demonstrated by using a face-hobbed spiral bevel gear example. Significant reductions in the maximum tooth contact pressure and in the transmission errors were obtained.

scholarly journals Precision of Spiral-Bevel Gears

1983 ◽  
Vol 105 (3) ◽  
pp. 310-316 ◽  
Author(s):  
F. L. Litvin ◽  
R. N. Goldrich ◽  
J. J. Coy ◽  
E. V. Zaretsky
Keyword(s):  
Tooth Surface ◽  
Surface Geometry ◽  
Spiral Bevel Gears ◽  
Gear Teeth ◽  
Bevel Gears ◽  
Bearing Contact ◽  
Order Of Magnitude ◽  
Gear Trains ◽  
Spiral Bevel ◽  
Kinematic Errors

An analytical method was derived for determining the kinematic errors in spiral-bevel gear trains caused by the generation of nonconjugate surfaces, by axial displacements of the gear assembly, and by eccentricity of the assembled gears. Such errors are induced during manufacturing and assembly. Two mathematical models of spiral-bevel gears were included in the investigation. One model corresponded to the motion of the contact ellipse across the tooth surface (geometry I) and the other along the tooth surface (geometry II). The following results were obtained: 1) Kinematic errors induced by errors of manufacture may be minimized by applying special machine settings. The original error may be reduced by an order of magnitude. The procedure is most effective for geometry II gears. 2) When trying to adjust the bearing contact pattern between the gear teeth for geometry I gears, it is more desirable to shim the gear axially; for geometry II gears, shim the pinion axially. 3) The kinematic accuracy of spiral-bevel drives is most sensitive to eccentricities of the gear and less sensitive to eccentricities of the pinion. The pecision of mounting accuracy and manufacture is most crucial for the gear, and less so for the pinion.

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