Mathematical Model of the Generated Gear Tooth Surfaces for the Function-oriented Design of Point-contact Tooth Surfaces of Spiral Bevel and Hypoid Gears

2011 ◽  
Author(s):  
Wu Xuncheng
Keyword(s):  
Mathematical Model ◽  
Gear Tooth ◽  
Point Contact ◽  
Hypoid Gears ◽  
Tooth Surfaces ◽  
Spiral Bevel

A Mathematical Model for the Generated Gear Tooth Surfaces of Spiral Bevel and Hypoid Gears

2011 ◽  
Vol 314-316 ◽  
pp. 384-388
Author(s):  
Xun Cheng Wu ◽  
Jing Tao Han ◽  
Jia Fu Wang
Keyword(s):  
Mathematical Model ◽  
Gear Tooth ◽  
Unit Vector ◽  
Bevel Gear ◽  
Position Vector ◽  
Hypoid Gears ◽  
Tooth Surfaces ◽  
Spiral Bevel ◽  
The Mathematical Model ◽  
Gear Machine

It is an important and fundamental work to establish a general mathematical model for the gear tooth surfaces of spiral bevel and hypoid gears. Based on the three-axis CNC bevel gear machine, a mathematical model with the equations of the radial position vector, the normal unit vector and the second order parameters for the generated gear tooth surfaces of spiral bevel and hypoid gears is established. The mathematical model can be used for the gear tooth surfaces generated in different types on both the three-axis CNC bevel gear machine and the cradle bevel gear machine. As an application example of the mathematical model, the generating motions of the cradle bevel gear machine are determined.

Function-Oriented Design and Verification of Point-Contact Tooth Surfaces of Spiral Bevel and Hypoid Gears with the Generated Gear

2010 ◽  
Vol 118-120 ◽  
pp. 675-680
Author(s):  
Xun Cheng Wu ◽  
Cong Li
Keyword(s):  
Mathematical Model ◽  
Point Contact ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Hypoid Gears ◽  
Contact Points ◽  
Tooth Surfaces ◽  
Spiral Bevel ◽  
The Mathematical Model ◽  
Gear Machine

Establishing a general technical platform for the function-oriented design of point-contact tooth surfaces of spiral bevel and hypoid gears is an important and fundamental work. Based on the three-axis CNC bevel gear machine, a general mathematical model for the generated gear tooth surfaces of spiral bevel and hypoid gears is established. According to the principle and the method for the function-oriented design of point-contact tooth surfaces, the locus of spatial tooth contact points on the tooth surface is described on the axial plane of the gear, and then the formulae for the design with the generated gear are derived from the mathematical model. The mathematical model and the formulae can be used in the function-oriented design of point-contact tooth surfaces with the gear generated in different types on both the three-axis CNC bevel gear machine and the conventional cradle one. A theoretical method for the verification of point-contact tooth surfaces is proposed and the formulae for the verification are presented. And lastly an example is given to demonstrate the function-oriented design of point-contact tooth surfaces of the hypoid gear drive with the generated gear.

A Study on Forging of Spiral Bevel Gear

2007 ◽  
Author(s):  
Masaki Watanabe ◽  
Minoru Maki ◽  
Sumio Hirokawa ◽  
Yasuhiro Kishimoto
Keyword(s):  
Gear Tooth ◽  
Conjugate Point ◽  
Point Contact ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Tooth Contact ◽  
Contact Points ◽  
Tooth Surfaces ◽  
Spiral Bevel

This study reports the method of forging of spiral bevel gear. Two ideas for crowning of tooth surface to obtain point contact for forging gears are proposed. By one idea, tooth surface of pinion meshes with the gear tooth surface by conjugate point contact. And the trace of contact points on the gear tooth surface is perpendicular to the lengthwise direction of gear tooth, namely becomes the “square contact” so called in gear technology. The trace can be set arbitrarily on the gear tooth, by setting the pitch point arbitrarily. By another idea, the trace of contact points lies along the tooth trace of the gear tooth. Both ideas proposed in this report, the numerical dataset of teeth surface of pinion and gear are given by the contact lines with the cutter cone. The dataset of teeth surface of pinion and gear are calculated to cut a pair of electrodes of spiral bevel gear. Tooth contacts of proposed gearing are confirmed by the 3D drawing of tooth surfaces. The tooth contact of the master pinion and gear were made and tested by tooth contact testing apparatus. The contact marks coincide well with the theoretical contact pattern estimated by 3D/CAD expression. The good results of running test of the performance of the master gear has been given. The authors completed the forging of spiral bevel gear pairs by two methods proposed in this report.

Function-Oriented Designing and Generating Technology for the Point-Contact Tooth Surfaces of Spiral Bevel and Hypoid Gears

2008 ◽  
Vol 44-46 ◽  
pp. 495-502 ◽  
Author(s):  
Xun Cheng Wu ◽  
Cong Li ◽  
Ruo Ping Zhang ◽  
Hai Bo Zhang
Keyword(s):  
Point Contact ◽  
Transmission Function ◽  
Tooth Surface ◽  
Free Form ◽  
Tooth Contact ◽  
Hypoid Gears ◽  
Contact Points ◽  
Tooth Surfaces ◽  
Spiral Bevel ◽  
Basic Equations

A function-oriented designing and generating technology for the point-contact tooth surfaces of spiral bevel and hypoid gears is introduced. The tooth surface parameters are determined directly with the designing variables of the instantaneous transmission function, the locus of tooth contact points and the tooth contact ellipse dimension to design the point-contact tooth surfaces with the expected performances. The formulae for designing are provided. The designed tooth surfaces are generated with the free-form bevel gear machine, and the basic equations and formulae for the four-axis generating of the tooth surfaces are presented. The generating motions are expressed as the functions of the work gear rotary angle, which is taken as a motion parameter. The methods to determine the motion functions and the other machine setting parameters are explained through an example.

scholarly journals Straddle Design of Spiral Bevel and Hypoid Pinions and Gears

1991 ◽  
Vol 113 (4) ◽  
pp. 422-426 ◽  
Author(s):  
F. L. Litvin ◽  
C. Kuan ◽  
J. Kieffer ◽  
R. Bossler ◽  
R. F. Handschuh
Keyword(s):  
Free Space ◽  
Gear Tooth ◽  
Main Difficulty ◽  
Hypoid Gears ◽  
Numerical Example ◽  
Cone Apex ◽  
Spiral Bevel

The design of spiral bevel and hypoid gears that have a shaft extended from both sides of the cone apex (straddle design) is considered. A main difficulty of such a design is determining the length and diameter of the shaft that might be undercut by the head cutter during gear tooth generation. A method that determines the free space available for the gear shaft is proposed. The approach avoids collision between the shaft being designed and the head cutter during tooth generation. The approach is illustrated with a numerical example.

Method for Generation of Spiral Bevel Gears With Conjugate Gear Tooth Surfaces

1987 ◽  
Vol 109 (2) ◽  
pp. 163-170 ◽  
Author(s):  
F. L. Litvin ◽  
Wei-Jiung Tsung ◽  
J. J. Coy ◽  
C. Heine
Keyword(s):  
Gear Tooth ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Straight Line ◽  
Parallel Motion ◽  
Tooth Surfaces ◽  
Spiral Bevel

The authors proposed a method for generation of spiral bevel gears that provides conjugate gear tooth surfaces. This method is based on a new principle for the performance of parallel motion of a straight line that slides along two mating ellispses with related dimensions and parameters of orientation. The parallel motion of the straight line, that is the contact normal, is performed parallel to the line which passes through the foci of symmetry of the related ellipses. The manufacturing of gears can be performed with the existing Gleason’s equipment.

Advanced Design and Manufacture of Face-Hobbed Spiral Bevel Gears

2009 ◽  
Author(s):  
V. Simon
Keyword(s):  
Gear Tooth ◽  
Numerical Control ◽  
Mutual Position ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Pinion Gear ◽  
Tooth Surfaces ◽  
Spiral Bevel ◽  
Tooth Flank ◽  
Crown Gear

The design and advanced manufacture of face-hobbed spiral bevel gears on computer numerical control (CNC) hypoid generating machines is presented. The concept of face-hobbed bevel gear generation by an imaginary generating crown gear is established. In order to reduce the sensitivity of the gear pair to errors in tooth-surfaces and to the mutual position of the mating members, modifications are introduced into the teeth of both members. The lengthwise crowning of teeth is achieved by applying a slightly bigger lengthwise tooth flank curvature of the crown gear generating the concave side of pinion/gear tooth-surfaces, and/or by using tilt angle of the head-cutter in the manufacture of pinion/gear teeth. The tooth profile modification is introduced by the circular profile of the cutting edge of head-cutter blades. An algorithm is developed for the execution of motions on the CNC hypoid generating machine using the relations on the cradle-type machine. The algorithm is based on the condition that since the tool is a rotary surface and the pinion/gear blank is also related to a rotary surface, it is necessary to ensure the same relative position of the head cutter and the pinion on both machines.

The Undercutting of Circular-Cut Spiral Bevel Gears

1992 ◽  
Vol 114 (2) ◽  
pp. 317-325 ◽  
Author(s):  
Zhang-Hua Fong ◽  
Chung-Biau Tsay
Keyword(s):  
Gear Tooth ◽  
Sufficient Conditions ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Tooth Surfaces ◽  
Sufficient And Necessary Conditions ◽  
Spiral Bevel

Undercutting is a serious problem in designing spiral bevel gears with small numbers of teeth. Conditions of undercutting for spiral bevel gears vary with the manufacturing methods. Based on the theory of gearing [1], the tooth geometry of the Gleason type circular-cut spiral bevel gear is mathematically modeled. The sufficient and necessary conditions for the existence and regularity of the generated gear tooth surfaces are investigated. The conditions of undercutting for a circular-cut spiral bevel gear are defined by the sufficient conditions of the regular gear tooth surface. The derived undercutting equations can be applicable for checking the undercutting conditions of spiral bevel gears manufactured by the Gleason Duplex Method, Helical Duplex Method, Fixed Setting Method, and Modified Roll Method. An example is included to illustrate the application of the proposed undercut checking equations.

scholarly journals Ideal Spiral Bevel Gears—A New Approach to Surface Geometry

1981 ◽  
Vol 103 (1) ◽  
pp. 127-132 ◽  
Author(s):  
R. L. Huston ◽  
J. J. Coy
Keyword(s):  
Differential Geometry ◽  
Gear Tooth ◽  
Parametric Representation ◽  
Spiral Bevel Gear ◽  
Surface Geometry ◽  
New Approach ◽  
Bevel Gears ◽  
Geometrical Characteristics ◽  
Tooth Surfaces ◽  
Spiral Bevel

This paper discusses the fundamental geometrical characteristics of spiral bevel gear tooth surfaces. The parametric representation of an ideal spiral bevel tooth is developed. The development is based on the elements of involute geometry, differential geometry, and fundamental gearing kinematics. A foundation is provided for the study of nonideal gears and the effects of deviations from ideal geometry on the contact stresses, lubrication, wear, fatigue life, and gearing kinematics.

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