A Mathematical Model for Grinding a Stick Blade Profile to Cut Hypoid Gears

2019 ◽  
Vol 142 (5) ◽  
Author(s):  
Yi-Hui Lee ◽  
Zhang-Hua Fong
Keyword(s):  
Mathematical Model ◽  
Cutting Parameters ◽  
Rake Angle ◽  
Industrial Applications ◽  
Grinding Wheel ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Relief Angle ◽  
Spiral Bevel ◽  
Cutter Head

Abstract Cutter head with multiple stick blades is widely used in the mass production of hypoid and spiral bevel gears because they allow more blades per revolution of the head cutter. However, the stick blade geometry for a head cutter that is used in face-milling and face-hobbing methods to produce hypoid and spiral bevel gears is complicated and difficult to describe. The geometry of a stick blade is defined in terms of cutting parameters such as the rake angle, the hook angle, and the side relief angle that are required to perform cutting and the theoretical cutter profile in the offset plane or the neutral cutter profile in the normal plane of an imaginary generating crown gear. This study uses a 5-axis profile grinder to grind the stick blade. The machine settings for the profile grinder and the corresponding grinding wheel geometry are derived for the grinding of each face of the stick blade. A sensitivity analysis for the machine settings is conducted to determine the accuracy of the profile grinder. The numerical example shows that the proposed mathematical model is sufficiently accurate for industrial applications.

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.

Mathematical model for generation of spiral bevel gears

1994 ◽  
Vol 44 (3-4) ◽  
pp. 327-334 ◽  
Author(s):  
B. Subba Rao ◽  
M.S. Shunmugam ◽  
V. Jayaprakash
Keyword(s):  
Mathematical Model ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Spiral Bevel

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.

General Mathematical Model of Internal Meshing Spiral Bevel Gears for Nutation Drive

2011 ◽  
Vol 101-102 ◽  
pp. 708-712 ◽  
Author(s):  
Zheng Lin ◽  
Li Gang Yao
Keyword(s):  
Mathematical Model ◽  
3D Modeling ◽  
Gear Tooth ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
General Mathematical Model ◽  
Spiral Bevel ◽  
The Mathematical Model ◽  
Crown Gear ◽  
Internal Meshing

The general mathematical model of internal meshing spiral bevel gears for nutation drive is studied. Based on conventional enveloping theory and transmission principle, the meshing of two spiral bevel gears in nutation drive was substituted by the meshing of an imaginary rotating crown gear engaging with the external and internal bevel gear respectively. The general mathematical model of crown gear was established. Then the general mathematical model of internal meshing spiral bevel gears is obtained by matrix transformation, which is suitable for a variety of gear tooth profiles. Finally, the mathematical model and 3D modeling of double circular-arc spiral bevel gears are developed.

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.

A Study on the Tooth Geometry and Cutting Machine Mechanisms of Spiral Bevel Gears

1991 ◽  
Vol 113 (3) ◽  
pp. 346-351 ◽  
Author(s):  
Z. H. Fong ◽  
Bill Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Spiral Bevel Gear ◽  
Helical Motion ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Gear Cutting ◽  
Cutting Machine ◽  
The Matrix ◽  
Spiral Bevel

The tooth geometry and cutting machine mechanisms of spiral bevel gears are investigated. Based on the kinematics of titled head cutter, machine cradle, sliding base and work head, the matrix presentation of spiral bevel gear’s tooth geometry are developed. The relations between the parameters of the proposed mathematical model and the machine settings of existing spiral bevel gear cutting machines are also investigated. The tilt of head cutter axis, motion of generation, helical motion of sliding base, and nongenerating cutting of spiral bevel gears are taken into consideration. An example is given to illustrate the application of the proposed mathematical model.

A Mathematical Model for the Tooth Geometry of Circular-Cut Spiral Bevel Gears

1991 ◽  
Vol 113 (2) ◽  
pp. 174-181 ◽  
Author(s):  
Z. H. Fong ◽  
Chung-Biau Tsay
Keyword(s):  
Mathematical Model ◽  
Cnc Machining ◽  
Tooth Contact Analysis ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Finite Element Stress Analysis ◽  
Tool Setting ◽  
Spiral Bevel ◽  
The Mathematical Model ◽  
Made In

A complete tooth geometry of the circular-cut spiral bevel gears has been mathematically modeled. The mathematical model has been divided into several independent modules, each representing an individual kinematic relation or tool-setting, with examples included. A comparison with the spiraloid model has also been made in this paper. The mathematical model can be applied to simulate and calculate the tooth profiles for the Duplex Method, Helical Duplex Method, Formate Method, and Modified Roll Method for circular-cut spiral bevel gears. It can also be applied to the computer numerical controlled (CNC) machining, computer-aided finite element stress analysis, and tooth contact analysis (TCA) for the spiral bevel gear.

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.

General mathematical model of spiral bevel gears of continuous pure-rolling contact

2014 ◽  
Vol 229 (15) ◽  
pp. 2810-2826 ◽  
Author(s):  
Rulong Tan ◽  
Bingkui Chen ◽  
Changyan Peng
Keyword(s):  
Mathematical Model ◽  
High Efficiency ◽  
Rolling Contact ◽  
Sliding Contact ◽  
Frictional Loss ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Pure Rolling ◽  
General Mathematical Model ◽  
Spiral Bevel

Sliding contact has negative influence for the performance of gears in most situations, such as frictional loss, pitting, micro-pitting, etc. To avoid sliding contact, this paper discusses the fundamental geometrical characteristics of spiral bevel gears of continuous pure-rolling contact. Differential geometry, gearing kinematics, and conjugate curve meshing method are used to derive the general mathematical model of spiral bevel gears of continuous pure-rolling contact. Simplified equations that continuous pure-rolling contact bevel gears must satisfy are derived. An example is represented to explain and verify this model. The theoretical results benefit further studies on design and manufacturing of spiral bevel gears of heavy loads and high efficiency.

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