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.

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.

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.

scholarly journals Lubrication characteristics of gear after shot peening base on 25-pellet model

2018 ◽  
Vol 10 (7) ◽  
pp. 168781401879065 ◽  
Author(s):  
Shuai Mo ◽  
Shengping Zhu ◽  
Guoguang Jin ◽  
Jiabei Gong ◽  
Zhanyong Feng ◽  
...  
Keyword(s):  
Shot Peening ◽  
High Speed ◽  
Target Material ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Spiral Bevel ◽  
Lubrication Characteristics

High-speed heavy-load spiral bevel gears put forward high requirement for flexural strength; shot peening is a technique that greatly improves the bending fatigue strength of gears. During shot peening, a large number of fine pellets bombard the surface of the metal target material at very high speeds and let the target material undergo plastic deformation, at the same time strengthening layer is produced. Spiral bevel gear as the object of being bombarded inevitably brought the tooth surface micro-morphology changes. In this article, we aim to reveal the effect of microtopography of tooth shot peening on gear lubrication in spiral bevel gear, try to establish a reasonable description of the microscopic morphology for tooth surface by shot peening, to reveal the lubrication characteristics of spiral bevel gears after shot peening treatment based on the lubrication theory, and do comparative research on the surface lubrication characteristics of a variety of microstructures.

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 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.

scholarly journals EHL Analysis of Spiral Bevel Gear Pairs considering the Contact Point Migration due to Deformation under Load

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Xiaoyu Sun ◽  
Yanping Liu ◽  
Yongqiang Zhao ◽  
Ming Liu
Keyword(s):  
Contact Point ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Gear Pair ◽  
Spiral Bevel Gears ◽  
Actual Contact ◽  
Bevel Gears ◽  
Contact Points ◽  
Spiral Bevel

The actual contact point of a spiral bevel gear pair deviates from the theoretical contact point due to the gear deformation caused by the load. However, changes in meshing characteristics due to the migration of contact points are often ignored in previous studies on the elastohydrodynamic lubrication (EHL) analysis of spiral bevel gears. The purpose of this article is to analyze the impact of contact point migration on the results of EHL analysis. Loaded tooth contact analysis (LTCA) based on the finite element method is applied to determine the loaded contact point of the meshing tooth pair. Then, the osculating paraboloids at this point are extracted from the gear tooth surface geometry. The geometric and kinematic parameters for EHL simulation are determined according to the differential geometry theory. Numerical solutions to the Newtonian isothermal EHL of a spiral bevel gear pair at the migrated and theoretical contact points are compared to quantify the error involved in neglecting the contact point adjustment. The results show that under heavy-loaded conditions, the actual contact point of the deformed gear pair at a given pinion (gear) roll angle is different from the theoretical contact point considerably, and so do the meshing parameters. EHL analysis of spiral bevel gears under significant load using theoretical meshing parameters will result in obvious errors, especially in the prediction of film thickness.

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.

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.

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.

Mathematical Model of Spiral Bevel Gears Manufactured by Generating Line Method

2010 ◽  
Vol 154-155 ◽  
pp. 103-108
Author(s):  
Zhao Jun Yang ◽  
Yan Kun Wang ◽  
Li Nan Li ◽  
Xue Cheng Zhang
Keyword(s):  
Mathematical Model ◽  
Contact Analysis ◽  
Tooth Surface ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Line Method ◽  
Spiral Bevel ◽  
The Mathematical Model

Generating line method for designing and manufacturing spiral bevel gears is proposed in this paper. The tooth surface of spiral bevel gears produced by generating line method is formed by exact spherical involutes, the mathematical model to describe tooth surface has been derived based on gear meshing theory and the cutting motion. This study can provide some fundamentals for manufacturing and contact analysis of spherical involutes spiral bevel gears.

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