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Mathematic model and tooth contact analysis of a new spiral non-circular bevel gear

新型螺旋非圆锥齿轮的数学模型和齿面接触分析

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Abstract

A novel spiral non-circular bevel gear that could be applied to variable-speed driving in intersecting axes was proposed by combining the design principles of non-circular bevel gears and the manufacturing principles of face-milling spiral bevel gears. Unlike straight non-circular bevel gears, spiral non-circular bevel gears have numerous advantages, such as a high contact ratio, high intensity, good dynamic performance, and an adjustable contact region. In addition, while manufacturing straight non-circular bevel gears is difficult, spiral non-circular bevel gears can be efficiently and precisely fabricated with a 6-axis bevel gear cutting machine. First, the generating principles of spiral non-circular bevel gears were introduced. Next, a mathematical model, including a generating tooth profile, tooth spiral, pressure angle, and generated tooth profile for this gear type was established. Then the precision of the model was verified by a tooth contact analysis using FEA, and the contact patterns and stress distributions of the spiral non-circular bevel gears were investigated.

摘要

本文结合非圆锥齿轮的设计原理和面铣螺旋锥齿轮的制造原理, 提出了一种可应用于相交轴变 速传动的新型螺旋非圆锥齿轮。与直齿非圆锥齿轮不同, 螺旋非圆锥齿轮有许多优点, 如高接触比、 高强度、良好的动态性能和可调节的接触区域。此外, 虽然制造直齿非圆锥齿轮很困难, 但螺旋非圆 锥齿轮可以用6 轴锥齿轮切削机床高效、精确地制造。首先, 介绍了螺旋非圆锥齿轮的产形原理。接 着, 建立了一个数学模型, 包括产形齿的齿廓、齿轮螺旋度、压力角和该齿轮的产形齿廓。然后, 利 用有限元进行齿轮接触分析, 验证了模型精度, 并研究了螺旋非圆锥齿轮的接触模式和应力分布。

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Abbreviations

W :

Point width of the cutter blade

α 1 :

Concave pressure angles of the cutter

α 2 :

Convex pressure angles of the cutter

e p :

Distance of the cutter center to the machine center

r v :

Mean radius of the cutter

r vd :

Outer radius of the cutter

r vx :

Inner radius of the cutter

r f :

Mean cone distance

β f :

Mean spiral angle

m f :

Module of the flat-top generator

A 0 :

Pitch cone of noncircular bevel gear

A f :

Root cone of noncircular bevel gear

A a :

Face cone of noncircular bevel gear

φ 0 :

Rotating angle of non-circular bevel gear

δ 0 :

Pitch angle of non-circular bevel gear

φ 1 :

Rotating angle of drive non-circular bevel gear

φ 2 :

Rotating angle of driven non-circular bevel gear

i 12 :

Gear ratio function

δ 1 :

The pitch angle of drive non-circular bevel gear

δ 2 :

The pitch angle of driven non-circular bevel gear

m F :

The mean module, or module on the mean pitch curve of noncircular bevel gear

t 1 :

Tangent vector of pitch cone in circumferential direction

t 1u :

The unit tangent vector of pitch cone in circumferential direction

t 2 :

Tangent vector of the pitch cone in radial direction

t 2u :

The unit tangent vector of pitch cone in radial direction

n 1u :

Unit normal vector of the pitch cone

A p :

Face cone of the flat-top generator (a plane)

t 3u :

Unit tangent vector of the root cone in circumferential direction

t 4u :

Unit tangent vector of the root cone radial direction

n 3u :

Unit normal vector of the root cone

M ij :

Homogeneous transformation matrix from coordinate system Sj to coordinate system Si

i i,j i,k i :

Base vector of coordinate system Si

θ p :

Angle of generator

t 1,t r :

Tangent vectors of the generator’s tooth profile in two different directions

n r :

Normal vector of the generator’s tooth profile

v c :

Relative velocity between the generator and generated gear

θ p :

Angle of generator

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Authors and Affiliations

  1. School of Automotive Engineering, Hubei Key Laboratory of Advanced Technology of Automotive Parts, Wuhan University of Technology, Wuhan, 430070, China

    Xing-hui Han  (韩星会), Xuan-cheng Zhang  (张轩诚), Fang-yan Zheng  (郑方焱), Man Xu  (徐曼) & Jun Tian  (田俊)

Authors
  1. Xing-hui Han  (韩星会)
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  2. Xuan-cheng Zhang  (张轩诚)
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  3. Fang-yan Zheng  (郑方焱)
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  4. Man Xu  (徐曼)
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  5. Jun Tian  (田俊)
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Corresponding author

Correspondence to Fang-yan Zheng  (郑方焱).

Additional information

Foundation item

Project(52175361) supported by the National Natural Science Foundation of China; Project (2019CFA041) supported by the Natural Science Foundation of Hubei Province, China; Project(WUT: 202407002) supported by the Fundamental Research Funds for the Central Universities, China

Contributors

HAN Xing-hui provided the concept and edited the draft of manuscript. ZHENG Fang-yan and TIAN Jun implemented the experiment and wrote the first draft of the manuscript. ZHANG Xuancheng and XU Man provided the experiment instrument and advised on the analysis of the experiment result.

Conflict of interest

HAN Xing-hui, ZHANG Xuan-cheng, ZHENG Fang-yan, XU Man and TIAN Jun declare that they have no conflict of interest.

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Han, Xh., Zhang, Xc., Zheng, Fy. et al. Mathematic model and tooth contact analysis of a new spiral non-circular bevel gear. J. Cent. South Univ. 29, 157–172 (2022). https://doi.org/10.1007/s11771-022-4898-8

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  • DOI: https://doi.org/10.1007/s11771-022-4898-8

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