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Analysis and formulation of spur gear stresses with different tip modifications

不同齿尖修正的直齿轮应力分析与计算

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Abstract

Spur gears are widely used in the power transmission mechanism of several machines. Due to the transmitted torque, spur gears experience high stresses which could cause gear tooth failure by surface pitting or root fracture. Tip relief and other gear profile modification have been considered for reducing the induced stresses in the gear tooth. In this work, the influence of tip relief on stresses on a pair of identical spur gear was analyzed using commercial FEA software ANSYS, and formulae for estimating contact and bending stresses were derived. Three cases of gear sets were analyzed; a non-modified pair and another two sets with linear and parabolic tip relief profiles. The non-modified gear set frictionless contact stress was validated against the calculated AGMA pitting resistance, Hertzian contact stress and a reported contact stress value in the literature. The four methods agreed well with each other. Similarly, bending stress was also compared with the AGMA bending strength and Lewis bending stress for validation. Then, friction coefficient was varied from 0.0 to 0.3 with increment of 0.1. The gear contact stress increased up to 11% relative to the frictionless case, whereas bending stress decreased by 6%. Linear tip relief modification was carried out for increasing normalised tip relief values of 0.25 to 1.0 with increment of 0.25. The gear frictionless contact and bending stresses decreased by a maximum of 4% and 2%, respectively. Frictional contact stress increased by up to 7.1% and the bending stress is almost identical with the frictionless case. Parabolic tip relief was also carried out with similar normalised tip relief values. Frictionless contact stress decreased by 5% while frictional contact stress increased by up to 11.5% and the bending stress is also almost identical with the frictionless case. Finally, four formulae were introduced for estimating the contact and bending stresses for a tip modified spur gear.

摘要

直齿轮广泛应用于多种机器的动力传动结构中。由于存在传递的扭矩, 直齿轮受到高应力作用, 这可能发生表面点蚀或根部断裂, 导致轮齿失效。通过齿尖减震和其他齿廓修形, 可以减小轮齿中的 感应应力。本文利用商用有限元分析软件 ANSYS 对一组相同直齿轮上的应力的尖端减震影响进行了 分析, 推导了接触和弯曲应力计算公式。分析了 3 个齿轮组:非调质齿轮组和另外两组具有线性和抛 物线的齿尖减震的齿轮组。根据计算的 AGMA 点蚀阻力、赫兹接触应力和文献中的接触应力, 验证 了非调质齿轮组无摩擦接触应力。通过这四种方法所得到的结论一致。同样的, 将弯曲应力与 AGMA 弯曲强度和 Lewis 弯曲应力进行比较, 以进行验证。摩擦系数以增量 0.1 从 0.0 增加到 0.3。相对于无 摩擦情况, 齿轮接触应力增加至11%, 而弯曲应力降低了6%。以增量0.25 将归一化齿尖减震从 0.25 增加到 1.0, 进行线性齿尖减震。无摩擦接触应力和弯曲应力分别下降了4% 和 2%; 摩擦接触应力增 加到 7.1%, 弯曲应力几乎与无摩擦情况相同。使用相似的归一化齿尖减震进行抛物线齿尖减震, 无 摩擦接触应力降低了 5%, 摩擦接触应力增加至11.5%, 弯曲应力与无摩擦情况几乎相同。最后, 引 入 4 个公式来估算齿尖修正直齿轮的接触应力和弯曲应力。

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Abbreviations

C a, C a,max :

Applied and maximum allowable amounts of tip relief, mm

C n, L n :

Normalized amount and length of tip relief, mm

L a, L a,max :

Applied and maximum allowable lengths of tip relief, mm

m t :

Transverse metric module (the same as module, m), mm

r T2 :

Length of the line connecting the origin and the ending point of modification, mm

S T1, S T2 :

Lengths of the lines connecting the base circle and a point on the tooth-tip and the point of modification, mm

ε :

Angle formed by the lines connecting the base circle and the starting and ending points of modification about the base circle radius, (°)

β 1, β 2 :

Angles formed by the lines from the origin to a point on the tooth tip and the ending point of modification with respect to the centreline, (°)

γ 1, γ 2 :

Angles formed by the starting and ending points of modification, the base circle and the gear origin, (°)

θ 1, θ 2 :

Angles formed by the lines connecting the base circle and a point on the tooth tip and the ending point of modification with respect to the centreline, (°)

Δ :

Nominal tip relief value for both the amount and length of modification, mm

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Authors and Affiliations

  1. Department of Mechanical Engineering, Universiti Teknologi PETRONAS, Bandar Seri, Iskandar, 32610, Perak Darul Ridzuan, Malaysia

    A. Maper & S. Karuppanan

  2. Department of Mechanical Engineering, Manipal University Jaipur, Jaipur, 303007, Rajasthan, India

    S. S. Patil

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  1. A. Maper
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  2. S. Karuppanan
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Correspondence to S. S. Patil.

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Maper, A., Karuppanan, S. & Patil, S.S. Analysis and formulation of spur gear stresses with different tip modifications. J. Cent. South Univ. 26, 2368–2378 (2019). https://doi.org/10.1007/s11771-019-4180-x

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