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An innovative tooth root profile for spur gears and its effect on service life

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Abstract

An innovative approach to the design of the gear-tooth root-profile, and its effects on the service life is reported in this paper. In comparison with the widely used trochoidal and the recently proposed circular-filleted root profiles, the optimum profile proposed here is a \(G^2\)-continuous curve that blends smoothly with both the involute of the tooth profile and the dedendum circle. Following the AGMA and ISO standards for fatigue loading, the von Mises stress at the critical section and stress distribution along the gear tooth root are studied. The process leading to gear-tooth failure is composed of the crack initiation phase, in number of cycles \(N_i\), and the crack propagation phase, in \(N_p\) cycles. The strain-life (\(\epsilon\)-N) method is employed to determine \(N_i\), where the crack is assumed to initiate at the critical section. Based on the ANSYS crack-analysis module, the effects of \(G^2\)-continuous blending on the stress intensity factor (SIF) are investigated for different crack sizes. Paris’ law, within the framework of Linear Elastic Fracture Mechanics, is used to correlate the SIF with crack size and, further, to determine \(N_p\). The optimum profile provides a significant reduction in SIF and improvement in both \(N_i\) and \(N_p\). Spur gears are made of high-strength steel alloy 42CrMo4, the effects of its properties and surface treatment on service life improvement not being included in this study.

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Notes

  1. Although N is an integer, its values lie in the order of \(10^6\), for which reason it is common practice to treat it as a real number.

  2. \(G^1\) continuity means point- and tangent-continuity; not curvature-continuity.

  3. According to Venkayya’s criterion, the optimum design can be interpreted as the one in which the strain energy per unit volume stays constant

  4. 42CrMo4 contains 0.43 % C, 0.22 % Si, 0.59 % Mn, 1.04 % Cr, and 0.17 % Mo.

Abbreviations

\(\alpha\) :

Crack propagation direction angle

\(\alpha _c\) :

Pressure angle

\(\beta\) :

Dimensionless geometric factor

\(\varDelta \epsilon\) :

Total cyclic strain range

\(\varDelta \epsilon _e\) :

Elastic strain range

\(\varDelta \epsilon _p\) :

Plastic strain range

\(\varDelta \sigma\) :

Stress range

\(\varDelta K\) :

Stress intensity factor range

\(\epsilon _f^\prime\) :

Fatigue ductility coefficients

\(\nu\) :

Poisson ratio

\(\sigma\) :

Uniform tensile stress in a direction normal to the plane of crack

\(\sigma _f^\prime\) :

Fatigue strength

\(\sigma _m\) :

Mean stress

\(\xi _{F}\) :

Angle between the fillet tangent and the tooth centerline

a :

Crack length

\(a_c\) :

Critical crack size

\(a_i\) :

Initial crack size

b :

Strength exponent

C :

Material constant

c :

Fatigue ductility exponent

E :

Young modulus

\(h_{Fe}\) :

Bending moment arm application at the tip

K :

Stress intensity factor

\(K^{\prime }\) :

Hardening coefficient

\(K_I\) :

SIF for fatigue mode I

\(K_{Ic}\) :

Fracture toughness

\(K_{II}\) :

SIF for fatigue mode II

m :

Material constant

\(m_t\) :

Gear module

\(n^{\prime }\) :

Cyclic strain-hardening exponent

\(N_i\) :

Crack initiation period

\(N_p\) :

Crack propagation period

\(N_t\) :

Number of teeth

\(r_a\) :

Radius of the addendum circle

\(r_b\) :

Radius of the base circle

\(r_d\) :

Radius of the dedendum circle

\(r_p\) :

Radius of the pitch circle

\(s_{Fn}\) :

Tooth thickness at the critical point

APDL:

ANSYS parametric design language

CAE:

Computer-aided-engineering

DOF:

Degrees of freedom

DTA:

Damage tolerance analysis

FEM:

Finite element method

HPSTC:

Highest point of single tooth contact

LEFM:

Linear elastic fracture mechanics

MTS:

Maximum tangential stress

OCS:

Optimum cubic-spline

ODA:

Orthogonal decomposition algorithm

SIF:

Stress intensity factor

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Acknowledgments

The research reported here was conducted with the support of Grant APCPJ418901-11 from NSERC’s Automotive Partnership Canada Project. NSERC is Canada’s Natural Sciences and Engineering Research Council

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Authors and Affiliations

  1. Department of Mechanical Engineering, Centre for Intelligent Machines, McGill University, Montreal, Canada

    Ting Zou, Jorge Angeles & Alexei Morozov

  2. MAYA Heat Transfer Technologies Ltd, Montreal, Canada

    Mathew Shaker

Authors
  1. Ting Zou
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  2. Mathew Shaker
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  3. Jorge Angeles
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  4. Alexei Morozov
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Corresponding author

Correspondence to Ting Zou.

Appendix: Matrices related to the \(G^2\)-continuity conditions

Appendix: Matrices related to the \(G^2\)-continuity conditions

$$\mathbf {A}=\varDelta \theta \left[\begin{array}{lllllll} 2&1&0&0&\cdots&0&0\\ 1&4&1&0&\cdots&0&0\\ 0&1&4&1&\cdots&0&0\\ \vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots \\ 0&0&\cdots&1&4&1&0\\ 0&0&0&\cdots&1&4&1\\ 0&0&0&\cdots&0&1&2 \end{array}\right]$$
(24)
$${\mathbf{C}}= \frac{1}{\varDelta \theta } \left[\begin{array}{ccccccc} c_{11}&1&0&0&\cdots&0&0\\ 1&-2&1&0&\cdots&0&0\\ 0&1&-2&1&\cdots&0&0\\ \vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots \\ 0&0&\cdots&1&-2&1&0\\ 0&0&0&\cdots&1&-2&1\\ 0&0&0&\cdots&0&1&c_{n^{\prime \prime }n^{\prime \prime }} \end{array}\right]$$
(25)

in which \(n^{\prime \prime }=n+2\), \(c_{11}=-1-\varDelta \theta / \tan (\gamma _0)\) and \(c_{n^{\prime \prime }n^{\prime \prime }}=-1-\varDelta \theta / \tan (\gamma _{n+1})\).

$$\begin{aligned}\mathbf {P}=& \varDelta \theta \left[\begin{array}{ccccccc} \frac{1}{\varDelta \theta }&0&0&0&\cdots&0&0\\ 1&4&1&0&\cdots&0&0\\ 0&1&4&1&\cdots&0&0\\ \vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots \\ 0&0&\cdots&1&4&1&0\\ 0&0&0&\cdots&1&4&1\\ 0&0&0&\cdots&0&0& \frac{1}{\varDelta \theta }\\ \end{array}\right]\\ \mathbf {Q}= &\frac{1}{\varDelta \theta } \left[\begin{array}{ccccccc} \frac{1}{\tan (\gamma _0)}&0&0&0&\cdots&0&0\\ -3&0&3&0&\cdots&0&0\\ 0&-3&0&3&\cdots&0&0\\ \vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots \\ 0&0&\cdots&-3&0&3&0\\ 0&0&0&\cdots&-3&0&3\\ 0&0&0&\cdots&0&0& \frac{1}{\tan (\gamma _{n+1})}\\ \end{array}\right]\end{aligned}$$
(26)

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Zou, T., Shaker, M., Angeles, J. et al. An innovative tooth root profile for spur gears and its effect on service life. Meccanica 52, 1825–1841 (2017). https://doi.org/10.1007/s11012-016-0519-7

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