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Analysis of Contact Stress and Fatigue Crack Growth of Transmission Shaft

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Abstract

The finite element model and fracture mechanic theory are applied to analyze the contact stress and fatigue crack growth (FCG) path in three dimensions for a transmission shaft, which owns the initial crack and runs with the periodic load. The study found that the periodic load generates a non-proportional stress intensity factor history (SIF) output in the root of the splines. A modified method based on Hertz theory and AGMA (American Gear Manufacture Association) standard is utilized to analyze contact stress with considering the crack growth. From this, it is feasible to predict FCG path in three dimensions for a cracked component. The comparison between the simulation and experiment illustrates that the crack growth path is sensitive to the location and magnitude of loads. Contact regions of stress and fracture mechanic parameters should be deeply analyzed and known before predicting FCG path in three dimensions.

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Abbreviations

\(T_{M}\) :

Main torque (77.51 N*m)

\(T_{S}\) :

Sub-cycle torque (7.75 N*m)

\(T_{M} /T_{S}\) :

Cyclic ratio (1:2667)

\(F_{{n_{i} }} (i = 1,2, \ldots ,24)\) :

Normal force on spline surface (N)

\(F_{{t_{i} }}\) :

Tangential force on spline surface (N)

\(F_{{r_{i} }}\) :

Radial force (N)

\(S_{i} (i = 1,2, \ldots ,9)\) :

Different stages of the FCG test

\(\sigma_{x} , \sigma_{y}\) :

Stress of crack front on the x-axis and y-axis (MPa)

\(\Delta K_{i} (i = {\text{I}},\; {\text{II}})\) :

Magnitude of \(K_{{\text{I}}}\) and \(K_{{{\text{II}}}}\)

\(\sigma_{\theta }\) :

Circumference stress (MPa)

\(\alpha_{i} (i = 1, \ldots ,9)\) :

Offset angle from contact stress to initial tangential force (degree)

θ :

Crack growth angle (radian)

\({\text{d}}a_{i}\) :

Increment of crack length

\({\text{d}}N_{i}\) :

Increment of cyclic times

\(a_{x} , a_{y}\) :

Semimajor and semiminor axes of the crack (mm)

r :

Radius of torque

\(T_{N}\) :

Teeth count of interior splines

C, m :

Material constant

\(\Delta K_{{{\text{eq}}}}\) :

Magnitude of equivalent SIF

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Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant No. 52075443). Basic Research Program of Natural Science of Shaanxi Province of China (No.2020JM-113).

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Authors and Affiliations

  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, China

    Long Li, Tianxiang Yu, Bolin Shang, Bifeng Song & Yijian Chen

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Appendix

Appendix

Hertz Theory

Hertz theory is based on the elastic mechanic, which can be used to describe the local stress of contact objects, it has three assumptions:(a) The contact objects are linear elastic bodies; (b)The contact area is smooth; (c)The size of the contact area is an infinitesimal variable compared with the cylinder curvature radius. This contact relation can illustrate in Fig. 

Fig. 22
figure 22

Schematic diagram of Hertz contact model

Full size image

22.

The contact stress (elliptic in dash area) of two cylinders is:

$$ \sigma_{\max } = \sqrt {\frac{{F_{s} }}{\pi b}\left( {\frac{{\frac{1}{{\rho_{1} }} + \frac{1}{{\rho_{2} }}}}{{\frac{{1 - \mu_{1}^{2} }}{{E_{1} }} + \frac{{1 - \mu_{2}^{2} }}{{E_{2} }}}}} \right)} $$
(17)

where \({\text{F}}_{{\text{s}}}\) is the normal stress of contact faces, b stands for the length of the contact line, \({\uprho }_{{\text{i}}} {\text{(i = 1,2)}}\), μ, and E represents the curvature radius, Poisson ratio, and elastic modulus of contact objects.

AGMA Standard

The definition and parameters of the AGMA standard are presented as follows [40]:

$$ \sigma_{\max } = C_{p} \sqrt {\frac{{F_{t} }}{{b\text{d}I}}\left( {\frac{\cos \beta }{{0.95CR}}} \right)K_{u} K_{v} (0.93K_{m} )} $$
(18)
$$ C_{p} = 0.564\sqrt {\frac{1}{{\left( {1 - \mu_{1}^{2} } \right)/E_{1} + \left( {1 - \mu_{2}^{2} } \right)/E_{2} }}} $$
(19)

where E and μ stand for Young’s modulus and Poisson ratio.

$$ I = \frac{\sin \alpha \cos \alpha }{2}\frac{i}{i + 1} $$
(20)

where \(i = \frac{{d_{2} }}{{d_{1} }} \) is the speed ratio, α is the transverse pressure angle.

$$ CR = \frac{{\sqrt {\left( {R_{1} + a} \right)^{2} - R_{b1}^{2} } + \sqrt {\left( {R_{2} + a} \right)^{2} - R_{b2}^{2} } - (R_{1} + R_{2} )\sin \alpha }}{\pi m\cos \alpha } $$
(21)

where R, \({\text{R}}_{{\text{b}}}\), and α represent the pitch circle radius, base circle radius, and addendum, respectively. Suffix 1 is for the interior spline and 2 is for the flange, m is the modulus that equals 2.5 mm.

$$ K_{u} = \left[ {\frac{{78 + \left( {200V} \right)^{\frac{1}{2}} }}{78}} \right]^{\frac{1}{2}} $$
(22)

V is rotation velocity or pitch line velocity.

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Li, L., Yu, T., Shang, B. et al. Analysis of Contact Stress and Fatigue Crack Growth of Transmission Shaft. J Fail. Anal. and Preven. 23, 465–483 (2023). https://doi.org/10.1007/s11668-023-01592-0

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