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Time-varying mesh stiffness calculation of a planetary gear set with the spalling defect under sliding friction

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Abstract

Time-varying mesh stiffness (TVMS) is an important excitation source of a planetary gear set. Sliding friction and spalling defects have significant effect on the TVMS. Accurate evaluation of the TVMS can help obtain the vibration characteristics and further detect the spalling defects of the planetary gear set. In this paper, an improved analytical model is proposed to calculate the TVMS of the planetary gear set with sliding friction by considering the tooth profile beginning with the root circle. Then the impact of spalling defects on the TVMS is investigated. The results show that the improved analytical model can increase the precision of the TVMS. Besides, the effect of sliding friction and spalling defects on the TVMS is significant. This study offers a basis for dynamic performance analysis of a planetary gear set with spalling defects under sliding friction.

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Abbreviations

F :

The normal contacting force

f :

The sliding friction force

\(\alpha _{m}\) :

The standard pressure angle

\(\mu\) :

The sliding friction coefficient

\(F_{a},F_{b}\) :

The tangential force and radial force acting on the gear tooth

\(U_{b},U_{s},U_{a}\) :

The bending, shear and axial compressive energies

\(k_{b},k_{s},k_{a}\) :

The bending, shear and axial compressive stiffness

\(k_{hi},k_{f}\) :

Hertzian stiffness and fillet-foundation stiffness

E, G :

Yang’s modulus and shear modulus

L :

The width of the gear tooth

A, I :

Area and area moment of inertia of the section

\(R_{r},R_{b}\) :

Root circle and base circle of the external gear

\(R_{rr},R_{rb}\) :

Root circle and base circle of the internal gear

\(l_{s},w_{s},h_{s}\) :

Length, width and depth of the spalling area

\(\alpha _{1},\beta _{1}\) :

The angle between normal contacting force and vertical decomposed force

\(\alpha _{2},\alpha _{3}\) :

Half angle of the base circle and root circle of the external gear

\(\alpha _{5}\) :

The angle between F and \(F_{a}\) at the root circle

\(\beta _{2},\beta _{3}\) :

Half angle of the base circle and root circle of the internal gear

\(LSR_{i}\) :

Load sharing ratio of the ith pair of gear teeth

\(\psi _{i}\) :

The position angle of the ith planet gear

\(\gamma _{si}\) :

The relative phasing relationships of the ith sun-planet gear pair

\(\gamma _{ri}\) :

The relative phasing relationships of the ith ring-planet gear pair

\(\gamma _{rs}\) :

The relative phasing relationships between the sun-planet gear pair and ring-planet gear pair

\(K_{spi}\) :

The TVMS of the ith sun-planet gear pair

\(K_{rpi}\) :

The TVMS of the ith ring-planet gear pair

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (Nos. 51705397 and 51875433), the National Key Basic Research Program of China (No. 2015CB057400) and China Postdoctoral Science Foundation (No. 2019T120900).

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Authors and Affiliations

  1. State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, 710061, Shaanxi, People’s Republic of China

    Wei Luo, Baijie Qiao, Zhixian Shen, Zhibo Yang & Xuefeng Chen

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  1. Wei Luo
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Luo, W., Qiao, B., Shen, Z. et al. Time-varying mesh stiffness calculation of a planetary gear set with the spalling defect under sliding friction. Meccanica 55, 245–260 (2020). https://doi.org/10.1007/s11012-019-01115-y

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