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Modeling and analyzing of torsional dynamics for helical gear pair considered double and three teeth drive-side meshing

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Abstract

Helical gears are generally considered to be more stable than spur gears. But rattling of the helical gear transmission is found in the engineering practice. The torsional dynamics equations of helical gear pair in high-speed railway gearbox are established in order to reveal the rattling mechanism of helical gear transmission. Double and three teeth pair drive-side meshing are considered. The multi-state meshing zone, load distribution rate and time-varying stiffness determined by contact ratio are analyzed and calculated. The dynamic characteristic transition process of the system is analyzed according to the bifurcation diagrams and the corresponding top Lyapunov exponent (TLE) diagrams, phase portraits, Poincaré maps and time history spectrums of dynamic meshing force based on the calculation of these parameters. The tooth disengagement, tooth back-side contact and their parameter range are found. This study can provide theoretical basis for rattling suppression and transmission stability improvement of helical gear pair.

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Abbreviations

R bp, R bg :

Base circle radius

R ap, R ag :

Addendum circle radius

T 0 :

The time of alternating meshing between three teeth and double teeth

T :

The time of a complete meshing cycle

B :

Tooth width

L dij(τ):

Load sharing ratio of drive-side tooth mesh

F Np31, F Np32, F Np33, F Ng31, F Ng32, F Ng33 :

Positive pressures acting on teeth along the line of action

F fp31, F fp32, F fp33, F fg31, F fg32, F fg33 :

Friction forces perpendicular to the line of action

F m :

Total dynamic meshing forces

F m1 :

Dimensionless total dynamic meshing force

μ :

The friction coefficient

λ dki(τ) (k  = 2, 3) (i = 1, 2, 3):

Direction coefficient of friction force of drive-side tooth mesh

v dki(τ) (k = 2, 3) (i = 1, 2, 3):

Slip velocity of drive-side tooth mesh

R dpi(τ), R dgi(τ) (i  = 1, 2, 3):

Distances from drive-side meshing point to the center of gears

α n,dpi(τ), α n,dgi(τ) (i = 1, 2, 3):

Normal pressures angle in the drive-side meshing points

α t,dpi(τ), α t,dgi(τ) (i = 1, 2, 3):

Transverse pressures angle in the drive-side meshing points

S dpki(τ), S dgki(τ) (k = 2, 3) (i = 1, 2, 3):

Friction moments of drive-side mesh

k bij :

Bending stiffness

k aij :

Axial compressive stiffness

k sij :

Shearing stiffness

k f :

Fillet-foundation stiffness

k h :

Hertzian contact stiffness

α 2 :

Half of the tooth angle measured on the base circle

σ n :

Coangle of the angle between meshing line and tooth centerline

l :

Variation along tooth width

k t :

Dimension time-varying meshing stiffness

k m(t):

Dimensionless time-varying meshing stiffness

μ 1(τ):

Increase of the end meshing length with time

p:

The pinion

g:

The gear

k :

Number of drive-side meshing teeth

i, j :

The codes of different meshing regions

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Acknowledgements

This investigation is financially supported by the Natural Science Foundation of Tianjin, China (Grant No. 18JCYBJC88800), and by the National Natural Science Foundation of China (Grant No. 51365025).

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Authors and Affiliations

  1. School of Mechanical Engineering, Tiangong University, Tianjin, 300387, China

    Xiang-Feng Gou, Hui Wang, Ling-Yun Zhu, Jian-Fei Shi & Zheng-Fa Li

  2. CRRC Qishuyan Institute Co., Ltd., Changzhou, 213011, China

    Hong-Bo Que

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  1. Xiang-Feng Gou
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Correspondence to Ling-Yun Zhu.

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Gou, XF., Wang, H., Zhu, LY. et al. Modeling and analyzing of torsional dynamics for helical gear pair considered double and three teeth drive-side meshing. Meccanica 56, 2935–2960 (2021). https://doi.org/10.1007/s11012-021-01435-y

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