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A dynamic calculation method of sliding friction losses for a helical gear pair

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Abstract

Sliding friction losses play a significant role in power losses of a gearbox. The author has proposed a calculation method of sliding friction losses considering the transmission errors. The tooth surface meshing force, relative sliding/entrainment velocity, and sliding friction coefficient are obtained under static or quasi-static conditions which restrict the accuracy of model in the middle- and high-speed gear transmission. The efficiency of dynamic state of a helical gear pair is studied, because it is widely used in the middle- and high-speed transmission. Therefore, a calculation method for helical gear pair of sliding friction losses based on dynamic characteristics is proposed. First, the dynamic model of helical gear pair considering friction and multiple backlashes is built and solved. The parameters associated with the calculation of sliding friction loss are obtained. Second, the tooth surface meshing force and relative sliding/entrainment velocity are calculated and brought into the formula of sliding friction coefficient. The formula of sliding friction losses is given, and the meshing efficiency considering the sliding friction losses in helical gear pair is calculated. Finally, by a comparison with Ref. [1] (Wang et al., J Mech Eng Sci, 230(9):1521–1531, 2016), we mainly propose an idea for calculating the sliding friction losses using dynamics instead of static method or quasi-static method.

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Abbreviations

m p , m g :

Equivalent mass of helical gear pair

ζ l (l = 1,2,…7):

Half backlash of the gear pair and support gaps

k px , k py , k pz , k gx , k gy , k gz :

Support stiffness

c px , c py , c pz , c gx , c gy , c gz :

Support damping

k :

Meshing stiffness

c :

Meshing damping

k y :

Meshing stiffness along the tangential direction

c y :

Meshing damping along the tangential direction

k z :

Meshing stiffness along the axial direction

c z :

Meshing damping along the axial direction

e y :

Meshing error along the tangential direction

e z :

Meshing error along the axial direction

λ = 1 or −1:

Subscript, the value is −1 from engagement to node segment; the value is 1 from node to disengagement segment

\(\bar{y}_{p}\), \(\bar{y}_{q}\), \(\bar{z}_{p}\), \(\bar{z}_{q}\) :

Generalized displacement of pinion and gear along the tangential direction and the axial direction

b i (i = 1, 2, …, 9):

Parameters in the formula of sliding friction coefficient

v e :

Sliding velocity

v s :

Entrainment velocity

S :

RMS composite surface roughness in µm

R :

Effective radius of curvature in meters

SR:

Slide-to-roll ratio, SR = v e /v s

P h :

Maximum hertzian pressure in Gpa

v 0 :

Absolute viscosity at oil inlet temperature in cPs

a :

Center distance between gear and pinion

r pb , r gb :

Radius of base circle of pinion and gear

r pa , r ga :

Radius of addendum circle of pinion and gear

Y p :

Displacement of torsion of pinion

w p t :

Theoretical angular position of pinion

r i (i = 1, 2):

Distance of meshing position to the center of gear i

β :

Helix angle of gear

W :

The gear power of a helical gear pair from mesh-in to mesh-out without considering power losses

w p :

Angular velocity of pinion

Y p :

Displacement of torsion

Ip :

Moment of inertia of pinion

Ig :

Moment of inertia of gear

Tp :

Input torque

Tg :

Output torque

rp :

Reference circle radius of pinion

rg :

Reference circle radius of gear

z 1 :

Number of teeth of pinion

z 2 :

Number of teeth of gear

\(\bar{v}_{p}\) :

Surface velocities of pinion

\(\bar{v}_{q}\) :

Surface velocities of gear

B :

Tooth width

e(t):

Integrated error of gear mesh

References

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Acknowledgments

The author would like to thank the National Natural Science Foundation of China, Shandong Province major key technologies of independent innovation, China Postdoctoral Science Foundation, and Beijing Postdoctoral Research Foundation for their financial support of the Research under Contract Nos. 51475210, 2014GJJS0401, 2014M550577, and 2014ZZ-27. The authors would also like to thank the editors and anonymous reviewers for their suggestions for improving the paper.

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Authors and Affiliations

  1. Beijing University of Technology, Beijing, China

    Cheng Wang & Zhaoyao Shi

  2. University of Jinan, Jinan, China

    Cheng Wang

Authors
  1. Cheng Wang
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  2. Zhaoyao Shi
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Correspondence to Cheng Wang.

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Technical Editor: Kátia Lucchesi Cavalca Dedini.

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Wang, C., Shi, Z. A dynamic calculation method of sliding friction losses for a helical gear pair. J Braz. Soc. Mech. Sci. Eng. 39, 1521–1528 (2017). https://doi.org/10.1007/s40430-016-0585-8

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  • DOI: https://doi.org/10.1007/s40430-016-0585-8

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