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

• Cheng Wang
• Zhaoyao Shi
Technical Paper

## 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.

## Keywords

Helical gear pair Efficiency Sliding power losses Dynamic characteristics Friction force

## Nomenclature

mp, mg

Equivalent mass of helical gear pair

ζl(l = 1,2,…7)

Half backlash of the gear pair and support gaps

kpx, kpy, kpz, kgx, kgy, kgz

Support stiffness

cpx, cpy, cpz, cgx, cgy, cgz

Support damping

k

Meshing stiffness

c

Meshing damping

ky

Meshing stiffness along the tangential direction

cy

Meshing damping along the tangential direction

kz

Meshing stiffness along the axial direction

cz

Meshing damping along the axial direction

ey

Meshing error along the tangential direction

ez

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

bi (i = 1, 2, …, 9)

Parameters in the formula of sliding friction coefficient

ve

Sliding velocity

vs

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

Ph

Maximum hertzian pressure in Gpa

v0

Absolute viscosity at oil inlet temperature in cPs

a

Center distance between gear and pinion

rpb, rgb

Radius of base circle of pinion and gear

rpa, rga

Yp

Displacement of torsion of pinion

wpt

Theoretical angular position of pinion

ri (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

wp

Angular velocity of pinion

Yp

Displacement of torsion

Ip

Moment of inertia of pinion

Ig

Moment of inertia of gear

Tp

Input torque

Tg

Output torque

rp

rg

z1

Number of teeth of pinion

z2

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

## Notes

### 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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