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

  • Cheng WangEmail author
  • Zhaoyao Shi
Technical Paper


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.


Helical gear pair Efficiency Sliding power losses Dynamic characteristics Friction force 


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


Meshing stiffness


Meshing damping


Meshing stiffness along the tangential direction


Meshing damping along the tangential direction


Meshing stiffness along the axial direction


Meshing damping along the axial direction


Meshing error along the tangential direction


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


Sliding velocity


Entrainment velocity


RMS composite surface roughness in µm


Effective radius of curvature in meters


Slide-to-roll ratio, SR = v e /v s


Maximum hertzian pressure in Gpa


Absolute viscosity at oil inlet temperature in cPs


Center distance between gear and pinion

rpb, rgb

Radius of base circle of pinion and gear

rpa, rga

Radius of addendum circle of pinion and gear


Displacement of torsion of pinion


Theoretical angular position of pinion

ri (i = 1, 2)

Distance of meshing position to the center of gear i


Helix angle of gear


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


Angular velocity of pinion


Displacement of torsion


Moment of inertia of pinion


Moment of inertia of gear


Input torque


Output torque


Reference circle radius of pinion


Reference circle radius of gear


Number of teeth of pinion


Number of teeth of gear


Surface velocities of pinion


Surface velocities of gear


Tooth width


Integrated error of gear mesh



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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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2016

Authors and Affiliations

  1. 1.Beijing University of TechnologyBeijingChina
  2. 2.University of JinanJinanChina

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