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Mesh stiffness model of a spur gear pair with surface roughness in mixed elastohydrodynamic lubrication

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Abstract

The calculation of time-varying mesh stiffness for gear meshing in mixed EHL regime is of great importance to the accurate evaluation of tooth damage, contact fatigue life and wear performance of a gear transmission system. In this work, the mesh stiffness of a spur gear pair in mixed elastohydrodynamic line (EHL) contact is established in conjunction with a revised contact stiffness to include the effect of surface roughness and oil film. The revised contact stiffness of gear tooth surface in EHL contact is developed by combining the stiffness of both the rough gear tooth and liquid film based on the load-sharing concept, which is used to replace the Hertzian contact stiffness of ideal smooth cylinders in traditional gear mesh stiffness. To include the effect of tooth curvature on the asperity distribution at the gear tooth surface, the cylindrical contact coefficient is introduced and incorporated into the statistical micro-contact Greenwood and Williamson model (GW model) to derive the stiffness of rough curved gear tooth contact. The film thickness equation for mixed EHL line contact is employed together with the lubricant bulk modulus to predict the liquid film stiffness at different mesh positions. Effects of surface roughness, input torque, rotating speed and lubricant on the contact stiffness and EHL mesh stiffness are analyzed. Results show that the lubricant film stiffness is much higher than the solid part, especially at tip or root position. The fluctuation of mesh stiffness in single-to-double teeth contact is smaller than that calculated using Hertzian contact model, indicating a better transmission stationarity.

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Abbreviations

A n :

nominal contact area between two rough flat surfaces, m2

A i :

equivalent cross-sectional area, m2

B :

bulk modulus, GPa

B 0 :

bulk modulus at ambient pressure, GPa

\(B_{0}^{\prime }\) :

pressure change rate at ambient pressure, GPa/s

B 00 :

bulk modulus at ambient pressure and absolute zero temperature, GPa

d d :

distance between mean of summit heights and that of surface heights, m

d i :

distance from the load point to the micro-element, m

d n :

dimensionless distance between the mean of summit heights and that of the surface heights, dn = dd/σ

E 1 E 2 E :

modulus of elasticity of first and second cylinder and the effective modulus of elasticity, GPa

e i :

cross-sectional width of the micro-element, m

F :

normal load, N

F C :

asperity load, N

F H :

fluid load, N

\(\overline{F}\) :

dimensionless normal load, \(\overline{F} = \sqrt {\frac{{4\pi L_{c} RE}}{F}}\)

G :

equivalent shear modulus, GPa

G 1 ,G 2 :

shear modulus of first and second cylinder, GPa

G c :

dimensionless material parameter

H :

separation of the mean line of the rough surface and the flat surface, m

h n :

dimensionless separation, hn = h/σ

\(\overline{h}\) :

dimensionless film thickness, \(\overline{h} = \frac{h}{R}\)

I i :

equivalent cross-sectional modulus, GPa

K a :

solid asperity contact stiffness, N/m

k a :

axial compressive stiffness, N/m

k b :

bending stiffness, N/m

k c :

contact stiffness of the gear pair, N/m

k f :

stiffness due to the fillet foundation deflection, N/m

k g :

stiffness of the contact asperities at gear tooth surfaces, N/m

k h :

Hertzian contact stiffness between ideal smooth cylinders, N/m

k l :

stiffness of the lubricant film, N/m

k s :

shear stiffness, N/m

L :

tooth width, mm

N :

total number of asperities

N c :

total number of asperities deformed at curved meshing surface

n :

rotating speed, r/min

n s :

asperity distribution density, m−2

n g :

dimensionless asperity density, \(n_{g} = n_{s} R\sqrt {\beta R}\)

R 1 R 2 R :

radius of first and second cylinder and the effective radius, m

T t :

Temperature, K

u :

relative motion velocity, m/s

v :

Poisson’s ratio

v 1 v 2 :

Poisson’s ratio of cylinders

W :

dimensionless load

z :

asperity height measured from the mean line of summit heights, m

α :

pressure–viscosity coefficient, GPa−1

α 1 :

pressure angle, deg

β :

asperity radius, μm

ω 1, ω 2 :

angular speed of first and second cylinder, rad/s

γ 1, γ 2 :

scaling factors for hydrodynamic part and asperity contact part

σ :

standard deviation of the surface heights distribution

σ s :

standard deviation of asperity heights distribution

σ sn :

dimensionless standard deviation of asperity heights, σsn = σs/R

β k :

Tait-Doolittle model constant

λ :

film thickness parameter, λ = h/σs

λ c :

cylindrical contact coefficient

η 0 :

inlet viscosity, Pa s

\(\phi (z)\) :

probability density function of Gaussian distribution

\(\phi_{n} (z_{n} )\) :

dimensionless standard normal distribution function

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Acknowledgements

This work was supported by the National Natural Science Foundation of China [Grant number 51775037].

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Huifang Xiao, Jinshan Gao & Jiazhou Wu

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  1. Huifang Xiao
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  2. Jinshan Gao
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Correspondence to Huifang Xiao.

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Technical Editor: Zilda de Castro Silveira.

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Xiao, H., Gao, J. & Wu, J. Mesh stiffness model of a spur gear pair with surface roughness in mixed elastohydrodynamic lubrication. J Braz. Soc. Mech. Sci. Eng. 44, 136 (2022). https://doi.org/10.1007/s40430-022-03397-y

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  • DOI: https://doi.org/10.1007/s40430-022-03397-y

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