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Thermoelastohydrodynamic Analysis of Spur Gears with Consideration of Surface Roughness

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Abstract

Thermoelastohydrodynamic lubrication (TEHL) analysis for spur gears with consideration of surface roughness is presented. The model is based on Johnson’s load sharing concept where a portion of load is carried by fluid film and the rest by asperities. The solution algorithm consists of two parts. In the first part, the scaling factors and film thickness with consideration of thermal effect are determined. Then, simplified energy equation is solved to predict the surfaces and film temperature. Once the film temperature is known, the viscosity of the lubricant and therefore friction coefficient are calculated. The predicted results for the friction coefficient based on this algorithm are in agreement with published experimental data as well as those of EHL simulations for rough line contact. First point of contact is the point where the asperities carry a large portion of load and the lubricant has the highest temperature and the lowest thickness. Also, according to experimental investigations, the largest amount of wear in spur gears happens in the first point of contact. Effect of speed on film temperature and friction coefficient has been studied. As speed increases, more heat is generated and therefore film temperature will rise. Film temperature rise will result in reduction of lubricant viscosity and consequently decrease in friction coefficient. Surface roughness effect on friction coefficient is also studied. An increase in surface roughness will increase the asperities interaction and therefore friction coefficient will rise.

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Abbreviations

b :

Half-width of contact (m)

B :

Roller’s width (m)

C pi :

Specific heat of roller i (J/kg/K)

d wp :

Pitch radius of pinion (m)

d wg :

Pitch radius of gear (m)

E i :

Modulus of elasticity of roller i (N/m2)

E′:

\( \frac{2}{{\frac{{1 - \nu_{1}^{2} }}{{E_{1} }}\, +\, \frac{{1 - \nu_{2}^{2} }}{{E_{2} }}}} \)

F T :

Applied force (N)

F C :

Load carried by asperities (N)

F H :

Load carried by fluid film (N)

F fC :

Asperity friction force (N)

F fH :

Hydrodynamic friction force (N)

f c :

Friction coefficient between asperities

G :

Material number (αE′)

h c :

Central film thickness (m)

H C :

Non-dimensional film thickness

K f :

Thermal conductivity of fluid (W/m/K)

K i :

Thermal conductivity of roller i (W/m/K)

L :

Thermal loading parameter

m i :

ith moment of the surface

n :

Asperity density

P c :

Pressure carried by asperities

R p :

Radius of curvature for pinion (m)

R g :

Radius of curvature for gear (m)

R :

Equivalent radius (m)

sr :

Slide-to-roll ratio

T 0 :

Oil inlet temperature (K)

T :

Temperature (K)

U :

Non-dimensional speed \( \eta_{0} \left( {V_{1} + V_{2} } \right)/E^{\prime}R \)

V i :

Rolling velocity of roller i (m/s)

w(x):

Elastic deformation

W :

Non-dimensional load \( F_{T} /BRE^{\prime} \)

Z :

Viscosity-pressure index

α :

Pinion pressure angle

β :

Radius of tip of asperities (m)

γ 1 :

Scaling factor for hydrodynamic part

γ 2 :

Scaling factor for asperity part

ν i :

Poisson ratio for roller i

σ :

Standard deviation of asperities (m)

μ 0 :

Viscosity at ambient condition (Pa·s)

μ :

Viscosity (Pa·s)

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Acknowledgement

This research was sponsored in part by Caterpillar Inc. and is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Louisiana State University, 2508 Patrick Taylor Hall, Baton Rouge, LA, 70803, USA

    S. Akbarzadeh & M. M. Khonsari

Authors
  1. S. Akbarzadeh
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  2. M. M. Khonsari
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Correspondence to M. M. Khonsari.

Appendices

Appendix A

1.1 Expressions for Radius of Curvature

At the pitch line, the radius of curvature for pinion is:

$$ \rho = \frac{{d_{wp} \sin \alpha }}{2} $$
(A1)

where α is the pressure angle and d wp is the pitch diameter for pinion. At any other diameter, such as d 1, the radius of curvature of pinion is:

$$ \rho_{1} = \frac{{d_{1} \sin \phi_{1} }}{2} $$
(A2)

The angle \( \phi_{1} \) can be found from the relation:

$$ \cos \phi_{1} = \frac{d\cos \alpha }{{d_{1} }} $$
(A3)

The radius of curvature for gear is:

$$ \rho_{2} = \frac{{d_{wp} + d_{wg} }}{2}\sin \alpha - \rho_{1} $$
(A4)

Appendix B

2.1 Expressions for Surface Spectral Moments

Surfaces profiles are usually explained by spectral moments of the surface. If z(x) is the profile in an arbitrary direction x and E[ ] denotes the statistical expectations, then the zeroth, second, and fourth profile moments are:

$$ m_{0} = E(Z^{2} ) = \sigma^{2} $$
(B1)
$$ m_{2} = E\left[ {\left( {\frac{{{\text{d}}z}}{{{\text{d}}x}}} \right)^{2} } \right] $$
(B2)
$$ m_{4} = E\left[ {\left( {\frac{{{\text{d}}^{2} z}}{{{\text{d}}x^{2} }}} \right)^{2} } \right] $$
(B3)

For an isotropic surface, the asperity density and the average radius of the spherical caps of the asperity can be calculated from [20]:

$$ {\text{asperity density}} = n = \frac{{m_{4} }}{{6\pi \sqrt 3 m_{2} }} $$
(B4)
$$ {\text{asperity radius}} = \beta = \frac{3}{8}\sqrt {\frac{\pi }{{m_{4} }}} $$
(B5)
$$ {\text{surface rms}} = \sigma = \sqrt {m_{0} } $$
(B6)

If the surface is anisotropic, the values of m 2 and m 4 will vary with the direction in which the profile is taken on the surface. The maximum and minimum values for m 2 and m 4 occur in two orthogonal principal directions. In this case, an equivalent isotropic surface is defined for which m 2 and m 4 are computed as harmonic mean of m 2 and m 4 along the principal directions.

Appendix C

3.1 Friction Coefficient Due to Asperity Contacts

Friction coefficient in rough contacts has two components: asperity part and hydrodynamic part. The hydrodynamic part as defined in Eq. 33 is a function of sliding velocity. Therefore around the pitch point when sliding is small, the hydrodynamic friction force is very small and at the pitch point where pure rolling occurs hydrodynamic friction force is nil. The asperity friction force, however, depends on the friction coefficient between asperities and the portion of load carried by asperities. Figure C1 illustrates the magnitude of two components of friction coefficient for the operating condition and surface properties reported in Table 4.

Fig. C1
figure 18

Components of friction coefficient

Full size image

The variation of asperity portion of friction coefficient with surface roughness is illustrated in Fig. C2. The rougher the surfaces, the higher the asperity heights and therefore the greater is the asperity–asperity contact. As a result, the asperity friction coefficient increases. Asperity portion of friction coefficient decreases as surface roughness decreases, but as long as the contact is in mixed lubrication regime, this quantity will have a value greater than zero.

Fig. C2
figure 19

Effect of surface roughness on friction coefficient due to asperity contact

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Akbarzadeh, S., Khonsari, M.M. Thermoelastohydrodynamic Analysis of Spur Gears with Consideration of Surface Roughness. Tribol Lett 32, 129–141 (2008). https://doi.org/10.1007/s11249-008-9370-x

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