Gearbox efficiency is becoming increasingly important for vehicle manufacturers to help achieve their overall fuel savings goals. Enhancing gearbox efficiency is also critical in saving the up-front cost and overall unreliability of gearbox cooling. It is well known that at high power levels, for current low-speed gears, gear sliding losses dominate the overall gearbox losses. Therefore, accurately predicting frictional losses is critical for increasing overall gearbox efficiency. Previous work by these authors has shown that available closed-form calculations do not provide the range of important inputs or accuracy required to perform reliable design estimates of the sliding losses that are so important to the thermal and efficiency characteristics of the gearbox. This paper documents an approach used to incorporate the effect of lubrication characteristics, gear geometry, surface finish, and operating conditions into an algorithm that accurately predicts sliding losses over a range of operating conditions for a standard set of gears. This study provides a method for accurately calculating gear sliding losses based on all the important design variables early in the process so that efficiency can be more easily assured. The methodology developed for simple contacts is used to predict gear sliding losses for much more complicated cases of spur and helical gears, where load and rolling and sliding speed of the contact patch varies at each roll angle during the mesh cycle.

### Introduction

The prediction of the coefficient of friction and resulting sliding losses in lubricated sliding-rolling contacts is a challenging problem. Under heavy loads and varying shear rates, the behavior of the shear stress, which determines the traction coefficient, is complex. The traction coefficient magnitude is highly influenced by the viscosity of the lubricant, which varies with pressure and temperature within the contact. To evaluate the traction coefficient, the shear thinning of the non-Newtonian lubricant with increasing shear rate under varying contact pressures, which dominates in the region of contact pressure studied, must be considered. Finally, the limiting shear stress of the lubricant because of its viscoelastic behavior must be calculated.

For typical operating loads, the contacting surfaces also undergo elastic deformation. If the materials of the contact surfaces are exposed to extreme pressures, the possibility of failure of the gear tooth because of pitting, scuffing, and wear exists. The contacts, by design, operate under the elasto-hydrodynamic (EHL) lubrication condition. In this piezo viscous-elastic regime, the lubricant behaves like a viscoelastic material with high magnitudes of viscosity because of large Hertzian pressures. In addition, the lubricant viscosity changes with temperature due to the viscous shear caused heat generation within the contact. Thus, there is a strong coupling between the traction force, sliding and film temperature.

This paper describes a methodology developed to incorporate the effect of lubrication characteristics, gear geometry, surface finish, and operating conditions to accurately predict sliding losses over a range of operating conditions.

Initially, the results of the methodology used to predict the traction coefficient obtained from a mini traction machine (MTM) are presented. A close agreement is observed between the experimental results and the predictions. Then, this methodology was used to predict the sliding loss for FZG Type-C gear sets at several speed-load combinations. Results from comparing predicted and experimental results are presented. The close agreement between predictions and experimental results provide confidence in using this technique in predicting gear sliding losses.

The initial calculations are focused on obtaining the friction coefficient for λ values larger than 3, defined as film thickness to mean surface roughness ratio, where the contact is in the thick film region.

The Ree-Eyring formula, incorporating lubricant viscosity at a low shear rate corrected for the contact pressure and temperature, is used to compute the shear stress. A model describing the pressure-temperature-viscosity relationship developed by Bair [1] was integrated into the traction loss prediction methodology.

After the traction loss prediction methodology was developed, it was used as a basis of calculating sliding losses in the FZG TYPE-C gear set at the various operating conditions. For the target gear set, the mesh cycle was broken in 25 segments. Then AGMA 925-A03 [2] was used to calculate the normal force, rolling speed, sliding speed, and effective tooth radius at each segment, These operating conditions and the oil rheology were used as input to the calculation of sliding losses at each segment of the mesh cycle, and processed to predict the sliding losses for the gear set at each of these conditions.

### Comparison of the Predicted Traction Coefficient with Mini Traction Machine (MTM) Results

Initial comparisons are conducted for simple contacts, i.e., ball on a disk and barrel on a disk of the MTM tests shown in Figure 1. For this machine, at a constant slide to roll ratio, the reduction in speed results in thinner EHL film. The thinner film may lead to the transition from full film to mixed and then boundary lubrication regimes. For contact pressures up to 1.25 GPa, a ball on a disk with a circular contact area is used. For contact pressures between 1.25 GPa and 3 GPa, a barrel on a disk with an elliptical contact area is used. These comparisons are focused on the region of the Stribeck curve corresponding to large values of λ, defined as film thickness to contact roughness ratio. In this region, the contact occurs in the thick lubricating film region, where there is no asperity contact between the two surfaces.

For tests in large values of λ region, smooth surfaces with an average roughness of σ_{x} = 20 nm are used. To calculate this average, the following equation is used:

In the above equation, Ra_{1x} , and Ra_{2x} , are measured average surface roughness for surfaces 1 and 2, respectively. For calculating the average surface surfaces, the filter cutoff wavelength L_{x} , comparable to the width of the Hertzian contact band was used. The value of specific film ratio λ, is defined by

where h is the central film thickness. For all the comparisons presented using the models in this section, λ is larger than 8, representing thick film lubrication conditions.

The traction coefficient μ_{f} (p_{m},T) at the mean contact pressure p_{m} and the contact temperature T is calculated by the following simple relationship:

The Ree-Eyring [3] formula is used to compute the shear stress τ:

Where:

τ_{E} is the Eyring stress.

η(p_{m},T) is the dynamic low-shear viscosity at the mean contact pressure and the temperature.

γ⋅ is the shear rate.

A viscosity relationship describing the variation of viscosity values η(p,T) with pressure and temperature was developed by Bair [1] using experimental results of a high-pressure viscometer. This model was generated for a lubricant that is used in all of the tests and calculations presented here.

The shear rate γ⋅ is determined by the following formula:

Where ∆U is the contact sliding speed, and is defined by:

Where U_{1} and U_{2}, are the rolling speeds of body “1” and “2” at the contact region, respectively.

For up to maximum contact pressures of 1.25 GPa, in the MTM tests, where a ball on disk is employed, the maximum pressure is obtained by the Hertzian formula:

Where:

W is the applied load.

E´ is reduced Young’s modulus.

R is the radius of the ball.

The reduced Young’s modulus is defined as:

Where:

ν_{1} and ν_{2} are Poisson’s ratios of the contacting bodies “1”, and “2”, respectively.

E_{1} and E_{2} are Young’s moduli of the contacting bodies “1”, and “2”, respectively.

The mean pressure “p_{m}” and the radius of the contact patch a_{c}, for this circular contact region, are defined as:

For maximum contact pressures between 1.25 GPa and 3.0 GPa in the MTM tests, where the barrel and disk are employed, the mean contact pressure is determined by the Hertzian formula:

Where:

a_{e} is the major half-width of the contact ellipse;

b_{e} is the minor half-width of the contact ellipse;

R_{x} and R_{y} are the equivalent radii;

and R´ is defined,

Where:

R_{x1}, and R_{x2} are radii of curvature for body “1”, and “2” in the x-direction, respectively;

R_{y1}, and R_{y2} are radii of curvature for body “1”, and “2” in the y-direction, respectively.

Hamrock and Dowson [4], [5], and [6] were among the first investigators to present numerical results for circular contacts. Based on their numerical results, they derived a formula to predict the central film thickness “h” as a function of operating conditions shown below. Chittenden, et al. [7] used a similar approach to derive the film thickness for elliptical contacts.

The Hamrock–Dowson equation for circular contact is given by:

and Chittenden for elliptical contact, long ellipse axis is along rolling/entrainment direction, R_{x }≥ R_{y} is:

Where:

U is the rolling or entrainment speed defined by surface or rolling speeds of body “1”, i.e., U_{1} and body “2”, i.e., U_{2} and is given by:

η_{0} is the dynamic viscosity at the temperature and pressure of the contact inlet;

α is the measured pressure viscosity coefficient of the oil.

The following relationship for the lubricant temperature rises at the contact ∆T, due to sliding relative to the inlet lubricant temperature T_{0} is derived by Olver, et. al. [3].

Where α_{h} his the proportion of heat entering body 1 and is given by the following equation:

B_{1} and B_{2} are transient thermal resistance due to the hot spot for bodies 1 and 2, respectively, and are given by:

Where:

A is the area of the contact.

K_{oil} is the thermal conductivity of the lubricant.

K is the thermal conductivity of steel bodies in contact.

ρ is the density of steel bodies in contact.

c is the specific heat of the steel bodies in contact.

For the calculations presented in this paper, the thermal responsivity of the surfaces of the two bodies defined by Olver [8], which is a function of the geometry of the bodies, their thermal conductivity, and the convective heat-transfer coefficient between the surface of each body and the lubricant is assumed to be negligible. This assumption was made as a first approximation for the prediction of the lubricant temperature at the contact. Although the temperature predictions for the cases presented in this paper provide a reasonable comparison with the test results, the thermal responsivity of the gear and shaft surfaces must be included in temperature predictions as described by Olver, et. al. [3]. The values of constants are given:

K_{oil} = 0.14 W/K-m

K = 59 W/K-m

ρ = 7,600 kg/m^{3}

c = 452 J/kgK

α = 0.0132667 mm^{2}/N, for a nominal viscosity of 28 mPa-sec at 80°C

The viscosity of the lubricant and its behavior with pressure and temperature is the prime determinant of the fluid-traction coefficient. The preceding equations for calculations of shear stress, mean contact pressure, central film thickness, shear rate, the temperature rise of the lubricant due to sliding losses, and the empirically derived equation between viscosity and pressure and temperature are used to calculate the lubricant traction coefficient μ_{f} (p_{m},T) at values of specific film ratios λ corresponding to the full fluid EHL regime.

The comparison of these predictions with those measured using a mini traction machine is presented in Figures 2 through 4. A comparison of the measured traction coefficient values at a maximum contact pressure of 0.9 GPa and rolling speeds of 0.88 m/sec, 2.62 m/sec, and 3.2 m/sec is shown in Figure 2. For these comparisons, the value of Eyring stress is calculated to be 7 MPa. A comparison of the measured traction coefficient with predictions at maximum contact pressure 1.25 GPa and rolling speeds 0.88 m/sec, 2.62 m/sec, and 3.2 m/sec is shown in Figure 3. For these comparisons, the value of Eyring stress is calculated to be 7.8 MPa. A comparison of the measured traction coefficient with predictions at maximum contact pressures 2, 2.5, and 3 GPa and rolling speed 2.5 m/sec is shown in Figure 4. For these comparisons, the value of Eyring stress is calculated to be 8.8 to 9.5 MPa. For large values of specific film ratios λ, in the range of the calculated shear rates, and different rolling speeds, and loads, there is an excellent agreement between measurement and predicted results as shown in Figures 2 through 4.

### Rough Surface Mean Coefficient of Friction Predictions

Experiments by Smeeth and Spikes [9] suggested that an effective mean coefficient of friction μ may be estimated from fluid generated traction using the following universal empirical relation with dependency on λ in the EHL region:

Where:

μ is effective mean friction coefficient.

μ_{f} is traction coefficient at high lambda.

μ_{b} is friction coefficient at λ=0.

m is exponent parameter.

Once μ_{f} was calculated and verified experimentally using the results of the tests obtained by MTM on the test lubricant for large values of specific film ratio, i.e., for values of λ in the full film EHD regime, the above relationship was curve fitted to the MTM friction coefficient test results obtained for rough surfaces at a wide range of λ values, i.e., 0.06 <λ<1.5 at maximum contact pressures of 1.25 GPa, and 2.5 GPa, separately.

The exponent m is a constant that determines the value of λ where μ approaches the value of μ_{f}, and is found to be m = 2 for the current MTM test results. This exponent is normally found to be between 2 and 3, depending on the value of λ being between 5 and 3 in the full fluid lubrication region, i.e., for larger magnitudes of m, lift-up is achieved at smaller values of λ.

The value of the friction coefficient for boundary lubrication, μ_{b}, corresponding to λ = 0 was found to be 0.116. This value was obtained by extrapolating the graph of traction coefficients, measured by MTM, versus calculated λ values to λ = 0. The value of 0.116 for friction coefficient at boundary lubrication was not obtained experimentally, i.e., it was not measured under pure sliding contact.

Because the above curve fitted equation with m = 2 provided a reasonable relationship describing the behavior of mean coefficient of friction for the loads and entrainment speeds used in the MTM tests, this relationship is used to predict the mean coefficient of friction for the FZG tests presented in the next section and the comparison of these predictions with test results.

### Comparison of the Predicted and Measured Sliding Power Loss for FZG Type-C Gear Set

The sliding power loss comparisons between the predictions and the actual measured values for an FZG Type-C gear set at three applied torques of 100, 200, and 300 Nm are presented here. For each applied torque values, the tests were conducted using the Ohio State University’s FZG test facility, at four driver speeds of 595, 1,785, 2,975, and 5,000 rpm. This test facility is described in detail by Moss, et.al. [10]. The parameters for this gear set are presented in the following table. The gears had no lead crown and mean surface roughness for the gear set was 220 nm. The contact ratio or average number of teeth in contact was calculated to be 1.427.

For each of the 12 tests, the gear mesh cycle was broken into 25 segments. The gear parameters of Table 1 and appropriate equations in AGMA 925–A03 [2] are used to calculate the normal force, rolling speed, sliding speed, and reduced radius of curvature at each segment in the mesh cycle. Hertzian contact pressure for contact parameters between two parallel cylinders provided by Stachowiak, et.al. [11] was used. Using the following formulas, the maximum contact pressure “p_{max},” mean contact pressure “p_{m},” and semi-width of the Hertzian contact rectangle “a” were calculated, assuming the load was distributed uniformly along the tooth face width in the mesh cycle,

where “l” is half length of the contact rectangle, which, for this case, is half of the face-width of 14 mm.

Once these were known, the film thickness and the resulting film temperature were calculated for each point in the mesh cycle.

Grubin’s equation [10] for film thickness was used for the line contact generated by these gears:

The temperature rise for the lubricant in each of the 25 segments of the mesh cycle was calculated similarly to what was described in the previous section. The surface finish input used for these gear calculations was the average of the measured surface roughness.

Finally, knowing the contact pressure and the temperature of the lubricant film, its viscosity for each segment was calculated, and the resulting traction coefficient was found, as per the same procedure as the MTM calculation, which involved iterating to a successful simultaneous calculation of temperature and heat flow. From there, using the calculated λ, a μ_{b} value of 0.116 and m value of 2, the power loss for each segment was calculated by obtaining μ from the rough surface mean coefficient of friction predictions, multiplying μ by the mean contact pressure at the segment to find the shear stress and multiplying it by the corresponding segment sliding speed and the contact area.

After the power loss for each segment of the mesh cycle was calculated, the average power loss was found by adding all the power losses for all the segments for a given running condition and dividing by 25. This average was multiplied by the contact ratio 1.427 for this gear set, to determine the total power loss. This is because for each mesh cycle, 42.7 percent of the time, there are two pairs of teeth in the mesh that are engaged. This was compared to the experimental power lost for each test.

Figures 5 through 8 show the normalized results comparing predicted- with measured-sliding losses. The normalization value for all of these figures is arbitrary chosen as a constant number. In the primary operating range of commercial gearing shown by the 1,785 and 2,975 rpm results, the predictions are within 15 percent. At high speed and high torque, there is an anomaly with a 26-percent difference between predicted results and measurements. There are also significant differences between calculations and measurements at low speed, although the absolute differences are not significant. To determine the sliding losses due to gears, the FZG test-stand losses were first measured with a torque cell, under a known test load and speed, and then at no load at the same speed. The no-load losses were then subtracted from the total losses to determine the net-loaded gear and bearing losses for each test condition. The loaded bearing losses were then calculated and subtracted from that loss to determine the gear sliding loss. The loaded bearing loss calculation was not verified and may be a source of error in the results.

### Conclusions

The gear sliding loss prediction technique discussed in this article relies on the use of existing appropriate parametric relationships for contact pressure, shear stress, heat conduction, and, most importantly, the use of experimentally determined dynamic low-shear viscosity at high pressure and temperature. This methodology yields comparable accuracy to the best methodology analyzed in Reference 12 – ISO-14179-2 (Hohn’s Modification). In addition, this prediction approach accounts for all relevant gear-design parameters, lubrication properties, and operating conditions. This technique eliminates the need for FZG gear testing as in ISO-14179-2, from which a lubrication factor XL is derived. The prediction algorithm can be extended to helical gearing by separately analyzing the individual lines of contact. Based on its effectiveness, it can serve as a basis for advancement in the accuracy and efficacy of calculation of gear-surface degradation, including scuffing and pitting, as all are significantly affected by the tribological factors used in this analysis.

### Acknowledgment

This work was supported by Western Michigan University’s Center for Advanced Vehicle Design and Simulation (CAViDS).

### Bibliography

- Bair, S., May 2017, A Characterization of the Pressure, Temperature and Shear Dependence of Viscosity of Transmission Oils, Technical Report.
- AGMA 925–A03, Effect of Lubrication on Gear Surface Distress.
- Olver, A.V., Spikes, H.A., 1998, “Prediction of Traction in Electrohydrodynamic Lubrication,” Proceedings of Institute of Mechanical Engineers, Vol. 212, Part J.
- Hamrock, B.J., and Dowson, D. ,1976, “Isothermal Electrohydrodynamic Lubrication of Point Contacts. Part 2-Elliticity Parameter Results,” Transactions of ASME Journal of Lubrication Technology, 98, pp. 375-383.
- Hamrock, B.J., and Dowson, D., 1977, “Isothermal Electrohydrodynamic Lubrication of Point Contacts. Part III-Fully Flooded Results,” Transactions of ASME, Journal of Lubrication Technology, Vol. 99, pp. 264-276.
- Hamrock, B.J., and Dowson, D., 1981, Ball Bearing Lubrication, The Electrohydrodynamic of Elliptical Contacts, John Willey & Sons.
- Chittenden, R.J., Dowson, D., Dunn, J.F., and Taylor, C.M., 1985, “A Theoretical Analysis of the Isothermal Electrohydrodynamic Lubrication of Concentrated Contacts 2. General Case, with Lubricant Entrainment Along Either Principal Axis of the Hertzian Contact Ellipse or at some Intermediate Angle,” Proceedings of the Royal Society London A, 397, (pp. 271-294.
- Olver, A.V., 1991,” Testing Transmission Lubricants: the Importance of Thermal Response,” Proceedings of Institute of Mechanical Engineers, Part G, Journal of Aerospace Engineering, Vol. 206, Part G1, pp. 35-44
- Smeth, M., and Spikes, H.A., 1995, “The Influence of Slide Roll Ratio on the Film Thickness of an EHD Contact Operating Within the Mixed Lubrication Regime” Presented at the Twenty-Second Leeds-Lyon Symposium on Tribology, The Third Body Concept, Lyon, France, 5-8 September 1995.
- Moss J, Kahraman A, Wink C, “An Experimental Study of Influence of Lubrication Methods on Efficiency and Contact Fatigue Life of Spur Gears” Journal of Tribology, 2018;140(5)
- Stachowiak G. W., and Batchelor, A.W., 2013, Engineering Tribology, Fourth Edition, Chapter 7, Butterworth-Heinemann.
- Gurd, C., Wink, C., Bair, J., Fajardo, C., 2019, Computing Gear Sliding Losses, AGMA Technical Paper, 19FTM11

^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2020 at the AGMA Fall Technical Meeting. 20FTM05}