## Abstract

A high-speed electric motor with a small reducer that has high-power transmission efficiency can be used to realize a high-power-density powertrain system because electric motors can be miniaturized to increase the rotational speed. A traction drive has low vibration noise due to its lack of meshing vibration, making it suitable as a transmission element for high-speed reducers. However, the traction coefficient, which greatly affects transmission performance, decreases with increasing rotational speed. In this study, to increase the traction coefficient using a surface texture, a model that takes into account transient temperature changes under high-speed conditions and the effects of micro-surface geometry was developed. The traction coefficient was measured using a high-speed test machine capable of operating at a maximum speed of 50,000 rpm. The model was able to predict the experimental values with an error of at most 6%. The high-pressure rheological properties of the oil were examined to develop design guidelines for the surface texture and a model was used to optimize the texture parameters. The designed texture was manufactured and evaluated. Experimental results show that the traction coefficient can be improved by up to 19%.

## 1 Introduction

Due to increasing demand for reductions in carbon dioxide emissions, many conventional vehicles are expected to be replaced by hybrid, electric, or fuel-cell vehicles. Motor miniaturization is important for reducing vehicle weight and maximizing occupant space. Motor output is the product of torque, which depends on the motor diameter and rotational speed, and thus motors can be made smaller while maintaining output by increasing the rotational speed. For example, the Toyota Prius increased the motor rotational speed from 5000 rpm to 17,000 rpm when it was first released, achieving a significant reduction in size while increasing output [1]. The European consortium DRIVEMODE has developed a unit for electric vehicles with a maximum rotational speed of 20,000 rpm [2]. Volk and Leighton have proposed a 30,000 rpm unit [3] and the Technical University of Munich studied the design of a 50,000 rpm unit [4]. Although increased losses, centrifugal force, vibration, cooling, and other factors must be taken into account, higher rotational speeds have become a major trend.

To provide sufficient driving force to a vehicle with a high-speed motor, a combination of reduction gears is required. Traction drives are an alternative to gears for high-speed rotation. They transmit power by shearing an oil film that forms between two rolling elements. Traction drives are suitable for high-speed rotation because they have no meshing vibration and low oil agitation resistance. However, the traction coefficient (ratio of transmission force to contact force on the transmission portion) decreases with increasing rolling speed of a traction drive because an increase in velocity causes the oil film to thicken in accordance with fluid lubrication theory [5], resulting in a decrease in shear strain rate and an increase in oil film temperature. To maintain the required transmission force, the contact force must be increased to compensate for a decreased traction coefficient, resulting in lower transmission efficiency and the requirement of a larger transmission. The author previously designed and built a high-speed traction drive testing machine (Fig. 1(a)) to investigate traction characteristics at high rotational speed, and experimentally confirmed that the traction coefficient drops by up to 20% at 75 m/s compared to that at several meters per second (Figs. 1(b) and 1(c)) [6]. Such a large drop in the traction coefficient has a significant impact on vehicle performance.

Fig. 1
Fig. 1
Close modal

Various methods have been developed to increase the traction coefficient, including those that modify the traction oil and those that apply a surface texture. Tsubouchi and Hata derived an equation that expresses the correlation between the molecular structure (e.g., stiffness and shape) and the traction coefficient [7]. One study attempted to calculate traction coefficients from molecular structures using molecular dynamics simulation [8]. However, there is a trade-off between the traction coefficient of oil and low-temperature flowability. It is thus difficult to increase the traction coefficient without reducing the low-temperature flowability of automotive oils, which are operated at extremely low temperatures. Nambu et al. applied microgrooves, a type of surface texture, to the transmission surface and thinned the oil film to increase the strain rate and improve the traction coefficient (an improvement of approximately 5% was observed) [9]. However, direct contact tends to occur at high loads, resulting in peeling, so the use of microgrooves is limited to low loads. The traction coefficient was measured at 15 m/s, so the effect at high speeds is unclear. There is thus a need for a texture that effectively improves the traction coefficient at high speeds and that can withstand high loads.

To design a surface texture that has both a high traction coefficient and high strength under high-speed operation, a traction model that agrees with experiments at high speeds is required. Itagaki et al. built a high-power traction drive testing machine and fit a nonlinear Maxwell model [10] to the obtained traction coefficients [11]. However, the model was fitted at peripheral velocities of 40 m/s or lower. The accuracy of the model at higher velocities has not been confirmed.

This study constructs a model for calculating the traction coefficient at high rotational speeds and uses it to create a texture that suppresses the decrease in the traction coefficient with increasing rotational speed. The model considers the non-Newtonian viscosity of the oil, changes in oil properties due to an increase in temperature, and the oil film shape and pressure distribution due to the surface microstructure. A test roller with a texture designed using the model is fabricated and its effectiveness is experimentally confirmed using a high-speed traction drive testing machine.

## 2 Traction Drive Model

### 2.1 Overall Structure.

Although the Reynolds equation can be used to accurately calculate the oil film shape and the pressure distribution, it is difficult to apply it to estimate the traction force because it assumes Newtonian viscosity. The nonlinear Maxwell model, which describes the non-Newtonian behavior of oil, is thus often used [10]. This model is employed in the present study.
$1Gdτdt+τ0ηpsinhττ0=γ˙$
(1)
where
$τ=τc(τ>τc)$
(2)

Traction drives generate heat due to micro slip, which is referred to as creep. Changes in rheological properties due to an increase in temperature affect the oil film thickness and the traction force. In order to account for the increase in temperature, traction calculations can be substituted into thermal elastohydrodynamic lubrication (EHL) equations to simultaneously calculate the oil film shape, pressure distribution, and traction force [12]. However, this method is highly nonlinear and requires a long calculation time. The objective of this research is to optimize texture design. Because many calculations are performed, a method with a small computational load is necessary. Therefore, the proposed method is based on an iterative calculation of the oil film thickness and pressure distribution, a traction calculation, and a temperature calculation. These calculations are repeated until the results become converge. For application to texture design, the oil film thickness and pressure are distributed based on EHL numerical calculations instead of using average values, and the traction force and increase in temperature at each location on the contact surface are calculated.

Under the operating conditions used in this study, the calculated minimum oil film thickness is about 0.2–1 µm and the surface roughness of the roller is less than Ra = 0.04, so the film thickness ratio is more than five. An analysis by Johnson et al. indicated that two rollers are sufficiently separated by an oil film if the film thickness ratio is greater than three [13]. Therefore, the friction due to direct contact is not considered here and the oil film is assumed to transmit all power.

### 2.2 High-Pressure Properties of Oil.

It is difficult to theoretically determine the high-pressure properties of oil given in Eqs. (1) and (2). Therefore, experimental equations were derived based on the experimental values for the operating conditions. The oil used in this study was Idemitsu Kosan’s traction fluid KTF-1, which was developed for automotive traction drive transmissions [14].

#### 2.2.1 Viscosity.

Nakamura et al. measured viscosity up to pressures of 2 GPa using a diamond anvil cell [15]. Their values well agree with those measured using a falling-ball tester [16] and are close to those calculated based on free-volume theory [17]. Based on these measurements, the following experimental equation was derived. This equation is a modified version of an equation derived by Hata and Tamoto [18], in which the coefficients are changed to match the measured values. A comparison of the measured and calculated values is shown in Fig. 2.
$ηp=η0exp(αpp)$
(3)
$αp=α01+α0[−0.6C+0.108]exp[{18C−3.8}p](2.256×10−3×T+0.8987)$
(4)
$C=0.0204(T+273)ρ0278ln(2.51×278ν0)$
(5)
Fig. 2
Fig. 2
Close modal

#### 2.2.2 Shear Modulus.

Ohno et al. measured the shear modulus under high-pressure using a pressure vessel [19]. Their value significantly differs from that calculated from a traction curve obtained using McCool’s equation [20]. Under the assumption that the effect of the shear modulus of oil is small and the stiffness of the roller is dominant, Evans and Johnson derived a conversion equation for the slope in the linear range [21]. This equation is used in the present study.
$1G=1Gf+a0.56E′h$
(6)

#### 2.2.3 Eyring Stress.

Watanabe et al. performed traction tests under isothermal conditions [22]. They measured the bulk temperature of the roller and calculated the Eyring stress. A higher oil temperature and a lower pressure led to a higher Eyring stress. The following equation was developed to represent this trend, where C1C5 are coefficients that are matched to the measured traction coefficients [6].
$τ0=C1T2+C2T+C3+C41+C5p2$
(7)

#### 2.2.4 Limiting Shear Stress.

Attempts to determine the limiting shear stress from oil properties have been made (e.g., Ref. [19]). However, it is difficult to predict the limiting shear stress with sufficient accuracy. Therefore, the following equation is derived, where the coefficients C6C9 are matched to the measured traction coefficients.
$τc=(C6+C7p+C8T+C9pT)p$
(8)

### 2.3 Oil Film Temperature Calculation.

The oil film temperature Tf is the sum of the roller bulk temperature Tb, the increase in roller surface temperature ΔTs, and the increase in oil film temperature ΔTf.
$Tf=Tb+ΔTs+ΔTf$
(9)

The temperatures were calculated as described below.

#### 2.3.1 Roller Bulk Temperature.

The roller bulk temperature is difficult to calculate because it depends on heat generation and cooling, heat capacity of the testing machine, heat conduction, and other factors. Therefore, the roller surface temperature at 90 deg before the contact point was measured using a thermocouple for moving surfaces, as shown in Fig. 3. This temperature was taken as the bulk temperature.

Fig. 3
Fig. 3
Close modal

#### 2.3.2 Increase in Roller Surface Temperature.

The equation for a moving heat source given by Carslow and Jaeger [23] is used.
$ΔTs(x,y,t)=14(πKs)3/2ρscs∫∫∫0tq(ξ,ζ)τ3/2exp{−(x−ξ+uχ)2+(y−ζ)24κsτ}dχdξdζ$
(10)
In general, Eq. (11), which integrates Eq. (10) from t = 0 to t = ∞ to simplify the calculation, is used. However, at the high rotational speeds considered in this study, the heat source may pass a certain point on the roller before the temperature rises.
$ΔTs(x,y)=12κsρscs∫∫q(ξ,ζ)exp{−u2Ks(x−ξ)2+(y−ζ)2}(x−ξ)2+(y−ζ)2dξdζ$
(11)

Therefore, the results of steady-state calculations were compared with those of transient calculations. The upper limit of time integration was defined as the time required to pass through one mesh, i.e., the peripheral velocity divided by the length of the mesh. Figure 4 shows the temperature increase rate distribution in the rolling direction for a heat source at a single point on the surface at a peripheral velocity of 75 m/s and a maximum Hertzian pressure of 3 GPa. In the unsteady-state calculations, the range over which the temperature increases is significantly narrower. When the temperature rise is integrated over the entire contact surface using Eq. (10) under the condition, the unsteady-state temperature rise is 5 °C compared with a steady-state rise of 26 °C, which is a difference of approximately five-fold, indicating that unsteady-state calculations are required at high speeds.

Fig. 4
Fig. 4
Close modal

#### 2.3.3 Increase in Oil Film Temperature.

Tomita et al. considered the shear position of the oil film in the thickness direction to be uniform in the viscous region and near the center in the plastic region, which is consistent with the temperature measurement results [24]. This consideration is used in the present study.

Let Q be the heat generated per unit volume of the oil film in the viscous region. Assuming that the oil film is thin, and thus has a small calorific value and can be treated as stationary, then
$Kf∂2ΔTf∂z2+Q=0$
(12)

For simplicity, let the temperatures of both rollers be equal and ΔTf = 0 at z = 0 and z = h. Then

$ΔTf=−qv2Kfhz2+qv2Kfz$
(13)
The mean and maximum values are respectively
$ΔTf_mean=qvh12Kf$
(14)
$ΔTf_max=qvh8Kf$
(15)
For simplicity, let z = h/2 be the shear position in the plastic region.
$qp2=Kf∂ΔTf∂z$
(16)
Then
$ΔTf_mean=qph8Kf$
(17)
$ΔTf_max=qph4Kf$
(18)
In the viscoelastic range, if the elastic component is considered to be elastic deformation, only the viscous component contributes to heat generation. Because the second term on the left-hand side of Eq. (1) is the viscous component of the strain rate, the viscous component of the slip rate is
$Δuv=Δuτ0ηpsinhττ0γ˙=ΔuhΔuτ0ηpsinhττ0$
(19)
Therefore
$qv=τhτ0ηpsinhττ0$
(20)
All shear components in the plastic regions generate heat, and thus
$qp=τΔu$
(21)

The thermal conductivity of oil under high pressure is calculated using an experimental equation [25] derived from Larsson’s equation [26].

$Kf=0.1172(1−0.00054T)ρ0{1+1.85p1+0.5p}$
(22)

The average oil film temperature is used in the EHL calculation and the maximum oil film temperature is used in the traction calculation.

### 2.4 Numerical Calculation Scheme

#### 2.4.1 Calculation Method.

The overall flow is shown in Fig. 5, where the oil film shape and pressure distribution were obtained from the EHL calculation, the traction force was calculated by substituting the obtained oil film shape and pressure distribution into the Maxwell model, the oil film temperature was calculated from the obtained traction force, and the EHL calculation was performed by modifying the temperature. The calculations were repeated until the traction force and temperature converged. The calculation methods and modifications made to stabilize and speed up the calculations are described below.

Fig. 5
Fig. 5
Close modal

#### 2.4.2 Elastohydrodynamic Lubrication Numerical Calculation.

Ichimaru’s method [25] was adopted in the present study. In the region where an oil film forms, the elastic deformation equation, high-pressure viscosity equation, and high-pressure density equation are substituted into the difference Reynolds equation to obtain the pressure distribution by the relaxation method. Because the Reynolds equation is not valid in a region where the oil film thickness is zero or negative, this is the direct contact region, and the pressure is obtained by setting the left-hand side of the elastic deformation equation to zero. In cavitation regions, where the pressure is negative, the pressure is set to zero. Methods for speeding up the calculations are described in the  Appendix.

#### 2.4.3 Traction Force and Oil Film Temperature.

The velocity in Eq. (1) is decomposed into the x and y directions and replaced by the derivative of the displacement. The strain rate is defined as the slip rate divided by the oil film thickness, which can be written as follows:
$1G(∂τ∂xu+∂τ∂yv)+τ0ηsinhττ0=Δu2+Δv2h$
(23)

This equation is integrated using the fourth-order Runge–Kutta method to obtain the shear stress. However, if τ > τc, Eq. (2) is used. The shear modulus G, Eyring stress τ0, and critical shear stress τc are calculated by considering the pressure and temperature at each node (Eqs. (6)(8)).

The heat generation rate is determined from the obtained shear stress and slip rate, and the temperature of the oil film is calculated. This temperature is substituted into the EHL calculation to obtain the pressure distribution and oil film shape, which are then substituted into the traction calculation. This process is repeated until convergence is reached. However, the Maxwell model contains a hyperbolic function in the viscosity term, which is highly nonlinear, and the temperature increase equation switches among elasticity, viscosity, and plasticity, causing the calculation to easily diverge, especially under high-speed conditions, which is the case considered in this study. Therefore, a Gaussian filter is applied to the calculated increase in temperature to smooth it out and improve convergence. The standard deviation is set at 0.5.

### 2.5 Comparison With Experimental Values.

The experimental results measured using the high-speed traction testing machine shown in Fig. 1(a) are compared with the calculation results obtained using the model developed in this study. The experimental and calculated values under typical conditions are shown in Figs. 6(a)6(c) and the calculated coefficients of the experimental equations for the Eyring stress and limiting shear stress are shown in Table 1. The calculations generally agree with the experiments regardless of the rotational speed and pressure. Figure 6(d) plots the maximum traction coefficient versus the rotational speed. The difference between the calculations and experiments is about 6% at maximum. The results indicate that the calculation model developed in this study can predict the traction coefficient with high accuracy.

Fig. 6
Fig. 6
Close modal
Table 1

Coefficient in equations of τc and τ0

C1C2C3C4C5C6C7C8C9
0.1277.24 × 10−32.12 × 10−42.87 × 10−57.79 × 10−55.99 × 10−20.56976.110
C1C2C3C4C5C6C7C8C9
0.1277.24 × 10−32.12 × 10−42.87 × 10−57.79 × 10−55.99 × 10−20.56976.110

## 3 Increase of Traction Coefficient Using Texture

### 3.1 Overview of Measures Used to Increase Traction Coefficient.

To reduce the decrease in the traction coefficient with increasing speed, a texture was designed using the developed computational model. In the design of the texture, the traction model and the high-pressure rheological properties of the oil were taken into consideration to determine a design policy to increase the traction coefficient.

#### 3.1.1 Policy and Method Considerations.

For simplicity, considering only motion in the x-direction, Eq. (23) becomes
$τ=∫Gu(Δuh−τ0ηsinhττ0)dx$
(24)
where G, τ0, and η are values determined for the oil, and u is an operating condition (it is thus excluded here). The traction coefficient can be increased by increasing the sliding velocity Δu and decreasing the oil film thickness h, which are the remaining terms. However, as shown in the following equations obtained from Eqs. (13) to (21), the increase in oil film temperature is proportional to the sliding speed, so increasing the sliding speed reduces the traction coefficient, as described below.
$ΔTf∝Δuτh$
(25)

Ohno et al. showed that the state transition and maximum traction coefficient for oil are determined by the temperature and pressure of the oil based on measurements of oil properties under high pressure and the results of a two-cylinder traction test [19]. A higher pressure and a lower temperature lead to a higher traction coefficient. In particular, when the oil transitions from viscoelastic to elastoplastic, it becomes solid and the traction coefficient significantly increases. Therefore, from Eq. (25), the temperature decreases as the oil film thickness h decreases. In addition, from Eq. (24), the shear force increases as the oil film thickness h decreases, resulting in a higher traction coefficient.

To increase the pressure, the radius of curvature of the roller can be reduced and the Hertzian contact pressure can be increased; however, this reduces the strength and life of the roller. Therefore, the aim is to equalize the pressure and increase the pressure in the low-pressure region within the contact surface without increasing the maximum pressure (Fig. 7(a)), and to increase the pressure at low loads without increasing the pressure at high loads (Fig. 7(b)). Table 2 summarizes the aforementioned studies. It shows that thinning the oil film and increasing the pressure in the low-pressure area effectively increase the traction coefficient.

Fig. 7
Fig. 7
Close modal
Table 2

Summary of methods used to increase traction coefficient

AimMeansReaction
Increases shear speedThins oil film
Increases slip speedTemperature rising
Decreases temperatureThins oil film
Increases pressureReduces contact areaStrength falls
Averages pressure distribution
Rises pressure at only low load
AimMeansReaction
Increases shear speedThins oil film
Increases slip speedTemperature rising
Decreases temperatureThins oil film
Increases pressureReduces contact areaStrength falls
Averages pressure distribution
Rises pressure at only low load

#### 3.1.2 Examination of Specific Measures.

To make the oil film thinner, fine grooves parallel to the rolling direction are formed on the roller surface to increase the flow coefficient [27]. However, in calculations performed at 43,750 rpm with various groove depths, widths, pitches, and angles, no combination could increase the traction coefficient. This is likely due to the oil film becoming thinner in flat areas but thicker in grooves, which reduces the traction coefficient by decreasing the pressure.

Next, consider the use of high-pressure in low-pressure regions. By changing the point contact to line contact, the pressure can be averaged in the direction perpendicular to the axis. The logarithmic function geometry used for cylindrical roller bearings can make the pressure distribution nearly constant in the perpendicular direction without generating edge loading [28].

Fig. 8
Fig. 8
Close modal

### 3.2 Design of Texture Specifications.

The proposed shape of the texture for increasing pressure in the low-pressure area, as described in the previous section, was examined in terms of the test roller specifications for evaluation in a high-speed traction testing machine.

#### 3.2.1 Multi-Stage Curved Surface Shape.

The texture consists of three convex surfaces, as shown in Fig. 9, with the center raised relative to the surfaces on the two sides. The central surface has a logarithmic function shape expressed by the following equation:
$z=−Kln{1−(yA)2}$
(26)
where K and A are parameters. K corresponds to the height of the projection, and A corresponds to the half-width of the projection. The surfaces on both sides have logarithmic function shapes with the same parameters, but offset by P in the y direction and D in the z-direction.
$z=−Kln{1−(|y|−PA)2}+D$
(27)
Fig. 9
Fig. 9
Close modal

The pressure at high and low loads is used as the evaluation index. All combinations are calculated for the parameters shown in Fig. 9. A smooth roller with a constant crowning radius of 23.5 mm is used for comparison. The Hertzian maximum pressure is about 1.5–4.0 GPa at pressing forces of 291–5510 N. With the texture, the pressure at a maximum load of 5510 N should be less than 4 GPa and the pressure at a minimum load of 291 N should be as high as possible.

Figure 10(a) shows the results of all calculations, where the horizontal axis represents the pressure at high load and the vertical axis represents the pressure at low load. The data points for the textured roller are located to the upper left of those for the smooth roller, which is the desired effect. Table 3 shows the results of the pressure and traction coefficient calculations for various parameters, where the pressure at high loads is near 4 GPa. The highest traction coefficient (an increase of 19%) is obtained for A = 600 and K = 16. A = 500 and K = 10 also increase the traction coefficient by 19%, with the advantage that the small K (i.e., low peak) makes processing easier. The latter parameters were thus selected.

Fig. 10
Fig. 10
Close modal
Table 3

Effect of multi-step logarithm texture on pressure and traction coefficients

A (μm)K (μm)D (μm)Pmax at 291 N (GPa)Pmax at 5510 N (GPa)Traction coefficient at 291 N
500491.433.95
891.693.93
1071.823.960.075
1251.863.980.069
6001491.793.950.075
1671.873.960.076
R23.51.483.940.063
A (μm)K (μm)D (μm)Pmax at 291 N (GPa)Pmax at 5510 N (GPa)Traction coefficient at 291 N
500491.433.95
891.693.93
1071.823.960.075
1251.863.980.069
6001491.793.950.075
1671.873.960.076
R23.51.483.940.063

Figure 10(b) shows the calculated pressure versus load and traction coefficient. There was concern that the pressure would drop when the projections on the two sides begin to make contact, but the pressure increases over the entire load range, and the traction coefficient also increases. At high loads, the traction coefficient increases even though the pressure remains the same. This is due to the logarithmic shape averaging the pressures and the increase in the pressure in the low-pressure region on the contact surface.

In the axial section through the center of the contact point, the radius of curvature R0 is constant when the distance from the center is less than y0, and it increases logarithmically beyond that distance. The shape is expressed by the following equation:
$z=y22R0−Kln{1−(|y|−y0A)2}$
(28)
At |y| ≤ y0
$z=y22R0$
(29)

The specifications for the texture and the calculated pressure and traction coefficient are shown in Table 4. For a roller with a constant crowning radius (23.5 mm), the pressure at low loads is increased and the pressure at high loads is maintained, resulting in a 5–8% increase in the traction coefficient.

Table 4

Effect of increasing radius on pressure and traction coefficient

R0 (mm)y0A (μm)K (μm)Pmax at 291 N (GPa)Pmax at 5510 N (GPa)μ at 291 Nμ at 5510 N
200.490010001.544.020.0660.082
170.4130050001.634.030.0680.080
23.51.483.940.0630.083
R0 (mm)y0A (μm)K (μm)Pmax at 291 N (GPa)Pmax at 5510 N (GPa)μ at 291 Nμ at 5510 N
200.490010001.544.020.0660.082
170.4130050001.634.030.0680.080
23.51.483.940.0630.083

### 3.3 Confirmation of Texture Effect.

The effect of a multi-stage logarithmic function shape, which is effective at increasing the traction coefficient, was experimentally confirmed using a high-speed traction drive testing machine. The texture was machined onto the low-speed rollers. The high-speed rollers were smooth. A photograph of the prototype roller and the designed and measured geometry of the texture are shown in Fig. 11. The measured convexities are somewhat flat, but the peak widths and height offsets are approximately equal to the designed values.

Fig. 11
Fig. 11
Close modal

The traction curves obtained at high rotational speeds and low load (maximum pressure of 1.5 GPa without texture) are shown in Fig. 12(a). The traction coefficient for the textured rollers is greatly increased compared with that for the smooth rollers. Figure 12(b) plots the maximum traction coefficient versus rotational speed. At a low load (291 N), the texture increases the traction coefficient over the entire speed range, with an increase of up to 19%. At a load of 689 N (2.0 GPa without texture), there is no difference between the cases with and without texture (i.e., there is no improvement effect). This may be because almost the entire area of the oil film had transitioned from viscoelastic to elastoplastic, as predicted by Fig. 10(b).

Fig. 12
Fig. 12
Close modal

Texture shape measurements made before and after the experiment are shown in Fig. 13. There is almost no change in shape. The amounts of plastic deformation and wear are considered to be extremely small because the experiments were conducted for a short period of time (approximately 2–3 min per condition, or several tens of minutes in total). Longer operation time would change the shape of the wheel, resulting in a decrease in the traction coefficient and a decrease in the life of the wheel. Confirmation of the change in the traction coefficient over time, for example during 100,000–200,000 km of driving (the life of an automobile), is a major issue for future research.

Fig. 13
Fig. 13
Close modal

## 4 Conclusion

To suppress the reduction of the traction coefficient at high rotational speeds, a model for predicting the traction coefficient was developed and used to design a texture, which was then evaluated experimentally. In summary:

1. A traction calculation model was created by coupling the nonlinear Maxwell model, EHL calculation, and oil film temperature calculation.

2. The model takes into account the pressure distribution, oil film shape, and temperature distribution, and can be used for texture design.

3. The calculated traction coefficients were in good agreement with the experimental results.

4. The nonlinear Maxwell model and the high-pressure rheological properties of oil were used to formulate design guidelines for textures.

5. The designed prototype texture greatly increased the traction coefficient.

## Acknowledgment

This research was supported by the “Project for Building Simulation Platforms to Accelerate Development of Next-Generation Vehicles” (TRAMI-Transmission Research Association for Mobility Innovation) grant of 2020.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

a =

h =

thickness of oil film, m

p =

pressure of oil film, m

q =

heat per unit area, W/mm2

t =

time, s

u =

rolling speed in x-direction, m/s

v =

rolling speed in y-direction, m/s

G =

elastic shear modulus, Pa

T =

temperature of oil, °C

cs =

specific heat capacity of roller, J/kg · K

qp =

heat per unit area in plastic region, W/mm2

qv =

heat per unit area in viscous region, W/mm2

Gf =

elastic shear modulus of oil film, Pa

Kf =

thermal conductivity of oil, W/mK

Ks =

thermal conductivity of roller, W/mK

Tb =

bulk temperature of roller, °C

Tf =

temperature of oil film, °C

E′ =

elastic modulus, Pa

x, y, z =

coordinates, m

αp =

pressure viscosity index under high pressure, Pa−1

α0 =

pressure viscosity index under atmospheric pressure, Pa−1

$γ˙$ =

shear rate of oil film, m

Δu =

slip speed in x-direction, m/s

Δv =

slip speed in y-direction, m/s

ΔTf =

temperature rise of oil film, °C

ΔTs =

temperature rise of roller surface, °C

ηp =

viscosity under high pressure, Pa · s

η0 =

viscosity under atmospheric pressure, Pa · s

ν0 =

kinematic viscosity under atmospheric pressure, m2/s

ρs =

density of roller, kg/m3

ρ0 =

density of oil, kg/m3

τ =

shear stress, Pa

τc =

limiting shear stress, Pa

τ0 =

Eyring stress, Pa

### Appendix: Speeding up of Elastohydrodynamic Lubrication calculations

The Reynolds equation is computed using the relaxation method, in which the pressure at a node is calculated from the values of the four surrounding nodes in the previous calculation cycle. Therefore, propagation of the effect of pressure modification is only transmitted to one neighboring node in one calculation, which requires an extremely large number of calculations when the mesh is fine. The multigrid method [29], which alternates between coarse and fine mesh calculations, is considered effective for this problem. However, if the texture cannot be represented by a coarse mesh, errors may occur when switching mesh size, increasing the number of calculations, and diverging the viscosity term in the highly nonlinear Maxwell model. Therefore, for a smooth shape with no texture, a coarse mesh is used to obtain a rough pressure distribution and oil film shape, and then a fine mesh is used to obtain the pressure distribution and oil film shape, taking texture into account. The traction, temperature, and EHL calculations are repeated using only the fine mesh.

The oil film shape equation is discretized as follows, where the elastic deformation term is the convolution sum
$hi,j=hc+xi22Rx+yj22Ry+2πE∑km∑lnpk,lKi−k,j−l$
(A1)
where K is the influence coefficient map (kernel) for elastic deformation, calculated using Love’s analytical formula [30]. The number of calculations is O(N2) for N nodes, which accounts for the majority of the total calculation time. The multilevel multi-integration method [29] is often used to speed up calculations. The number of calculations is O(NlnN). For example, for a 500 × 500 mesh, the number of calculations is reduced by a factor of 20,000. However, the calculation procedure is extremely complex and cumbersome. In addition, when the mesh size is changed, the interpolation approximation causes an error in the texture shape.
Liu took advantage of the fact that a Fourier transform of the convolutional sum is equal to the product of the Fourier transforms of each element, and used a fast Fourier transform to speed up the calculation of elastic deformation [31]. The number of calculations is also O(NlnN). Compared to the multilevel multi-integration method, the computation is simpler and there is no risk of interpolation errors. This method is used in the present study. Liu’s elastic deformation kernel has a wrap-around order, which makes the linear convolution equivalent to the circular convolution. However, the procedure is rather complicated. In this study, a simpler method is used. If the mesh size for the pressure is (m, n), the size of the kernel is (2m + 1, 2n + 1). Now, both the pressure and the kernel are extended to (3m + 1, 3n + 1) by adding zeros (Fig. 14). The following equation holds for expanded pressure ppad and kernel Kpad.
$ppad*Kpad=F−1F(ppad)F(Kpad)$
(A2)
Fig. 14
Fig. 14
Close modal

The center of the calculated result is the original convolution sum.

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