Journal of Advanced Mechanical Design,
Systems, and
Manufacturing
Vol. 5, No. 4, 2011
A Method to Derive Friction and Rolling Power
Loss Formulae for Mixed Elastohydrodynamic
Lubrication*
Sheng LI** and Ahmet KAHRAMAN**
**Department of Mechanical and Aerospace Engineering, The Ohio State University
201 W. 19th Avenue, Columbus OH 43210, USA
E-mail: [email protected]
Abstract
Prediction of mechanical power losses of gear pairs requires elastohydrodynamic
lubrication (EHL) analyses to be carried out at every discrete contact position.
This makes gear efficiency models computationally demanding. As a remedy to
this problem, a method is proposed in this study to derive EHL-based friction
coefficient and rolling power loss formulae to be used in gear efficiency models.
This method employs a large number of EHL analyses covering a full matrix of all
key contact parameters, namely normal load, rolling and sliding velocities, radii of
curvature, lubricant parameters and surface roughness amplitudes, within typical
ranges dictated by gear contacts. Linear regression of the results of the EHL
analyses yields friction coefficient and rolling power loss formulae suitable for gear
efficiency models. In the EHL model, the hydrodynamic fluid and asperity
contact zones are treated in a unified manner. The sliding friction is computed as
the sum of the viscous shear within the lubricant and the boundary friction at the
local asperity contact spots. The rolling power loss induced by fluid pressure
gradient is formulated to include the entire contact zone instead of only the inlet
region, since the pressure gradient is substantial within the contact for rough
surface condition. The proposed method is demonstrated using an example
turbine fluid. Typical measured gear surface roughness profiles are used in the
analyses. At the end, the results from the regression formulae are compared to
actual EHL predictions to assess their accuracy under various contact conditions.
Key words: Mixed Elastohydrodynamic Lubrication, Friction Coefficient, Rolling
Power Loss, Linear Regression, Gear Design
1. Introduction
*Received 25 June, 2011 (No. 11-0341)
[DOI: 10.1299/jamdsm.5.252]
Copyright © 2011 by JSME
Mixed elastohydrodynamic lubrication (EHL) conditions are common in many gear
applications where high levels of torque are transmitted at relatively low speed and high
temperature under rough surface condition (as a result of the finishing processes such as
grinding, honing or shaving). Under such condition where hydrodynamic fluid film and
local asperity interactions coexist, both components of the mechanical power loss at the
gear mesh interfaces, i.e. sliding and rolling losses, are important(1). For the sliding
component, a number of studies assumed constant friction coefficient µ along the contact
zone to calculate the friction torque from which the power loss was determined (for instance
Ref. (2)). Recognizing that µ actually varies as gears roll owing to the variations of the
contact parameters such as radius of curvature, normal load, sliding and rolling velocities,
the works such as (3) relied on published empirical µ formulae. However, the experimentally
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collected data that were used to derive these formulae had relatively narrow ranges of
operating temperature, speed and load, and surface roughness, limiting their applications (4).
The third group of studies (1),(5),(6) adapted a physics-based approach to compute the friction
traction from EHL analyses of the contacts. These studies varied in the way they included
the surface roughness and gear specific transient effects (7). While some of these models (5)
assumed that the spur gear contact surfaces are smooth, the others (1),(6) included the impact
of surface roughness in the EHL model. In order to further take into account the gear
transient lubrication effects due to the variation of tooth force, contact radii, and sliding and
rolling velocities, Li and Kahraman (1) employed a transient gear mixed EHL model (7)
which simulates the tribological behavior from the start of active profile (SAP) to the tip in
a continuous way to determine the power losses. Most of the other studies in literature used
independent discrete EHL analyses at a series of the contact positions along the line of
action.
Studies in this third group can be viewed to be more accurate since they do not rely on
any externally defined µ formula. Yet, they suffer from the extensive computational
demand, making them less desirable for design optimization studies where efficiency must
often be weighed against noise and durability metrics. Xu et al (4) and Li et al (8) developed
customized gear contact friction formulae using full film EHL theory and mixed EHL
theory, respectively, to remedy this shortcoming. The parametric EHL simulations in their
works covered gear specific ranges of key parameters such as temperature, normal load,
rolling speed, slide-to-roll ratio, radius of curvature and surface roughness amplitude. The
traction data from the EHL predictions were then reduced into µ formulae using the general
linear regression technique. Coupling the sliding friction formulae with a computationally
efficient load distribution model (9), each gear mechanical power loss computation was
reduced to less than one second of CPU time. The work of Li et al (8) considered the
lubrication regime ranging from boundary to full film conditions. The transition from the
regime where boundary friction dominates to the regime where fluid shear dominates
introduced distinct slope change in the Stribeck-curve. Although a relatively high R-square
value was obtained by dividing the whole data set into subsets representing different
lubrication regimes and performing regression separately, discontinuities were introduced at
the seams of these regimes.
The rolling power loss component was either completely neglected or determined
approximately using an empirical rolling traction formula derived under smooth line contact
condition (10), such as the works (4),(6),(8). Physics-based formulations for rolling power loss
are quite limited in the literature. Li and Kahraman (1) proposed a formulation for rolling
loss, which integrates the product of the fluid shear and the sliding velocity within the
lubricant across the film thickness, covering the entire contact zone instead of only the inlet
region (10). The formulation was incorporated with a transient gear mixed EHL model to
demonstrate that the rolling power loss is rather significant and should be modeled
accurately in gear efficiency predictions. Due to the heavy computational efforts, however,
this real-time mixed EHL analysis is not realistic for design optimization purposes.
In this work, a methodology is proposed to develop formulae for both the sliding
friction and rolling power loss using a line contact mixed EHL model. The friction
coefficient is formulated as the sum of the boundary friction component and the fluid shear
component, such that the discontinuities at the lubrication regime boundary as experienced
by the earlier studies (8),(11) can be eliminated meanwhile better regression accuracy can be
achieved. The rolling power loss is determined by using the earlier formulation of Li and
Kahraman (1). The methodology is applied to a typical aerospace lubricant to (i) demonstrate
the proposed approach and (ii) assess the accuracy of the sliding friction and rolling power
loss formulae.
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2. Prediction of Sliding Friction and Rolling Power Loss
Vol. 5, No. 4, 2011
2.1 Mixed EHL Model
For one-dimensional line contacts, the pressurized hydrodynamic fluid flow within the
contact zone is dictated by the transient Reynolds equation
∂  ∂p  ∂ (ur ρ h) ∂ (ρ h)
+
f
=
∂x  ∂x 
∂x
∂t
(1.a)
where p, h and ρ represent the contact pressure, film thickness and density of the fluid at
local position x and time instant t. The rolling velocity ur is the average of the tangential
surface velocities of the contacting body 1 ( u1 ) and body 2 ( u2 ), i.e. ur = ( u1 + u2 ) 2 .
Assuming an Eyring fluid, the flow coefficient is approximated as (12)
f =
τ 
ρh3
cosh  m 
12η
 τ0 
(1.b)
where η is the lubricant viscosity, and the shear stress τm is derived as
τm = τ0 sinh −1 [ηus (τ0 h)] ( us = u1 − u2 is the sliding velocity) considering shear flow
alone. Here, τ0 is the lubricant reference stress which was found to be linearly dependent
on pressure (13),(14) within a certain range of ps < p < pe (Typically ps ≈ 0.7 GPa and
pe ≈ 2.5 GPa). Below ps , τ0 varies very limitedly with pressure and can be assumed to
be constant. Above pe , the relationship between τ0 and pressure becomes nonlinear. In
this work, it is assumed that the dependence of τ0 on p can be described within a wide
loading range in the form of
τ0 = (τ0 ) amb +
(τ0 )lim − (τ0 )amb
2

 2ωp

− ω 
 tanh(ω) + tanh 

 ps + pe
 
(2)
where (τ0 )amb is the reference shear at ambient pressure, (τ0 )lim is the limiting value of
τ0 and ω = κ( ps + pe ) [ (τ0 )lim − (τ0 )amb ] . Here, κ is the slope of the reference shear
pressure relationship within the linear range. Experimental measurements (13),(14) showed
(τ0 )amb is on the order of 1 MPa, (τ0 )lim is on the order of 100 MPa, and κ is on the
order of 0.05.
For any local spot where the fluid film breaks down due to surface asperities, the
reduced Reynolds equation (1),(7),(8) is applied to describe the contact behavior as
∂ (ur h) ∂ h
+
=0
∂x
∂t
(3)
Equations (1) and (3) together allow the solution of the hydrodynamic pressure and the
asperity contact pressure simultaneously without additional efforts searching for the asperity
interaction regions.
The instantaneous film thickness profile across the contact zone is defined as
h( x, t ) = h0 (t ) + g0 ( x) + V ( x, t ) − R1 ( x, t ) − R2 ( x, t )
(4)
where h0 (t ) is the reference film thickness, and R1 ( x, t ) and R2 ( x, t ) are the roughness
height distributions of the two surfaces that move at velocities of u1 and u2 , respectively.
g0 ( x) = x 2 (2req ) represents the unloaded geometry gap between the two surfaces ( req is
the equivalent contact radius). The elastic deformation V ( x, t ) is the convolution of the
normal pressure distribution and the influence kernel function K ( x) = − 4ln x (πE ′) (15)
x
such that V ( x, t ) = ∫x e K ( x − x′) p( x ′, t )dx′ . Here xs and xe are the start and end points
−1
s
of the computational domain, and E ′ = 2 (1 − υ12 ) E1 + (1 − υ22 ) E2 
where υi and Ei


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are the Poisson’s ratio and the Young’s modulus of contact body i ( i = 1, 2 ).
The two-slope model is used here to describe the pressure-viscosity relationship (1,7,8)
η0 exp(α1 p),

η = η0 exp(c0 + c1 p + c2 p 2 + c3 p3 ),

η0 exp [ α1 pt + α 2 ( p − pt )] ,
p < pa
pa ≤ p ≤ pb
(5)
p > pb
where α1 and α 2 are the pressure-viscosity coefficients for the low ( p < pa ) and high
( p > pb ) pressure ranges, respectively, and pt is the transition pressure between these
two ranges. The coefficients ci ( i = 0,1, 2,3 ) are determined such that both η and
d η dp are continuous at p = pa and p = pb . The lubricant compressibility is
approximated using the Dowson-Higginson equation as (16)
ρ = ρ0
(1 + γ1 p)
(1 + γ 2 p )
(6)
where γ1 = 2.266 × 10−9 Pa -1 and γ 2 = 1.683 × 10−9 Pa -1 .
Additionally, a load balance equation is written by equating the total contact force due
to the pressure distribution over the entire contact area to the normal load applied
x
W = ∫x e p( x, t )dx
s
(7)
where W is the force per unit width. Equation (7) acts as the check of the load balance
convergence for the pressure prediction. The reference film thickness h0 (t ) in Eq. (4) is
adjusted within a load iteration loop until Eq. (7) is satisfied.
Fig. 1 Mixed EHL model flow chart.
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Figure 1 describes the numerical method for the solution of the line contact mixed EHL
problem. The model starts with a guess of the initial contact pressure (Hertzian pressure),
which is used to compute the initial elastic deformation and film thickness. According to the
film thickness distribution, the asperity contact spots can be located and the fluid viscosity
and density are computed for the areas where h > 0 . The unified equation system of Eqs.
(1) and (3) are then solved for p over the entire contact zone. The pressure solution is
checked for both the load balance convergence and pressure convergence. The converged
solutions are used for the initial guess of the next time step. In this work, the size of the
computational domain is defined as L = δa , where a is the half Hertzian width and the
coefficient δ ≥ 3 . The origin of the x coordinate is selected such that
−0.625L ≤ x ≤ 0.375 L . The computational domain is discretized into N elements and the
mesh size ∆x = L N is controlled to be on the order of microns by varying δ and N
under different loading conditions. The detailed discretization and linearization of the
governing equations have been fully presented (7),(8) and are omitted here.
2.2 Sliding Friction
The relative motion between the lubricant layers (where h( x, t ) > 0 ) and between the
contacting surfaces (where h( x, t ) = 0 ) causes friction forces onto the surfaces.
Considering both the Poiseuille and Couette flows, the fluid shear stress varies linearly
across the fluid film ( z direction pointing from surface 1 to surface 2) as
 ∂u  2 z − h  ∂p  ∗ (u2 − u1 )
q ( x, z , t ) = η∗   =
 +η
2  ∂x 
h
 ∂z 
(8)
where η∗ = η cosh(τm τ0 ) is the effective viscosity in the direction of rolling. The first
shear component on the right hand side of Eq. (8) is caused by the relative motion induced
by the fluid pressure gradient and is commonly referred as rolling friction, since it occurs
even no sliding of the surfaces exists, i.e. u1 = u2 . The second component is the sliding
viscous shear and is usually assumed to be the dominant one.
In order to include the thermal effects on lubricant viscosity and film thickness
reduction, the correction factor proposed by Gupta et al (17) for film thickness is adopted
φT =
 13.2 ph LT0.42 
1− 



E'


(
1 + 0.213 1 + 2.23 SR
0.83
)
LT0.64
(9)
Here, LT = η0βur2 k is the thermal loading parameter, where β and k are the
temperature-viscosity coefficient and the fluid thermal conductivity, respectively, and
SR = us ur is the slide-to-roll ratio. Assuming the reduction in film thickness is mainly
due to the decrease in viscosity, a viscosity thermal correction factor is derived as
(1)
Cη = φ1.445
, such that the thermally corrected fluid shear exerted on surface 1 has the
T
form of
q = −φT
h  ∂p  0.445 ∗ (u2 − u1 )
  + φT η
2  ∂x 
h
(10.a)
For local spots where film thickness breaks down, the friction shear is approximated as
q = µb p
(10.b)
where µb is the coefficient of friction for boundary lubrication. A typical value of
µb = 0.15 (18) is used in this study. With these, the friction coefficient at a given time
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instant tn can be found as
µ tn =
1 xe
∫ q( x, tn )dx
W xs
(11)
Due to the movement of the surface roughness profiles on both the contact surfaces, the
lubrication behavior is highly transient. The friction force at one single time instant cannot
effectively reflect the shear traction when one surface is sliding over the other. Thus, each
simulation is carried out for a sufficiently large number of time steps of Nt ( Nt = 1000 in
this study) and the average friction coefficient for the contact condition considered is
1
N
defined as µ =
∑ t µt .
Nt n =1 n
2.3 Rolling Power Loss
The mechanical power loss due to rolling was conventionally estimated by multiplying
(4),(6),(8),(10)
the
rolling
velocity
by
the
approximate
rolling
traction
0.658 0.0126
Fr = 4.318 ( GU )
W
req α1 , where the non-dimensional parameters G = α1 E′ ,
U = η0 ur ( E ' req ) and W = W ( E ′req ) . This equation assumes full film, smooth contact
conditions where the rolling friction is mostly contributed from the inlet region
xs < x < − a . This is because ∂p ∂x > 0 when − a < x < 0 and ∂p ∂x < 0 when
0 < x < a , resulting in the cancellation of the rolling friction within the Herzian zone,
which is a vector. However, the power loss computation must include the entire contact
zone. Especially for more realistic rough surface contacts, the pressure gradient can be
extremely large within the Hertzian zone owing to the local surface roughness profiles.
Thus, simply using this empirical rolling traction formula (10) might significantly
underestimate the rolling power loss.
In this work, the power loss intensity (power loss per unit area) across the fluid film at
position x and time t is determined by integrating the product of the local shear stress
q and sliding velocity along the film thickness as
h
Q( x, t ) = ∫0 q ( x, z , t )
∂u
dz
∂z
(12.a)
Substituting Eq. (8) into Eq. (12.a), and applying the thermal corrections, the power loss
intensity (including both rolling and sliding components) reads
Q( x, t ) = φ1.555
T
2
2
h3  ∂p 
0.445 ∗ u s
+
φ
η
T
 
h
12η∗  ∂x 
(12.b)
The two terms on the right hand side of Eq. (12.b) represent the rolling component Qr and
the sliding component, respectively. It is apparent that Qr exists across the entire
hydrodynamically lubricated contact zone, and is not subject to any cancellation,
highlighting the unsuitability of the rolling traction formula (10) in estimating the rolling
loss. For any regions where film thickness breaks down, Qr = 0 . Likewise, the total rolling
loss density (rolling loss per unit width) at a given time instant tn is found as
x
( Pr )tn = ∫x e Qr ( x, tn )dx
s
(13)
Each simulation is carried out for a total of Nt time steps as well, and the average total
rolling power loss density for the corresponding contact condition is defined as
1 Nt
Pr =
∑ ( Pr )tn .
Nt n =1
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3. Linear Regression of Mixed EHL Results
Vol. 5, No. 4, 2011
The derivation of friction and rolling loss formulae is demonstrated in this section
through an example. The mixed EHL analyses are carried out within typical ranges dictated
by gear contacts of all key contact parameters, including Hertzian pressure, radius of
curvature, rolling and sliding velocities, lubricant temperature and surface roughness
amplitudes as listed in Table 1. The roughness profile is measured from a shaved gear
surface with root-mean-square (RMS) amplitude of Rq = 0.5 µm and the roughness
profiles with different RMS amplitudes are obtained from this baseline profile by
multiplying it by a constant. It is also assumed that the combination of Rq1 = e1 and
Rq2 = e2 is equivalent to the combination of Rq1 = e2 and Rq2 = e1 . A typical turbine
fluid Mil-L23699 whose basic properties are shown in Table 2 is considered. The transition
pressure for its pressure-viscosity relationship is curve-fitted as pt = 79.0163 + 0.91401T
with the units of pt in MPa and temperature T in Celsius. The low and high pressure
range thresholds pa = 0.7 pt and pb = 1.4 pt , respectively. The pressure-viscosity relation
defined by Eq. (5) with these parameter values matches the measured pressure-viscosity
data (19) well.
A total of 21,000 sliding friction coefficient data points (Fig. 2(a)) and 21,000 rolling
power loss density data points (Fig. 2(b)) covering a wide range of contact conditions
experienced by typical gear applications are obtained from the parametric mixed EHL
analyses defined in Table 1. Figure 2(a) illustrates the variation of µ (predicted using the
formulation of §2.2) versus the lambda ratio λ (ratio of smooth surface minimum film
thickness to composite surface roughness Sq = Rq2 + Rq2 ), exhibiting a Stribeck-curve
1
2
shape. Two distinct regimes are observed in this figure. When λ > 1 , the fluid shear
dominates and the friction coefficient increases gradually with λ while µ decreases at a
faster pace for λ < 1 where boundary lubrication dominates, leading to a very difficult
regression problem. In order to improve the regression accuracy, Li et al (8) and Kolivand et
Table 1 Parametric design for the development of friction coefficient and rolling power loss
density formulae.
Lubricant
Mil-L23699
Hertzian Pressure, ph [GPa]
0.5, 1, 1.5, 2, 2.5
Radius of Curvature, req [mm]
5, 20, 40
Rolling Velocity, ur [m/s]
1, 5, 10, 15, 20
Slide-to-roll Ratio, SR
0.025, 0.05, 0.1, 0.25, 0.5, 0.75, 1
Inlet Lubricant Temp., T [ DC ]
50, 75, 100, 125
Surface 1 roughness , Rq1 [µm]
0.1, 0.4, 0.7, 1
Surface 2 roughness, Rq2 [µm]
0.1, 0.4, 0.7, 1
Table 2 Basic parameters of the lubricant Mil-L23699.
Temperature
T [ DC ]
Dynamic
Viscosity
η0 [Pa ⋅ s]
Pressure-viscosity
Coefficient
α1 [GPa -1 ]
Pressure-viscosity
Coefficient
α 2 [GPa -1 ]
Density
ρ0 [kg/m3 ]
50
0.01502
15.8
8.83
977.80
75
0.00703
13.7
7.17
962.80
100
0.00398
12.2
6.01
947.80
125
0.00256
11.1
5.17
932.80
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Fig. 2 Variation of (a) µ and (b) Pr with λ for all the contact combinations of Table 1.
al (11) divided the whole data set into several subsets based on λ and performed the linear
regression for each of them independently. Although relatively high R-square values were
achieved for each subset, discontinuities were evident at the boundaries of the subsets under
certain operating conditions, potentially leading to large jumps in sliding power loss
predictions.
It is recognized that although the total friction changes its trend when the dominant
component switches from boundary shear to fluid shear, each component itself behaves
monotonically (either decreases or increases continuously). With this recognition, the two
components of the friction are regressed separately such that the total friction coefficient is
simply the sum of these two. Using this approach, very high R-square values can be
achieved without introducing discontinuities.
The non-dimensional parameters selected for the general linear regression model are
SR , V = (ur η0 ) ( E′ A 0 ) , G , req = req r0 , P = ph E′ , and two roughness parameters
Sq = S q A 0
and
Se = Se A 0 , where
ph
represents Hertzian pressure,
2
2
Sq = Rq + Rq and Se = Rq1 Rq2 ( Rq1 + Rq2 ) . The reference parameters A 0 = 1 µm
1
2
and r0 = 5 mm. The friction coefficient is expressed as µ = µ a + µv where the boundary
friction component µa and the fluid shear component µv are regressed for this example
lubricant as


VP
µ a = exp  a0 + a1
+ a2 ( SR) Sq + a3 ( SR) Se 
Sq


⋅ ( SR)a4 ln Se V
a5V Sq
G a6 ( SR ) P
a7 + a8 req
req
a9 V Sq + a10 ln(V )
(14.a)
Se
a11V Sq + a12 P
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µ p
sinh  v h
 τe
Vol. 5, No. 4, 2011
V

 = exp b0 + b1VP + b2 GP  

 Sq
⋅ ( SR )b5G +b6 ln(G ) + b7 ln( P ) V




b3 P + b4 ln(V )
b8 ln(G ) + b9 ln( P ) +b10 ln( req )
reqb11P
(14.b)
with the adjusted R-square values of 94.1% and 98.6%, respectively. Here, τe = 10 MPa
and the regression coefficients are listed in Table 3. In Fig. 3, the regressed formula
predictions are compared with the EHL predictions for the operating ranges of
0.5 ≤ ph ≤ 2 GPa , 1 ≤ ur ≤ 20 m/s , 0.025 ≤ SR ≤ 1 , req = 20 mm, Rq1 = Rq2 = 0.4
µm and lubricant temperature T = 100 DC (black color represents ur = 1 m/s; red color
represents ur = 5 m/s; orange color represents ur = 10 m/s; blue color represents
ur = 15 m/s; purple color represents ur = 20 m/s), showing good agreement between the
regressed and mixed EHL results. It is seen, the friction coefficient decreases as the rolling
velocity increases since the thicker fluid film effectively reduces the portion of boundary
lubrication. After the contacting surfaces are fully separated by the hydrodynamic film, µ
varies very limitedly with the speed. The influence of pressure on µ is subjected to the
lubrication regime. When the boundary lubrication dominates (low speed and high
pressure), the increase of p leads to the decrease of µ . The opposite is true when the
hydrodynamic lubrication prevails. The thermal effect that the frictional heat reduces the
lubricant viscosity and consequently µ is observed under the condition of high sliding and
high contact pressure.
The rolling power loss density in Fig. 2(b) shows a bell shaped distribution. Pr peaks
when λ ≈ 0.2 where sufficient area of the contact zone is hydrodynamically lubricated and
the local pressure gradients within the fluid film are large due to the surface asperities. As
λ decreases, the fluid regions decrease, resulting in the decrease of rolling loss. On the
other hand, when λ increases, the contacting surfaces are more separated by the fluid film,
reducing the fluid pressure gradient and Pr . Using the same non-dimensional parameters,
the rolling loss density formula is regressed as
Table 3 Regression coefficients.
i
ai
0
-6.43447
-0.715642
12.5992
1
96052930
-56505652
-173516.68
2
1.55957
-0.0461565
64360.97
3
-3.14657
-13.1599
-0.00884031
4
-0.0530553
-0.0253092
-0.0868522
5
90263.3
-0.000193947
0.0000850709
6
-0.115511
-0.107263
-0.117314
7
-0.538303
-0.332885
2.38870
8
0.0120564
-0.122517
-0.511883
9
-211931
-0.101177
0.414829
10
0.0175699
0.0130492
0.198814
11
292992
35.2044
0.000191988
12
-24.4704
13
bi
di
118.6686
-64197.84
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Fig. 3 Comparison of µ between regression formula Eq. (14) (solid curves) and mixed
EHL predictions (solid dots) for (a) ph = 0.5 GPa, (b) ph = 1 GPa, (c) ph = 1.5 GPa
and (d) ph = 2 GPa.
Fig. 4 Comparison of Pr between regression formula Eq. (15) (solid curves) and mixed
EHL predictions (solid dots) for (a) ph = 0.5 GPa, (b) ph = 1 GPa, (c) ph = 1.5 GPa
and (d) ph = 2 GPa.
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
V ( SR )   V
Pr = 1×10 exp  d0 + d1

Sq   Sq

5
⋅V d 4 ( SR ) + d5G + d6 ln( P ) req




d 2V + d3 ln(V )
d7 + d8 S q + d9 ln( Sq ) + d10 ln( P )
(15)
Sq d11G + d12 P Se
d13V Sq
The adjusted R-square value is 98.9% and the corresponding coefficients are listed in Table
3.
Comparisons of the predicted rolling loss between regression formula (Eq. (15)) and
EHL analyses are also performed and reasonably good agreement is observed in Fig. 4.
When SR is small (namely SR < 0.1 ), the rolling loss yielded from the EHL analyses
varies non-linearly with SR . Within this low sliding range, Eq. (15) shows certain
deviation from the EHL prediction, pointing to the requirement of higher order SR terms
in the general linear regression model. For SR ≥ 0.1 , both Eq. (15) and the EHL analyses
predict Pr increases linearly with SR . The influence of sliding on rolling power loss can
be due to the shear thinning effect that increases the pressure gradient within the fluid. It is
also shown Pr increases with ur and ph .
4. Conclusions
The mechanisms of sliding friction and rolling power loss under mixed lubrication
condition were examined. The friction coefficient was formulated as the sum of the fluid
shear component within the lubricant and the boundary friction component at the local
asperity contact spots. The conventional way for rolling loss estimation was shown to be
inappropriate and can lead to substantially underestimation. In this work, the rolling power
loss was derived through integrating the product of the local fluid shear stress and the
sliding velocity across the film thickness. Considering mixed lubrication of line contacts, a
parametric study was performed covering a full matrix of all key contact parameters,
including normal load, rolling and sliding velocities, radius of curvature, lubricant
parameters and surface roughness amplitudes within typical ranges of gear contacts. The
mixed EHL predictions of friction coefficients and rolling loss density were regressed to
yield easy-to-use formulae suitable for gear efficiency models, circumventing the heavy
computational effort required for real time mixed EHL analyses. The high R-square values
and the direct comparisons between the regression formula results and the EHL predictions
show reasonable accuracy under various contact conditions.
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