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Thermal EHL analysis of the inner
ring rib and roller end in tapered
roller bearings with the Carreau
model
Xiaoling Liu*, Tao Long, Xinming Li and Feng Guo
School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao, China
The roller end/rib contact of tapered roller bearings significantly affects lubricating
condition and power loss. To improve the lubrication performance of the inner ring
rib and the large end of the roller in tapered roller bearings used in railway coaches,
based on the structural analysis of the inner rib and the large end of the roller and
considering spin–slide effects between the rib and the large end of the roller, a
thermal elastohydrodynamic lubrication model with a Carreau rheological model
was established in a tapered roller bearing. Two kinds of rib structures were provided:
the tapered rib and spherical rib. Under different conditions, variations in the friction
coefficient versus the ratio of curvature radius of the large end of the roller to that of
the rib were compared, and the film thickness and film temperature varied with the
rotational speed and the effect of load was compared between the two rib structures.
Results showed that spinning motion has little effect on the lubrication at the contact
point between the inner ring rib and the large end of the tapered roller. There exists
an optimal ratio of the curvature radius between the large end of the roller and the
spherical or tapered rib; moreover, the friction coefficient corresponding to this
optimal ratio value is the smallest. With the increase in the inner ring speed, both film
thickness and temperature increase for the two rib structures. Different from the
spherical rib, the difference between the minimum and the central film thickness is
almost unchangeable, and the tapered rib shows a slight temperature rise. As the load
increases, the difference between the minimum and the central film thickness
becomes larger, and the temperature in the contact zone gradually increases for
the two ribs. Different from the tapered rib, the lower frictional coefficient and lower
minimum film thickness are generated for the spherical rib because of higher film
temperature.
KEYWORDS
tapered roller bearing, Carreau model, inner rib structures, roller end, thermal EHL
1 Introduction
A tapered roller bearing has the ability to carry combinations of large radial and thrust
load or to carry only a thrust load. Moreover, due to large stiffness and convenient
installation, it has wide applications in automobile, mining, metallurgy, and mechanical
industries.
Because the contact angle is different between the inner and outer raceway, a contact force
exists between the large end of the roller and the inner ring rib, and relatively large sliding
friction will be generated at the contact zone (Jamison et al., 1976;Majdoub et al., 2020)in
tapered roller bearings. Therefore, wear and sliding friction happen easily at the rib, and the
load-carrying capacity and service life will be affected. It is indicated that the main failure mode
OPEN ACCESS
EDITED BY
Yebing Tian,
Shandong University of Technology, China
REVIEWED BY
Aqib Mashood Khan,
Nanjing University of Aeronautics and
Astronautics, China
Lixiao Wu,
Lanzhou University of Technology, China
Van-Canh Tong,
Samsung Display Vietnam, Vietnam
Bing Liu,
Shandong University of Technology, China
*CORRESPONDENCE
Xiaoling Liu,
liu_xiaoling06@126.com
SPECIALTY SECTION
This article was submitted to Material
Forming and Removal, a section of
the journal
Frontiers in Manufacturing Technology
RECEIVED 28 August 2022
ACCEPTED 16 December 2022
PUBLISHED 10 January 2023
CITATION
Liu X, Long T, Li X and Guo F (2023),
Thermal EHL analysis of the inner ring rib
and roller end in tapered roller bearings
with the Carreau model.
Front. Manuf. Technol. 2:1029860.
doi: 10.3389/fmtec.2022.1029860
COPYRIGHT
© 2023 Liu, Long, Li and Guo. This is an
open-access article distributed under the
terms of the Creative Commons
Attribution License (CC BY). The use,
distribution or reproduction in other
forums is permitted, provided the original
author(s) and the copyright owner(s) are
credited and that the original publication in
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accepted academic practice. No use,
distribution or reproduction is permitted
which does not comply with these terms.
Frontiers in Manufacturing Technology frontiersin.org01
TYPE Original Research
PUBLISHED 10 January 2023
DOI 10.3389/fmtec.2022.1029860
of the inner rib is wear (Gadallah and Dalmaz, 1984), and the tapered
roller bearing is not suitable for high-speed operation without paying
special attention to cooling and lubrication (Harris and Kotzalas,
2009).
To improve the lubrication between the rib of the inner race
and the large end of the roller, it is necessary to analyze the
structure and the lubrication between the rib and the roller end.
Jamison et al. (1976) found that there are good lubricating effects
between the rib of the inner race and the large end of the roller,
when the nominal contact lies in the middle part of the inner race
rib. Zhang et al. (1988) investigated the lubrication between the
rib of the inner race and the large end of the roller and obtained
the numerical solution corresponding to different structures of
the rib in tapered roller bearings. Jiang et al. (1994),basedonthe
method proposed by Zhang et al. (1988), considered the thermal
and non-Newtonian effects and showed that the thermal and
non-Newtonian effects are negligible under moderate- and low-
speed operation and load. Taking into account the complex
geometry pairings and kinematics, the tribological behavior of
the roller end/face rib contact of tapered roller bearings was
predicted by EHL contact simulations, and it is mainly
determined by the basic geometric pairing and the radii
(Wirsching et al., 2021). Through finite element analysis, the
temperature distribution of railway double-row tapered roller
bearings under test conditions was investigated (Gao et al.,
2022). A computational fluid dynamics (CFD) model was
developedtoconsidereffectsofaerationonthelubricant
behavior in tapered roller bearings (Maccioni et al., 2022).
Under angular misalignment, two types of logarithmic
crowning models for the roller profile were theoretically
designed (Wu et al., 2020). To disclose the mechanism of
defect effects, contact states of roller–raceways and roller–rib
were analyzed (Liu et al., 2020).
A rheological model based on power function was put
forward by Carreau (1972). The Carreau–Yasuda model was
developed by Yasuda (1979) and Yasuda et al. (1981), and the
shear stress of its constructive equation can be expressed by
generalized Newtonian viscosity and strain rate (Bair, 2004;Bair
and Khonsari, 2006;Bair, 2009;Kumar and Khonsari, 2009;
Katyal and Kumar, 2012;Kumar and Kumar, 2014). Liu et al.
(2007) analyzed the point contact EHL and found that numerical
results for polyalphaolefin (PAO) obtained with the
Carreau–Yasuda rheological model are coherent with the
experimental ones. Therefore, the Carreau rheological model
is well suited to the majority of synthetic lubricating oil
widely used at present.
Aboveall,theoptimumdesignoftheribandthelargeendof
the roller in tapered roller bearings based on EHL can improve the
lubricating performance of the contact and prolong the life of the
bearing. To simulate the engineering practice, the non-Newtonian
effects, the thermal effects, and the spinning motion should be
combined. Therefore, with the aim of designing and manufacturing
the rib surface of the inner race in tapered roller bearings (Tian
et al., 2021a;Tian et al., 2021b), a thermal EHL model for the
rib–roller end contact is built with the Carreau rheological model,
which is different from the previous model in the literature, and
thermal EHL analysis for the contact between the inner ring rib and
the roller end in tapered roller bearings used in railway coaches is
carried out.
2 Velocity analysis and rib structures
2.1 Velocity analysis
The geometry structure is important for the lubricating status in
bearings. As shown in Figure 1, the schematic of a tapered roller
bearing is given. The rib is called Solid 1, and the roller is called Solid 2.
FIGURE 1
Schematic diagram of the tapered roller bearing.
FIGURE 2
Analysis of velocity.
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The coordinate system is built at the contact point B.Ois the center of
the pitch circle of the bearing, and O
1
is the center of the spherical
surface of the roller’s large end. O
2
is the conical point of the raceway,
and O
3
is the center of the spherical surface of the rib. Ais the vertex of
the rib cone. R
1
is the curvature radius of the rib, and R
2
is the
curvature radius of the roller end. R
i
is the radius at the point of
tangency for the large end of the tapered roller and the rib of the inner
race, and Ristheconedistanceoftheroller.α
i
is the angle between
the inner race and the axis of the bearing, and α
o
is the angle
between the outer race and the axis of the bearing. ψis the cone
angle of the rib.
In this analysis, parameters of a double-row tapered roller bearing
used in railway coaches (Hu et al., 2013) are as follows: α
i
= 11.11°,
α
o
= 15.11°, the diameter for the large end of the roller D
max
= 17.56 mm,
the diameter for the small end of the roller D
min
=16.15mm,thelength
of the roller l= 18.97 mm, and the pitch diameter d
m
= 155 mm.
When the tapered roller bearing is running, pure roll exists
between the roller and the race. The outer race is stationary, and
the inner race rotates with angular speed ω
i
, assuming that the
direction of the angular speed is positive. According to the kinetic
principle and the structure analysis of tapered roller bearings, the
rotational speed ω
b
and revolution speed ω
c
(Jiang et al., 1994) are
given, respectively, as follows:
ωb−
2RiωiDmax cos 0.5αi+αo
()()
+Ri
()
2RiDmax +D2
max cos 0.5αi+αo
()()
.(1)
Note that the minus sign in Eq. 1denotes the opposite direction of
ω
b
to that of the angular speed of the inner race ω
i
,
ωcRiωi
2Ri+Dmax cos 0.5αi+αo
()()
.(2)
The spinning speed of the roller relative to the inner race is
ωs− ωi−ωc
()
cos αi+α
()
+ωbβ−α
,(3)
where αis the angle between the direction of the spinning speed and
the center axis of the roller, and βis the half conical angle, i.e., β= 0.5
(α
o
−α
i
).
Defining h
c
as the contact height from the nominal contact point B
to the inner raceway, the velocity of the rib of the inner race in the
contact region is written as
u1Ri+hc
()
ωi.
The velocity of the large end of the tapered roller in the contact
area is
u20.5Dmax −hc
()
ωb+Ri+hc
()
ωc.
Velocity analysis shown in Figure 2 defines that
u0u1+u2
()2,(4)
v0v1+v2
()2,(5)
where u
1
and v
1
are velocities of the rib in the direction of xand y,
respectively; u
2
and v
2
are velocities of the roller in the direction of x
and y, respectively.
The entrainment velocities u
R
and v
R
in the direction of xand yare
uRu0+yωs2,(6)
vRv0−xωs2. (7)
2.2 Structures of the rib
2.2.1 Spherical rib
To guarantee the wedge gap and surface tangent between the rib
face and the roller end, the relationship of R
1
>R>R
2
should be
satisfied in tapered roller bearings.
Provided that the curvature radii of the rib on vertical and axial
planes are R
x1
and R
y1
and curvature radii of the roller end on vertical
and axial planes are R
x2
and R
y2
, respectively, then R
x1
=R
y1
and R
x2
=
R
y2
are suitable for the spherical rib and spherical roller end.
The contact height h
c
(Zhang et al., 1988) between the nominal
contact point Band the inner raceway can be written as follows:
hcR2sin ψ−αi
−1
cos ψ−αi
cot β+sin β
+R
cot β+tan ψ−αi
.(8)
If βis small, then
hcRsin β−R2sin β−sin ψ−αi
.(9)
From Figure 1, the following relationship can be obtained:
R1R2+Rcos β+R2
2−R2sin 2β
1/2
sin αi+β
sin ψ. (10)
2.2.2 Tapered rib
Different from the spherical rib, the generatrix of the tapered rib is
a straight line on the axial plane; therefore, the curvature radius on the
axial plane is R
y1
=∞.R
x2
=R
y2
and R
x1
>R
x2
are still satisfied. Eqs. 8,
9, and 10 are also suitable for the tapered rib and spherical roller end.
3 Mathematical model
3.1 Rheological model
The Carreau rheological model is used, and its constitutive
equation (Yasuda et al., 1981;Maccioni et al., 2022) is as follows:
TABLE 1 Parameters of the lubricant.
Parameter Squalane oil
Thermal conductivities, k,W/(m·K) 0.14
Environmental density, ρ, kg/m
3
864
Specific heat, c, J/(kg·K) 2000
Environmental viscosity, η
0
,Pa·s 0.0276
Environmental temperature, t
0
, K 298
B5.258
V
occ
/V
0
0.6633
K′
0
11.74
K
0
, GPa 1.312
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Liu et al. 10.3389/fmtec.2022.1029860
η*η1+η_
γG
2
n−1
()
/2,(11)
where η* is the effective viscosity of the non-Newtonian fluid. Gis the
shear modulus, and it is the demarcation point between the Newtonian
fluid and Carreau–Yasuda fluid. nis the power exponent, and it can be
obtained by the experiment or the non-equilibrium molecular
dynamics simulation.
3.2 Reynolds equation
The Reynolds equation is simplified from the generalized
Reynolds equation proposed by Yang and Wen (1990).
z
zxρη
eh3zp
zx
+z
zyρη
eh3zp
zy
12 z
zxρ*
xuRh
+12 z
zyρ*
yuRh
,(12)
where pis the film pressure and his the film thickness.
The boundary conditions of Eq. 12 are
px
in,y
px
out,y
px,y
in
px,y
out
0,
px,y
≥0xin <x<xout,y
in <y<yout
.
(13)
3.3 Temperature control equations
Thermal convection can be neglected as the entraining speed is not
very high in spinning EHL. The temperature equation for the oil film is
simplified as
kz2t
zz2−t
ρ
zρ
ztuzp
zx+vzp
zy
+η*zu
zz
2
+zv
zz
2
0,(14)
where ρis the density of the lubricant, kis the thermal conductivity of
the lubricant, tis the temperature, and u,vare the flow velocities in the
direction of xand y, respectively.
The equation of thermal conduction for moving Solid 1 is
c1ρ1u1
zt
zxk1
z2t
zz2
1
,(15)
where c
1
,ρ
1
,k
1
, and z
1
are the specific heat, the density, the thermal
conductivity, and the coordinate of Solid 1, respectively.
The equation of thermal conduction for stationary Solid 2 can be
written using the Laplace equation as follows:
z2t
zx2+z2t
zy2+z2t
zz2
2
0,(16)
where z
2
is the coordinate for Solid 2.
Different from the previous model, Solid 2 is assumed as a half-
infinitely large solid, and the surface temperature can be derived from
Eq. 16, i.e.,
t2x, y
t0−k
2πk2zt
zzzh
dx′dy′
x−x′
2+y−y′
2
,(17)
where t
0
is the environmental temperature and k
2
is the thermal
conductivity of Solid 2.
3.4 Film thickness equation
The film thickness equation between the roller and the rib can be
expressed as
FIGURE 3
Typical numerical solutions for the spherical rib for R
x2
/R
x1
= 0.3,
w= 320 N, and n
i
= 1,000 r/min. (A) Film thickness, (B) pressure, and (C)
mid-layer film temperature.
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Liu et al. 10.3389/fmtec.2022.1029860
hx,y
h00 +x2
2Rx
+y2
2Ry
+2
πE′
Ω
px′,y′
x−x′
2+y−y′
2
dx′dy′,(18)
where Ωis the computing domain and R
x
and R
y
are equivalent
curvature radii along the xand ydirection, respectively.
3.5 Density–pressure–temperature
relationship
If the Tait state equation (Liu et al., 2006) is amended by the
thermally modified function, then the relationship of
density–pressure–temperature is described as
FIGURE 4
Typical numerical solutions for the tapered rib for R
x2
/R
x1
= 0.3, w=
320 N, and n
i
= 1,000 r/min. (A) Film thickness, (B) pressure, and (C) mid-
layer film temperature.
FIGURE 5
Variations in the frictional coefficient versus R
x2
/R
x1
with various
rotation speeds for w=560N.(A) n
i
= 700 r/min, (B) n
i
= 1,000 r/min,
and (C) n
i
= 1,300 r/min.
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Liu et al. 10.3389/fmtec.2022.1029860
ρ
ρ0
V0
V1−1
1+K0
′ln 1 +p
K0
1+K0
′
−1
1−Ct−t0
()[]
,(19)
where ρ
0
and V
0
are the density and volume of the lubricant at the
atmospheric pressure. Vis the volume of the lubricant at pressure p.K
0
and K′
0
are constants, which are determined by the experiment. Cis
the coefficient of thermal expansion, and C= 0.0008 K
−1
is used in the
analysis.
3.6 Viscosity equation
The viscosity of the lubricant is calculated by the Doolittle
equation. Because the Doolittle equation is solved with the
density equation, if the density equation under thermal
conditions is substituted into the viscosity equation, the
viscosity–temperature relationship of the lubricant can be
expressed as (Liu et al., 2006)
ηη0exp BVocc
V0
1
V
V0−Vocc
V0
−1
1−Vocc
V0
⎡
⎣⎤
⎦
⎧
⎨
⎩⎫
⎬
⎭,(20)
where V
occ
/V
0
and Bare unknown constants, and V/V
0
is determined
using Eq. 19.
4 Numerical method
In tapered roller bearings, the position of the nominal contact
point or the curvature radius of the rib and the roller large end can be
determined by the optimal design. Based on α
i
,α
o
,β, and R
i
of the
bearing, if h
c
/Rand R
2
/R
1
are given, R
1
,R
2
, and ψwill be obtained by
solving Eqs. 8,9, and Eq. 10. The numerical methods such as the secant
method (Zhang et al., 1988) are used to solve the problem in this
article. The details are as follows.
Define the residual error aas follows:
aR2R
Ⅰ
−R2R
Ⅱ
,(21)
where
R2R
Ihc
R−1
cot β+tan φ
sin φ−1
cos φcot β+sin β
,(22)
R2R
II R2
R1
sin αi+β
sin αi+φ
1−R2
R1
1−sin αi+β
sin αi+φ
,(23)
where φ=ψ−α
i
.
The secant method is employed, and the range of φis defined as
φ∈e
R−1
R2R1
−1
tan β,e
R
.
FIGURE 6
Effect of rotating speeds on film thickness and temperature of two rib structures for R
x2
/R
x1
= 0.7 and w= 320 N. (A) Spherical rib/spherical roller end; (B)
tapered rib/spherical roller end.
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Liu et al. 10.3389/fmtec.2022.1029860
The iteration will stop after |a|<ε, and εis set at 0.0001 to ensure
accuracy. φand R
2
/Rare output at last. R
1
and R
2
can be obtained by
calculating R
1
/R=(R
2
/R)/(R
2
/R
1
).
After R
1
,R
2
,andψare computed, the equivalent curvature radii R
y
and R
x
can be obtained for the two kinds of rib structures. The numerical
calculation for TEHL is carried out based on the dimensionless form.
Different from the previous model, the film pressure is obtained with the
multi-grid method (Venner et al., 1990), the elastic deformation is
computed with the multi-level multi-integration method (Brandt and
Lubrecht, 1990), and the temperature fieldiscalculatedwiththecolumn-
by-column technique (Qu et al., 2000). Five levels are used, and the node
number is 257 and 257 along the X-andY-direction, respectively, at the
finest level. A total of 16 layers are arranged in the Z-direction, of which
10 layers are arranged across the film and six layers are arranged for
moving Solid 1. The temperature of stationary Solid 2 is calculated with
the multi-level multi-integration method. The convergence criterion is
thattherelativeerrorislessthan1×10
–4
for the pressure and
temperature,andtherelativeerrorislessthan1×10
–3
for the load.
5 Results and discussion
Squalane oil with a low pour point is selected as the lubricant, and
its parameters are shown in Table 1. The shear modulus G= 6 MPa,
and the power exponent n= 0.42 for the Carreau model. The nominal
contact height h
c
= 4 mm.
5.1 Typical solutions
Surface plots of the film thickness, the pressure, and the mid-layer
film temperature for the spherical rib are shown in Figure 3 for R
x2
/
R
x1
= 0.3, w= 320 N and n
i
= 1,000 r/min. From Figures 3A, B, it can be
found that typical characteristics of EHL are exhibited by the spherical
rib, i.e., the exit constriction of the film thickness and the second spikes
for the pressure distributions. Moreover, the surface plots of the film
temperature in Figure 3C are similar to those of the pressure.
With the same input parameters as inFigure 3, the surface plots of the
film thickness, the pressure, and the mid-layer film temperature are
shown in Figure 4 for the tapered rib. It can be seen that the characteristics
of EHL for the tapered rib are weaker than those for the spherical rib. This
is because the contact region is so large that the maximum pressure is
small with the same load, and it contributes less to the elastic deformation.
Compared to the spherical rib, the film temperature is lower than that for
the tapered rib. Hence, the film thickness for the tapered rib is larger. It
can be observed that the spinning motion has little influence on the film
thickness, the pressure, and the temperature for the two rib structures in
the present analysis. The reason is that the maximum film temperature
occurs near the center of the contact region, which is similar to that of the
maximum pressure, and illustrates the primary effect of pressure on the
temperature. Therefore, the contribution of the spinning velocity is small
near the contact center, and the effects of spinning on the maximum
temperature are minimal.
5.2 Effects of rotational speed
Figure 5 shows variations in the frictional coefficient versus the
ratio of the curvature radius of the roller large end to that of the rib
with various rotational speeds for the two rib structures for w= 560 N.
It can be found that the variation tendency of the frictional coefficient
is similar for the two types of ribs, i.e., decreasing at first and then
increasing with the ratio of the curvature radius of the roller large end
to that of the rib. Therefore, there exists an optimal ratio of the
curvature radius. For the three rotational speeds, the optimal ratio of
the curvature radius is 0.75 for the spherical rib. However, the optimal
ratio with three speeds is 0.55, 0.5, and 0.45, respectively, for the
FIGURE 7
Variations in the friction coefficient versus R
x2
/R
x1
with different
loads for n
i
= 1,000 r/min. (A) w= 160 N, (B) w= 360 N, and (C) w=
560 N.
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tapered rib. Hence, the ratio for the tapered rib/spherical roller R
x2
/R
x1
ranges from 0.45 to 0.55 to obtain the least frictional coefficient. When
the ratio approaches unity for the tapered rib, the frictional coefficient
increases remarkably, and the reason is that the film breakdown occurs
due to the lack of wedge effect. In addition, effects of R
x2
/R
x1
on the
frictional coefficient for the spherical rib are less sensitive than those
for the tapered rib. Different from the tapered rib, the frictional
coefficient for the spherical rib is always low with the increase in
R
x2
/R
x1
. This is because the Hertzian contact region is circular for the
spherical rib/spherical roller large end, whereas it is elliptical for the
tapered rib/spherical roller large end which produces lower maximum
pressure and film temperature.
It can be concluded that the optimal ratio of the curvature radius is
determined by the rotational speed for the tapered rib, so it can be
adjusted according to the bearing speed when the tapered rib is
designed. Additionally, it can be observed that when the speed is
low, the difference in the frictional coefficient is small corresponding
to the optimal ratio of the curvature radii for the two ribs, whereas
when the speed is high, the frictional coefficient for the spherical rib is
less than that for the tapered rib corresponding to the optimal ratio of
the curvature radii. This can evidently be attributed to the reduction of
viscosity caused by thermal effects under higher speed conditions.
Additionally, the maximum film temperature for the spherical rib is
higher than that for the tapered rib.
Effects of the rotational speed of the inner race on the film
thickness and temperature for the two ribs are shown in Figure 6
for R
x2
/R
x1
= 0.7 and w= 320 N; Figure 6A represents the spherical rib,
and Figure 6B represents the tapered rib. As the rotational speed
increases, both the minimum film thickness and central film thickness
increase for the two ribs. This clearly demonstrates the dominant effect
of speed on the film thickness in EHL. In addition, for the spherical rib,
the difference between the central and minimum film thickness
increases with the increasing speed. However, the difference is
almost unchangeable for the tapered rib. The reason is that,
compared to the tapered rib, the temperature of the film and the
roller large end rises obviously for the spherical rib, and thermal effects
cause an obvious reduction in film thickness. Moreover, due to the
sliding effect, the highest temperature of the film and the roller large
end increases with the increasing rotational speed, but the rib
temperature changes negligibly. This is because the rib is running
more quickly than the roller, and the heat transfer from the film to the
rib is difficult. Therefore, when the ratio of the curvature radius and
load are fixed, the minimum film thickness is larger for the tapered rib.
The optimal lubricating performance is determined by the dealing of
the frictional coefficient and the film thickness.
5.3 Effects of the load
When n
i
= 1,000 r/min, variations in the frictional coefficient
versus the ratio of curvature radius of the roller large end to that of the
rib with various loads for the two ribs are seen, as shown in Figure 7.It
FIGURE 8
Effects of load on film thickness and temperature of two rib structures for R
x2
/R
x1
= 0.7 and n
i
= 1,000 r/min. (A) Spherical rib/spherical roller end; (B)
tapered rib/spherical roller end.
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Liu et al. 10.3389/fmtec.2022.1029860
can be observed that the frictional coefficient decreases first and then
increases with the ratio of curvature radius when the load is different,
and the difference between the two types of rib becomes large when the
load increases. This is because the position of the nominal contact
point changes with the ratio of curvature radius for the tapered rib/
spherical roller. Moreover, large load will lead to high temperature and
thus lower viscosity of the lubricant and low frictional coefficient.
With the increase in the load, the optimal ratio of the curvature radius
is always 0.75 for the spherical rib, while the optimal ratio for the
tapered rib is 0.35, 0.45, and 0.5, respectively. Therefore, the optimal
ratio of the curvature radius is determined by both the load and the
velocity for the tapered rib. However, the optimal ratio is hardly
influenced by the velocity and the load for the spherical rib. This can
be attributed to the characteristics of the spherical rib/spherical roller
end, and thermal effects on the frictional coefficient are more
significant than those for the tapered rib/spherical roller end.
Figure 8 shows effects of the load on the film thickness and
temperature for the two rib structures for R
x2
/R
x1
= 0.7 and n
i
=
1,000 r/min; Figure 8A represents the spherical rib, and Figure 8B
represents the tapered rib. It can be found that the spherical rib is
much more sensitive to the load, and the film thickness varies more
widely than that for the tapered rib. In addition, the minimum film
thickness decreases remarkably, and the central film thickness
increases first and then decreases slightly with the increase in the
load. This is because the central film thickness is the function of the
load. The central film thickness decreases slightly as the load surpasses
a value. This is coherent with the film thickness formula proposed by
Hamrock and Downson. Moreover, the minimum film thickness
decreases because of the high pressure and temperature in the
contact region with large load. However, with the increase in the
load, the central film thickness increases for the tapered rib. There may
be two reasons: one is that the thermal expansion caused by the film
temperature rises in the contact region; the other is that the special
contact between the rib and the roller end leads to the low pressure and
small elastic deformation in the contact region, resulting in weak
thermal effects. As the load increases, the central elastic deformation
and film thickness increase to balance the uneven distribution of the
pressure. It can be concluded that variations in the central film
thickness versus the load are abnormal for the tapered rib/spherical
roller end (Jiang et al., 1994).
It can be seen from the temperature profiles that, as the load
increases, for the spherical rib, higher temperature for the film and the
roller will be generated because the contact between the rib and the
roller end prevents the heat loss. However, the rib temperature
increases insignificantly due to high rotational speed and slow heat
transfer. Different from the spherical rib, there is a little change in
temperature for the film, the rib, and the roller for the tapered
rib–roller contact. Therefore, to improve the lubricating
performance, the decisive effects on the minimum film thickness
are the rib structures, followed by the speed and the load.
Conclusion
Based on the structure analysis for the rib of the inner race and the
large end of the roller in tapered roller bearings, a thermal EHL model
considering spin–slide effects is established with Carreau fluid. The
main conclusions are as follows:
1. Effects of spinning motion on the lubricating performance are minimal
between the rib of the inner race and the large end of the tapered roller.
2. The optimal ratio of the curvature radius of the roller large end to that of
the rib exists both for the tapered rib and the spherical rib, and the
frictional coefficient corresponds the least to the optimal ratio.
Moreover, the optimal ratio changes with the rotational speed and
the load for the tapered rib. Different from the tapered rib, the frictional
coefficient is low for the spherical rib with the increase in R
x2
/R
x1
.
3. With the increase in the rotational speed of the inner race, the film
thickness and the temperature increase for the two rib structures.
Different from the spherical rib, the difference between the
minimum and the central film thickness is nearly unchangeable,
and the minimum film thickness is larger due to low film
temperature rise for the tapered rib.
4. With the increase in the load, the difference between the minimum
and central film thickness becomes large for the two rib structures,
and the film temperature rises. Compared to the tapered rib,
variation in the temperature is significant, and the minimum
film thickness is lower for the spherical rib.
5. Further study on this problem is needed for practical engineering.
Data availability statement
The original contributions presented in the study are included in
the article/Supplementary Material;, further inquiries can be directed
to the corresponding author.
Author contributions
XLiu and TL carried out the theoretical analysis. XLiu wrote the
manuscript with support from XLi. FG and XLiu conceived the
original idea and supervised the project.
Funding
This work was supported financially by the National Natural
Science Foundation of People’s Republic of China (grant no.
52275196 and 51475250) and the “Taishan Scholar”Talents Project
from Shandong Province (No. TS20190943).
Conflict of interest
The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be
construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated organizations,
or those of the publisher, the editors, and the reviewers. Any product
that may be evaluated in this article, or claim that may be made by its
manufacturer, is not guaranteed or endorsed by the publisher.
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Liu et al. 10.3389/fmtec.2022.1029860
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