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A new logarithmic profile model and optimization design of cylindrical roller bearing

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Industrial Lubrication and Tribology
A new logarithmic profile model and optimization design of cylindrical roller bearing
Li Cui Yafei He
Article information:
To cite this document:Li Cui Yafei He , (2015),"A new logarithmic profile model and optimization design of cylindrical roller bearing", IndustrialLubrication and Tribology, Vol. 67 Iss 5 pp. 498 - 508
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D o w n l o a d e d b y T e c h n i s c h e U n i v e r s i t a e t B e r l i n A t 1 5 : 1 8 2 3 S e p t e m b e r 2 0 1 5 ( P T )
A new logarithmic proﬁle model andoptimization design of cylindricalroller bearing
Li Cui and Yafei He
Faculty of Engineering, Shanghai Second Polytechnic University, Shanghai, China
Abstract
Purpose
– The purpose of this paper is to ﬁnd a new logarithmic proﬁle model of cylindrical roller bearing, which is expected to avoid edge effectand allow a straight portion on the roller considering uniform pressure distribution and easier manufacturing.
Design/methodology/approach
– A new logarithmic cylindrical roller proﬁle model using three parameters is proposed. Contact model betweenroller and rings and quasi-static model of roller bearing are given to obtain contact pressure distribution and solved by multi-grid andNewton–Raphson method. Optimization of modiﬁed reference rating life model of the roller bearing is proposed by using genetic algorithms.
Findings
– Under heavy load or tilting moment conditions, modiﬁed reference rating life of cylindrical roller bearing may increase greatly byoptimization of three design parameters using the new logarithmic proﬁle model.
Originality/value
– The results of the present paper could aid in the design of logarithmic proﬁle of cylindrical roller bearing and increase fatiguelife of cylindrical roller bearing.
Keywords
Design, Optimization, Bearings, Fatigue life, Cylindrical roller bearing, Logarithmic proﬁle, Optimization design
Paper type
Research paper
1. Introduction
Cylindrical roller bearing is used widely for its compactstructure and large carrying capacity. Concentrated stressoccurs at the end of contact area when a cylindrical rollercontacts with raceways, which results in early fatigue failure of the bearing, especially when the bearing is working undermisaligned loads condition. Therefore, the crowning design of roller to improve the contact condition between roller andrings has attracted substantial attention to increase the fatiguelife of cylindrical roller bearing.The early work by Lundberg (1939) introduced alogarithmic function to crowning of roller, which can ensureuniform contact pressure distribution between roller and ringsby using Lundberg’s proﬁle. The function used very widelybecause of uniform contact pressure distribution and itsconvenience and simplicity. However, this curve isdiscontinuous at the end of roller because of an inﬁnite dropat the end of the effective contact length and a ﬁnite value atthe end of roller, which results in difﬁcult manufacturing. Johns-Gohar ( Johns and Gohar, 1981) improved Lundberg’s
function by avoiding the inﬁnite drop to obtain continuouslogarithmic curves; however, it was difﬁcult to obtain uniformcontact pressure distribution, especially under heavy load andtilting moment conditions. Horng
et al.
(2000) provided adeformation formula for a circular crowned roller. Acomparison between crowned and non-crowned parts wasanalyzed. Zhu
et al.
(2012) gave a mixed lubrication model toinvestigate roller contact elastohydrodynamic lubrication(EHL) problems. Effect of roller length, crown radius andround corner radius were studied. Roller proﬁle was optimizedto obtain uniform pressure distribution and ﬁlm thicknessdistribution. Chen
et al.
(2001) studied the EHL characteristics in logarithmic proﬁled roller contact and theeffect of crown value. Urata (2000) proposed a crowningshape formed by combining two or more circular arcs. Edgestress disappeared in the crowning shape, but misalignmentcould not be considered in the model. Hiroki
et al.
(2009)proposed a new logarithm proﬁle function. Three designparameters were introduced into the proﬁle, and this proﬁlecould prevent edge stress due to misalignment. However,these parameters are not easy to choose and also theparameters can only be optimized in a small range, thus theproﬁle is not used widely in the roller bearing design.Optimization of fatigue life of rolling bearing has beenstudied for its importance. Chakraborty
et al.
(2003)optimized ball bearing and roller bearings using geneticalgorithms. The results showed that the fatigue life of thebearing improved marginally compared with the fatigue lifegiven in standard catalogs. Rao and Tiwari (2007) improved
The current issue and full text archive of this journal is available onEmerald Insight at:
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Industrial Lubrication and Tribology67/5 (2015) 498–508© Emerald Group Publishing Limited [ISSN 0036-8792][DOI 10.1108/ILT-01-2015-0006]
This work was ﬁnancially supported by Shanghai Natural ScienceFoundation (No. 14ZR1416800), Shanghai Alliance Plan (No.LM201408) and Foundation of Shanghai Second Polytechnic University(EGD14XQD12).Received 1 February 2015Revised 22 May 201523 June 2015Accepted 25 June 2015
498
D o w n l o a d e d b y T e c h n i s c h e U n i v e r s i t a e t B e r l i n A t 1 5 : 1 8 2 3 S e p t e m b e r 2 0 1 5 ( P T )
the design optimization of deep-groove ball bearings usinggenetic algorithms, and the major improvement was informulating realistic constraints. However, these attempts foroptimum design of bearings were concentrated on the basicdesign geometric parameters of the bearings, such as diameterand number of rolling elements. Few works of literature areavailable that focus on the proﬁle of rollers for theoptimization. Kamamoto
et al
. (2001) optimized the crowningproﬁle of cylindrical roller bearing. Results showed that thepressure became axially uniform and fatigue life wasimproved, but the effect of misalignment was not taken intoaccount. Kumar
et al.
(2009) optimized fatigue life of rollerbearing by genetic algorithm; geometric parameter and twologarithmic proﬁle parameters of roller were chosen as designvariables. Results showed that the logarithmic proﬁleparameters had a signiﬁcant effect on the fatigue life.In the present paper, a new logarithmic roller proﬁle modelon the basis of Johns-Gohar’s function is proposed. Themodel can be used in heavy load and tilting momentconditions, as well as is convenient in engineering application.Three design parameters of the proﬁle are introduced, and theeffect of these parameters on the contact pressure and fatiguelife is analyzed. A procedure for optimizing fatigue life of the roller bearing is given by using genetic algorithm, theoptimized bearing life show improvement as compared with Johns-Gohar’s result.
2. Calculating of contact pressure of crownedroller
2.1 Three design parameters logarithmic proﬁle
Crowned roller in a cylindrical roller bearing is shown inFigure 1.
s
x
is crowning proﬁle of roller.Lundberg gave a shape of crowning expressed in alogarithmic function; however, this curve had an inﬁnite dropat the end of the effective contact length. Johns-Goharproposed an improved function based on Lundberg’s formula(Kabus
et al.
, 2012):
s
x
2
Q
l
we
E
=
ln
11
1
0.3033
ba
2
xl
we
2
(1)where
Q
is load on roller,
E
=
is equivalent Young’s modulus,
x
is position in the axial direction,
l
we
is effective roller length,
a
is half length of the effective contact and
b
is half width of contact.However, edge stress cannot be avoided completely by Johns-Gohar model, especially under heavy load conditionand tilting moment conditions; also in addition, it is difﬁcultto obtain uniform contact pressure distribution of roller. Tosolve these problems, a new logarithmic proﬁle modelconsidering three design parameters is proposed:
s
x
2
k
1
Q
l
we
E
=
ln
11
1
0.3033
k
2
ba
2
xl
we
2
k
3
(2)where
k
1
stands for multiple factor of the applied load,
k
2
stands for roller radius reduction at rolling contact surfaceends and
k
3
stands for ratio of straight portion to the proﬁle.
2.2 Contact calculation model
The contact between roller and ring is shown in Figure 2.EHL and dry contact are considered in contact calculationmodel.For EHL contact condition, Reynolds equation for aNewtonian lubricant can be expressed as follows:
Figure 1
Crowned roller in a cylindrical roller bearing
j
ZY
O
X
Roller
s
(
x
)
Raceway
(a)(b)Notes:
(a) Cylindrical roller bearing view; (b) crowned roller and raceway
Figure 2
Line contact between roller and ring
C o n t a c t a r e a
Roller Ring
A new logarithmic proﬁle model
Li Cui and Yafei He
Industrial Lubrication and Tribology
Volume 67 · Number 5 · 2015 · 498–508
499
D o w n l o a d e d b y T e c h n i s c h e U n i v e r s i t a e t B e r l i n A t 1 5 : 1 8 2 3 S e p t e m b e r 2 0 1 5 ( P T )
x
h
3
12
p
x
y
h
3
12
p
y
u
m
h
y
0 (3)where
p
is oil ﬁlm pressure,
h
is ﬁlm thickness,
u
m
is the meansurface velocity of roller and ring in the ﬂow direction,
isviscosity and
is density of the lubricant.Film thickness can be calculated by the following:
h
(
x
,
y
)
h
00
s
x
x
2
2
R
x
y
2
2
R
y
2
E
=
·
A
p
x
=
,
y
=
dx
=
dy
=
x
x
=
2
y
y
=
2
(4)where
h
00
is minimum ﬁlm thickness;
R
x
and
R
y
are theeffective radius of curvature between roller and ring in the xand y direction; and elastic deformation due to the pressuredistribution is considered in the last item.Viscosity is varied as contact pressure (Harris, 2001):
p
0
exp
ln
0
9.67
1
1
p
1.96 · 10
8
0.6
(5)Density is given by Dowson–Higginson relation:
p
0
5.9 · 10
8
1.34
p
5.9 · 10
8
p
(6)For dry contact condition, the distance between roller andring can also be calculated by the formula given in equation(4) because pressure occurs at the area where two surfacescome in contact with each other and no pressure occurs wherethere is no contact; the following relationships are obtained:
h
(
x
,
y
)
0,
p
(
x
,
y
)
0 Non-contact area
h
(
x
,
y
)
0,
p
(
x
,
y
)
0 Contact area (7)Pressure in dry contact condition is given by a semi-ellipticalpressure distribution based on the Hertz solution (Harris,2001):
p
(
x
,
y
)
p
0
1
xa
2
yb
2
(8)
p
0
3
A
p
x
,
y
dxdy
2
ab
(9)Multi-grid algorithm is used to calculate the equations(Venner and Lubrecht, 2000). Considering calculationaccuracy and speed, ﬁve-level grid is used and the ﬁnest gridhas 256
128 points.
2.3 Quasi-static model of cylindrical roller bearing
Figure 3 shows geometry of roller–inner ring interaction of roller bearing, where inner ring is moved and rotated relativeto the outer ring. OXYZ is coordinate of the bearing.Considering tilting moment acting on the bearing, assumingload on the bearing is
F
y
,
F
z
,
M
y
, displacement of inner ringis
y
,
z
,
y
, tilting angle of
j
th roller is
j
and angularlocation of the
j
th roller is
j
.
j
2
Z
(
j
1)
o
t
(10)where
Z
is the number of rollers and
o
is rotating speed of roller.Assuming roller is divided into
n
s
slices along the axialdirection, the minimum ﬁlm thickness between inner ring and
k
th slice of
j
th roller can be obtained:
h
00
ijk
u
r
4
12
y
cos
j
z
sin
j
x
k
12
D
i
tan
12
y
cos
j
tan
y
cos
j
j
(11)The minimum ﬁlm thickness between outer ring and
k
th sliceof
j
th roller can be given as follows:
h
00
ejk
u
r
4
12
y
cos
j
z
sin
j
x
k
12
D
i
tan
12
y
cos
j
tan
j
(12)where
u
r
is radial clearance of roller bearing,
D
i
is diameter of the inner raceway, cos
j
0 for upper and cos
j
0 for lowersign,
x
k
k
0.5
l
we
/n
s
0.5
l
we
.By solving the contact calculation model,
p
ijk
and
p
ejk
, whichare contact pressure between inner rings, outer ring and
k
thslice of
j
th roller can be obtained. Then, contact force can beobtained as follows:
q
ijk
A
ijk
p
ijk
x
,
y
dA
ijk
(13)
q
ejk
A
ejk
p
ejk
x
,
y
dA
ejk
(14)The centers of the inner ring must satisfy force equationconditions as follows:
Figure 3
The geometry of roller and inner ring interaction
X
Y
ZO
D
w
Δ
θ
y
α
j
k
t h s l i c e
A new logarithmic proﬁle model
Li Cui and Yafei He
Industrial Lubrication and Tribology
Volume 67 · Number 5 · 2015 · 498–508
500
D o w n l o a d e d b y T e c h n i s c h e U n i v e r s i t a e t B e r l i n A t 1 5 : 1 8 2 3 S e p t e m b e r 2 0 1 5 ( P T )
F
y
1
n
s
j
1
Z
·
k
1
n
s
q
ijk
sin
j
0 (15)
F
z
1
n
s
j
1
Z
·
k
1
n
s
q
ijk
cos
j
0 (16)
M
y
j
1
Z
1
n
s
·
k
1
n
s
q
ijk
·
k
1
n
s
q
ijk
·
k
0.5
l
we
n
s
0.5
l
we
k
1
n
s
q
ijk
· cos
j
0 (17)In radial direction, the
j
th roller equilibrium is as follows:1
n
s
k
1
n
s
q
ijk
0.5
m
r
o
2
D
pw
1
n
s
k
1
n
s
q
ejk
(18)where
m
r
is mass of roller and
D
pw
is pitch diameter of rollerbearing. Newton–Raphson method is used to solve theequations.
3. Fatigue life model of and optimization designof cylindrical roller bearing
3.1 Modiﬁed rating life model
The basic rating life of rolling bearing
L
10
is used widely.However, for many applications, it has become desirable tocalculate the life more accurate under speciﬁed lubricationand contamination conditions. Accordingly, the modiﬁedrating life model of rolling bearing is given as follows:
L
nm
a
1
a
iso
L
10
(19)where
a
1
is reliability factor,
a
iso
is life modiﬁcation factor and
n
means probability of failure.
3.2 Modiﬁed reference rating life model
In equation (19), the inﬂuence of tilted or misaligned bearings
and the inﬂuence of bearing clearance during operation cannot be considered in modiﬁed rating life calculation. To makethese improvements, modiﬁed reference rating life model isgiven according to contact load distribution results of rollingbearing.According to ISO/TS 16281 (2008), the basic dynamiccapacity between inner ring, outer ring and
k
th slice of rollerare as follows:
q
kci
10.83 ·
C
r
0.378
Z
1
1.038
1
1
143/108
9/2
2/9
·
1
n
s
7/9
(20)
q
kce
10.83 ·
C
r
0.364
Z
1
1.038
1
1
143/108
9/2
2/9
·
1
n
s
7/9
(21)where
D
we
/
D
pw
,
D
we
is roller diameter and
C
r
is dynamicload rating of bearing.Assuming outer ring is stationary and inner ring is rotatingrelative to the load, the dynamic equivalent load betweeninner ring, outer ring and
k
th slice roller are as follows:
q
kei
1
Z
j
1
Z
p
ijk
271
2
D
we
1
l
we
n
s
4
14
(22)
q
kee
1
Z
j
1
Z
p
ejk
271
2
D
we
1
l
we
n
s
4.5
14.5
(23)The modiﬁed reference rating life is described as follows:
L
nmr
a
1
k
1
n
s
a
iso
–9/8
q
kci
q
kei
9/2
q
kce
q
kee
9/2
8/9
(24)In equations (19) and (24), reliability of bearing is deﬁned as
90 per cent, reliability factor is 1.0 and
L
10
m
and
L
10
mr
arestudied.
3.3 Optimization design of crowned cylindrical roller bearing
The single optimization objective function is maximizedmodiﬁed reference rating life:max
f
X
max
L
10
mr
(25)where
f
X
represents the objective function and
X
is thedesign variable vector.Assuming the basic geometric parameters of roller bearinghave been chosen, the three design parameters of roller proﬁleare chosen as design variables for the optimization:
X
k
1
,
k
2
,
k
3
(26)To reduce the solution range, the strict upper and lowerbounds of the parameters are given as constraint:1
k
1
50
k
2
11
k
3
5 (27)Genetic algorithm is used as optimization design algorithms of bearing fatigue life (Deb and Tiwari, 2008). Initial populationsize is set as 500 and number of generations as 100, crossoverprobability is 0.90 and mutation probability is 0.01.Figure 4 gives ﬂow chart for calculation and optimization of cylindrical roller bearing. First, the proﬁle parameters areinputted. Then, the contact pressure and contact force arecalculated by multi-grid algorithm. The quasi-static equationsare solved by Newton–Raphson method. Finally, the bearingfatigue life is calculated and optimized by genetic algorithm.
4. Results and discussions
The speciﬁcations of the roller bearing and parameter valuesused in the analysis are listed in Table I. Rotating speed is
2,000 rpm; it is assumed that only rollers are proﬁled, boththree design parameters roller proﬁle model proposed in thispaper and Johns-Gohar’s proﬁle model are calculated and
A new logarithmic proﬁle model
Li Cui and Yafei He
Industrial Lubrication and Tribology
Volume 67 · Number 5 · 2015 · 498–508
501
D o w n l o a d e d b y T e c h n i s c h e U n i v e r s i t a e t B e r l i n A t 1 5 : 1 8 2 3 S e p t e m b e r 2 0 1 5 ( P T )

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