TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 8, August 201
4, pp. 6009 ~ 6016
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.559
3
6009
Re
cei
v
ed
Jan
uary 6, 2014;
Re
vised Ma
rch 15, 2014; A
c
cepted Ap
ril 3, 2014
Modeling and Analyzing for the Friction Torque of a
Sliding Bearing Based on Grey System Theory
Wang Baomi
n
g
1
*
，
Xu Jinxin
1
，
Chen S
h
engShen
g
2
，
Wu Zaixin
1
1
School of Mec
han
ical El
ectro
n
ical E
ngi
ne
eri
ng La
nzh
ou Un
iversit
y
of T
e
chnol
og
y
Gansu La
nzh
o
u
in Ch
ina, 0
9
3
1
-29
738
60
2
Lanzh
ou Instit
ute of Chemic
a
l
Ph
y
s
ics Ch
in
a Academ
y of
Scinec
es,
Gansu La
nzh
o
u
730
00
0, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
b
m
2
0
07@
1
63.com
A
b
st
r
a
ct
Based on the
grey system theory, the gr
ey
rela
tional analysis m
e
thod
is
proposed
and used in
ana
ly
z
i
n
g
th
e i
n
flue
nce
of var
i
ous f
a
ctors o
n
the fricti
on tor
que
of a
sli
d
i
n
g b
eari
ng. On
the b
a
sis
of th
e
grey relati
on
al
analysis th
e mu
ltidi
m
ens
ion
a
l grey
mo
d
e
l
GM(1,N,D) fo
r the friction torqu
e
of a sli
d
in
g
bear
ing is b
u
ilt
up. T
a
king Al-
base
d
all
o
y sli
d
in
g bear
i
ng a
s
an exa
m
pl
e, the calcu
l
atio
n results show
that,
compar
ed w
i
th
other
influ
enc
e factors, friction co
e
fficie
n
t, loa
d
, temper
ature a
nd r
o
tatio
nal s
p
e
ed
hav
e
mor
e
si
gn
ifica
n
t infl
ue
nce
o
n
the
b
eari
n
g
friction
torq
u
e
. Co
mpari
n
g
exp
e
ri
me
ntal
resu
lts an
d
the
calcul
ated va
l
ue of the GM(1,N,D) mo
del
based o
n
these i
m
p
o
rtant
influe
nc
e factors, the maxi
mu
m
relativ
e
resid
u
a
l
s is 9.09%, th
e avera
ge rel
a
t
i
ve resid
u
a
l
s is
7.9% and th
e accuracy is 9
2
.1%. It verify th
at
GM(1,N,D) mo
del h
a
s go
od a
ccuracy an
d is app
lica
b
le for
p
r
edicti
ng frictio
n
torque of a sl
idin
g be
ari
ng.
Ke
y
w
ords
: Grey system the
o
r
y, slidin
g bear
i
ng,
friction torq
ue, influ
enc
e factors
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
The fri
c
tion
t
o
rqu
e
i
s
an i
m
porta
nt pe
rf
orma
nce p
a
rameter of b
e
a
ring.
It rep
r
e
s
ent
s a
n
energy lo
ss a
nd
cau
s
e
s
a
retardatio
n of
motion,hen
ce
it is
witne
s
se
d a
s
a te
mpe
r
ature in
crea
se
[1-3]. In past
several yea
r
s, the beari
ng
friction to
rqu
e
ha
s re
ceiv
ed wi
de atte
ntion from m
any
resea
r
chers, signifi
cant a
m
ounts
of re
sea
r
ch hav
ed
bee
n d
e
vote
d to it. In
19
59 Palm
gren
[4]
gave a cal
c
ul
ation formul
a to determin
e
the fric
tion to
rque value of
a rolling b
e
a
r
i
ng. Lin Gua
n
y
u
[5] studied th
e cha
r
a
c
teri
st
ics of the dee
p groove
b
a
ll beari
ng un
de
r different vacuum. In rece
nt
years
study o
n
the f
r
ictio
n
torque
of
a rolling
bea
ring
ha
s m
ade
great p
r
og
re
ss
in term
s
of th
e
energy and scuffing failure
[6-8]. Howev
e
r, there
ha
s
been little research on the friction to
rque
of
a slidin
g bea
ring. So far t
here
ha
s bee
n no sy
stemi
c
analy
s
e
s
a
nd re
se
arche
s
on the fri
c
ti
on
torque
of a
sliding
bea
ring
[9]. The mai
n
rea
s
o
n
i
s
th
at the fri
c
tion
torqu
e
of
a
sliding
bea
ring
is
determi
ned b
y
a numberof factors (s
uch as loa
d
, ro
tational sp
ee
d,
stru
cture, etc),
un
ce
rtai
nty
factors
(such
as l
u
b
r
icatio
n
,
material, l
u
b
r
icatio
n, et
c)
and
other
co
mplex fa
ctors. The
s
e fa
cto
r
s
were inte
ra
ctional, the fri
c
tion torq
ue of
a slid
i
ng be
aring sho
w
s g
r
eat ran
dom
n
e
ss an
d chan
ce
s
[10].
Furthe
rmore, due
to stru
cture con
s
train
s
it
is dif
f
icult to
obtai
n a
ma
ss of
e
x
perime
n
tal d
a
ta
in actual
con
d
itions.
Grey
system
theory was
founde
d by a
Ch
in
ese sci
entist, Profe
s
sor
De
ng
Ju
Long i
n
1982 [11], takin
g
the un
certai
n sy
ste
m
of “sm
a
ll
sampl
e
”, “p
oor info
rmati
on”, i.e. “pa
r
tial
informatio
n known, partial
informatio
n unkn
o
wn”
a
s
the re
sea
r
ch obje
c
t. This theory hol
ds t
hat
for a grey
system just some informa
t
ion is
kn
own, some inf
o
rmatio
n is
uncertain, ev
en
compl
e
tely u
n
kn
own, ho
wever the
r
e i
s
a ne
ce
ssary
intrin
sic
rel
a
tion between
them.Throu
gh
exploiting an
d excavating
the limited informatio
n,
gre
y
system theory can h
e
lp us to unde
rstand
the re
al worl
d and
to an
a
l
yze its i
nhe
rent ch
ar
acte
ristics. Becau
s
e g
r
ey p
r
e
d
iction m
odel
can
comp
ute a
n
d
p
r
esume
according
to
the limit
ed
inform
ation
and
do
es
not have
sp
ecial
requi
rem
ent and limit to the ob
servatio
nal data,
it has be
en wi
de
ly used in pe
troleum d
o
m
a
in,
geolo
g
ical fiel
d, medical fie
l
d, automotive fi
eld and m
anag
ement fi
eld, and b
r
ou
ght rem
a
rkab
le
eco
nomi
c
an
d so
cial be
ne
fits [12-14].
The f
r
iction
t
o
rqu
e
of a
slidin
g b
eari
ng h
a
s the
cha
r
a
c
teri
stics of
no
nline
a
rity an
d
rand
omn
e
ss,
in addition, it
is un
reali
s
tic t
o
an
slyz
e
usi
ng a ma
ss of
origin
al data.
This
con
d
itio
n
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 600
9 –
6016
6010
just a
c
cord
s
with the g
r
ey
system
the
o
ry. Therefore, it is feasibl
e
to analy
z
e the
friction to
rqu
e
of
a sli
d
ing
bea
ring by u
s
in
g
grey
system
theory.In
thi
s
pape
r the
infl
uen
ce
of ma
terial h
a
rdne
ss,
rotational
sp
eed, loa
d
, te
mperature
ri
se, friction
co
efficient an
d
themal di
spla
ceme
nt on t
he
friction to
rqu
e
of a slidi
ng b
earin
g is
stud
ied. seve
ral
key factors a
r
e
sele
cted to
e
s
tabli
s
he
d th
e
multidimen
sio
nal grey mod
e
l GM(1,
N
,D) of the bear
in
g friction torq
ue, this mod
e
l is verified
by
an appli
c
atio
n example
s
. This pa
per
will provide the
o
retical ba
sis for analyzin
g and pre
d
ict
i
ng
the friction torque of a slidi
ng bea
ring.
2. Gre
y
Relational Analy
s
is Model for
the Fric
tion Torque o
f
a Sliding Bearing
Affected by many kin
d
s
o
f
factors, the
fric
tion to
rqu
e
cha
r
a
c
teri
sti
cs
of a slidi
n
g bea
ring
is ge
nerally rand
om a
n
d
nonline
a
r. T
herefo
r
e,
u
s
i
ng conventio
nal metho
d
s it is difficult
to
determi
ned
the influ
e
n
c
e
degree
of the
s
e fa
cto
r
s on
bea
ring
fri
c
tion to
rqu
e
with limited
ori
g
i
n
a
l
data. Grey relative analy
s
is metho
d
i
s
mai
n
ly u
s
ed to a
nalyse the
relatio
n
bet
ween
the
behavio
ral v
a
riabl
e an
d i
n
fluen
cing fa
ctors, it ca
n
determi
ne th
e influen
ce
d
egre
e
of a
g
r
ey
factor o
n
the behavio
ral va
riable. Th
us,
throug
h co
m
parin
g the influen
ce de
gre
e
of the material
hard
n
e
ss,
rot
a
tional
spee
d
,
load, tempe
r
ature ri
se, fri
c
tion
coeffici
e
n
t and the
r
m
a
l displa
cem
ent
on the friction
torque of a sl
iding be
arin
g,
we ca
n determine key influ
enci
ng facto
r
s.
2.1. Dete
rmining the Infl
uence Spa
c
e
@
INU
T
h
e
in
flue
nc
e s
p
ac
e @
INU
i
s
a
data
set i
n
clu
d
ing th
e
behavio
ral va
riable
and
infl
uen
cing
factors, the behavioral vari
able is the fri
c
tion torq
ue
of a sliding b
earin
g, influe
ncin
g factors
are
beari
n
g
mate
rial hardne
ss,
rotatio
nal sp
eed,
lo
ad,
te
mperature
ri
se, frictio
n
coe
fficient, therm
a
l
displ
a
cement
, etc. So the influen
ce spa
c
e @
INU
can
be expre
s
sed
as follow.
IN
U
i
={
ω
|=
{
}
}
@
i
I
1
,
2
,3
,4
,5
,
6
,
7
,8
(1)
Whe
r
e
1
ω
is the
data serie
s
of
beari
ng fri
c
ti
on torq
ue,
2
is the data
seri
es of b
eari
n
g
materia
l
hard
n
e
ss,
3
is the data
seri
es of b
eari
n
g
rotational
sp
eed,
4
ω
is the d
a
ta se
rie
s
of
beari
n
g
load,
5
ω
is the d
a
ta se
rie
s
of beari
ng temp
eratu
r
e ri
se,
6
ω
is the data
se
ries
of beari
n
g frictio
n
c
oeffic
i
ent,
7
ω
is the data
se
ries of b
e
a
r
in
g inne
r ri
ng t
herm
a
l di
spla
ceme
nt,
8
ω
is th
e data
seri
es of be
aring outer
ring
thermal di
spl
a
cem
ent.
2.2. Dete
rmining the Gr
e
y
Relational
Factor Spac
e @
GRF
In orde
r to m
a
ke the
s
e
da
ta seri
es
co
mparable in t
he influen
ce
spa
c
e, we m
u
st initial
value treat them as follo
ws:
）
（
(1)
ω
(k)
ω
(1)
ω
(3)
ω
(1)
ω
(2)
ω
(1)
ω
(1)
ω
k
i
i
i
i
i
i
i
i
,
,
))
(
X
)
3
(
X
),
2
(
X
),
1
(
X
(
@
i
i
i
i
GRF
(2)
Whe
r
e
k
is the total number
of serie
s
ele
m
ents, and
we call
i
X
initial value series, a data set
GRF
@
includi
ng the
s
e initial valu
e seri
es i
s
cal
l
ed the grey relational fa
ctor sp
ace.
2.3. Dete
rmining the Gr
e
y
Rela
tional
Differen
t Information Sp
ace
∆
GR
The g
r
ey rela
tional differe
n
c
e info
rmatio
n sp
ace in
clu
des th
e difference informat
ion set
, discriminatio
n coeffici
ent
ζ
and environm
ental paramet
ers
set
1i
.I
t c
an be writen as
follow:
GR
1
i
1
i
=(
,
ζ
,
(
m
a
x
),
(m
i
n
)
)
(3)
The diffe
ren
c
e information
is th
e differe
nc
e se
r
i
es
b
e
tw
e
e
n
fr
ic
tion to
r
q
ue
s
e
r
i
es
1
x
X
1
and
sev
en inf
l
u
e
n
c
e f
a
ct
o
r
s se
r
i
es X
i
after intial value treat
ment. It can be written a
s
follow:
)
k
(
x
)
k
(
x
)
k
(
i
1
i
1
, i=
2,3……8
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Modelin
g and
Analyzi
ng for the Friction T
o
rqu
e
of a Sliding Bea
r
ing
…
(Wang Ba
om
ing)
6011
The differen
c
e inform
ation
set
inclu
de
all differe
nce
informatio
n el
ements. It
ca
n be
written as
follow:
1i
ΔΔ
(
k
)
i
2,
3,
4,
5,
6,
7
,
8
(5)
The di
scrimi
nation
coefficient
ζ
i
s
a we
ighted coefficient
of
enviro
n
mental
para
m
eter,
gene
rally the value of
ζ
is 0.5.
The envi
r
on
mental pa
ra
meters set in
clud
es m
a
ximum value a
nd minm
um
value in the
differen
c
e inf
o
rmatio
n set
. It can be writ
ten as follo
w:
1i
i
k
1i
1i
i
k
1i
(m
ax)
m
ax
m
a
x
(k
)
(
m
in
)
m
in
m
i
n
(
k
)
(6)
2.4. Calculati
ng the Gr
e
y
Relatio
n
al Coefficient
The grey rel
a
tional coefficient provid
es a cr
iteri
a
to measure the influen
ce d
egre
e
of
influen
cing f
a
ctors(mate
ri
al ha
rdne
ss,
rotation
al
spe
ed, loa
d
, temperature ri
se, fri
c
tion
coeffici
ent, thermal
di
splacement, etc)
on the
behavi
o
ral variabl
e
(the fri
c
tion torque).It can
be
written as
follow:
1i
1i
1i
1i
1i
(mi
n
)
+
ζ
(m
ax)
γ
(
x
(k),x
(k)
)
=
(k)+
ζ
(ma
x
)
(7)
The grey re
lational grad
e
)
x
,
x
(
i
1
is an a
v
erage valu
e
of the gre
y
relational
c
oeffic
i
ent. It
c
an be written as
follow:
1i
1
i
1
1
γ
(x
,x
)
=
γ
(x
(k
)
,
x
(
k
))
n
n
k
(8)
2.5. Determining the Grey
Relational
Order
The grey rela
tional order i
s
the seq
uenti
a
l arr
ang
eme
n
t of the grey
relation
al co
efficient.
comp
ari
ng the grey relati
on gra
d
e
s
)
x
,
γ
(x
2
1
,
)
x
,
γ
(x
3
1
,
)
x
,
γ
(x
4
1
,
)
x
,
γ
(x
5
1
,
)
x
,
γ
(x
6
1
,
)
x
,
γ
(x
7
1
and
)
8
x
,
γ
(x
1
,the infl
uen
cing
deg
ree of fa
ctors on b
e
a
r
ing
friction to
rq
u
e
can b
e
determi
ned.S
o
, the im
port
ant influe
nci
n
g facto
r
s o
n
t
he fri
c
tion
torque
ca
n b
e
f
ound, m
ean
while
it is applicabl
e for buildin
g GM(1,
N
,D) model an
d furth
e
r analy
s
is.
3. GM(1,N,
D
) Model for th
e Friction To
rque of a Sliding Bearing
GM(1,
N
,D) m
odel i
s
multivariabl
e g
r
ey
model,
Its form in
clud
es o
ne b
ehavio
ral
varia
b
l
e
and N-1
fa
ct
or
va
riabl
es. In
this pap
er the
be
haviora
l varia
b
le i
s
the fri
c
tion
torqu
e
a
n
d
fa
ctor
variable
s
a
r
e
material h
a
rd
ness, rotatio
n
a
l spe
ed, loa
d
, temperatu
r
e, friction coe
fficient, therm
a
l
displ
a
cement
,etc. Based
on the grey relational
a
nalysi
s
of the friction torque of a sli
d
ing
beari
ng,
som
e
key facto
r
s
influen
cing f
r
i
c
tion to
rqu
e
a
r
e
sele
cted
to
build
GM(1,N,D) mo
del a
nd
predi
ct the fri
c
tion torq
ue o
f
a sliding be
aring.
Th
e G
M
(1,N,
D
) mo
del is built up
as follo
w.
One-accu
mul
a
ting the g
r
e
y
relational f
a
ctor space
@
GRF
, for instance,
)
1
(
i
x
is the
one-
acc
u
mulate series
of
i
x
, i=1,2,3….
(1)
(1)
(
1)
(1
)
(
1
)
()
(
1
)
(
)
2,
3
,
4
.
.
.
ii
ii
i
xx
x
kx
k
x
k
kn
(9)
)
1
(
1
z
is
th
e
n
e
i
gh
bo
r
me
an
s
e
r
i
es
o
f
)
1
(
1
x
.
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046
TELKOM
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KA
Vol. 12, No. 8, August 2014: 600
9 –
6016
6012
(1
)
(
1
)
(1
)
11
1
()
0
.
5
(
)
0
.
5
(
1
)
zk
x
k
x
k
(10)
Then, the GM
(1,N,D) mod
e
l
can be exp
r
essed a
s
follo
w:
(0)
(
1
)
(
1
)
11
2
()
()
()
N
ii
i
x
kb
x
k
a
z
k
(1
1)
Whe
r
e,
a
is development co
efficient ,
i
b
is driving co
effici
ent.
Suppo
se a p
a
ram
e
ter pa
cket P
N
is
:
[]
T
N2
3
N
P
a
,
b
,
b
,
...,
b
(12)
Usi
ng lea
s
t square metho
d
, P
N
can be got.
1
1
()
TT
NN
N
a
b
PB
B
B
y
b
(13)
Whe
r
e,
(1
)
(
1
)
(1
)
12
(1
)
(
1
)
(1
)
12
(1
)
(
1
)
(1
)
12
(2
)
(
2
)
(2
)
(3
)
(
3
)
(3
)
()
(
)
()
N
N
N
zx
x
zx
x
B
z
n
xn
xn
(14)
(0
)
1
(0
)
1
(0
)
1
(2
)
(3
)
()
N
x
x
y
x
n
(15
)
There a
r
e
three te
st meth
ods to te
st th
e p
r
e
c
is
io
n of
the gray pre
d
iction
mod
e
l
,
that
is
resi
dual e
r
ror examination,
poste
rior diff
eren
ce
exami
nation an
d rel
a
tive degree
examination.
In
this pa
per,
we ta
ke the
resi
dual te
st. Note
re
sidual
s and
the pe
rcenta
ge of re
sid
u
a
ls
r
e
spec
tively as
e(k)
and
e(avg)
. Then,
(0
)
(
0
)
(0
)
()
(
)
(
)
1
00%
()
ii
i
xk
x
k
ek
xk
(16)
1
1
()
(
)
N
k
ea
v
g
ek
N
(17)
The preci
s
io
n
of the model
p
is
:
(1
0
0
(
)
)
%
pe
a
v
g
(18)
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TELKOM
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046
Modelin
g and
Analyzi
ng for the Friction T
o
rqu
e
of a Sliding Bea
r
ing
…
(Wang Ba
om
ing)
6013
4. Example Analy
s
is
Takin
g
an Al-based alloy sliding bea
rin
g
as an
exam
ple, its frictio
n
torque i
s
a
nalyse
d
usin
g the grey system th
eory. The ex
perim
ental
d
a
ta of a slidi
ng bea
ring friction torque
is
obtaine
d on
FALEX-5 te
st-bed
produ
ct
ed by FE
LA
X from Engl
a
nd. The
inne
r an
d oute
r
ring
diamete of te
sted be
arin
g is 19mm a
n
d
23mm re
sp
ectively, bearing sle
e
ve h
e
ight is 19m
m.
Acco
rdi
ng to
the experi
m
e
n
t, the testin
g rotation
al speed i
s
fro
m
24rp
m
to 30
00rp
m
, the test
maximum temperature i
s
70
℃
, the radial load is from 0 to 3600N. The stru
cture di
agra
m
of
experim
ental
installatio
n
is
sho
w
n in Fig
u
re 1.
1.Measureme
n
t and co
ntrol
holder 2. Friction coeffi
ci
ent instru
men
t
3. Temperature sen
s
o
r
4.Load
weig
h
t
s 5.Fulcru
m
6. Tested bearin
g 7.
Clamping
sup
p
ort 8. Spindle 9. Belt
10. Motor
Figure 1. The
Structure Dia
g
ram
of Expe
rimental In
sta
llation
In this expe
riment, the e
x
perime
n
tal
datas
of the
friction to
rq
ue, tempe
r
at
ure
rise
,
friction
coeffi
cient, therma
l displa
ce
me
nt of
beari
n
g
inner and o
u
tside ring are
tested,
te
sted
beari
n
g
s
h
a
ve four differe
nt materi
als, t
e
mpe
r
ature
ri
se
and
frictio
n
coefficie
n
t
of tested
be
a
r
ing
is real
-time m
easure
m
ent value, the thermal di
spl
a
ce
ment value of inner dia
m
et
er and the
r
m
a
l
displ
a
cement
are te
sted
a
fter shutdo
wn. the
ex
peri
m
ental
data i
s
sho
w
n
in
Table
1. A
s
the
origin
al data,
the expe
rim
ental value i
s
u
s
ed
to bu
ild the g
r
ey relational
anal
ysis m
odel
a
n
d
GM(1,
N
,D) model for the fri
c
tion torq
ue o
f
a sliding be
aring.
Table 1. The
Data of GM(1
,N,) Model
1
x
/N
·m
2
x
/ HV
3
x
/rpm
4
x
/ N
5
x
/
Ԩ
6
x
7
x
/
μ
m
8
x
/ um
2.520
180
100
88.96
66
0.297
1.83
11.49
3.447
180
100
88.96
101
0.407
3.16
19.87
2.94 180
100
88.96
103
0.347
3.24
20.35
0.407
700
100
22.24
18
0.192
0.04
0.24
0.554
700
100
44.48
23
0.131
0.19
1.20
0.870
700
100
66.72
31
0.137
0.50
3.11
1.209
700
100
88.96
40
0.143
0.84
5.27
1.503
700
100
111.21
45
0.142
1.03
6.46
1.887
700
100
133.45
54
0.148
1.37
8.62
2.294
700
100
155.69
64
0.155
1.75
11.01
2.644
700
100
177.93
75
0.156
2.17
13.65
2.836
700
100
200.17
84
0.149
2.52
15.80
3.379
700
100
222.4
94
0.159
2.90
18.19
3.899
700
100
244.65
107
0.167
3.39
21.31
1.684
250
200
111.21
100
0.16
3.13
19.63
1.571
250
200
111.21
88
0.148
2.67
16.76
2.102
250
200
111.21
109
0.198
3.47
21.79
1.379
300
400
111.21
142
0.13
4.73
29.69
2.158
300
400
111.21
178
0.204
6.10
38.30
2.339
300
400
111.21
226
0.221
7.93
49.80
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Vol. 12, No. 8, August 2014: 600
9 –
6016
6014
4.1. Results and Disc
uss
i
ons of Grey
Relatio
n
al Analy
s
is
Based o
n
the grey theory, the grey relat
i
onal analy
s
is method is u
s
ed in an
alyzing the
influen
ce
de
gree
of th
e
materi
al
ha
rdne
ss, rotational
sp
eed,
load,
temp
eratu
r
e, fri
c
ti
on
coeffici
ent a
nd therm
a
l d
i
spla
cem
ent on the fric
tio
n
torqu
e
of a slidin
g be
aring.T
a
ki
ng
the
experim
ental
data sho
w
n i
n
Table
1 a
s
the ori
g
inal
da
ta, the grey relational
coef
ficient of vari
ous
factors ca
n b
e
obtained by
analyzing th
e grey rela
tio
nal ord
e
r an
d
solving the Equation (1)-(8).
The cal
c
ul
ation re
sult of grey relati
onal
grad
e is represe
n
ted in Ta
ble 2.
Table 2. The
Grey Relation
al Grad
e of the Frictio
n
Torque
Influence
factor
s
Material
hardness
Rotational
speed
Load
Temper
ature
Friction
coefficient
Hermal
displacement of
inner diameter
Hermal
displacement of
outter diamete
r
Gre
y
relational
grade
0.65 0.83
0.85
0.87
0.95
0.71
0.68
As sh
own i
n
Table
2, comp
ared
with other
infl
uen
ce fa
ctors, friction
co
efficient,
rotational
spe
ed, load and t
e
mpe
r
ature ri
se have mo
re
significa
nt influen
ce on the
friction torqu
e
of a sliding b
earin
g, amon
g them frictio
n
coeffici
e
n
t has mo
st sig
n
ificant influe
nce. The rea
s
on is
that, the increase of rotational
spe
ed, l
oad, temp
era
t
ure
a
nd
the
friction co
efficient red
u
ce the
film thickne
s
s of lubri
c
ant
ion oil and h
i
nder the
fo
rmation of ela
s
tohydrodyna
mic lub
r
icatio
n,
leadin
g
to the incre
a
se in friction torq
ue.
4.2. Results and Disc
uss
i
ons of GM
(1
,N,D)
Dodel
Based
on
th
e grey relati
onal
analy
s
i
s
, fri
c
tion
co
efficient, rota
tional
spe
ed,
load,
temperature
have mo
re
si
gnifica
nt effect upo
n the
fri
c
tion torque
o
f
a slidin
g be
aring. T
herefore,
the four key
factors are selecte
d
as fa
ctor
vari
able
s
and the frict
i
on torqu
e
is the behavioral
variable
s
. Ta
king the exp
e
r
imental d
a
ta
sho
w
n in
Ta
ble 1 as the
origin
al data,
n is 20, and
P
N
can b
e
obtain
ed by solvi
ng the
Equation (9)-(18).
T
N
0.4618
0.4981
0.2877
0.2979
-
P
，
，
，
，
0433
.
0
(19
)
The predi
ctio
n model of
controllin
g frict
i
on
torq
ue of
a slidin
g bea
ring ca
n be b
u
ilt. The
model is:
(
0
)
(
1
)
(1
)
(
1
)
(1
)
12
3
4
1
(
)
0
.
1280
(
)
0.
0152
0.
2437
(
)
0
.
6432
(
)
x
kx
k
x
x
k
z
k
(20)
Whe
r
e
)
0
(
1
x
is the predi
ctive value of bea
ri
ng friction torque,
)
1
(
2
x
is one-accumul
a
te serie
s
of
rotation sp
e
ed,
)
1
(
3
x
is one
-accumul
ate
seri
es of lo
ad,
)
1
(
4
x
is one
-accumul
ate
seri
es of
temperature,
)
1
(
5
x
is one-accu
mulate se
rie
s
of friction co
efficient,
)
1
(
1
z
is one-a
c
cum
u
lat
e
seri
es
of bearin
g friction torque.
The re
sid
ual test is represe
n
ted as follo
w.
Accordi
ng
to Equation (13)-(1
8), the
averag
e
resi
dual
s e
(
a
v
g) is 7.9%, the pre
c
i
s
ion
P is 92.1%.
As sh
own in the Equatio
n (19), the b
eari
ng
rotatio
nal
spe
ed ha
s th
e negative inf
l
uen
ce
on the fri
c
tio
n
torqu
e
, that is, the fri
c
tion
torque
of a slidin
g
beari
ng d
e
creases
with the
increa
sing
of rotation
spe
ed. Ho
weve
r, load, te
mpe
r
ature a
nd fri
c
tion co
efficie
n
t have po
siti
ve
influen
ce
on
beari
ng f
r
ictio
n
torque,
na
mely, the fri
c
tion torque
in
cre
a
se
with t
he in
crea
sing
of
load, tempe
r
ature an
d friction coeffici
ent. Moreov
er, amon
g the po
sitive influen
ce fact
ors,
friction coefficient ha
s mo
st signifi
cant
influence.
T
h
is con
c
lusi
o
n
is con
s
i
s
te
nt with the grey
relation
al ana
lysis.
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TELKOM
NIKA
ISSN:
2302-4
046
Modelin
g and
Analyzi
ng for the Friction T
o
rqu
e
of a Sliding Bea
r
ing
…
(Wang Ba
om
ing)
6015
Table 3. The
Re
sidu
al Test
of of GM(1,N,D) Mod
e
l
k
)
0
(
1
x
)
0
(
1
x
e(k)
2
2.4076
2.3677
-1.68%
3
3.7691
3.5336
-6.66%
4
4.0063
3.6951
-8.42%
5
4.2378
3.9148
-8.25%
6
4.6150
4.2601
-8.33%
7
5.1469
4.7399
-8.59%
8
5.7924
5.3363
-8.55%
9
6.5929
6.0852
-8.34%
10
7.5584
6.9955
-8.05%
11
8.6870
8.0448
-7.98%
12
9.9492
9.1704
-8.49%
13
11.3807
10.5112
-8.27%
14
13.0037
12.0583
-7.84%
15
13.8190
12.7265
-8.58%
16
14.5051
13.3498
-8.65%
17
15.4310
14.1839
-8.79%
18
15.9028
14.7309
-7.96%
19
16.7618
15.5874
-7.53%
20
18.0178
16.5157
-9.09%
As sho
w
n in
Table
3, the
maximum residual i
s
9.09
%, the avera
ge relative re
sidu
als
i
s
7.9%, the accuracy is 92.1%.It verify that
GM(1,N,D) m
odel
has
good accuracy and is
appli
c
able fo
r predi
cting fri
c
tion torq
ue o
f
a sliding be
aring.
4. Conclusio
n
(1) In this pa
per, the
grey relatio
nal a
n
a
ly
sis
metho
d
is
used i
n
analyzi
ng the
frictio
n
torque
of
a
sliding
bea
ring
and
its i
n
flu
ence fa
ct
or.
Based
on
th
e g
r
ey relatio
nal a
nalysi
s
,four
main influ
e
n
c
e facto
r
were
sele
cted,
GM
(1,N,D) m
ode
l for the f
r
ictio
n
torq
ue of
a
slidin
g be
arin
g
was
built.
(2) An
alysi
s
sho
w
that
comp
ared
wi
th other
influenc
e
fac
t
ors
,
fric
tion c
o
effic
i
ent,
rotational
spe
ed, load and t
e
mpe
r
ature ri
se have mo
re
significa
nt influen
ce on the
friction torqu
e
of a
sliding
b
earin
g; Fu
rth
e
rmo
r
e, the
b
earin
g ro
tatio
nal
spee
d h
a
s
the
neg
ative influen
ce
o
n
the
friction to
rqu
e
, load, tem
p
eratu
r
e a
nd f
r
iction
c
oefficient have
po
sitive influen
ce on th
e fri
c
tion
torque; Amon
g the positive
influence fa
ctors, fricti
o
n
coefficient ha
s most significant influence on
slidin
g bea
rin
g
torque.
(3)
Com
pari
n
g experim
ent
al re
sults
an
d the
cal
c
ul
ated value of t
he GM
(1,N,
D
) model
based on the
s
e impo
rtant i
n
fluen
cing fa
ctors
,
the ma
ximum resi
du
al of GM(1,N,
D
) is 9.0
9
%, the
averag
e
relat
i
ve re
sidu
als
is 7.9%, the
accuracy
is
92.1%. It verify t
hat GM(1,N,D) mo
del h
a
s
good a
c
curacy and is appli
c
abl
e for pred
icting the fricti
on torqu
e
of a slidin
g bea
ring.
Ackn
o
w
l
e
dg
ements
The auth
o
rs gratefully
wish to ack
nowl
edge th
e su
ppo
rts
of National
Scien
c
e
Found
ation (5116
5024
), Gan
s
u
P
r
ovi
n
ce
Fun
dam
ent
al Re
sea
r
ch Fund
s
f
o
r
the Universi
ties
(120
2ZT
C
06
0) a
nd P
r
oj
ect
sup
porte
d by
Devel
opment P
r
o
g
ram
for O
u
tstandi
ng Y
oun
g
Teache
rs in L
anzhou
Unive
r
sity
of Tech
n
o
logy (10
02Z
CX004
).
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 600
9 –
6016
6016
Referen
ces
[1]
Eser D, D
e
re
li
Y. Com
paris
o
n
s of
Rotord
yn
amic
C
oe-ffici
e
n
ts in
Step
ped
La
b
y
rinth
Se
a
l
s b
y
Usin
g
Cole
bro
o
k-
w
h
ite Friction Factormod
e
l.
Mecc
anic
a
. 200
7; 42: 177-1
6
8
.
[2]
Yang C
h
a
o
ju
n, Su Yue
y
u
e
, Sun Jia
n
ro
ng. Improvem
ent a
nd App
lic
ation
of F
r
iction F
a
ctor Equati
on for
a Slidi
ng Be
ari
ng.
T
r
ansactio
n
s of the Chin
ese Soci
ety for Agricultura
l
Machi
nery.
20
04; 35(6):1
67-
170.
[3]
Zhang
X
i
nbao, Geng Baolong, Ya
ng Yif
an.
Optimum Design of
Load
Ratio bet
w
een Bear
ings
in
Rational Alignment
of Propulsion Shaft.
Journa
l of Hua
z
h
ong U
n
ivers
i
ty
of Science a
n
d
T
e
chno
lo
gy
(Natural Sc
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