3rd International Conference on Mechanical Engineering and Intelligent Systems (ICMEIS 2015)
A dynamics modeling research for angular contact ball bearing of
high-speed spindle under rigid preload
ZhifengLiu1, a,Bohua Zhang2, b
1
School of Beijing University of Technology,Beijing 100124,China;
2
School of Beijing University of Technology,Beijing 100124,China.
a
[email protected], [email protected]
Keywords:angular contact ball bearings; Hertz contact theory; rigid preload; high-speed spindle;
dynamic model.
Abstract. Determining the parameters of angular contact ball bearings under preload is the key
issue to improve the accuracy and performance of high-speed spindle. The impact of gyro force and
centrifugal force were considered, a static and a dynamic model for angular contact ball bearings
under rigid preload were presented. By using Newton iterative method, dynamic parameters of the
angular contact bearings were obtained, including eccentricity of contact ellipse, contact angle,
contact force, maximum contact stress, contact ellipse radii and stiffness, etc. According to
numerical simulation, relationship between the parameters, versus preload and versus rotation speed
were given. The results reveal that the stiffness increases as the preload increases appropriately, the
contact stress and contact ellipse radii increase as the preload increases excessively. In addition,
under rigid preload this model has no bearing softening effect. This method provides valuable
theoretical basis for further researches, such as calculating bearingscalorific value, calculating
bearings fatigue life and optimizing bearings structure, etc.
Introduction
Angular contact ball bearings are commonly used in high-speed spindles. The performance of
angular contact ball bearings effect the static and dynamic performance of high-speed spindles
seriously [1]. These bearings require preloading in order to improve the rotational accuracy and
sufficient stiffness, and reduce vibration noise, etc. Basically, there are two types of bearing preload:
rigid and constant. Constant preload is using helical springs, disc springs, and other preload devices
to make the bearings obtain appropriate preload force. Rigid preload is using sleeves and washers to
make the inner ring and outer ring in an appropriate distance, so that the bearings have appropriate
preload force [2]. In actual manufacturing process, because of lacking accurate modeling and
analyzing method, it is a key problem to determine the preload force or preload distance of the
bearings. Consequently, it is significant to establish a model of bearings under preload.
There are many studies make researches about bearings in modeling, analyzing, testing fields.
Altintas considered gyro force and centrifugal force of rotation parts, and established a model by
using the finite element method [3, 4]. Lim derived 5 degree of freedom analysis model and
stiffness matrix based on Hertz contact theory [5, 6]. Royston derived 5 degree of freedom vibration
model based on Lim’s work. And studied the change of natural frequency and stiffness in different
preload force [7]. LigangCai presented a modeling theory of angular contact ball bearings under
constant preload, and verified the model through an experiment [8]. Wang Hong-jun introduced
bearing softening effect under high speed rotation [9]. Wensing calculated elastic deformation
according to the generalized shape function, and used software ANSYS to verify the result [10].
BaominWang analyzed mechanical properties of bearings by using Pseudo dynamic method [11].
Ali investigated the stiffness of bearings under different preload force and rotation speed through
hammer test [12]. TaipingHuang used eccentric mass excitation method to obtain the equivalent
stiffness of the bearings [13].
Few investigations have focused on the modeling method of angular contact ball bearings under
rigid preload. This paper based on Hertz contact theory, and considered the impact of gyro force
© 2015. The authors - Published by Atlantis Press
890
and centriffugal force. This paper presented a new modeel of angularr contact baall bearings under rigidd
preload, inncluding conntact area model,
m
prelooad model in
i static staate, and prelload modell in rotatingg
state. Accoording to Newton
N
iteraative methodd, importan
nt parameterrs of the m
model in versus preloadd
force and vversus rotation speed were
w obtaineed (such as eccentricity
y of contactt ellipse, con
ntact angle,,
contact forrce, maxim
mum contactt stress, conntact ellipsee radii and stiffness). This metho
od providess
valuable thheoretical baasis for furtther optimizzing research, fatigue liife researchh, etc.
Dynamic m
model of an
ngular conttact ball beearings
This ppaper basedd on Hertz contact theeory, establlished contaact area moodel, and obtained thee
parameter equations of eccentriicity of conntact ellipsee. Then, prreload moddel in staticc state wass
establishedd and force balance
b
equ
uations was obtained. According
A
to the equatiions, the acttual contactt
angle
ccan be solveed. And theen, the impaact of gyro force and centrifugal
c
fforce were considered,,
preload m
model in rottating state was estabblished, and
d nonlinear equations about ecceentricity off
contact elliipse(ei,eo), contact
c
angle(θik,θok), ccontact forcce(Qik,Qok) were
w presennted. These parameterss
can be sollved by New
wton iteratiive methodd. At last, stiffness
s
mo
odel was esstablished. Making
M
thee
previous soolutions(ei, eo, θik, θok, Qik, Qok) iinto stiffnesss model, th
he stiffness of axial direction andd
radial direcction can bee given.
Contact area mod
del of angu
ular contacct ball bea
arings. According to thheBoussineesq solutionn
placement aat a point caan be expresssed as:
and Hertz ccontact theoory, the disp
ω x, y
∬
′
′
(1)
Here, p  x, y   is the
t distributtion pressurre, p0 is the maximum stress on thhe center off the contactt
ellipse,the elliptical coontact area has
h semi-m
major axis off length a an
nd semi-minnor axis of lengthb,and
l
d
2
the eccentrricity e is giiven by: e  1   b / a  , b  a .
After cooordinate traansformatio
on:
2
ω
Using:
L
M
N
(2
2)
(3)
(4)
(5)
By integgrating the distributed
d
pressure
p
onn the contactt area, the external
e
loaddingQ can be
b given:
∬
1
/
/
2//3πab
(6)
Fig. 1 Point con
ntact diagram
am
Fig
g.2 Contact surface disttance
891
Fig.3 Deformation
D
n coordinatesdiagramd
diagram
The poiint O is thee contact zone betweeen the two elastic bod
dies V1 and V2 before the load iss
applied, R111, R12, R21, and R22 aree the main rradii of curv
vature, as sh
hown in Figg 1. DD’ is the
t distancee
between pooints D(x1,yy1) and D’(x
x2y2) as show
wn in Fig 2.which
2
can be expresseed as:
z
(7))
For an aangular conntact ball beaaring, the roolling body has a spherrically symm
metric struccture, so thee
rolling elem
ment and itts contact object’s
o
maiin curvaturees are coinccident. Hencce, z can bee expressedd
as:
z
A
(8))
Here:
B
A
B
A
/
4
(99)
(10)
Accordiing to the deformatio
on coordinaates shown
n in Fig 3,
x, y ,
x, y , x, y ,
x, y arre the contact plane dissplacementss, the follow
wing equatio
on can be obbtained:
δ
(11)
Combinning Eq.(9) to Eq. (11),, the equatioon of eccenttricity of co
ontact ellipsee e can be obtained:
o
F ρ
Here:
/
a
b
δ
1(12)
√
/
/
(13)
/
/
Σρ
(14
4)
/
(15)
(16)
(17)
Preload
d model in static statee of angulaar contact ball
b bearings.With a ppreload forcce acting onn
the bearingg, the bearinng contact angle
a
will chhange, and the inner riing of the b earing will produce ann
axial displlacement. Assuming
A
the
t bearing is isotropiic in the raadial directtion,there is no radiall
displacemeent. In the preloaded
p
case
c
there iss just an ax
xial displaceement produuced in the inner ring..
The outer iis fixed andd the geomeetric relationnships betw
ween the inneer and outerr rings are as
a shown inn
Fig4.
892
Fig. 4D
Displacemennt diagram under
u
static preload
With the axial foorce Fa acting on a beearing, each
h ball will be subject too the same deformation
d
n
under the same load. The centerr of curvatuure will mo
ove from Oi to Oi . Oi aand Oe aree the initiall
groove currvature cennters of outter and innner rings respectively;  and   aare the inittial bearingg
contact anggle and the actual contaact angle.
Accordiing to the Geometrical
G
nce of
and
can be pressented:
relationshipp, the distan
1
(18)
(19)
The diffference betw
ween Oi’Oe and OiOe iss δn, which
h is the norm
mal contactt deformatio
on:
(220)
The norrmal contactt load:
(21)
Where, Z is the num
mber of steeel balls, Kn is the Stiffn
ness coefficient.
∗
2.1343 ∗ 10 ∗ Σ
Σ (22)
nted:
Force balance equaation of the bearings caan be presen
(23)
Z
By subsstituting Eq..(18) ~(22) into Eq. (233):
′
1 . (2
24)
.
The actuual contact angle can be obtaineed by solvin
ng Eq. (24)
The cennter of curvaature of the steel ball:
(25)
The cennter of curvaature of the inner ring:
(26)
(27)
The cennter of curvaature of the outer ring:
(28)
(29)
By subsstituting Eq..(25) ~(29) into Eq. (122), equation
n f1, f2were obtained:
o
1
0 (30))
1
0(31))
Preload
d model in rotating state of angu
ular contacct ball bearings.When the bearing
g in rotatingg
state, the oouter ring is fixed and the inner rinng is fixed relative
r
to it as well. W
With high speeed rotationn,
the centrippetal force Fckand gyroscopic mom
mentof theb
bearing Mgk must be coonsidered, as
a shown inn
Fig5. As caan be seen in the Fig 6,
6 O’ is the iinitial posittion of ball center, O’’ is the final position off
ball center, D is the cuurvature cen
nter of innerr ring, and B is the curv
vature centeer of outer ring
r
893
Fig.5 Bearings
B
forrce relationsship
Fig. 6 Disp
placement ddiagram
The cenntripetal forcce Fckand gy
yroscopic m
momentof th
hebearing Mgk can be ppresented:
Ω
(32)
Ω
(33)
The equuilibrium eqquations in the verticall and horizo
ontal directiions for thee ball can bee expressedd
as:
0(34)
0(3
35)
Accordiing to the geeometrical relationship
r
p, equation f5 and f6 weere obtainedd:
∗
∗
∗
0(36)
∗
∗
∗
0 (37)
Where, θik and θok are conttact angle oof inner ring
g and outer ring,
r
Qik and
nd Qok are co
ontact forcee
of inner rinng and outerr ring, and DO , BO can be exp
pressed as:
DO
0.5
(38
8)
0.5
(39
9)
BO
Combinned equationn f1to f6 in together, a nonlinear equations
e
caan be givenn, where ei, eo, θik ,θok,
Qik, Qok arre unknownn parameterrs. Accordinng to Newto
on iterative method, thhe nonlineaar equationss
can be solvved.
1
1
（40）
∗
∗
∗
∗
The iterrative methood is describ
bed as folloows:
J x
J
∗
∗
（41）
（42）
（43）
894
′
′