The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015
DOI Number: 10.6567/IFToMM.14TH.WC.OS6.012
Analysis of Influence of External Load Variation of High Speed Ball Bearings
on Cage Dynamic Performance
Ye Zhenhuan1,2
Zunyi Normal College
Zunyi, China
Abstract: According to the geometry and force
relationship of bearing elements, the dynamic model of
high speed ball bearing was built. Integration method of
Runge-Kutta and Newton-Raphson is proposed to
efficiently solve the nonlinear equations. Computation
program is developed and the cage stability in different
bearing external loads is studied by taking type 7004
bearing as an example. It is shown that the increase of
ratio of axial load to radial load makes both the cage
stability and cage sliding ratio improved. In addition, the
cage stability will reduce gradually during the loading
process of radial load. Finally, accurateness of the
dynamic model constructed in this paper was verified by
the test and computation results of Gupta example.
Keywords: High speed ball bearings, Dynamic, Cage stability,
Loading process
I.
Introduction
The dynamic performance of high-speed ball bearings
not only affects its own life and stability, but also relates to
the vibration and reliability of the rotor system. In order to
understand the dynamic performance of the bearings
during operation, a fully dynamic analysis of the bearings
is needed. Over the years, many researchers pay them
attentions to the dynamic analysis of high-speed ball
bearings, and producing calculation procedures such as
BRAIN, BEAST, and COBRA [1~3]. In particular, with
regard to the force and motion analysis of cage, Gupta
developed the influence of cage clearance [4] and the cage
unbalance mass [5] on cage stability, while Weinzapfel et
al. [6] also analyzed and compared the effects of rigid and
flexible cage on cage stability. Targeting surface defects,
Ashtekar et al. [7] analyzed the impact of pit size and
location on the dynamic performance of the bearings. In
addition, Bai [8] and Liu [9] also studied the dynamic
characteristics of the bearings in terms of the waviness and
cage clearance ratio of angular contact bearings.
However, previous studies on high-speed ball bearings
are limited to the basic dynamic performance analysis of
different steady state conditions and have rarely ever
investigated the dynamic properties of the bearing in the
variable transition process. Frequent impact of the cage
and the rolling elements during transmission operation
directly affects the dynamic stability of the cage and
ultimately influences the stability and reliability of
machinery transmission system.
In this paper, through the establishment of a dynamic
analysis model and development of an efficient dynamics
program, the effect of load changes of high-speed ball
bearings on the dynamic performance of the cage was
analyzed and discussed. The results support the
improvement of the design computation and failure
Wang Liqin2
Harbin Institute of Technology
Harbin, China
analysis of high-speed ball bearings.
II. Dynamic Model of High-Speed Ball Bearings
Study the high-speed ball bearings in the following
assumptions while the geometry parameters and working
conditions of the bearings are known.
1) The centroid and the geometry center of the bearing
elements are overlapped.
2) All elements are rigid, without regard to flexible
deformation.
3) Outer ring revolves around the center of the bearings.
Inner ring revolves around its own centroid and also moves
horizontally. Ball makes revolution around the center of
the bearings, revolves on its own axis and also moves
horizontally. Centroid of the cage moves horizontally and
also revolves along its own axis.
4) The parameter of lubrication and its traction
performance have been identified.
A.
Interaction Between Ball and Raceway
The contact load Q between the raceways and the balls
are calculated by the Hertz contact theory [10]. Thus, the
contact load between the jth ball and the raceways is
expressed as:
Q j = K jδ j 3/2
(1)
where the stiffness K is calculated according to the study
by Johnson [10], and the contact deformation δ is
determined by the positional relationship between the
rolling element and the raceway as shown in Fig. 1. Thus,
the contact deformation δj between the jth rolling element
and the raceway is expressed by the vector OrjObj which
means that the center of the raceway to the center of the
ball, the groove curvature coefficient f of the ring, and the
diameter of the ball Db are expressed as:
δ j = ( f − 0.5) ⋅ Db − OrjObj
Fig. 1 The sketch of the displacement of raceway and ball
(2)
The traction force Fμ of the contact area is equal to the
traction coefficient μ multiplied the contact load Q. The
traction coefficient formula [11] is:
μ = ( A + Bs )e − Cs + D
(3)
that rotational speed of the element. Subscripts 1, 2, and c
represent the outer ring, inner ring, and cage, respectively.
The sign of the equations is negative while the cage with
outer race guidance.
where s is the slip-roll ratio which could be obtained from
the kinematic analysis. Coefficients A, B, C, and D are
calculated according to the formula in study by Wang [11].
B.
Interaction Between Ball and Cage Pocket
Fig. 2 shows the relative position relationship between
the ball and the cage pocket. According to the vector
OpjObj between the center of cage pocket and the center of
the ball and the initial clearance between the cage pocket
and the ball Δbp, the minimum clearance δbp can be
expressed as follows:
δ bh = Δ bh − OpjObj
(4)
⎡ Fly ⎤ ⎡cos ϕ d
⎢ F ⎥ = ⎢ sin ϕ
d
⎢ lz ⎥ ⎢
⎢⎣ M l ⎥⎦ ⎢⎣ 0
Fig. 2 The sketch of the displacement of the cage pocket and the ball
According to the amplitude of the clearance between the
cage pocket and the ball bearing δbp, taking into
consideration both the roughness of the ball bearing εb and
the cage εp, two interaction models are used.
If the inequality δbp ≤ εb2 + ε p2 is true, the interaction is
assumed to be Hertz contact. The load between the cage
pocket and ball bearing is calculated by Hertz point contact
theory [10].
Otherwise, if the inequality δbp > εb2 + ε p2 is true, the
interaction is assumed to be hydrodynamic lubrication.
Reynolds equations [12] are used to calculate the
interaction force.
C.
Interaction Between Cage and Guiding Ring
Fig. 3 shows the interaction between the cage and the
guiding ring, and the force between the two can be
approximated as the fluid dynamic pressure of the journal
bearing. As the contact surface of the ring flange and
cylinder surface of the cage is relatively small, the short
journal bearings model [13] is adopted.
WM = ±
Wφ = ±
Wρ =
η RB
3
ε
2
Cg 2 (1 − ε 2 ) 2
πη RB3
4Cg
2
(ω1,2 + ωc )
ε
(ω1,2 + ωc )
(1 − ε 2 )3/2
Fig. 3 The interaction between cage and guiding ring
According to the interaction angle between the cage and
the ring φd, the interaction force in the inertial coordination
could be expressed as equation (6) through a coordination
transformation.
− sin ϕ d
0⎤ ⎡WM ⎤
⎢ ⎥
0⎥ ⎢ Wφ ⎥
⎥
1⎥⎦ ⎢⎣ Wρ ⎥⎦
cos ϕd
0
(6)
D. Solution of Dynamic Model
In addition to the interaction force, the bearing lubricant
retarding force FD and retarding moments Me, which can be
calculated by Schlichting’s fluid theory [14], should also
be taken into consideration. Forces of the loaded rolling
element are shown in Figure 4. In the figures, Mμ and Mbp
are the traction moment between the raceway and the
rolling element and friction moment between the cage and
the rolling element, respectively. The motion equation of
the jth rolling element centroid and of its revolving around
the centroid is determined by its resultant force and the
moment as shown in equation (7).
Fμ j − FDj + Fbpjy = mb rbjθj
M bpjx − M ejx + M μ jx = I bω xj
(7)
M bpjy − M ejy + M μ jy = I bω yj
M bpjzx − M ejz + M μ jz = I bω zj
Subscript j represents the parameter jth ball. Subscripts x, y,
and z represent the components in each of the three
directions.
zb
Fbpz
Q1
Fz
ob F
Fbpx x
zb
M ex
xb
M ey
yb
F1μ
ob
F2 μ
FD
yb
M ez
Fbpy
Fy
ob
xb
Fbpx
Q2
(5)
2πη R3 B 1
(ω1,2 − ωc )
Cg
1− ε 2
where WM, WΦ, and Wρ are the normal force, tangential
force, and moment between the cage and the guiding ring.
η is the viscosity of the oil. Rg and Bg are the radius and the
width of the guiding surface. Cg is initial clearance of the
bearings. ε is the eccentricity ratio of the cage center. ω is
Fig. 4 Forces and moments acting on ball
To eliminate high-frequency vibrations generated by
contact between a ball bearing and rings, a balance
constraint is adopted for the contact between the ball and
the ring. First, nonlinear equilibrium equations with the
quasi-dynamic method are established, and then the
Newton-Raphson method is used to solve the equations
[15].
Similarly, based on the force condition, the differential
equations of the cage can be listed as follows:
n
∑ (− F
bpjx
j =1
) = mc xc
Elements
Density
[kg/m3]
Ball
Rings
Cage
3200
7750
1500
Elastic
modulus
[GPa]
310
200
1.73
n
Fly + ∑ (− Fbpjy ⋅ cos θ j − Fbpjz ⋅ sin θ j ) = mc yc
j =1
n
Flz + ∑ (− Fbpjy ⋅ sin θ j − Fbpjz ⋅ cos θ j ) = mc zc
(8)
j =1
TABLE 2. Material parameters of ball bearings [4]
Dc ⎞
⎛
∑
⎜ − Fbpjy ⋅ ⎟ + M l = I cθ c
2 ⎠
⎝
j =1
n
A.
Based on simultaneous motion differential equations of
the ball bearing and cage, one should use non-dimensional
quantities to the parameters of the equation in formula (9),
and resolve the non-dimensional differential equations
with the adaptive step size 4th order Runge-Kutta method.
A stationary solution is obtained by the quasi-dynamic
method as the initial values of whole solution process. The
calculation procedure is shown in Fig. 5.
F=
F
Fx
m=
m
mb
r=
r
Db / 2
t=
Poisson's
ratio
[-]
0.260
0.250
0.300
t
mb ⋅ Db / (2 Fx )
Effect of Bearing Load on Cage Stability
A1. Effect of axial load on cage stability on steady
conditions
Fig. 6 and Fig. 7 show the cage whirl track and
deviation-ratio of cage whirl speed on different axial
loads with radial load 400 N. Where, the whirl track
indicates dimensionless displacement, which is defined
as the ratio of actual displacement of cage mass to the
guiding clearance of cage. And the deviation-ratio of
cage whirl speed is used to evaluate the cage stability
according to Ghaisas's definition [16] as follows:
(9)
n
σ=
∑ (v − v
i =1
i
m
) 2 / (n − 1)
(10)
vm
It can be seen that cage whirl track become regular
gradually with the increase of the axial load.
Meanwhile, deviation ratio of cage whirl speed
decreases when the axial load rises. Therefore, it can be
concluded that cage stability could be improved by
increase the axial load when the ball bearings in steady
condition.
Fig. 5 Calculation flow chart of dynamic equations
III. Results and Discussions
Based on the dynamic model of high-speed ball bearings,
the effect of changes in the load on the dynamic
performance of the bearing is studied with the example of
7004 angular contact ball bearings (MIL-L-7808 lubricants
are used). Geometry and material parameters are shown in
Table 1 and Table 2. The velocity of inner ring is 120000
r/min, and axial load and radial load are 2000 N and 400 N,
respectively.
Geometry parameter
Pitch diameter
Ball diameter
No. of balls
Contact angle
Inner curvature factor
Outer curvature factor
Cage inner diameter
Cage outer diameter
Cage pocket clearance
Cage guiding clearance
u.m.
mm
mm
°
mm
mm
mm
mm
Value
31
8
6
24
0.56
0.52
30
35
0.1
0.25
TABLE 1. Geometry parameters of ball bearings [4]
Fig. 6 Cage whirl motion on different axial loads (radial load: 400 N)
Fig. 7 Stability of cage on different axial load (radial load: 400 N)
The reason is that the traction interaction between
ball and raceway is improved with the raising of the
axial load, and then the fluctuation of ball speed
decreases (rotation speed of ball on different axial load
are shown in Fig.8). With the fluctuation of ball speed
decreases, the random collision between cage pocket
and ball becomes weaker (Fig.9 show the spectrum of
the cage pocket/ball impact on different axial loads),
meanwhile the guiding function between ring and cage
enhanced (Fig.10 show the spectrum of the cage
pocket/ball impact on different axial loads).
critical load which makes high speed ball bearing not
skidded. Based on the Cui's evaluate method [17], the
critical load is about 1500 N according to the cage
sliding ratio on different axial loads shown in Fig. 11.
Fig. 11 Effect of axial loads on cage sliding ratio (radial load: 400 N)
Fig. 8 Effect of axial loads on ball speed (radial load: 400 N)
A2. Effect of radial load on cage stability on steady
conditions
Fig. 12 and Fig. 13 show the cage whirl track and
deviation-ratio of cage whirl speed under different
radial loads with axial load 2000 N. It can be seen that
the cage whirl track gradually ranges from
approximately circular to an irregular whirl in a small
area as the radial load increases. Meanwhile, the
deviation ratio of cage whirl speed increases when the
radial load rises. According to the cage whirl track and
the deviation ratio of cage whirl speed based on
Ghaisas's evaluate method [16], it can be concluded
that cage stability could be improved by decreasing the
radial load when the ball bearings are in steady
condition.
Fig. 9 Spectrum of the cage pocket/ball force on different axial loads
(radial load: 400 N)
Fig. 12 Cage whirl motion on different radial loads (axial load: 2000 N)
Fig. 10 Spectrum of the cage/guiding ring force on different axial loads
(radial load: 400 N)
In addition, it can be seen from Fig.7 that there are
minor difference between deviation-ratio of cage whirl
speed for low level while axial loads greater than 2000
N. The reason is that axial loads play limited role in ball
speed fluctuation when the axial load greater than
Fig. 13 Stability of cage on different radial load (axial load: 2000 N)
In addition, the Poincarè sections of the cage mass
center vibrations on the radial plane under different
radial loads are shown in Fig.14. There are four
point-groups on the figures of the Poincarè section
while the radial loads are 400 N, 1000 N and 1500 N. In
those cases, the cage mass center motions exist as
periodic vibrations with narrow frequency band noise,
and the vibration period is the four times of the rotation
period of inner ring. When the radial load was increased
to 2000 N, the Poincarè section of the cage mass center
vibration became a series of disordered points,
indicating the vibration of cage mass center was chaos.
Fig. 16 Effect of radial loads on cage sliding ratio
(axial load: 2000 N)
B.
Fig. 14 The Poincarè section of cage vibration on different radial loads
(axial load: 2000 N)
In this case, the load distribution of rolling elements
became gradually uneven as the radial load of the
bearings increased. Uneven load distribution led to a
difference in the traction force between the rolling
elements and rings, which made the rotation speed of
the rolling elements appear to fluctuate, while the
rolling elements rotated to a different azimuth. The
significant fluctuation of rolling elements increased the
impact between the rolling element and the cage pocket,
decreasing cage stability. Fig. 15 shows the load
distribution and rotation speed distribution of rolling
elements working under the different radial loads.
Effect of Loading Process on Cage Stability
Bearing loading process is a transient motion stage while
the bearing ranges from normal working condition to limit
working condition. According to the analysis results in the
last section, it is clear that the effect of axial load and radial
load on cage stability is different, so the loading process of
axial load and loading process of radial load should be
treated differently in terms of their impact on the cage
stability.
B1. Cage stability in loading process of axial load
Fig. 17 shows the cage rotation speed and cage
sliding ratio when the axial load of bearings ranges
from 2000 N ~ 2500 N with the loading speed 5000 N/s.
As the figure shows, in the whole loading process of
axial load, the cage rotation speed decrease gradually
but the cage sliding ratio has no obviously change. The
main reasons is that distribution of all of the contact
angle between ball and raceway become even with the
increase of axial load, and the maximum contact angle
in the inner loading zone of bearings decreases
gradually which result in the both theoretical rotation
speed and actual rotation speed of cage decrease.
Fig. 17 Cage speed and sliding ratio in the bearing loading process of axial
load
Fig. 15 Effect of radial load on loads and revolution speeds of ball
(axial load: 2000 N)
In addition, the cage-sliding ratio decreased as the
radial load increased. Fig. 16 shows the cage sliding
ratio with different radial loads and the critical radial
load based on Cui's evaluate method [17].
Thus, it is possible to decrease the cage sliding ratio
and improve cage stability through increase the ratio of
the axial load to radial load when the bearings worked
under steady conditions.
Fig. 18 shows the cage whirl track and the ratio of
cage whirl speed to rotation speed during the loading
process of axial load. At the onset of the loading
process, the cage whirl track is disorder and the whirl
speed ratio presents accelerated raise. And then, cage
whirl track becomes regular gradually and the whirl
speed ratio presents a steady fluctuation state. It can be
concluded that the loading process of axial load will
affect the cage stability in a certain extent but not lead
to the cage lose stability.
Fig. 21 shows the cage whirl track and the ratio of
cage whirl speed to rotation speed during the loading
process of radial load. It can be seen from the figure
that the radius of cage whirl track decreases gradually
with the increase of radial load, and then the cage whirl
track focus on a small certain region. Combine with the
cage whirl speed ratio, it can be concluded that the cage
stability reduces gradually during the loading process
of radial load.
Fig. 18 Cage whirl motion and speed ratio in the bearing loading process
of axial load
Fig. 19 shows the time response and frequency
spectrum of the cage force during the bearing loading
process of axial load. As can be seen from Figure 19 b)
and d), both the frequency spectrums of the force
between cage and ball and the force between cage and
guiding ring are exist broadband noise, it means cage
bears many random impact force which will affect the
motion stability of cage mass center. Meanwhile, apart
from the broadband noise in the frequency spectrums,
the interaction between cage and guiding ring presents
the characteristic of single-cycle vibration which means
that the cage not lose stability. The interaction between
cage pocket and ball presents the characteristics of
double-cycle vibration.
Fig. 21 Cage whirl motion and speed ratio in the bearing loading process
of radial load
Fig. 22 shows the time response and frequency
spectrum of the cage force during the bearing loading
process of radial load. As can be seen from Figure 22 b)
and d), both the frequency spectrums of the force
between cage and ball and the force between cage and
guiding ring are exist obviously broadband noise,
compare with the loading process of axial load, there is
more influence on the motion stability of cage mass
center during the loading process of radial load.
Meanwhile, apart from the broadband noise in the
frequency spectrums, the interaction between cage and
guiding ring also presents the characteristic of
single-cycle vibration. The interaction between cage
pocket and ball presents the characteristics of
multi-cycle vibration.
Fig. 19 Load vibration response of cage in the bearing loading process of
axial load
B2. Cage stability in loading process of radial load
Fig. 20 shows the cage rotation speed and cage
sliding ratio when the radial load of bearings ranges
from 400 N ~ 900 N with the loading speed 5000 N/s.
As the figure shows, in the whole loading process of
radial load, the cage rotation speed is basically stable
but the cage sliding ratio increases gradually. The main
reasons is that distribution of all of the contact angle
between ball and raceway become uneven with the
increase of radial load which makes the cage theoretical
rotation speed increased.
Fig. 20 Cage speed and sliding ratio in the bearing loading process of
radial load
Fig. 22 Load vibration response of cage in the bearing loading process of
radial load
IV. Model Validations
Taking a certain type of angular contact ball bearings as
used in Gupta’s experiment, the reliability verification on
the bearing dynamic model is carried out. The axial load of
the bearing is 4448 N, the working velocity is 20000 r/min,
and other specific parameters are those used in the
reference [18].
Fig. 23 shows the results of the cage mass center orbit
simulated by Gupta's experimental and theoretical work.
Fig. 24 shows the results of the cage mass center orbit and
the ratio of the cage epicycle speed and bearing speed
simulated in this study. It can be seen that there is good
agreement between the shape of the experimental orbit and
the theoretical prediction. In addition, according to the
ratio of cage epicycle speed and bearing speed simulated in
this study, the range of the cage speed ratio is 0.4-0.65
when the cage whirl tends to be stable. Also, the range of
the cage speed ratio determined by Gupta's experiment and
theory are 0.37-0.50 and 0.31-0.35, respectively [18]. It
can be seen that errors exist between the experimental
work and theoretical work on the ratio of the cage epicycle
speed and bearing speed, but the errors are in the
acceptable range.
the cage stability during the loading process of axial load
while the cage stability reduces gradually during the
loading process of radial load. But no matter the axial
loading process or radial loading process, the interaction
between cage and guiding ring always present the
characteristics of single-cycle vibration, and the interaction
between cage pocket and ball always present the
characteristics of multi-cycle vibration.
Acknowledgements
The work has received support from the National
Natural Science Foundation of China (51375108)，Science
and Technology Foundation of GuiZhou Province of China
(Qian Ke He J Zi [2014]2172), Natural Science Research
Project of Education Department of GuiZhou Province of
China (Qian Jiao He KY Zi [2014]294) and Doctoral
Research Foundation of Zunyi Normal College
(2013BJ06).
References
Fig. 23 Cage mass center orbit determined by Gupta [18]
Fig. 24 Cage mass center orbit and cage speed ratio predicted in this study
According to the comparison results, it can be concluded
that the ball bearing dynamic model developed by this
paper is reliable, and this model can be used to simulate the
cage instability.
V.
Conclusions
In this paper, the dynamics analysis model of the
high-speed ball bearings is established based on the
interaction among the bearing elements. The hybrid of
Runge-Kutta and Newtown-Raphson’s method is adopted
to achieve high efficiency for solving nonlinear equations.
And the effects of load changes on the stability of the
dynamic performance of high-speed ball bearings are
analyzed, with the following conclusions:
(1) Bearing loads have obviously influence on the cage
sliding ratio and cage stability with the bearings under
steady conditions. As the axial loads increased and the
radial loads decreased, the cage sliding ratio decreased and
the cage stability increased. Thus, it is necessary to control
the ratio of the radial load to the axial load to ensure cage
stability as well as the sliding ratio when the ball bearings
worked under steady conditions.
(2) Cage dynamic characteristics which presents in the
loading process of axial load and the loading process of
radial load are different. There is no obviously influence on
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