CALCULATION OF ADDITIONAL AXIAL FORCE OF ANGULAR-CONTACT BALL
BEARINGS IN ROTOR SYSTEM
Zhenhuan Ye1,2 , Zhansheng Liu1 and Liqin Wang3
1 School
of Energy Science and Engineering, Harbin Institute of Technology, Harbin, P.R. China
of Engineering and Technology, Zunyi Normal College, Zunyi, P.R. China
3 School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, P.R. China
E-mail: [email protected]
2 School
IMETI 2015 SA5026_SCI
No. 16-CSME-15, E.I.C. Accession 3901
ABSTRACT
Based on a loading-deformation relationship of bearing elements and the coordination of displacement between bearings in the rotor system, a model for calculating the additional axial force of angular-contact ball
bearings in a single-rotor system is established. Nonlinear equations of this model are solved through
the Rapid Descent method and Newton–Raphson method. The simulation results which are based on
Gupta’s example verify that both the model and solving methods in this paper are reliable. A pair of
276927NK1W1(H) angular-contact ball bearings in symmetry in the single-rotor system is used as the example, calculation results of the additional axial force of bearings from the model in this paper and from
the ISO method are compared and the influence of bearing geometry parameters and working conditions on
the additional axial force is further studied. This model and its conclusions could provide the basic data and
reference for analyzing the carrying ability and dynamic properties of rolling bearings.
Keywords: single-rotor system; angular-contact ball bearing; displacement-deformation coordination;
additional axial force.
CALCUL DE LA FORCE AXIALE ADDITIONNELLE D’UN ROULEMENT À BILLES À
CONTACT OBLIQUE DANS UN ENSEMBLE ROTOR
RÉSUMÉ
Basé sur la relation de chargement et de déformation des éléments composants des billes, et la coordination
de leur déplacement dans l’ensemble rotor, un modèle est établi pour calculer la force axiale additionnelle
d’un roulement à billes dans un ensemble rotor. Les équations nonlinéaires de ce modèle sont résolues par
la méthode de la plus forte pente et la méthode Newton–Raphson. Les résultats de simulation dans cette
recherche, basés sur l’exemple de la méthode de Gupta pour vérifier aussi bien le modèle que les méthodes
de solution, sont fiables. Dans l’exemple, on utilise une paire de roulement à billes à contact oblique de
type 276927NK1W1(H) en symétrie dans un ensemble monorotor. Les résultats du calcul de la force axiale
additionnelle à partir du modèle et de la méthode ISO sont comparés, et l’influence des paramètres de la
géométrie des billes et du régime de marche sur la force axiale, sont étudiés plus à fond. Ce modèle et les
conclusions pourraient permettre aux données de base et à la référence d’être utilisés pour l’analyse de la
capacité de charge et des propriétés dynamiques des roulements à billes.
Mots-clés : ensemble monorotor; roulement à billes à contact angulaire; coordination déplacement
déformation; force axiale additionnelle.
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1. INTRODUCTION
A high-speed angular-contact ball bearing is commonly used in important fields such as aviation, railway
and precision equipment. Its load-carrying properties cannot only relate to the fatigue life, temperature rise
and stability of the bearing, but also affect the stability of the rotor system through bearing-rotor coupling
[1]. That is why the analysis on its properties and geometry design has attracted a great concern. Prediction
on the numerical calculation of rolling bearings is an indispensable analyzing method. There are four developments in the model for analyzing rolling bearings: statics analysis, quasi-statics analysis, quasi-dynamic
analysis and dynamic analysis [2] and calculation programs such as BRAIN, BEAST, COBRA and ADORE
have been developed in succession [3–6].
However, what can be observed is that the analysis model works with given external loads of the bearings,
so the distribution of load on the bearings in the rotor system (especially the additional axial force produced
when the angular-contact ball bearing carries the radial loading) plays a vital role in predicting the dynamic
properties of single bearing. Currently, the calculation of additional axial force is usually estimated based
on the product of load coefficient of initial contact angle and radial load [7]. However, only values of load
coefficient of standard contact angles of 15, 25 and 40◦ are presented, so the interpolation method should
be adopted since the geometry parameters of sophisticated bearings are not standard. In addition, because
the combined load of bearings and the high-speed centrifugal force can change the initial contact angle, a
big error in external load of single bearings will occur in the calculation of additional axial force through
the product of load coefficient and radial loading. The inaccurate external load of bearing will render all
innovations on analysis methods, including the improved algorithm and analysis model, insignificant.
In addition, many scholars have considered the effect of bearing-rotor coupling in a system dynamics
model to analyze the dynamic properties of bearings [8–10]. This analysis focuses more on the rotor system
than one single bearing, thus the bearing has been changed into varying stiffness spring for establishing
dynamic equations [11–13]. However, the varying stiffness of springs is difficult to calculate because the
nonlinear relationship between the load and th displacement in bearings is still not described accurately in
those models.
In conclusion, from the aspect of the rotor system, the model of calculating additional axial force of
high-speed angular-contact ball bearing based on loading-deformation-displacement relationship in dynamic
analysis model is indispensable in analyzing the practical working condition of bearings. The more accurate analysis methods of predicting loading-deformation-displacement relationships include quasi-dynamic
analysis and dynamic analysis.
Harris [14] established the first quasi-dynamic analysis model after abandoning the control theory of
quasi-statics ferrule according to the definition of quasi-dynamic analysis by SKF [15]. Afterwards,
Poplawski and Rumbarger [16, 17] set up an improved quasi-dynamic analysis model for rolling bearings
in which frictional force between cage and guiding ring and that between rolling element and cage pocket
are taken into consideration. For a completely precise simulation of bearing operation and analysis on the
transmission of load and speed changing, many scholars have turned to dynamic analysis. In 1971, Walters
[18] pioneered the analysis on high-speed ball bearings in the respective of kinetics and analyzed for the
first time the dynamic change between rolling element and cage. Afterwards, Gupta published a series of
dynamic studies on ball and rolling bearings [19, 20], and systemically studied the dynamic time-varying
performance of rolling bearings, analyzed the motion and strained condition of angular-contact ball bearing
with elastohydrodynamic lubrication and verified the correctness of model and calculation method through
experiment on angular-contact ball bearings [21]. In 1985, Meeks [22] first put forward his dynamic model
which is not universally used because only axial load is considered and the model is only applicable with
solid lubrication. Then, Meeks [23] improved his model in 1996. Although all factors are considered in the
model, it is seldom used in practice because of extensive calculation.
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Fig. 1. Model of single rotor system.
Because of the large and complicated calculation, dynamic analysis is mainly used to explore the dynamic
instability of high-speed bearing elements and transient process of motion and stress of bearing elements
when researching how the working condition changes. Dynamic method is not widely used in the practical
analysis of bearings’ dynamic performances while the quasi-dynamic method has been widely applied in
analyzing high-speed bearings with high precision and easy calculation.
Therefore, in this paper we wish to establish an accuracy calculation method of additional axial load based
on the displacement-deformation relationship of bearings and quasi-dynamic method. The influence of
geometry parameters of bearing and working conditions of the rotor on the additional axial load of bearings
is further analyzed.
2. CALCULATION MODEL OF ADDITIONAL AXIAL LOAD
The rotor supported by two angular-contact ball bearings is the simplest bearing-rotor system whose structure is shown in Fig. 1. Here, Fx and Fz are the external axial and radial load of rotor, respectively. Ll and Lr
are the distance from load application point to left pivot bearings and the distance from the load application
point to right pivot bearings.
2.1. Bearing Load Distribution
According to the external load on rotor system, the force equilibrium equations of the rotor system can be
expressed as Eq. (1). M is the gravity of the rotor. Fa and Fr are the axial and radial load of bearings.
Subscripts l and r are the relevant parameters of left and right bearings, respectively.
Far − Fal = Fx
Frl + Frr = Fz − M
Frl · (Ll + Lr ) = (Fz − M) · Lr
(1)
The radial load of left and right pivot bearings can be direct obtained through
Frl = (Fz − M) ·
Fr
(Ll + Lr )
Frr = (Fz − M) ·
Ll
(Ll + Lr )
(2)
While exerting more radial load, contact deformation and contact load will be produced between the ball
and raceway. There is an axial component of contact load because of the existence of contact angle. For the
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Fig. 2. Model of angular contact ball bearing with external loads.
whole bearing, the sum of axial component of contact load on all contact points is the actual axial load of
bearings. But in the rotor system, the actual axial load of each bearings is not equal to external axial load of
rotor for the displacement constrain between the two bearings, the difference between maximum axial load
of the two bearings and external axial load of the rotor is defined as the additional axial force of bearings.
With external load and additional axial load, the left and right pivot bearings in the axial direction satisfy
the following conditions:
(a) load equilibrium condition:
Far − Fal = Fx
(3)
∆ar + ∆al = 0
(4)
(b) displacement coordination condition
where ∆ is the relative displacement between the inner and outer rings.
However, in the ISO calculation method, the displacement coordination condition is not a condition
strictly by deformation calculation. For the ISO calculation method [24], considering the load equilibrium
condition, the additional axial force is only calculated by the radial load and the values of load coefficient
of standard contact angles of 15, 25 and 40◦ . The calculation method is expressed as follows:
Sr = f p · Frr ,
Sl = Fp · Frl
(5)
where f p is the coefficient of load distribution, which is 0.4, 0.68 and 1.14 when the initial contact angles
are 15, 25 and 40◦ , respectively.
If Sr > Sl + Fx , then Far = Sr and Fal = Far − Fx . If Sr < Sl + Fx , then Fal = Sl and Far = Fal + Fx .
2.2. Displacement Calculation of Bearing Rings
A geometric model of angular-contact ball bearing is shown in Fig. 2. When the load on bearing is [Fa Fr ],
the sum of contact loads between ball and inner raceway should satisfy the requirement of external load
equilibrium. The equilibrium equation can be expressed as
∑ Q2 j sin α j = Fa
∑ Q2 j cos α j cos θ j = Fr
588
(6)
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where Q j is the contact load between ball and raceway. α j and β j are the contact angle between jth ball and
the inner raceway and contact angle between jth ball and the outer raceway, respectively. θ j is the azimuth
angle of jth ball. Subscripts 1 and 2 represent the outer and inner rings, respectively.
According to Hertz point contact theory [25], the deformation of contact load between the ball and the
raceway can be calculated through
s
π 3 m3a δ j3
1 0
Qj = E
(7)
3
K(e)3 ∑ ρ
p
where E 0 is the equivalent elastic modulus of ball and raceways, ma = 3 2L(e)/πν 2 . K(e) and L(e) are the
first and second elliptic integral, respectively. ν is the ellipticity of the contact region. ∑ ρ is the sum of the
curvature of the ball and the raceways.
There is a relationship of displacement-deformation coordination between the ball and rings. That is, the
axial relative displacement between the inner and outer ring equals the sum of axial displacement of ball’s
center and the axial deformation of the ball on the contact zone.
∆a = ∆ab + δab = Db /2 · (sin β j − sin α j ) + δ j · (sin β j + sin α j )
(8)
where Db is the diameter of the ball.
For certain external bearing loads, the displacement of bearing ring can be worked out according to
Eqs. (6) to (8).
2.3. Modeling and Solving
First of all, radial load of bearing can be calculated according to the external loads of the rotor system and the
initial value of additional load which can be obtained through the ISO calculation method. The calculation
equation is shown in Eq. (5), where with other contact angles except 15, 25 and 40◦ , the coefficient of load
distribution f p can be worked out through calculus of interpolation. The relative displacement of bearing
ring is calculated through Eqs. (6)–(8) and the additional axial load is adjusted according to the constraint
conditions presented in Eqs. (1) and (2). The calculating process is shown in Fig. 3.
3. MODEL VERIFICATION
In order to verify the reliability of model developed in this paper, the typical example in Gupta’s dynamic
analysis [6] is used as an example. Parameters of the bearing are displayed in Table 1, elasticity modulus
and Poisson’s ratio of each elements are 200 Gpa and 0.25, respectively. The axial load, radial load and
working speed of the bearing are 2000 N, 400 N and 120000 r/m respectively.
Comparison of results from Gupta’s model and this model is presented in Table 2. This table reveals that
the calculation results in this paper are quite close to the results in Gupta’s model and the prediction error is
within 10%.
The above analysis indicates that the established calculation model for load distribution and its solution
is accurate, thus it can be used in calculating the additional axial force of bearings.
Table 1. Geometry parameters of example ball bearings [6].
Parameters
Values Parameters
Values
Pitch diameter (mm)
31
Ball diameter (mm)
8
No. of balls
6
Initial contact angle (◦ )
24
Inner curvature factor
0.56
Outer curvature factor
0.52
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589
Fig. 3. Calculation flow chart of axial load of angular contact ball bearings in rotor system.
Table 2. Comparison of the results of the maximum loading ball calculated by two different models.
Parameters
Gupta’s results Current paper’s results Error (%)
◦
Contact angle (ball/inner raceway) ( )
27.87
30.00
7.6
Contact angle (ball/outer raceway) (◦ )
14.63
15.43
5.5
Contact stress (ball/inner raceway) (GPa)
2.942
2.935
0.2
Contact stress (ball/outer raceway) (GPa)
2.323
2.304
0.8
Axial displacement of inner ring (µm)
14.88
15.94
7.1
Radial displacement of inner ring (µm)
4.147
3.816
8.0
4. RESULTS AND DISCUSSIONS
In this paper, the rotor symmetrically supported by 276927NK1W1(H) angular-contact ball bearing is used
to analyze the bearing characteristics. Figure 1 shows the layout of the bearing-rotor system: Ll equals Lr ,
face-to-face layout mode and back-to-back layout mode of bearings are used in the rotor system, respectively. Total mass of the rotor and the bearing inner ring is 20 kg with a rotor axial force of 20000 N, a
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Table 3. Geometry parameter of angular contact ball bearings.
Geometry parameter
Values Geometry parameter
Values
Inner ring diameter (mm) 133.35 Outer ring diameter (mm)
201.725
Ball diameter (mm)
22.225 Number of balls
20
Initial contact angle (◦ )
25
Cage guiding clearance (mm)
0.4
Inner curvature factor
0.52
Outer curvature factor
0.515
Fig. 4. Load distribution of bearings with the axial load which is calculated by two different methods.
radial force of 5000 N and a speed of 10000 r/min. The basic parameters of rolling bearing are presented
in Table 3. Elasticity modulus, Poisson’s ratio and density of bearing rings and balls both are 218 GPa, 0.3
and 7870 kg/m3 , respectively. Results of predicting axial load through the ISO calculation method and the
method based on the relationship of displacement coordination are displayed in Table 4.
Since the two calculation methods are different, the axial loads obtained are different, which causes a
difference in bearing load and load distribution in the contact area. Bearing load and its distribution will
influence the fatigue life of bearings through contact stress and friction power consumption of bearings
caused by cage slip. Take the back-to-back layout mode as an example, the inner load distribution, friction
power consumption and fatigue life calculated by two different methods are presented in Fig. 4 and Table 3, where the calculation method for friction power consumption and fatigue life could be obtained from
[26]. Comparison of results reveals that the load distribution, contact stress, fatigue life and friction power
consumption worked out through the two methods are quite in line with each other, but there is still some
difference. Furthermore, the ISO method is just aimed at the three certain contact angles while the structure
parameters, such as curvature coefficient and working condition parameters including speed and external
load of bearings, are not taken into consideration. This will lead to a difference in results compared with the
results calculated by the displacement-deformation coordinate method.
4.1. Influence of Structure Parameters on Calculation of Additional Axial Load of Bearing
4.1.1. Influence of initial contact angle on additional axial load
Left pivot bearing is used as an example to analyze the influence of contact angle on the additional axial load
of bearings in the single-rotor system. Based on the displacement-deformation coordinate method, influence
of initial contact angle of bearing on additional axial load is shown in Fig. 5 and the additional axial load
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Table 4. Calculation results of axial load of supporting ball bearings in singular rotor system.
Layout type
Calculation method
Axial load of
Axial load of
left pivot bearings (N) right pivot bearings (N)
Face to face
ISO method
21700
1700
Dis.-def. coordinate
21988
1988
Back to back ISO method
1700
21700
Dis.-def. coordinate
1988
21988
Table 5. Bearing properties with the axial load which calculated by two different method.
Bearings
Calculation method Max. contact Fatigue Cage sliding
Friction power
of axial load
stress (Gpa)
life (h)
ratio (%)
consumption (W)
Left pivot
Dis.-def. coordinate
1.081
6685
7.9
1294
ISO method
1.094
8029
7.3
1230
Right pivot Dis.-def. coordinate
1.723
297
0.4
7148
ISO method
1.716
306
0.3
7055
Fig. 5. Effect of initial contact angle on additional axial load of bearings.
worked out through ISO calculation method is presented as well. It can be seen from this figure that with
the increase in initial contact angle, the additional axial load will increase gradually (in the displacementdeformation coordination method, the load can be calculated with any initial contact angle, but in the ISO
method, there are only three contact angles). The main reason is that when the radial load keeps constant,
the increase in initial contact angle will directly raise the axial component of load in the loading zone.
4.1.2. Influence of curvature factor on additional axial load
The influence of the curvature factor on the additional axial load of a left pivot bearing is shown in Fig. 6. It
can be seen that the additional axial load on the left pivot bearing calculated by the ISO method is constant
since the curvature factor is not considered in the ISO method. In the method of displacement-deformation
coordination, the influence of curvature factor on adaptation between ball and raceway (contact area between
ball and raceway) is taken into consideration, so it can be seen that the increase in the outer curvature factor
and decrease in the inner curvature factor will lead to the increase in additional axial load. Meanwhile, outer
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Fig. 6. Effect of curvature factor on additional axial load of bearings.
Fig. 7. Effect of rotor axial load on additional axial load of bearings (initial contact angle: 25◦ ).
curvature factor exerts more influence than inner curvature factor on the additional axial load. The reason is
that the influence of inner curvature factor on the additional axial load mainlys results from the adaptation
between ball and raceway while the influence of outer curvature factor on the additional axial load is mainly
result from the displacement coordination between two bearings in the rotor system.
4.2. Influence of Working Condition of Rotor on Calculation of Additional Axial Load
4.2.1. Influence of rotor axial load on additional axial load of bearings
When the radial load on rotor keeps constant, the influence of rotor axial load on the additional axial load
of the left pivot bearing is shown in Fig. 7. This figure shows that the axial load on the left pivot bearing
calculated through the ISO method is constant. This is because the ISO calculation formula is the function
of radial load of bearings and the initial contact angle of balls as shown in Eq. (5), and the axial load is not
considered. In the displacement-deformation coordination method, the axial load on the left pivot bearing
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Fig. 8. Effect of rotor radial load on additional axial load of bearings (initial contact angle: 25◦ ).
decreases with an increase of the axial load on the rotor. The main reason is that the left pivot bearing is in
the state of “relaxation” when the bearing is assembled as “back-to-back” layout type on the rotor system.
Therefore, with the increase of axial force of rotor system, the axial load on left pivot bearing will decline.
Meanwhile, when there is axial load on the rotor system, the right pivot bearing is in the state of “being
pressed” and the deformation of the inner loading zone of the right pivot bearing will directly affect the
degree of “relaxation” of the left pivot bearing. Finally, based on constraint of displacement in the rotor
system, the axial load of the left pivot bearing takes on a nonlinear trend.
4.2.2. Influence of rotor radial load on additional axial load of bearings
When the axial force of rotor keeps constant, the influence of rotor radial force on the axial load of the left
pivot bearing is shown in Fig. 8. It can be seen that with the increase of radial load on rotor, the axial load
of bearings will increase as well. The axial load calculated through the ISO method appears as a linear
growth with the increase of radial force of rotor, which results from the fact that the additional axial force
can be simply described as the product of radial force of bearing and proportionality coefficient which has
constant value with certain initial contact angle. The additional axial load obtained through displacementdeformation coordination has a nonlinear growth with the increase of radial force of rotor. This is mainly
because an increase in radial force of rotor can lead to an increase in radial force of bearing, reducing
the contact angle when the bearing is constrained by axial displacement and lowering the ratio of axial
load component to radial load component on the loading zone. The axial load on bearing can simply be
described as the product of radial load on bearing and proportionality coefficient as well. However, in the
method of displacement-deformation coordination, the proportionality coefficient is not a constant value,
but will decrease with an increase of radial load.
4.2.3. Influence of rotor speed on additional axial load of bearings
Apart from the influence of rotor load, the speed of rotor also exerts some influence on additional axial load
of bearings. In Fig. 9, the axial load on the left pivot bearing with an increase of rotor speed calculated
by two different methods is presented, while the axial load on bearings obtained through the ISO method
is constant. This is because the ISO calculation formula is the function of radial load of bearings and the
initial contact angle of balls as shown in Eq. (5), and the rotation speed is not considered. In the method of
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Fig. 9. Effect of rotor speed on additional axial load of bearings (initial contact angle: 25◦ ).
displacement-deformation coordination, the additional axial force of bearings increases nonlinearly with the
rise of rotor speed. This is because the rise of rotor speed will raise the centrifugal load inside the bearings,
and then increase the contact angle between the inner ring and the ball. As a result, the axial load will
increase with unchanged radial load, which will finally affect the load distribution in the rotor system.
5. CONCLUSIONS
In this paper, the model of calculating additional axial force of angular-contact ball bearings in the singlerotor system is established, it can be used to precisely predict the additional axial force of bearings with
different structure parameters and working conditions, providing reliable prediction of external load for
analyzing the carrying ability and dynamic properties of rolling bearings. Effects of structure parameters of
bearings and working conditions of rotor on additional axial force of bearings are analyzed. The following
conclusions are produced:
1. The additional axial force and inner load distribution of bearings calculated according to
displacement-deformation coordination are more accurate and have a wider range of application than
those obtained through the ISO method.
2. With unchanged working condition, the additional axial load of bearings will increase gradually with
the increase of initial contact angle. Meanwhile, an increase in the curvature coefficient of the outer
ring and a decrease in curvature coefficient of the inner ring can raise the additional axial force.
However, the curvature coefficient of inner ring exerts smaller influence on the additional axial load
than that of outer ring.
3. With unchanged structure parameters of bearings, the bearing’s additional axial load will (a) decrease
gradually with an increase of the rotor’s axial load, (b) increase with a rise of rotor’s radial force and
(c) increase nonlinearly with the rise of the rotor’s speed.
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
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595
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).
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