Journal of Sound and Vibration 332 (2013) 5898–5916
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsvi
Stiffness matrix formulation for double row angular contact
ball bearings: Analytical development and validation
Aydin Gunduz, Rajendra Singh n
Acoustics and Dynamics Laboratory, Department of Mechanical & Aerospace Engineering and Smart Vehicles Concepts Center,
The Ohio State University, Columbus, Ohio 43210, USA
a r t i c l e in f o
abstract
Article history:
Received 10 July 2012
Received in revised form
12 March 2013
Accepted 6 April 2013
Handling Editor: H. Ouyang
Available online 13 July 2013
Though double row angular contact ball bearings are widely used in industrial, automotive,
and aircraft applications, the scientific literature on double row bearings is sparse. It is also
shown that the stiffness matrices of two single row bearings may not be simply superposed
to obtain the stiffness matrix of a double row bearing. To overcome the deficiency in the
literature, a new, comprehensive, analytical approach is proposed based on the Hertzian
theory for back-to-back, face-to-face, and tandem arrangements. The elements of the fivedimensional stiffness matrix for double row angular contact ball bearings are computed
given either the mean bearing displacement or the mean load vector. The diagonal elements
of the proposed stiffness matrix are verified with a commercial code for all arrangements
under three loading scenarios. Some changes in stiffness coefficients are investigated by
varying critical kinematic and geometric parameters to provide more insight. Finally, the
calculated natural frequencies of a shaft-bearing experiment are successfully compared
with measurements, thus validating the proposed stiffness formulation. For double row
angular contact ball bearings, the moment stiffness and cross-coupling stiffness terms are
significant, and the contact angle changes under loads. The proposed formulation is also
valid for paired (duplex) bearings which behave as an integrated double row unit when the
surrounding structural elements are sufficiently rigid.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Double row bearings offer certain advantages over single row bearings, as they are capable of providing higher axial and
radial rigidity and carrying bi-directional or combined loads. Consequently, double row bearings are widely used in machine
tool spindles, industrial pumps, and air compressors, as well as in automotive, helicopter, and aircraft applications such as
gear boxes, wheel hubs, and helicopter rotors [1–3]. In certain problems, a double row bearing may be simply represented
by two single row bearings attached next to each other. However, in many static and dynamic problems, a double row
bearing must be considered as an integrated unit. The stiffness matrices of two single row bearings may not be simply
superposed to obtain the stiffness matrix of a double row bearing.
This study focuses on double row angular contact ball bearings which can be categorized into three arrangements
according to the organization of their rolling elements as shown in Fig. 1: (i) back-to-back (DB) or ‘O’ arrangement (Fig. 1a),
in which the load lines (the lines that pass through the contact point of the rolling elements) meet outside of the bearing;
(ii) face-to-face (DF) or ‘X’ arrangement (Fig. 1b), where the load lines converge toward the bore of the bearing; and
(iii) tandem (DT) or series arrangement (Fig. 1c), where the load lines act in parallel so they never meet each other as
opposed to the other two arrangements. In general the effective load center (spread) of the back-to-back arrangement is
n
Corresponding author. Tel.: +1 614292 9044; fax: +1 614292 3163.
E-mail address: [email protected] (R. Singh).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jsv.2013.04.049
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5899
Fig. 1. Three different arrangements of double row angular contact ball bearings: (a) back-to-back (DB) or ‘O’ arrangement; (b) face-to-face (DF) or
‘X’ arrangement; (c) tandem (DT) arrangement.
larger; thus, it has higher moment stiffness terms and a higher moment load carrying capacity. The face-to-face
arrangement has a smaller load center; however, it has a larger misalignment angle and allows larger misalignments.
The tandem arrangement can carry heavier axial loads but only in one direction due to the fact that all rolling elements are
organized in the same direction. The vast majority of commercial double row angular contact ball bearings are found in the
back-to-back arrangement; however, other arrangements are also utilized [1].
A new, theoretical bearing stiffness matrix (Kb) formulation is proposed in this article for double row angular contact ball
bearings through a novel extension of the well-known theory for single row [4] and self-aligning (spherical) bearings [5].
Unlike self-aligning bearings [5], the moment stiffness terms of double row angular contact ball bearings are significant and
highly dependent on the configuration of the rolling elements [6,7]. Therefore, tilting moments and angular motions bring a
major complexity to the proposed formulation.
2. Literature review
2.1. Bearing stiffness
To understand the interfacial characteristics of rolling element bearings in rotating machinery, a wide range of bearing stiffness
models of varying complexity have been proposed [1,8–16]. Some of these models describe the bearing as time-invariant
translational springs in the axial and radial directions, which can predict only in-plane motions transmitted through the bearing
while neglecting out-of-plane or flexural motions. This may result in an inadequate understanding of the bearing as a vibration
transmitter, as experimental results have shown that the casing vibrations are typically out-of-plane [13–15]. Although Jones [17]
did not define a bearing stiffness matrix, his load–deflection formulations for ball and roller type bearings under static loading
conditions could be used to define a fully populated stiffness matrix for single row bearings. Lim and Singh [4] developed a five
dimensional symmetric bearing stiffness matrix (which is in fact a six dimensional matrix with the last column and row being all
zeros, corresponding to free torsion) for single row ball-type and roller-type bearings. The main advantage of Lim and Singh's [4]
stiffness model over previous models was the introduction of flexural and out-of-plane type motions through cross-coupling
stiffness terms which clearly explained the vibration transmission through rolling element bearings. Lim and Singh showed the
merits of their model in their series of papers [4,18–21] through parametric studies and comparisons with previous analytical and
experimental results [12,13]. The importance of moment stiffnesses and cross-coupling stiffness terms are clearly pointed out in
these studies. De Mul et al. [22] outlined a general theory for the numerical calculation of the stiffness matrices of loaded bearings
by extending Jones' [17] equations. Hernot et al. [23] calculated the stiffness matrix of angular contact ball bearings by integration
techniques.
Cermelj and Boltezar [24] used Lim and Singh's [4] stiffness model to further investigate the dynamics of a structure containing
ball bearings. Further, Royston and Basdogan [5] studied double row self-aligning (spherical) bearings by proposing a stiffness
matrix. As the moment stiffness of self-aligning bearings is negligible, Royston and Basdogan [5] did not consider the effects of
angular displacements and tilting moments. Thus, their three dimensional stiffness matrix was, in fact, a simplified version of Lim
and Singh's five dimensional model [4], where the last two rows and columns of moment stiffness terms are neglected, but the
three dimensional translational motions of each rolling element of both rows are included.
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A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
Guo and Parker [25] have recently proposed a numerical method for calculating the bearing stiffness matrix of singlerow bearings with a commercial finite element based contact mechanics code [26]. Their numerical approach, however,
is not attractive since at least 50 executions of the finite element code (hence excessive computing times) are needed to
estimate Kb; and also, utilization of finite difference approximation [25] may create problems. Additionally it is not clear if
the double row bearings can be analyzed with this method. Thus, analytical methods must be preferred over such numerical
methods for the theoretical calculation of Kb as well as insight into the significance of various terms.
Experimental techniques have also been utilized to measure radial and axial bearing stiffness coefficients. For example, Walford
and Stone [27,28] used a two-degree-of-freedom model to extract representative stiffness values from measurements. Stone [29]
reviewed various efforts to measure the stiffness and damping coefficients of rolling element bearings with changes in preload,
speed, or lubrication. Kraus et al. [13] designed an in-situ measurement test to determine the translational bearing stiffness
measured from vibration spectra using a single-degree-of-freedom model. Marsh and Yantek [30] extracted translational bearing
stiffness coefficients by measuring the resulting responses under known excitation forces. Tiwari and Vyas [31] suggested a method
for estimating nonlinear bearing parameters without making explicit force measurements. Finally, Pinte et al. [32] experimentally
minimized coupling between horizontal and vertical directions by using hinges and leaf springs in order to block out-of-plane
forces when they developed a piezo-based bearing for active noise control.
2.2. Double row bearings
Although single row bearings have been extensively studied, publications specific to double row bearings are sparse. For
instance, Bercea et al. [6] formulated the relative displacement between the bearing rings (also termed as the ‘ring
approach’) for various double row bearing types such as tapered, spherical, cylindrical roller, and angular contact ball
bearings. Their bearing deflection formulation is only valid for a back-to-back arrangement and did not include any stiffness
formulation. Then Nelias and Bercea [7] used their double row tapered rolling bearing model for case studies. Cao and Xiao
[33] developed a dynamic model for double row spherical roller bearings based on energy principles. Later Cao [34]
improved this model by including the effects of rotational motions and shaft misalignments. Choi and Yoon [35] proposed
a method for determining discrete design variables of an automotive wheel assembly that contained a double row angular
contact ball bearing. Their optimization algorithm attempted to maximize the bearing service life while satisfying various
design constraints including limited mounting space.
3. Stiffness formulations of a double row bearing vs. two single row bearings
Two approaches may be utilized in the stiffness modeling of double row or duplex angular contact ball bearings as
shown in Fig. 2(a–b). The first approach (with two single row bearings) essentially yields 2 separate mean displacement
i
qim ¼ fδixm ; δiym ; δizm ; βixm ; βiym gT and load vectors f m ¼ fF ixm ; F iym ; F izm ; M ixm ; M iym gT (here z-axis is the rotational axis) and two
i
separate stiffness (Kb ) matrices (i.e. one for each individual bearing) where
2 i
3
i
i
i
i
kxθy
kxx kxy kxz kxθx
6
7
6 i
7
i
i
i
i
6 kyx kyy kyz kyθx kyθy 7
6
7
6
7
i
i
i
i
i
7
k
k
k
k
k
Kib ¼ 6
ði ¼ 1; 2Þ
(1)
zx
zy
zz
zθ
zθ
x
y
6
7
6 i
7
i
i
i
i
6k
7
k
k
k
k
θx y
θx z
θx θx
θx θy 7
6 θx x
4 i
5
i
i
i
i
kθy x kθy y kθy z kθy θx kθy θy
D
D
D
D T
D
D
D
Conversely, the second approach results in single mean displacement qD
m ¼ fδxm ; δym ; δzm ; βxm ; βym g and load f m ¼ fF xm ;
D
vectors. Note that a single stiffness matrix (Kb ) needs to be defined for the double row bearing where
D
D
D T
FD
ym ; F zm ; M xm ; M ym g
(2)
K
Translational stiffness elements of two side by side bearings with the same shaft and housing are in parallel [36]; thus,
D
1
2
they can be superposed to obtain the stiffness values of the integrated unit (i.e. kpq ≈kpq þ kpq where p,q¼x, y, z). However,
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5901
lSHAFT
zG
x
Bearing 1
Fzext
Bearing 2
z
O
G
y
ext
Fx
zBRG1
zBRG2
zBRG1
lSHAFT
zG
Outer ring
x
Inner ring
ext
Fz
z
O
G
y
Double row angular contact ball
bearing
ext
Fx
(DB, DF or DT configuration)
Fig. 2. Illustration of two approaches that can be practiced in modeling of double row/duplex angular contact ball bearings: (a) Modeling two single row
ext
bearings, and (b) modeling an integrated double row bearing. Here, the rigid shaft is subjected to radial ðF ext
x Þ and axial ðF z Þ external loads at point O.
Table 1
Kinematic properties of the example case: Double row angular contact ball bearing.
Symbol
Value
Description
Z
rL (mm)
Kn (N/mm1.5)
R (mm)
Ao (mm)
e (mm)
αo (deg)
n
14
0
395,000
34.75
0.52
10.0
30
1.5
Number of rolling elements in one row
Radial clearance
Hertzian stiffness constant
Radius of the inner raceway groove curvature center (pitch radius)
Unloaded distance between inner and outer raceway centers
Axial distance between the geometric center and bearing one row
Unloaded contact angle
Load–deflection exponent
the rotational stiffness or rotational coupling terms ðkrs where r or s ¼ θx ; θy Þ of two single row bearings and a double row
D
i
D
bearing may not be easily correlated. In fact, there is simply no way of obtaining krs from krs (i ¼1,2), as krs terms depend on
geometric parameters and the organization of rolling elements; thus, they must be derived from basic load–deflection
relations.
To illustrate the above-mentioned issues, consider an illustrative example with a 148 mm long uniform solid shaft with a
diameter of 50 mm as shown in Fig. 2(a–b). First, two single row bearings (in all three arrangements) are placed on the shaft
with z-coordinates zBRG1 ¼64 mm and zBRG2 ¼84 mm, respectively. These two bearings are then replaced with a double row
bearing with its geometric center located at zG ¼ 74 mm (zG ¼ ðzBRG1 þ zBRG2 Þ=2). The kinematic properties of the bearing are
ext
ext
given in Table 1. The external load is applied at point O with magnitudes F ext
x ¼ 1 kN and F z ¼ 3 kN. Observe that F x
imposes a bending moment (Miym ) on both bearings. A combined asymmetrical loading case ensures that the analysis
presents a general case. The diagonal stiffness elements of the bearings (i.e. kpp where p ¼x,y,z,θx ; θy ) are calculated using a
commercial code [8]. This particular code [8] contains a large bearing library of commercial single and double row bearings
from major manufacturers, and yields only the diagonal elements of Kb which is sufficient and convenient for the
comparative analysis. The calculated elements of Kb under the combined load are listed in Table 2(a–c). It is clear from the
D
1
2
tables that kpp ≈kpp þ kpp is valid for p¼x,y,z but not for p ¼ θx ; θy for all arrangements. Note that the overall tilting
D
D
i
i
characteristics of a shaft-bearing system are dictated by kθx θx and kθy θy terms but not the kθx θx and kθy θy (i¼1, 2) terms
D
individually. Since it is not possible to obtain the rotational stiffness or rotational coupling terms of a double row bearing (krs
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A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
Table 2
Single row versus double row bearing stiffness analyses for the example case of Fig. 2. Bearings are organized in (a) back-to-arrangement; (b) face-to-face
arrangement; and (c) tandem arrangement.
(a)
Back-to-back arrangement
kxx (kN/mm)
kyy (kN/mm)
kzz (kN/mm)
kθxθx (MN mm/rad)
kθyθy (MN mm/rad)
Bearing 1
Bearing 2
Sumn
Double Row
290.5
293.7
238.4
153.5
154.2
110.6
33.9
44.2
13.1
43.9
401.9
327.6
282.6
166.6
198.1
401.1
327.6
282.6
342.2
410.1
144.2
53.6
86.1
28.3
82.9
297.8
287.7
256.2
155.1
175.6
441.9
341.3
342.3
183.4
258.5
442.3
341.7
342.2
65.7
100.6
166.2
128.8
134.7
70.2
103.7
162.6
144.5
82.5
56.2
50.3
328.8
273.3
217.2
126.4
153.9
329.0
272.8
216.9
156.1
211.4
(b)
Face-to-face arrangement
kxx (kN/mm)
kyy (kN/mm)
kzz (kN/mm)
kθxθx (MN mm/rad)
kθyθy (MN mm/rad)
(c)
Tandem arrangement
kxx (kN/mm)
kyy (kN/mm)
kzz (kN/mm)
kθxθx (MN mm/rad)
kθyθy (MN mm/rad)
n
: Arithmetic sum of the stiffness elements of bearings 1 and 2
Fig. 3. Mean loads and moments on the bearing and resulting translational and rotational displacements. The coordinate system is also shown.
where r or s ¼ θx ; θy ) from those of two single row bearings, a separate analytical formulation for double row bearings is
absolutely essential. Yet another advantage of using a double row formulation is that it could eliminate the need for the
solution of multidimensional statically indeterminate problems for simple cases (such as in Fig. 2(b)) by assuming the entire
external load is carried by the double row bearing.
4. Problem formulation: scope, assumptions and objectives
There are two primary differences between the ball and roller type rolling element bearings. First, the ball type bearings
have a point contact under unloaded conditions and an elliptical contact under loaded conditions between the bearing races
and the rolling elements, whereas this contact is a line contact under unloaded conditions and a rectangular contact under
loaded conditions for roller type bearings. Due to this difference in contact geometry, the load–deflection relationship (defined
by the Hertzian contact theory) is expressed by different exponential coefficients for ball and roller type bearings [1]. The
second primary difference is that the contact angle (α) of ball type bearings changes under a load, whereas it remains relatively
constant for roller type bearings. Thus, a change in α under loaded conditions must be taken into account. Besides, the moment
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5903
stiffnesses and the cross-coupling stiffness coefficients of the angular contact ball bearings are significant. Therefore, some of
the simplifying assumptions made for some other bearing types do not hold for angular contact ball bearings.
For the sake of analytical development, consider the double row angular contact ball bearing with a mean bearing load
vector f m ¼ fF xm ; F ym ; F zm ; M xm ; Mym gT and the resulting mean displacement vector qm ¼ fδxm ; δym ; δzm ; βxm ; βym gT , as illustrated
in Fig. 3 (beyond this point, all symbols are relevant to a double row bearing, and thus the superscript ‘D’ is simply omitted
for the sake of convenience). Here δxm ; δym ; δzm and F xm ; F ym ; F zm are the mean displacements and loads in the x, y, and z
directions, and βxm , βym and M xm , M ym are the mean angular displacements and tilting moments about the x and y
coordinates, respectively. The shaft is allowed to rotate freely about the z-axis so the corresponding angular displacement
and torsional terms are zero.
In general, both the inner and outer rings of a rolling element bearing may deflect under a load. However, for the purposes of
this study it is sufficient to consider only the relative displacement between the bearing rings. Thus, assuming the outer ring is
fixed in space and the inner ring is displaced under a load permits a tractable approach which is also pointed out by previous
investigators [4,22]. The mean bearing load vector (fm) will be considered as an effective point load on the inner ring and acting
along the geometrical center of the bearing. Similarly, the mean displacement vector qm corresponds to the displacement of the
bearing center. The total displacement vector is defined as q(t)¼qm+qa(t), where qa(t) is the alternating displacement vector that
fluctuates about the mean point qm. For small mean loads or sufficiently high dynamic displacements, the bearing stiffness
is time-varying, and nonlinear models should be used for the dynamic analysis. In this analysis it is assumed that |qa(t)| is much
smaller than qm; thus, linearized and time-invariant values of bearing stiffness can be determined about an operating point. In
this study, it is assumed that both rows of the bearings are identical in terms of the structural and kinematic parameters. Since
the bearings are often mounted on rigid shafts, and within robust housings the structural deformation of bearing rings is
assumed to be negligible, and only the elastic deformation of the rolling elements at contact points is considered. It is assumed
that the elastic deformation of the rolling elements occurs according to the Hertzian contact stress theory [1], and the load
experienced by each rolling element is defined by its angular position in the bearing raceway. The internal friction is also
assumed to be negligible compared with the normal loads on rolling elements. Further, the angular position of each rolling
element relative to one another remains the same due to rigid cages and retainers. It is also assumed that the bearing does not
operate at overcritical speeds; therefore, the gyroscopic moments and centrifugal effects can be neglected. It is assumed that the
lubrication does not directly affect the bearing stiffness coefficients except at very high speeds, as explained in the earlier work
[4,5]; this assumption is also utilized in some commercial codes [8]. Thus, tribological issues are beyond the scope of this paper.
Nevertheless, the authors have shown that the effect of lubrication can be successfully included via bearing (viscous) damping
elements in their recent work [36,37]. Finally the proposed Kb is also valid for duplex (paired) bearings mounted in face-to-face,
back-to-back, or tandem arrangements, assuming all structural elements (such as the shaft and bearing rings) are sufficiently
rigid; thus, the inner rings of the two rows do not move independently under external loads (except initial preloading).
The specific objectives of this study can be listed as follows: (1) develop a systematic and comprehensive approach to
determine the fully populated five dimensional stiffness matrix (Kb) for double row angular contact ball bearings of back-toback, face-to-face, and tandem arrangements (which may be extended for other types of double row bearings in later
studies); (2) develop a numerical scheme to compute Kb given the mean displacement (qm) or the mean loading (fm)
vectors; (3) verify the diagonal elements of the proposed stiffness matrix with a commercial code for all arrangements for
various load cases; (4) conduct parametric studies to observe the effect of different loading scenarios, unloaded contact
angle, and angular position of the bearing on the stiffness coefficients; and (5) validate the proposed model by comparing
the natural frequencies of a vehicle wheel bearing assembly with the measurements on a double row angular contact ball
bearing (in the back-to-back arrangement).
Fig. 4. Elastic deformation of a rolling element of the bearing. Elastic deformation of the rolling element is defined as the relative displacement between
the inner and outer raceway groove curvature centers due to mean bearing loads and displacements.
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A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5. New analytical formulation: load–deflection relations of a rolling element
To define the bearing stiffness, the relationship between fm and qm will first be derived. Consider the jth rolling element
of the ith row of a double row angular contact ball bearing shown in Fig. 4. Assuming the outer ring is fixed, the total elastic
deformation δðψ ij Þ of this rolling element can be calculated by
8
< Aij −Ao for δðψ ij Þ 4 0
i
δðψ j Þ ¼
(3)
:0
for δðψ ij Þ≤0
Here, ψ ij is the angular distance of the rolling element from the x-(horizontal) axis, and Ao and Aij are the unloaded and
loaded relative distances between the inner and outer raceway groove curvature centers, respectively. If Ao is greater than Aij ,
the rolling element is unloaded, and the elastic deformation is zero. To calculate Aij , the radial and axial displacements
ððδr Þij and ðδz Þij Þ of the rolling element are first expressed in terms of the elements of bearing displacement vector qm
ðδr Þij ¼ ½δxm þ c1 βym e cos ðψ ij Þ þ ½δym −c1 βxm e sin ðψ ij Þ−r L
(4a)
ðδz Þij ¼ δzm þ R½βxm sin ðψ ij Þ−βym cos ðψ ij Þ
(4b)
Here R is the radius of the inner raceway groove curvature center (pitch radius), αo is the unloaded contact angle, r L is the
radial clearance, and e is the axial distance between the geometric center of the bearing and bearing rows. The effective load
center (E), which controls the tilting stiffness terms of a double row arrangement, can be defined as E ¼ 2ðe þ c2 R tan ðαo ÞÞ,
and coefficients c1 and c2 are given as follows:
(
−1 for i ¼ 1 ðLeft rowÞ
(5)
c1 ¼
1 for i ¼ 2 ðRight rowÞ
8
>
<1
c2 ¼ −1
>
:0
back to back ðDBÞ arrangement
face to face ðDFÞ arrangement
(6)
tandem ðDTÞ arrangement
The net (effective) radial ðδnr Þij and axial ðδnz Þij displacements of the rolling element are then expressed including the effects
of initial displacement due to the unloaded contact angle αo
ðδnr Þij ¼ ðδr Þij þ Ao cos ðαo Þ
(7a)
ðδnz Þij ¼ ðδz Þij þ c3 ðAo sin ðαo Þ þ ðδz0 Þi Þ
(7b)
Here ðδz0 Þi defines an axial displacement preload on the ith row obtained by bringing the inner and outer raceways closer
together by a distance ðδz0 Þi (for instance in the case of split inner rings [35]). ðδz0 Þi can have a positive value only if the radial
clearance is eliminated (i.e. r L ¼ 0). Here, c3 is a constant dependent on the type of rolling element bearing
(
1
for i ¼ 1 ðLeft rowÞ
Back to back ðDBÞ arrangement : c3 ¼
(8a)
−1 for i ¼ 2 ðRight rowÞ
(
Face to face ðDFÞ arrangement : c3 ¼
−1
for i ¼ 1 ðLeft rowÞ
1
for i ¼ 2 ðRight rowÞ
Tandem ðDTÞ arrangement : c3 ¼ 1ðnÞ
ðBoth rowsÞ
(8b)
(8c)
(*): Assuming the tandem arrangement is axially loaded in the direction it can hold.
The loaded relative distance between the inner and outer raceway groove curvature center Aðψ ij Þ is finally obtained by
vectoral addition of the net displacements
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(9)
Aðψ ij Þ ¼ ððδnr Þij Þ2 þ ððδnz Þij Þ2
Then, the elastic deformation δðψ ij Þ of the rolling element can be determined by Eq. (3) to be used in conjunction with the
Hertzian contact stress theory in order to obtain the resultant normal load (Q ij ) on the element
Q ij ¼ K n ðδðψ ij ÞÞn :
(10)
Here Kn is the rolling element load–deflection stiffness constant, which is a function of material properties and geometry [1].
The exponent n is equal to 1.5 for point/elliptical type contact (for ball type bearings). The loaded contact angle αij for the
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5905
same element is determined by trigonometry
tan ðαij Þ ¼
ðδnz Þij
n i
ðδr Þj
¼
ðδz Þij þ c3 ðAo sin ðαo Þ þ ðδz0 Þi Þ
(11)
ðδr Þij þ Ao cos ðαo Þ
where ðδz Þij and ðδr Þij are given by Eq. (4a and b).
6. Stiffness matrix
6.1. Formulation
The bearing stiffness matrix is a comprehensive representation that combines the bearing's kinematic and elastic
properties and the effect of each rolling element [4]. To apply the mathematical definition of the stiffness matrix, the mean
load vector fm is first represented through vectoral sums of Q ij ði ¼ 1; 2; j ¼ 1; …; ZÞ for each loaded ðδðψ ij Þ 4 0Þ rolling element
as follows:
9
8
9
8
>
>
cos ðαij Þ cos ðψ ij Þ
>
>
>
>
F
>
>
>
> xm >
>
>
>
i
i
>
>
>
>
>
>
>
>
Þ
sin
ðψ
Þ
cos
ðα
>
>
>
>
F
j
j
>
>
>
> ym =
=
<
<
2
Z
i
i
Þ
sin
ðα
F
zm
(12)
¼ ∑ ∑ Qj
fm ¼
j
>
>
>
>
> i¼1j¼1 >
>
>
>
>
>
>
> ½R sin ðαij Þ−c1 e cos ðαij Þ sin ðψ ij Þ >
> M xm >
>
>
>
>
>
>
>
>
;
>
:M >
>
>
> ½−R sin ðαi Þ þ c e cos ðαi Þ cos ðψ i Þ >
ym
;
:
1
j
j
j
Expressing Q ij and αij in Eq. (12) in terms of mean deflections gives the explicit expressions between fm and qm
9
8
F xm >
>
>
>
>
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
>
> F ym >
>
>
>
>
=
<
ððδr Þij þ Ao cos ðαo ÞÞ2 þ ððδz Þij þ c3 ðAo sin ðαo Þ þ ðδz0 Þi ÞÞ2 −Ao
2
Z
F
zm
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ Kn ∑ ∑
fm ¼
>
>
>
>
i¼1j¼1
ððδr Þij þ Ao cos ðαo ÞÞ2 þ ððδz Þij þ c3 ðAo sin ðαo Þ þ ðδz0 Þi ÞÞ2
>
>
> M xm >
>
>
>
>
:M ;
ym
9
8
ððδr Þij þ Ao cos ðαo ÞÞ cos ðψ ij Þ
>
>
>
>
>
>
>
>
>
>
i
i
>
>
>
>
ððδr Þj þ Ao cos ðαo ÞÞ sin ðψ j Þ
>
>
>
>
=
<
i
i
Þ
þ
c
ðA
sin
ðα
Þ
þ
ðδ
Þ
Þ
ðδ
(13)
z j
o
3 o
z0
>
>
>
>
>
>
i
i
i
i
>
>
ðRððδ
Þ
þ
c
ðA
sin
ðα
Þ
þ
ðδ
Þ
ÞÞ−c
eððδ
Þ
þ
A
cos
ðα
ÞÞÞ
sin
ðψ
Þ
>
>
z
o
o
r
o
o
3
z0
1
j
j
j
>
>
>
>
>
>
>
;
: ½−Rððδz Þi þ c3 ðAo sin ðαo Þ þ ðδz0 Þi ÞÞ þ c1 eððδr Þi þ Ao cos ðαo ÞÞ cos ðψ i Þ >
j
j
j
Substituting ðδz Þij and ðδr Þij terms from Eq. (4a–b) into Eq. (13), and assuming qa ðtÞ⪡qm , the five dimensional stiffness
matrix Kb around the operating point can be defined
2
3
kxx
kxy kxz kxθx kxθy
6
2 ∂F pm ∂F pm 3
kyy kyz kyθx kyθy 7
6
7
6
7
∂δqm
∂βsm
kzz kzθx
kzθy 7
(14)
Kb ¼ 4 ∂Mrm ∂Mrm 5 ¼ 6
6
7
6 symmetric
∂δqm
∂βsm
kθ x θ x kθ x θ y 7
4
5
qm
kθy θy
qm
Here, p; q ¼ x; y; z and r; s ¼ x; y. The explicit expressions of each stiffness term are symbolically calculated and given in
their simplest form as follows:
2
Z
kxx ¼ K n ∑ ∑
ðδij Þn cos 2 ðψ ij ÞðnAij ððδnr Þij Þ2 =ðAij −Ao Þ þ ððδnz Þij Þ2 Þ
ðAij Þ3
i¼1j¼1
2
Z
kxy ¼ K n ∑ ∑
ðδij Þn sin ðψ ij Þ cos ðψ ij ÞððnAij ððδnr Þij Þ2 =ðAij −Ao ÞÞ þ ððδnz Þij Þ2 Þ
ðAij Þ3
i¼1j¼1
2
Z
kxz ¼ K n ∑ ∑
i¼1j¼1
ðδij Þn ðδnr Þij ðδnz Þij cos ðψ ij ÞððnAij =ðAij −Ao Þ−1Þ
ðAij Þ3
(15a)
(15b)
(15c)
5906
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
2
2
ðδij Þn
Z
sin ðψ ij Þ cos ðψ ij Þ4
kxθx ¼ K n ∑ ∑
2
ðδij Þn
Z
cos
2
ðψ ij Þ4
kxθy ¼ K n ∑ ∑
þðRðδnz Þij ðδnr Þij −c1 eððδnz Þij Þ2 Þ
2
2
Z
(15g)
ðAij Þ3
2
ðδij Þn
Z
(15f)
ðδij Þn ðδnr Þij ðδnz Þij sin ðψ ij Þð1−ðnAij =ðAij −Ao ÞÞÞ
i¼1j¼1
sin
kyθx ¼ K n ∑ ∑
2
ðψ ij Þ4
ðnAij =ðAij −Ao ÞÞðRðδnz Þij ðδnr Þij −c1 eððδnr Þij Þ2 Þ
−ðRðδnz Þij ðδnr Þij −c1 eððδnz Þij Þ2 Þ
3
5
(15h)
ðAij Þ3
i¼1j¼1
2
ðδij Þn
sin ðψ ij Þ cos ðψ ij Þ4
kyθy ¼ K n ∑ ∑
−ðnAij =ðAij −Ao ÞÞðRðδnz Þij ðδnr Þij −c1 eððδnr Þij Þ2 Þ
þðRðδnz Þij ðδnr Þij −c1 eððδnz Þij Þ2 Þ
3
5
(15i)
ðAij Þ3
i¼1j¼1
2
ðδij Þn ððnAij ððδnz Þij Þ2 =ðAij −Ao ÞÞ þ ððδnr Þij Þ2 Þ
Z
kzz ¼ K n ∑ ∑
i¼1j¼1
2
ðδij Þn sin ðψ ij Þ4
Z
(15e)
ðAij Þ3
kyz ¼ K n ∑ ∑
2
5
ðδij Þn sin 2 ðψ ij ÞððnAij ððδnr Þij Þ2 =ðAij −Ao ÞÞ þ ððδnz Þij Þ2 Þ
Z
i¼1j¼1
Z
3
ðAij Þ3
kyy ¼ K n ∑ ∑
2
5
(15d)
−ðnAij =ðAij −Ao ÞÞðRðδnz Þij ðδnr Þij −c1 eððδnr Þij Þ2 Þ
i¼1j¼1
2
−ðRðδnz Þij ðδnr Þij −c1 eððδnz Þij Þ2 Þ
3
ðAij Þ3
i¼1j¼1
2
ðnAij =ðAij −Ao ÞÞðRðδnz Þij ðδnr Þij −c1 eððδnr Þij Þ2 Þ
kzθx ¼ K n ∑ ∑
(15j)
ðAij Þ3
RðAij Þ2 −ðRððδnz Þij Þ2 þ c1 eðððδnz Þij Þ3 =ðδnr Þij ÞÞ þ c1 eððδnz Þij =ðδnr Þij ÞðAij Þ2
þðRððδnz Þij Þ2 −c1 eðδnz Þij ðδnr Þij ÞðnAij =ðAij −Ao ÞÞ
3
5
(15k)
ðAij Þ3
i¼1j¼1
2
2
3
−RðAij Þ2 þ Rððδnz Þij Þ2 þ c1 eðððδnz Þij Þ3 =ðδnr Þij Þ −c1 eððδnz Þij =ðδnr Þij ÞðAij Þ2
6
7
ðδij Þn cos ðψ ij Þ4
5
−ðRððδnz Þij Þ2 −c1 eðδnz Þij ðδnr Þij ÞðnAij =Aij −Ao Þ
Z
kzθy ¼ K n ∑ ∑
2
ðδij Þn sin ðψ ij Þ4
2
2
Z
kθ x θ x ¼ K n ∑ ∑
R2 ðAij Þ2 þ c1 eRððδnr Þij ðδnz Þij −ðððδnz Þij Þ3 =ðδnr Þij Þ þ ðAij Þ2 ððδnz Þij =ðδnr Þij ÞÞ
þðe2 −R2 Þððδnz Þij Þ2 þ ðnAij =ðAij −Ao ÞÞðRðδnz Þij −c1 eðδnr Þij Þ2
2
Z
ðδij Þn
sin ðψ ij Þ cos ðψ ij Þ4
kθx θy ¼ K n ∑ ∑
5
(15m)
−R2 ðAij Þ2 −c1 eRððδnr Þij ðδnz Þij −ðððδnz Þij Þ3 =ðδnr Þij Þ þ ðAij Þ2 ððδnz Þij =ðδnr Þij ÞÞ
−ðe2 −R2 Þððδnz Þij Þ2 −ðnAij =ðAij −Ao ÞÞðRðδnz Þij −c1 eðδnr Þij Þ2
2
Z
kθ y θ y ¼ K n ∑ ∑
i¼1j¼1
ðδij Þn
cos 2 ðψ ij Þ4
R2 ðAij Þ2 þ c1 eRððδnr Þij ðδnz Þij −ðððδnz Þij Þ3 =ðδnr Þij Þ þ ðAij Þ2 ððδnz Þij =ðδnr Þij ÞÞ
þðe2 −R2 Þððδnz Þij Þ2 þ ðnAij =ðAij −Ao ÞÞðRðδnz Þij −c1 eðδnr Þij Þ2
ðAij Þ3
kyx ¼ kxy ; kzx ¼ kxz ; kzy ¼ kyz
3
5
(15n)
ðAij Þ3
i¼1j¼1
2
3
ðAij Þ3
i¼1j¼1
2
(15l)
ðAij Þ3
i¼1j¼1
3
5
(15o)
(15p–r)
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5907
kθx x ¼ kxθx ; kθx y ¼ kyθx ; kθx z ¼ kzθx
(15s–u)
kθy x ¼ kxθy ; kθy y ¼ kyθy ; kθy z ¼ kzθy ; kθy θx ¼ kθx θy
(15v–y)
6.2. Numerical estimation of Kb
If the mean displacement vector qm is known, the stiffness coefficients kpq ðp; q ¼ x; y; z; θx ; θy Þ can be calculated by direct
substitution into Eq. 15(a–y). However, in general, fm is known, and the resulting displacement vector qm is unknown. In this
case, the coupled nonlinear equations as described by Eqs. (12) and (13) are numerically solved to determine qm for a given
fm. To implement this method, Eq. (12) is rearranged as follows:
9
8
cos ðαij Þ cos ðψ ij Þ
>
>
9 8
9
8
>
>
>
>
>
>
F
G
xm >
1>
>
>
>
>
i
i
>
>
>
>
>
>
Þ
sin
ðψ
Þ
cos
ðα
>
>
>
>
>
>
j
j
>
> >
>
>
>G >
> F ym >
>
>
>
>
>
>
>
=
= >
= 2 Z
<
< 2>
<
i
Þ
sin
ðα
i
j
F
G
zm
¼0
(16)
¼
− ∑ ∑ Qj
g¼
3
h
i
>
>
> >
> i¼1j¼1 >
> R sin ðαi Þ−c e cos ðαi Þ sin ðψ i Þ >
>G >
>M >
>
>
>
>
xm >
>
>
>
>
>
>
4>
1
j
j
j
>
>
>
>
>
>
>
> :
>
>h
>
>
>
;
>
:
i
>
>
M ym ;
G5
>
>
>
;
: −R sin ðαij Þ þ c1 e cos ðαij Þ cos ðψ ij Þ >
Here g ¼ fG1 G2 G3 G4 G5 gT are the functions to be minimized. In this study, a secant method as described by Xu et al. [38] is
employed to solve the multidimensional minimization problem due to its strong convergence characteristics. A summary of
the proposed method, with focus on model inputs and outputs, is illustrated in Fig. 5.
7. Computational verification of proposed stiffness model
The proposed model is verified with a commercial code [8]. Since the algorithm used by the code is not published, the
verification will be limited to a numerical comparison of the diagonal elements of Kb for an example case under various
loading scenarios. All three configurations of the rolling elements (DF, DB, and DT) are analyzed.
Consider the commercial double row angular contact ball bearing with properties given in Table 1; this example case will
be used in the whole article. The bearing is initially unpreloaded, and its outer ring is fixed. First, the shaft is subjected to an
axial load (F ext
z ) that is increased from 1 kN to 10 kN in 1 kN increments (refer to Fig. 2(b)). Assuming the shaft is rigid, the
entire axial load is supported by the double row bearing (i.e. F ext
z ¼Fzm). In the absence of any radial or moment load, the
radial stiffness terms (kxx and kyy ) as well as the tilting stiffness terms (kθx θx and kθy θy ) of Kb are equal, yielding three
independent diagonal stiffness elements ðkxx ; kzz and kθy θy Þ that are illustrated in Fig. 6 for all three configurations (the
proposed model is given by discrete markers, and solid lines denote predictions by the commercial code). As seen from the
figure, all stiffness elements of the proposed model show an excellent match with those of the commercial code.
Fig. 5. Summary of the proposed stiffness matrix formulation with focus on model inputs and outputs.
5908
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
Radial Stiffness (kxx)
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
10
0
10
1
1
0.8
0.8
0.8
0.6
0.4
0.6
0.4
0.2
0
0
5
0
10
5
10
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
Axial Force (kN)
0
5
10
5
10
0
0
0.8
10
10
0.2
1
5
5
0.4
1
0
0
0.6
1
0
DT
5
1
kzz [MN/mm]
kxx [MN/mm]
5
0.2
DB
0
0
0
kθyθy [GNmm/rad]
DF
Tilting Stiffness (kθyθy)
Axial Stiffness (kzz)
1
0
0
5
0
10
Axial Force (kN)
Axial Force (kN)
Fig. 6. Comparison of axial, radial and tilting stiffness coefficients of the proposed model and the commercial code [8] for the example case of Table 1 with
respect to axial load. Key: Discrete points, proposed model; ( ), face-to-face arrangement; (◯), back-to-back arrangement; ( ), tandem arrangement;
solid line, commercial code.
Table 3
Comparison of diagonal stiffness elements of the proposed model and the commercial code [8] for DF, DB, and DT arrangements for the example case given
fm ¼ {1000 N, 0, 0,0, −74,000 N mm}T.
Bearing arrangement
Face-to-face (DF)
Back-to-back (DB)
Tandem (DT)
Stiffness element
Commercial code
Proposed model
Commercial code
Proposed model
Commercial code
Proposed model
kxx (kN/mm)
kyy (kN/mm)
kzz (kN/mm)
kθxθx (MN mm/rad)
kθyθy (MN mm/rad)
407
302
311
55
95
401
306
318
50
89
360
267
226
264
354
375
279
235
263
353
297
36
50
18
161
313
38
51
18
159
Next, a radial shaft load of F ext
x ¼1 kN is applied in positive x-direction with an axial distance of dz ¼74 mm away from the
bearing center (according to Fig. 2(b)) which imposes a moment load on the bearing Mym ¼ −74 k Nmm and results in
a mean load vector f m ¼ f1000 N; 0; 0; 0; −74; 000 N mmgT . Due to the absence of axial load, the solution for qm in this case
is quite sensitive to the selection of the initial guess, especially for the tandem arrangement. Thus, a more stable loading case
with f m ¼ f1000 N; 0; 3000 N; 0; −74; 000 N mmgT is also considered. Since kxx ≠kyy and kθx θx ≠kθy θy for both cases, five
distinct diagonal stiffness terms are calculated and presented in Tables 3 and 4, respectively. As seen from both tables, the
stiffness elements of the proposed model show an excellent match with the commercial code, with minor errors for both
loading cases. In the absence of axial loading (Table 3), stiffness elements in unloaded directions (y, z, and θx for this
particular case) are unconventionally small, especially for a tandem arrangement (while both models still being consistent).
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5909
Table 4
Comparison of diagonal stiffness elements of the proposed model and the commercial code [8] for DF, DB, and DT arrangements for the example case given
fm ¼{1000 N, 0, 3000 N,0, −74,000 N mm}T.
Bearing arrangement
Face-to-face (DF)
Back-to-back (DB)
Stiffness element
Commercial code
Proposed model
Commercial code
Proposed model
Commercial code
Proposed model
kxx (kN/mm)
kyy (kN/mm)
kzz (kN/mm)
kθxθx (MN mm/rad)
kθyθy (MN mm/rad)
442
341
342
66
100
465
357
359
61
96
402
328
283
342
410
420
346
297
345
410
330
273
217
156
211
345
274
221
151
209
Axial Stiffness (MN/mm)
Axial Force (kN)
25
20
15
10
5
0
0
0.01 0.02 0.03 0.04 0.05
Axial Displacement (mm)
1
1
0.8
0.8
Axial Stiffness (MN/mm)
30
Tandem (DT)
0.6
0.4
0.2
0
0
0.01 0.02 0.03 0.04 0.05
Axial Displacement (mm)
0.6
0.4
0.2
0
0 1 2 3 4 5 6 7 8 9 10
Axial Load (kN)
Fig. 7. Relationships between the axial load, axial displacement and axial stiffness coefficients for the example case. The stiffness values of DB and DF
arrangements are coincident with those of a single row bearing. Key: (◯), back-to-back arrangement; ( ), face-to-face arrangement; ( ), tandem
arrangement; (), single row bearing. (a) F zm vs. δzm ; (b) kzz vs. δzm ; (c) kzz vs. F zm .
After the application of F ext
z ¼ 3 kN, the values become more conventional, and the agreement between the two models is
still excellent.
8. Examination of stiffness coefficients
8.1. Effect of mean bearing loads on stiffness coefficients
Typical variations in the stiffness elements under various loads are analyzed with the proposed model and compared
with those of the single row bearing with identical kinematic properties as described by Lim and Singh's model [4]. First, the
effect of pure axial load (δzm ≠0; F zm ≠0 while other elements of qm and fm are zero) on kzz is analyzed. The relationship
between F zm and δzm is shown in Fig. 7(a). Here, the slope of the tangent (about an operating point) defines kzz , which is
plotted with respect to δzm (Fig. 7(b)) and F zm (Fig. 7(c)). As seen from Fig. 7(b and c), kzz of the DT arrangement is
significantly higher than those of the DF and DB arrangements, as expected. In fact, the DF and DB arrangements tend to act
as a single row bearing, showing a behavior identical to Lim and Singh's model [4], since only one row of these
arrangements is loaded under a pure axial load [1]. Note that at a given δzm , kzz of the DT arrangement is twice that of the DB
or DF arrangements (i.e. DT arrangement acts like two parallel springs in the axial direction). However, at a given F zm , kzz of
the DT arrangement is less than twice the DB or DF arrangements due to the stiffening nature of the load–deflection curve.
Second, the bearing is loaded in a radial (x) direction (δxm ≠0; F xm ≠0 while all other qm and fm elements are zero). The
relationship between F xm and δxm is shown in Fig. 8(a), and its slope (kxx ) is plotted with respect to δxm in Fig. 8(b). Here, the
load distribution for all configurations are identical; thus, DF, DB, and DT arrangements show an identical kxx behavior,
which is exactly twice that of a single row bearing for a given δxm (i.e. all double row configurations act like two parallel
springs in the radial direction). Next, a misalignment about the y-axis (βym ) is applied, and kθy θy elements are investigated.
Plots of M ym vs. βym and kθy θy vs. βym are given in Fig. 9(a) and (b), respectively. As expected, kθy θy of the DB arrangement is
significantly higher than those of the DF or DT arrangements due to its larger effective load center.
Since a bearing load affects all diagonal and some off-diagonal elements of Kb, one can generate a large number of plots
similar to Figs. 7–9 considering different loading and stiffness coefficient combinations. In general, it is more difficult to
5910
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
25
0.7
0.6
Radial Stiffness (MN/mm)
Radial Force (kN)
20
15
10
5
0.5
0.4
0.3
0.2
0.1
0
0
0.01
0.02
0.03
0.04
0
0.05
0
0.01
0.02
0.03
0.04
0.05
Radial Displacement (mm)
Radial Displacement (mm)
4
1.4
3.5
1.2
3
kθyθy (GNmm/rad)
Bending Moment (MNmm)
Fig. 8. Variation of the radial force and the radial stiffness with respect to radial displacement of the bearing for the example case. Key: (◯), back-to-back
arrangement; ( ), face-to-face arrangement; ( ), tandem arrangement; (), single row bearing. (a) F xm vs. δxm ; (b) kxx vs. δxm .
2.5
2
1.5
1
0.8
0.6
0.4
1
0.2
0.5
0
0
0
1
2
3
4
5
0
1
Angular Displacement (mrad)
2
3
4
5
Angular Displacement (mrad)
Fig. 9. Variation of the bending moment and the tilting stiffness with respect to angular displacement of the bearing for the example case. Key: (◯),
back-to-back arrangement; ( ), face-to-face arrangement; ( ), tandem arrangement; (), single row bearing. (a) M ym vs. βym ; (b) kθy θy vs. βym .
1.4
25
25
20
20
1.2
0.8
0.6
kzθy (MN/rad)
kxθy (MN/rad)
kxz (MN/mm)
1
15
10
15
10
0.4
5
5
0.2
0
0
0
2
4
6
8
Angular Displacement (mrad)
10
0
0
0.01
0.02
0.03
0.04
Axial Displacement (mm)
0.05
0
0.01
0.02
0.03
0.04
0.05
Radial Displacement (mm)
Fig. 10. Variation of some off-diagonal elements of Kb under various loads for the example case. Key: (◯), Back-to-back arrangement; (
arrangement; ( ), tandem arrangement; () single row bearing. (a) kxz vs. βym ; (b) kxθy vs. δzm ; (c) kzθy vs. δxm .
), face-to-face
Translational Stiffness (MN/mm)
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5911
1
0.8
kxx
0.6
kyy
0.4
kzz
0.2
0
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0.7
kθxθx (GNmm/rad)
0.6
0.5
0.4
0.3
0.2
0.1
0
0.8
kθxθx (GNmm/rad)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Unloaded Contact Angle (deg)
Fig. 11. Variations in the diagonal stiffness elements of Kb with α0 given δxm ¼ 0.05 mm for the example case. (a) kxx ; kyy ; and kzz , (b) kθx θx , (c) kθy θy . Key: (◯),
Back-to-back arrangement; ( ), face-to-face arrangement; ( ), tandem arrangement.
observe clear stiffness trends under combined loading cases. It is also rather difficult to predict which bearing loads might
specifically affect off-diagonal elements of Kb. For instance, for a pure axial load, the only significant cross-coupling terms
are kxθy and kyθx , whereas the matrix becomes fully populated if all elements of qm are nonzero. Fig. 10(a–c) illustrates
sample variations in some cross-coupling terms under various loading cases. Note that off-diagonal terms are also highly
dependent on the organization of the rolling elements.
8.2. Effect of unloaded contact angle (under radial load) on stiffness coefficients
A radial displacement of δxm ¼0.05 mm is applied to the bearing of Table 1, and the effect of unloaded contact angle (α0 )
on the stiffness coefficients are investigated and shown in Figs. 11 and 12 for diagonal and off-diagonal elements of Kb,
respectively. These figures are plotted up to α0 ¼851 as numerical issues seem to occur around α0 ¼901 in some cases. The
translational stiffness coefficients ðkxx ; kyy ; kzz Þ of all three arrangements are found to be identical under a pure radial load as
shown in Fig. 11(a). Here, kxx and kyy follow a similar trend, as they have a maximum value for α0 ¼01 (deep groove ball
bearing), which nonlinearly converge to zero as α0 approaches 901. On the other hand, kzz is minimum when α0 ¼01, reaches
its maximum value around α0 ¼651, and then decreases for higher α0 values. Recall that bearings with α0 ≥451 are often
viewed as “thrust bearings”, and thus, an increase in kzz for higher α0 is an expected result. Fig. 11(b and c) illustrates kθx θx
and kθy θy coefficients, which depend on the bearing configuration. As expected kθx θx and kθy θy of the DB arrangement are
higher than the other two arrangements at all α0 values. Also kθy θy is greater than kθx θx for all arrangements and α0 values.
The dominant off-diagonal terms of Kb (kxz , kxθy , kyθx , and kzθy ) for a given δxm are plotted with respect to α0 in Fig. 12(a–d).
Other off-diagonal terms are negligible under pure radial load. As seen from Fig. 12(a), kxz is identical for all three
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A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
0.5
kxz (MN/mm)
0.4
0.3
0.2
0.1
0
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
5
kxθy (MN/rad)
0
-5
-10
-15
-20
kyθx (MN/rad)
15
10
5
0
-5
0
kzθy (MN/rad)
-5
-10
-15
-20
-25
Unloaded Contact Angle (deg)
Fig. 12. Variations in dominant off-diagonal stiffness elements of Kb with α0 given δxm ¼0.05 mm for the example case. (a) kxz , (b) kxθy (c) kyθx , (d) kzθy .
Key: (◯), back-to-back arrangement; ( ), face-to-face arrangement; ( ), tandem arrangement.
arrangements at all α0 and has a maximum value when α0 ¼451. On the other hand, kxθy , kyθx , and kzθy are dependent on the
bearing arrangement and α0 . One can also observe that kxθy and kzθy are negative, and kyθx ¼ −kxθy .
8.3. Effect of angular position of the bearing on stiffness coefficients
So far, the stiffness elements have been calculated by assuming that the rolling elements are equally spaced along the
bearing races, and the first rolling element of both rows are coincident with the x-axis (i.e. ψ i1 ¼0 rad, i¼1,2). This
assumption should be verified since the load distribution among the rolling elements changes with angular position under
radial and moment loads; thus, the bearing may exhibit considerable stiffness variations.
Again, consider the same example case (of Table 1) subjected to δxm ¼0.05 mm. Three diagonal stiffness elements of Kb
ðkxx ; kzz ; kθy θy Þ are normalized with respect to their values at ψ i1 ¼0 rad and plotted against the angular position of the
bearing over a complete ball passage period as shown in Fig. 13(a–c). These results are illustrated for the DB arrangement,
though all arrangements show a similar behavior. As seen from Fig. 13(a–c), the stiffness elements vary between minimum
and maximum values over a ball passage period. In particular, kzz shows the maximum variation (about 6 percent) over the
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
1.0015
5913
1.0015
1
0.99
1.001
1.0005
kθyθy/(kθyθy)ψ=0
kzz/(kzz)ψ=0
kxx/(kxx)ψ=0
1.001
0.98
0.97
0.96
1.0005
1
1
0.95
0.9995
0
0.5
0.9995
0.94
1
0
Ball Passage Period
0.5
1
1.8
1
0.98
1.7
0.98
0.96
1
0.96
kzz/(kzz)ψ=0
0.92
0.9
0.88
kθyθy/(kθyθy)ψ=0
1.6
0.94
kxx/(kxx)ψ=0
0.5
Ball Passage Period
Ball Passage Period
1
1.5
1.4
1.3
0.86
1.2
0.84
0
0.5
Ball Passage Period
1
0.94
0.92
0.9
0.88
0.86
0.84
1.1
0.82
0.8
0
0.82
0.8
1
0
0.5
Ball Passage Period
1
0
0.5
1
Ball Passage Period
Fig. 13. Variations in normalized kxx ; kzz and kθy θy over a ball passage period for back-to-back configuration given δxm ¼ 0.05 mm for the example case.
(a–c), with Z ¼14 rolling elements; (d–f), with Z ¼4 rolling elements.
ball passage period. The variations in kxx and kθy θy on the other hand are negligible. These results suggest that the angular
position does not have a significant effect on the stiffness elements, as the maximum variation in a stiffness coefficient is less
than 6 percent even for a fairly large radial displacement (such as δxm ¼0.05 mm).
When the variations in the stiffness elements are not small, the accuracy of the linear stiffness model decreases, and
certain kinematic parameters have a significant role in such variations. For instance, the number of rolling elements is
especially critical as it changes the ball passage period and affects the load zone of a bearing. A reduction in the number of
(loaded) rolling elements extends the ball passage period and results in higher variations in the stiffness elements. To
illustrate this issue, consider an extreme case with only 4 rolling elements in each row (while all other parameters of the
example are kept the same), and calculate kxx ; kzz ; kθy θy for δxm ¼0.05 mm. These results are shown in Fig. 13(d–f). As seen
from the figures, kzz now shows a variation of almost 70 percent over a ball passage period, whereas the variations of kxx and
kθy θy are now around 18 percent and 16 percent, respectively. These results clearly show that the angular position is an
important parameter; however, its effect can often be neglected in the presence of a sufficient number of rolling elements.
Although the variation of the stiffness elements of all arrangements are similar for a given radial load, they occur
differently for a given moment load. Fig. 14 shows the variations in kxx ; kzz and kθy θy for βym ¼0.03 rad (for Z¼14). As seen
from the figure, the variations are arrangement specific for a given βym ; however, they are all within 2 percent; hence, they
are negligible.
9. Experimental validation
An experiment consisting of a vehicle wheel bearing assembly with a double row angular contact ball bearing (in the
back-to-back arrangement) is designed, instrumented, and tested to experimentally validate the proposed formulation;
details of this experimental study are described by the authors in a recent article [37]. The shaft-bearing experiment is
analytically described by a five degree-of-freedom model that consists of a rigid shaft (with three translational (x, y, z) and
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A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
Face-to-Face Arrangement
1.015
1.01
1.02
1.004
1.005
1
1.01
0.995
1
0.998
0
0.5
0.99
1
0.5
0.995
1
1.005
0.995
0.996
0.99
0
0.5
0.985
1
1
0
0.5
1
1.015
1.01
1
0.997
0.5
1.005
1
k
0.998
0
=0
0.999
0.995
0
y y /(k y y)
1.01
k zz /(k zz) =0
1
1.005
1
Back-to-Back Arrangement
1.015
1.001
k xx /(k xx) =0
1.015
k
1.002
=0
1.006
y y /(k y y)
1.01
k zz /(k zz) =0
k xx /(k xx) =0
1.008
1.025
0.995
0
0.5
0.99
1
Tandem Arrangement
1.007
1.006
1.004
1.005
1.002
1.004
1.002
=0
1.003
1
0.998
y y /(k y y)
1.004
k zz /(k zz) =0
0.996
1.002
1.001
1
1.001
0.994
1
0.999
1.003
k
k xx /(k xx) =0
1.005
0
0.5
1
Ball Passage Period
0.992
0.999
0
0.5
1
Ball Passage Period
0.998
0
0.5
1
Ball Passage Period
Fig. 14. Variations in normalized kxx ; kzz and kθy θy over a ball passage period given βym ¼ 0.03 rad for face-to-face, back-to-back, and tandem configurations
of the example case.
two rotational (θx , θy ) dimensions). The double row angular contact ball bearing is modeled using the proposed fivedimensional bearing stiffness matrix (Kb ) with associated viscous damping (Cb ). An impulse hammer test is conducted at an
intermediate axial preload. Table 5 summarizes measured and predicted natural frequencies of the system (where r is the
A. Gunduz, R. Singh / Journal of Sound and Vibration 332 (2013) 5898–5916
5915
Table 5
Measured and predicted natural frequencies of the experiment, with several formulations of the analytical model.
Natural frequencies (Hz)
Modal index (r)
Experiment
5 Dof model (original)
5 Dof model (with radial load)
5 Dof model (with diagonal Kb)
1
2
3
4
5
773
894
1636
2049
2246
763
763
1583
2046
2046
711
864
1617
2059
2127
1422
1422
1565
1657
1657
modal index). Observe that predictions match well with measurements, with small errors. When there is no external radial
or moment load applied in the model, natural frequencies at r ¼1 and r ¼2, as well as r¼ 4 and r ¼5, are repeated. With an
application of a slight amount of load in the radial (x) direction (Fxm ¼0.3 kN), these repeated roots separate as shown in the
fourth column of Table 4. Now, the second, third, and fifth natural frequencies show a better match with the measurements,
but the estimation of the first natural frequency deviates further from experiments.
The last column of Table 5 shows a case where a diagonal Kb (with zero cross-coupling terms) is employed in the
analytical model. In this case, the natural frequency calculations deviate significantly from measurements. These results
clearly highlight the importance of cross-coupling stiffness terms of Kb and verify the need for a bearing stiffness matrix
when analyzing the vibration transmission paths [4].
10. Conclusion
The chief contribution of this study is the analytical development of the fully populated five dimensional stiffness matrix
(Kb) for double row angular contact ball bearings of back-to-back, face-to-face, and tandem arrangements. First, calculations
show that it is not possible to obtain the rotational stiffness or rotational coupling terms of a double row bearing from those
of two single row bearings, and thus a separate analytical formulation for double row bearings is absolutely essential. Using
the proposed analytical expressions, the diagonal and non-diagonal (cross-coupling) stiffness coefficients of a double row
angular contact ball bearing are determined given either the mean displacement vector qm (by direct substitution) or the
mean load vector fm (by numerical solution of the nonlinear system equations). A secant method is utilized to solve the
multidimensional minimization problem and the convergence issues seen in prior work [4] are minimized. The diagonal
elements of the stiffness matrix are verified with a commercial code through a detailed comparison of the stiffness elements
for an example case. Excellent agreement between the proposed model and the commercial code has been obtained under
three different loading scenarios. Then, some changes in Kb elements are further investigated with the proposed model
by varying bearing loads, unloaded contact angle, and angular position of the bearing to provide some insight. The
experimental results (as briefly presented in this article) clearly suggest that the proposed stiffness model is valid and can be
confidently utilized for modeling double row angular contact ball bearings; further validation at this stage is not possible
since the stiffness elements (especially the off diagonal terms) cannot be directly measured.
Considering the lack of publications on double row bearings, the new formulation provides a useful tool in the static and
dynamic analyses of double row angular contact ball bearings, such as the vibration analysis of shaft-bearing assemblies.
The proposed formulation is also valid for paired (duplex) bearings which behave as an integrated (double row) unit when
the surrounding structural elements (such the shaft and bearing rings) are sufficiently rigid. The proposed theory could be
extended to the analyses of double row cylindrical and tapered roller bearings as a part of the future work. In fact, the
mathematical formulation of angular contact ball bearings (as presented) is the most comprehensive as some of the
simplifying assumptions made for other bearing types (e.g. constant contact angle for roller type bearings) do not hold
for angular contact ball bearings.
Acknowledgments
We are grateful to the member organizations (such as the Army Research Laboratory and Honda R&D) of the Smart
Vehicle Concepts Center (www.SmartVehicleCenter.org) and the National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for supporting this work.
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