MULTI-DIMENSIONAL STIFFNESS CHARACTERISTICS OF DOUBLE ROW
ANGULAR CONTACT BALL BEARINGS AND THEIR ROLE IN
INFLUENCING VIBRATION MODES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
The Degree Doctor of Philosophy in
the Graduate School of The Ohio State University
By
Aydin Gunduz, B.S.
Graduate Program in Mechanical Engineering
The Ohio State University
2012
Dissertation Committee:
Professor Rajendra Singh, Advisor
Professor Marcelo Dapino
Professor Ahmet Kahraman
Professor Ahmet Selamet
Copyright by
Aydin Gunduz
2012
ABSTRACT
A new analytical stiffness model for the double row angular contact ball bearings
is proposed since the current methods do not provide stiffness matrix formulations for
double row bearings except for self-aligning (spherical) bearings in which angular
deflections and tilting moments are negligible. The moment stiffness terms and the crosscoupling stiffness elements in double row angular contact ball bearings are significant;
and the stiffness coefficients are highly dependent on the configuration of the rolling
elements. Also, unlike roller-type bearings, the contact angle of ball-type bearings
depends on the static load(s). The five-dimensional bearing stiffness matrix is first
developed for three configurations (face-to-face, back-to-back, and tandem) from basic
principles. The diagonal and off-diagonal (cross-coupling) elements of the matrix are
calculated from the explicit expressions given the mean bearing load or displacement
vector. Modeling approaches between a double row bearing vs. two single row bearings
are also analyzed from statics and stiffness perspective. The proposed stiffness matrix is
valid for duplex (or paired) bearings assuming all structural elements (such as the shaft
and bearing rings) are sufficiently rigid.
Next, a new modal experiment consisting of a vehicle wheel bearing assembly
with a double row angular contact ball bearing in a back-to-back arrangement is
designed. The bearing is subjected to axial or radial preloads in a controlled manner.
ii
Modal experiments with two preloading mechanisms (under non-roatating conditions)
show that the nature and extent of bearing preloads considerably affect the natural
frequencies and resonant amplitudes, thus influencing the vibration behavior of the shaftbearing assembly. A five-degree-of freedom vibration model of the shaft-bearing
assembly (including the proposed bearing stiffness matrix) is developed to describe the
modal experiment. Two alternate (preload-dependent and preload-independent) viscous
damping models are then proposed to describe the effect of bearing preload on the
resonant amplitudes, similar to those observed experimentally. The proposed bearing
stiffness model is then validated by comparing predicted natural frequencies and
accelerance spectra with modal measurements.
Finally, four calculation methods are comparatively evaluated by critically
examining bearing loads, deflections and stiffness elements; predicted modal properties
of the shaft-bearing assembly using each method are also compared with measurements.
In particular, the diagonal elements of the proposed stiffness matrix are compared with a
commercial code; and, the effects of critical geometric and kinematic parameters on the
stiffness coefficients are explored. A finite element based contact mechanics tool is
employed to verify certain assumptions of the new matrix formulation. Preliminary
modal experiments with a faulty bearing are included to motivate further research.
iii
DEDICATION
Dedicated to my mom and dad
iv
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my advisor Prof. Rajendra Singh
for his assistance, guidance and patience in the preparation of this dissertation. His
tremendous insight, knowledge and experience has helped me to improve myself and
encouraged me throughout my doctoral study. I also thank my committee members, Prof.
Marcelo Dapino, Prof. Ahmet Kahraman and Prof. Ahmet Selamet for their time to
review my works and dissertation. I would like to express my sincere appreciation to the
members of Smart Vehicle Concepts Center, especially to Dr. Tim Krantz (Army/NASA),
for sponsoring this work and making this research possible. I would like to thank Dr.
Jason Dreyer for the technical help and mentorship he provided. I also would like to
thank Caterina Runyon-Spears for her careful review of this dissertation.
I am most grateful to my parents and brother for their tremendous support,
encouragement, and guidance; their continuous faith in me has been the primary source
of my motivation. I am also very thankful to my girlfriend for being a great support and
encouragement during the most stressful years of my doctoral study. Finally, I would like
to thank all of my friends, including the past and current members of ADL, for their
friendship and assistance throughout this process.
v
VITA
October 29, 1984············································Born – Ankara, Turkey
June 2006 ······················································B.S. Mechanical Engineering
Middle East Technical University
Ankara, Turkey
September 2006 – Present······························Graduate Research Associate and Instructor
The Ohio State University
Columbus, OH
PUBLICATIONS
1. A. Gunduz, A. Inoue, and R. Singh, Estimation of interfacial forces in time domain
for linear systems, Journal of Sound and Vibration, 329 (13) (2010) 2616–2634.
2. A. Gunduz, J.T. Dreyer, R. Singh, Effect of Preloads on Vibration Transmission
through Double Row Angular Contact Ball Bearings, Paper # DETC2011/PTG-47759,
11th ASME International Power Transmission and Gearing Conference, Washington,
DC, Aug. 29-31, 2011
FIELDS OF STUDY
Major Field: Mechanical Engineering
Specialty Areas: Dynamics and Vibration, Acoustics, Rolling Element Bearings
vi
TABLE OF CONTENTS
ABSTRACT ........................................................................................................................ ii
ACKNOWLEDGMENTS .................................................................................................. v
VITA .................................................................................................................................. vi
LIST OF TABLES ............................................................................................................. xi
LIST OF FIGURES ......................................................................................................... xiv
LIST OF SYMBOLS ...................................................................................................... xxii
CHAPTER 1 INTRODUCTION ........................................................................................ 1
1.1 Motivation ............................................................................................................. 1
1.2 Literature Review.................................................................................................. 2
1.2.1 Bearing stiffness ......................................................................................... 2
1.2.2 Double row bearings .................................................................................. 4
1.2.3 Preload issues ............................................................................................. 5
1.3 Problem Formulation ............................................................................................ 5
1.3.1 Bearing deflections, loads and stiffnesses .................................................. 5
1.3.2 Scope, assumptions and objectives .......................................................... 10
References for chapter 1 ............................................................................................ 18
CHAPTER 2 ANALYTICAL DEVELOPMENT OF A NEW STIFFNESS MATRIX
FOR DOUBLE ROW ANGULAR CONTACT BALL BEARINGS .............................. 24
2.1 Introduction ......................................................................................................... 24
2.2 Literature Review................................................................................................ 27
2.3 Problem Formulation: Scope, Assumptions and Objectives .............................. 29
2.4 New Analytical Formulation: Load-Deflection Relations of a Rolling Element 34
2.5 Stiffness Matrix ................................................................................................... 38
2.5.1 Formulation .............................................................................................. 38
vii
2.5.2 Numerical estimation of Kb ..................................................................... 43
2.6 Numerical Verification of Diagonal Stiffness Elements ..................................... 45
2.7 Examination of Stiffness Coefficients................................................................. 52
2.7.1 Effect of mean bearing loads on stiffness coefficients ............................ 52
2.7.2 Effect of unloaded contact angle (under radial load) on stiffness
coefficients ........................................................................................................ 58
2.7.3 Effect of angular position of the bearing on stiffness coefficients .......... 61
2.8 Conclusion........................................................................................................... 65
References for chapter 2 ............................................................................................ 67
CHAPTER 3 EFFECT OF BEARING PRELOADS ON MODAL CHARACTERISTICS
OF A SHAFT-BEARING ASSEMBLY AND VALIDATION OF THE STIFFNESS
MODEL USING A MODAL EXPERIMENT ................................................................. 72
3.1 Introduction ......................................................................................................... 72
3.2 Literature Review ................................................................................................ 73
3.3 Problem Formulation: Scope, Assumptions and Objectives ............................... 75
3.4 Analytical Model of a Rigid Shaft Supported by Double Row Bearing ............. 77
3.5 Computational Study ........................................................................................... 83
3.6 Experimental Study ............................................................................................. 93
3.6.1 Development of a modal experiment ....................................................... 93
3.6.2 Effects of axial preloads........................................................................... 96
3.6.3 Effects of combined radial and moment loads ....................................... 100
3.6.4 Summary of experimental studies .......................................................... 102
3.7 Validation of the Bearing Stiffness Matrix ....................................................... 109
3.7.1 Comparison of natural frequencies ........................................................ 109
3.7.2 Comparison of radial accelerance spectra.............................................. 111
viii
3.8 Conclusion......................................................................................................... 115
References for chapter 3 .......................................................................................... 118
CHAPTER 4 CRITICAL EXAMINATION OF STIFFNESS CALCULATIONS USING
ANALYTICAL AND COMPUTATIONAL METHODS ............................................. 121
4.1 Introduction ....................................................................................................... 121
4.2 Problem Formulation......................................................................................... 122
4.3 Analytical and Computational Methods ............................................................ 124
4.3.1 Method A: Gunduz and Singh’s model ................................................. 124
4.3.2 Method B: Lim and Singh’s model ........................................................ 124
4.3.3 Method C: Finite element/contact mechanics based commercial code
(Calyx) ............................................................................................................ 125
4.3.4 Method D: A commercial code for gearing/bearing systems
(RomaxDesigner) ............................................................................................ 128
4.4 Justification of Some Assumptions of Analytical Model (Method A) using Finite
Element Model (Method C) .................................................................................... 133
4.4.1 Justification of rigid ring assumption ..................................................... 133
4.4.2 Justification of rigid shaft assumption ................................................... 135
4.5 Analysis of a Double Row bearing vs. Two Single Row Bearings .................. 138
4.5.1 Examination of bearing deflections........................................................ 141
4.5.1.1 Two single row bearings.............................................................. 141
4.5.1.2 Double row bearing ..................................................................... 141
4.5.1.3 Example case ............................................................................... 142
4.5.2 Examination of bearing forces and moments................................................. 145
4.5.2.1 Two single row bearings.............................................................. 145
4.5.2.2 Double row bearing ..................................................................... 145
ix
4.5.2.3 Example case (cont’d) ................................................................. 146
4.5.3 Examination of multi-dimensional stiffness coefficients ....................... 148
4.5.3.1 Two single row bearings vs. a double row bearing ..................... 148
4.5.3.2 Example case (cont’d) ................................................................. 149
4.6 Methods for Calculating Bearing Stiffness Coefficients .................................. 151
4.6.1 Analytical methods ................................................................................. 151
4.6.2 Finite difference approximation ............................................................. 151
4.7 Investigation of Bearing Loads, Deflections and Stiffness Elements with Four
Calculation Methods ............................................................................................... 155
4.7.1 Pure axial load ........................................................................................ 155
4.7.2 Combined load: Back-to-back arrangement ........................................... 160
4.8 Comparison of Predicted Natural Frequencies with Experimental Results ...... 168
4.9 Preliminary Modal Experiments with a Faulty Bearing ................................... 174
4.10 Conclusion ...................................................................................................... 179
References for chapter 4 ......................................................................................... 180
CHAPTER 5 CONCLUSION......................................................................................... 184
5.1 Summary ........................................................................................................... 184
5.2 Contributions..................................................................................................... 187
5.3 Future Work ...................................................................................................... 188
References for chapter 5 ......................................................................................... 191
BIBLIOGRAPHY ........................................................................................................... 192
x
LIST OF TABLES
Table
2.1
Page
Kinematic properties of the example case: Double row angular contact ball
bearing................................................................................................................... 47
2.2
Comparison of diagonal stiffness elements of the proposed model and the
commercial code [2.8] for DF, DB and DT arrangements for the example case
given fm  {1000 N, 0, 0, 0, 74000 Nmm}T ............................................................ 50
2.3
Comparison of diagonal stiffness elements of the proposed model and the
commercial code [2.8] for DF, DB and DT arrangements for the example case
given fm  {1000 N, 0, 3000 N, 0, 74000 Nmm}T ................................................... 51
3.1
Kinematic properties of the double row angular contact ball bearing utilized for
computational and experimental studies ............................................................... 80
3.2
Non-zero elements of the stiffness matrix calculated by the proposed stiffness
model (of chapter 2) with alternate bearing configurations for Fz 0 = 0.5 kN or 5
kN .......................................................................................................................... 86
3.3
Measured natural frequencies and peak amplitudes of the first vibration mode at
various axial preloads ......................................................................................... 106
3.4
Measured natural frequencies and peak amplitudes of the first vibration mode at
various radial preloads ........................................................................................ 107
3.5
Measured and predicted natural frequencies of the experiment (of Figure 3.8),
with several formulations of the analytical model of Figure 3.2. ....................... 110
xi
4.1
Single row versus double row bearing displacement analyses with Method D for
the example case of section 4.5.1.3. Bearings are organized in (a) back-toarrangement; (b) face-to-face arrangement; and (c) tandem arrangement.......... 144
4.2
Single row versus double row bearing load analyses with Method D for the
example case of section 4.5.1.3. Bearings are organized in (a) back-toarrangement; (b) face-to-face arrangement; and (c) tandem arrangement.......... 147
4.3
Single row versus double row bearing stiffness analyses with Method D for the
example case of section 4.5.1.3. Bearings are organized in (a) back-toarrangement; (b) face-to-face arrangement; and (c) tandem arrangement.......... 150
4.4
Comparative evaluation of the four calculation methods for a shaft-bearing system
of three different bearing configurations that is under pure axial load. (a) Fzm
calculated Methods C and D; (b) δzm calculated by four calculation methods; and
(c) kzz calculated by four calculation methods. ................................................... 157
4.5
Calculation of axial deflections, forces, and stiffness elements around the
operating point with Method C for the back-to-back arrangement. (a) Calculation
i
of  zm
and Fzmi (i=1,2) around the operating point. (b) Calculation of k zz with
finite difference approximation using two step sizes and accuracy orders. ........ 159
4.6
Comparative evaluation of the four calculation methods of a shaft-bearing system
of back-to-back configuration that is under combined load. (a) Bearing loads
calculated by Methods C and D; (b) Bearing deflections calculated by four
calculation methods; and (c) Bearing stiffness elements calculated by four
calculation methods ............................................................................................ 161
4.7
Calculation of bearing deflections and loads at multiple points around the
i
i
i
operating point. (a) Calculation of  xm
,  zm
and  ym
, and (b) calculation of Fxmi ,
i
(i = 1,2) around the operating point. ............................................ 163
Fzmi and M ym
xii
4.8
Calculated bearing stiffness elements for the combined load case with finite
difference approximation. (a) Calculation of kxx, kzz and kθyθy using Methods C and
D, and (b) Calculation of kxx and kθyθy by Method C using three different step sizes
and two different accuracy orders. ...................................................................... 166
4.9
A comparison of measured and predicted natural frequencies obtained by three
calculation methods (A, C and D) using the five degree-of-freedom model. ..... 172
xiii
LIST OF FIGURES
Figure
1.1
Page
Illustration of bearing loads and displacements. (a) A typical shaft-bearing system
where the shaft is supported by a double row angular contact ball bearing, and is
subjected to external forces and moments. (b) Illustration of mean bearing load
and moment components resulting in translational and rotational displacements. . 7
1.2
A typical nonlinear load-deflection relationship of a rolling element bearing. The
slope of the tangent of the load-deflection curve around the operating point
defines the bearing stiffness. ................................................................................... 8
1.3
Illustration of the three configurations of the duplex (paired) angular contact ball
bearings and double row angular contact ball bearings. ....................................... 12
1.4
Generic vibration problem of a shaft-bearing system and a simple representative
vibration model. (a) Schematic of the vibration problem of a shaft supported by a
double row angular contact ball bearing. (b) Analytical vibration model (of
dimension 5) of a rigid shaft supported by double row bearing. .......................... 17
2.1
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. .............................................................................. 26
2.2
Mean loads and moments on the bearing and resulting translational and rotational
displacements. The coordinate system is also shown. .......................................... 33
2.3
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
xiv
raceway groove curvature centers due to mean bearing loads and displacements.
............................................................................................................................... 35
2.4
Summary of the proposed stiffness model in terms of inputs and outputs ........... 44
2.5
The schematic to illustrate the static shaft loads utilized in the numerical analysis
of example case of Table 2.1. ............................................................................... 48
2.6
Comparison of axial, radial and tilting stiffness coefficients of the proposed model
and the commercial code [2.8] for the example case of Table 2.1 with respect to
axial load. Key: Discrete points, proposed model; (
(
), back-to-back arrangement; (
), face-to-face arrangement;
), tandem arrangement; solid line,
commercial code ................................................................................................... 49
2.7
Relationships between the axial load, axial displacement and axial stiffness
coefficients for the example case. Key: (
face-to-face arrangement; (
), Back-to-back arrangement; (
),
), tandem arrangement; ( ), single row bearing. (a)
Fzm vs.  zm ; (b) k zz vs.  zm ; (c) k zz vs. Fzm ....................................................... 53
2.8
Variation of the radial force and the radial stiffness with respect to radial
displacement of the bearing for the example case. Key: (
arrangement; (
), face-to-face arrangement; (
), Back-to-back
), tandem arrangement; ( ),
single row bearing. (a) Fxm vs.  xm ; (b) k xx vs.  xm .. .......................................... 55
2.9
Variation of the bending moment and the tilting stiffness with respect to angular
displacement of the bearing for the example case. Key: (
arrangement; (
), face-to-face arrangement; (
), Back-to-back
), tandem arrangement; ( ),
single row bearing. (a) M ym vs.  ym ; (b) k y y vs.  ym .. ..................................... 56
2.10
Variation of some off-diagonal elements of Kb under various loads for the
example case. Key: (
), Back-to-back arrangement; (
xv
), face-to-face
arrangement; (
), tandem arrangement; ( ), single row bearing. (a) kxz vs.  ym ;
(b) k x y vs.  zm ; (c) k z y vs.  xm .. ........................................................................ 57
2.11
Variations in the diagonal stiffness elements of Kb with 0 given  xm = 0.05 mm
for the example case. (a) k xx , k yy , and k zz , (b) k x x , (c) k y y . Key: (
back arrangement; (
2.12
), face-to-face arrangement; (
), tandem arrangement ... 59
Variations in dominant off-diagonal stiffness elements of Kb with 0 given  xm =
0.05 mm for the example case. (a) k xz , (b) k z y . Key: (
arrangement; (
2.13
), Back-to-
), face-to-face arrangement; (
), Back-to-back
), tandem arrangement ........... 60
Variations in normalized k xx , k zz and k y y over a ball passage period for backto-back configuration given  xm = 0.05 mm for the example case. (a-c), with Z =
14 balls; (d-f), with Z = 4 rolling elements ........................................................... 63
2.14
Variations in normalized k xx , k zz and k y y over a ball passage period given  ym
= 0.03 rad for DF, DB and DT configurations of the example case. .................... 64
3.1
Schematic of the vibration transmission problem. Here a shaft is subjected to
dynamic loads and moments fsa (t)  {Fxsa (t), Fysa (t), Fzsa (t ), M xsa (t ), M ysa (t )}T ,
resulting in vibratory motions qsa (t )  { xsa (t ),  ysa (t ),  zsa (t ),  xsa (t ),  ysa (t )}T .
Here, subscript a implies alternating or vibratory. x and y are radial directions,
and z is the axial direction. ................................................................................... 76
3.2
Analytical model (of dimension 5) of a rigid shaft supported by double row
bearing................................................................................................................... 78
xvi
3.3
Natural frequencies vs. axial preload for the computational example. (a) Face-toface arrangement, (b) Back-to-back arrangement, (c) Tandem arrangement. Key
for modes: (
3.4
), r = 1, 2; (
), r = 3; (
), r = 4, 5 ............................................. 85
Effect of axial preload on the radial accelerance magnitude spectra A xx ( ) of the
computational example with preload-independent damping model described by
Eq. (3.9). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c)
Tandem arrangement. Key: Fz 0 ; (
(
3.5
), 0.5 kN; (
), 1 kN; (
) 2 kN; (
) 3 kN;
) 5 kN. ............................................................................................................. 89
Effect of axial preload on the axial accelerance magnitude spectra A zz ( ) of the
computational example with preload-independent damping model described by
Eq. (3.9). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c)
Tandem arrangement. Key: Fz 0 ; (
(
3.6
), 0.5 kN; (
), 1 kN; (
) 2 kN; (
) 3 kN;
) 5 kN ……………………………………………………………….………………………………..90
Effect of axial preload on the radial accelerance magnitude spectra A xx ( ) of the
computational example with preload-dependent damping model described by Eq.
(3.10). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c) Tandem
arrangement. Key: Fz 0 ; (
), 0.5 kN; (
), 1 kN; (
) 2 kN; (
) 3 kN; (
)5
kN. ......................................................................................................................... 91
3.7
Effect of axial preload on the axial accelerance magnitude spectra A zz ( ) of the
computational example with preload-dependent damping model described by Eq.
(3.10). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c) Tandem
arrangement. Key: Fz 0 ; (
), 0.5 kN; (
), 1 kN; (
) 2 kN; (
) 3 kN; (
)5
kN. ......................................................................................................................... 92
xvii
3.8
Experiment with wheel-hub assembly and illustration of preloading mechanisms
utilizing hydraulic jack (for radial direction) and shaft with threaded rod (for axial
preloading). ........................................................................................................... 94
3.9
Effect of axial preload on the measured axial accelerance magnitude spectra at
location 1. (a) 1-2500 Hz, (b) 300-1000 Hz, (c) 2000-3800 Hz. Key: Fz 0 ; (
0.45 kN; (
kN; (
3.10
), 0.67 kN; (
), 1.78 kN; (
), 2.00 kN; (
), 1.11 kN; (
), 2.22 kN; (
), 1.33 kN; (
) 2.45 kN; (
), 1.56
) 2.67 kN; .... 98
Effect of axial preload on the measured axial accelerance magnitude spectra at
location 2. Key: Fz 0 ; (
(
), 1.33 kN; (
2.45 kN; (
3.11
), 0.89 kN; (
),
), 0.45 kN; (
), 1.56 kN; (
), 0.67 kN; (
), 1.78 kN; (
), 0.89 kN; (
), 2.00 kN; (
), 1.11 kN;
), 2.22 kN; (
)
) 2.67 kN; .......................................................................................... 99
Effect of radial preload (and imposed moment load) on the measured axial
accelerance spectra at location 1. (a) Magnitude spectra, (b) Phase spectra. Key:
Fx 0 ; ( ), 0.89 kN; ( ), 1.33 kN; ( ) 1.78 kN; ( ) 2.22 kN; ( ) 2.67 kN; ( ),
3.11 kN; (
3.12
), 3.55 kN; (
) 4.00 kN; (
) 4.44 kN; ......................................... 103
Effect of radial preload (and imposed moment load) on the measured axial
accelerance magnitude spectra. (a) At location 2, (b) At location 3. Key: Fx 0 ; (
0.89 kN; (
(
3.13
), 1.33 kN; (
), 3.55 kN; (
) 1.78 kN; (
) 4.00 kN; (
) 2.22 kN; (
) 2.67 kN; (
),
), 3.11 kN;
) 4.44 kN; ........................................................ 104
Effect of preloads on the measured natural frequencies (between 500-2500 Hz).
(a) Under axial preload, (b) Under radial preload (while the bearing is also axially
preloaded). Key: Discrete points, measurements at, ( ), r = 1; (
= 3; (
3.14
), r
), r = 4; continuous line, a second order curve fit of measured data.. 108
Comparison of radial accelerance magnitude spectra
Key: (
), r = 2; (
A
), Measured at accelerometer location 1; (
xviii
xx
( )  for Fz 0 =1.56 kN.
), predicted by two
degree-of-freedom model; (
), predicted by five degree-of-freedom model
including participation of the r = 3 mode. (a) Prediction with original
I yy  0.015 kgm 2 , (b) Prediction with modified................................................. 114
4.1
A screenshot from the 3D visualizer of Calyx [4.21] that illustrates a shaft
supported by two single row angular bearings found in a back-to-back
arrangement subjected to a combined load as shown on the . The stress
distribution on the rolling elements and the load lines that pass through the
contact points may be seen. The outer rings of the bearings are not shown for
illustrative purposes. ........................................................................................... 127
4.2
Illustration of the contact of a rolling element of an angular contact ball bearing.
(a) Formation of the contact ellipse under loaded conditions. (b) Parabolic stress
distribution along both dimensions of the contact surface.................................. 129
4.3
Illustration of the parabolic contact pressure distribution on the rolling elements
with Calyx© [4.21]. (a) Several rolling elements shown within the inner and outer
races. (b) A zoomed view to one rolling element. .............................................. 130
4.4
Illustration of contact grid variables (M, N, and s ) with an example case with
N=3 and M=1, resulting in 7 grids in axial and 3 grids in profile direction. Blue
dots represent contact grids, and elliptical area represents the contact zone. (a)
Grid width is sufficiently large. (b) Grid width is too small, contact zone is
truncated. ............................................................................................................. 131
4.5
Several screenshots from the 3D visualizer of RomaxDesigner© [4.22] that
illustrates a shaft supported by a double row angular bearing. ........................... 132
4.6
Locations and magnitudes of elastic deformations (in colors) of rolling elements
and bearing rings for a combined load case. (a-b) Elastic deformation of rolling
elements; (c) deformation of outer rings; and (d) deformation of inner rings. ... 134
xix
4.7
Elastic deformation and maximum shear stresses distribution along the shaftbearing system for a combined load case including the shaft. (a) Total elastic
deformations, and (b) maximum shear stress distribution. ................................. 136
4.8
Rigid body deflections of two bearings in radial and angular directions with rigid
1
and elastic shafts with respect to time assuming a shaft speed of 1000 rpm. (a)  xm
2
2
vs. t; (b)  xm
vs. t; and (c)  1ym   ym
vs. t. Key: (
MPa); (
4.9
), elastic (steel) shaft (E = 205
), rigid shaft (E = 2050 MPa)... ......................................................... 137
Illustration of two approaches that can be practiced in modeling of double
row/duplex angular contact ball bearings: (a) Modeling two single row bearings,
and (b) modeling an integrated double row bearing. Here, the rigid shaft is
subjected to radial ( Fxext ) and axial ( Fzext ) external loads at point O. ................ 139
4.10
The typical load-deflection relationship of a rolling element bearing illustrating
the operating point and bearing stiffness. Multiple points around the operating
point can be analyzed to approximate bearing stiffness by finite difference
approximation. .................................................................................................... 152
4.11
Application of a second order curve fit to the data collected by Method C and
1
1
2
vs.  xm
; (b) Fxm2 vs.  xm
, and
numerical differentiation to approximate kxx. (a) Fxm
(c)
1
Fxm
x
1
vs.  xm
; (d)
x  1xm
Fxm2
x
2
vs.  xm
. Key: (a-b) ( ), actual data; (
),
2
x  xm
second order curve fit. (c-d) ( ), finite difference approximation; (
), numerical
differentiation. ..................................................................................................... 167
4.12
Modeling of experimental preload utilizing system symmetry. (a) Axial preload (
Fz 0 ) applied in experiment; (b) External axial load utilized in the simulations of
xx
section 4.4-4.7; (c) Modeling of experimental preload with Methods C and D
making use of the system symmetry. .................................................................. 171
4.13
Comparison of radial accelerance magnitude spectra
A
xx
( )  for Fz 0 =1.56 kN
with three methods of calculation. Key: (
), Measured from the modal
experiment; (
), predicted by Method C; (
), predicted by Method A; (
)
predicted by Method D.. ..................................................................................... 173
4.14
Experiment with wheel-hub assembly using faulty and healthy bearings.
Locations of the accelerometers, sensors, and impact are illustrated. ................ 177
4.15
Comparison
A
0z
(
5.1
measured
cross-point
accelerance
magnitude
spectra
/ F6 z ( )  for faulty and healthy bearings at various Fz 0 . Key: Fz 0 , faulty
bearing: (
kN; (
of
), 0.45 kN; (
), 0.67 kN; (
), 2.22 kN; healthy bearing: (
), 1.33 kN; (
), 1.78 kN; (
), 0.89 kN; (
), 0.45 kN; (
), 1.33 kN; (
), 0.67 kN; (
), 1.78
), 0.89 kN;
), 2.22 kN.. ..................................................... 178
Proposed methodology for vibration based preload and stiffness estimation. …190
xxi
LIST OF SYMBOLS
Ao
Aij
Unloaded distance between the inner and outer raceway groove curvature
centers
Loaded distance between the inner and outer raceway groove curvature
A ( )
ai
centers of the jth rolling element of the ith row
Accelerance matrix
Location of the inner raceway groove curvature center
ao
C
Cb
c
c1
c2
c3
Location of the outer raceway groove curvature center
System damping matrix
Bearing damping matrix
Damping
A coefficient dependent on the row index (i)
A coefficient dependent on the bearing configuration
A coefficient dependent on the bearing row index (i) and bearing
F0
Fx 0
Fxm
Fym
configuration
Axial distance from the bearing center
Ball diameter
Effective load center (spread) of a double row bearing
Elastic modulus
Axial distance between the geometric center of the bearing and one
bearing row
Mean bearing load vector around the operating point
Radial preload in x-direction
Mean bearing load component in x-direction
Mean bearing load component in y-direction
Fz 0
Fzm
fm
f a (t)
g
G
Axial preload in z-direction
Mean bearing load component in z-direction
Mean bearing load vector
Generalized alternating load vector
Error vector
Center of mass
dZ
D
E
Emod
e
xxii
G1,G2,…,G5
I
j
Kn
K
Elements of the error vector
Moment of inertia
1
Hertzian contact stiffness constant
Kb
[K b ]6 x 6
k
k xx
k yy
System stiffness matrix
Proposed bearing stiffness matrix of dimension 5
Six-dimensional bearing stiffness matrix
Stiffness
Radial stiffness of the rolling element bearing in x-direction
Radial stiffness of the rolling element bearing in y-direction
k zz
k x x
Axial stiffness of the rolling element bearing
Tilting stiffness of the rolling element about x-axis
k y y
Tilting stiffness of the rolling element about y-axis
k pq
An off-diagonal stiffness coefficient of the rolling element bearing
( p , q  x, y , z ,  x ,  y , p  q )
l
M
M
M xm
M ym
m
N
Length
System mass (or inertia) matrix
User defined input that controls the number of grid cells in profile
direction
Mean bearing moment about x-axis
Mean bearing moment about y-axis
n
nd
Qij
Mass
User defined input that controls the number of grid cells in the axial
direction
Load-deflection exponent (n =1.5 for ball type bearings)
Dimension of the analytical model
Total load on the jth rolling element of the ith row
qa(t)
qm
Generalized alternating displacement vector
Mean bearing displacement vector
q0
R
Mean bearing displacement vector around the operating point
Pitch radius
xxiii
Modal index
Radial clearance
Separation tolerance
Time
Eigenvector
State vector
x dimension in Cartesian coordinates (radial dimension)
y dimension in Cartesian coordinates (radial dimension)
z dimension in Cartesian coordinates (axial direction)
Number of rolling elements in one row
Unloaded contact angle
r
rL
Stol
t
v
xa(t)
x
y
z
Z
o
 ij
 xm
 ym
Loaded contact angle of the jth rolling element of the ith row
c
 uc
Elastic displacement of the finite element grids around the contact zones
 ij
Total elastic deformation of the jth rolling element of the ith row
Mean bearing displacement in  x -direction
Mean bearing displacement in  y -direction
Elastic displacement of the finite element grids that are far away from the
contact zones
( )
Radial elastic deformation of the jth rolling element of the ith row
( )
Net radial elastic deformation of the jth rolling element of the ith row
( )
Axial elastic deformation of the jth rolling element of the ith row
i
r j
* i
r j
i
z j
( )
* i
z j
( z 0 )
 xm
 ym
 zm



x
i
Net axial elastic deformation of the jth rolling element of the ith row
Axial displacement preload on the ith row
Mean bearing displacement in x-direction
Mean bearing displacement in y-direction
Mean bearing displacement in z-direction
Very small number defined by the user
An empirically obtained constant
Eigenvalue
Rotational dimension about x-axis (Tilting dimension)
xxiv
y
Rotational dimension about y-axis (Tilting dimension)
z

 ij
Rotational dimension about z-axis (Torsional dimension)
Proportionality constant
Angular position of the jth rolling element of the ith row
ω
ωn
ζ
Frequency
Natural frequency
Damping ratio
Subscript
a
b
BRG1
BRG2
j
m
p, q
r
S
x
y
Alternating
Property of the bearing
Property of bearing 1
Property of bearing 1
Rolling element index ( j  1, 2,  Z )
Mean (or a static) value
Dummy subscripts that may replace x, y, z ,  x or  y .
Modal index
Property of the shaft
Property of x direction
Property of y direction
x
y
Property of z direction
Property of  x direction
Property of  y direction
2
11,12,21,22
Property of 2-dimensional reduced order matrix
Denotes the elements of 2-dimensional reduced order matrix
z
Superscript
D
ext
i
1
Property of the double row bearing
External force
Row index (i  1, 2)
Property of bearing 1
xxv
2
T
.
..
~
Property of bearing 2
Transpose
1st derivative with respect to time
2nd derivative with respect to time
Complex value indicator
Operators



Increment (or disturbance)
Partial derivative
Summation
Absolute value
sgn
Sign operator
Abbreviations
DB
DF
DT
DOF
LTI
Back-to-back
Face-to-face
Tandem
Degree of freedom
Linear time-invariant
xxvi
CHAPTER 1
INTRODUCTION
1.1 Motivation
Double row bearings are often utilized to increase the radial, axial or moment
rigidity of shaft-bearing systems, and they may also offer excellent bi-directional or
combined load accommodating capabilities that cannot be provided by single row
bearings. Consequently, double row bearings are used in automotive, helicopter, and
aircraft applications such as gear boxes, wheel hubs, and helicopter rotors; as well as in
machine tool spindles, industrial pumps, and air compressors. [1.1-1.3]. In certain
problems, a double row bearing may be simply represented by two single row bearings
attached next to each other. However, a double row bearing must be considered as an
integrated unit in many static and dynamic problems. The stiffness matrices of two single
row bearings may not simply be superposed to obtain the stiffness matrix of a double row
bearing.
Prior researchers have not developed stiffness formulations for double row
bearings except Royston and Basdogan [1.4], who studied self-aligning bearings and
neglected all angular displacement and tilting moment effects due to the unique
1
properties of self-aligning bearings. In double row angular contact ball bearings, however,
the moment stiffnesses and the cross-coupling stiffness elements are significant; and the
stiffness coefficients are highly dependent on the configuration of the rolling elements.
Also, the contact angle (  ) of ball-type bearings change under a load, unlike roller-type
bearings in which the contact angle remain relatively constant.
To overcome a void in the literature, this dissertation proposes a new, analytical
bearing stiffness matrix (Kb) formulation for double row angular contact ball bearings
that may be found in face-to-face, back-to-back, or tandem arrangements. The proposed
bearing stiffness matrix is a global representation that combines the kinematic and elastic
properties of the double row bearing by considering each rolling element of both rows.
This dissertation also aims to extensively investigate the effect of bearing preloads on
system modal characteristics, as well as to experimentally validate the new Kb
formulation. Finally, further verification for the proposed Kb using analytical and
computational methods (including finite element based techniques) is needed.
1.2 Literature Review
1.2.1 Bearing stiffness
A wide variety of models of varying complexity have been proposed to characterize
the stiffness properties of rolling element bearings [1.5-1.13]. Most of these models
describe the bearing as translational stiffness elements in radial and axial dimensions
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
2
bearing as a vibration transmitter, as experimental results have shown that the casing
vibrations are typically out-of-plane [1.7-1.9]. Although Jones [1.14] did not define a
bearing stiffness matrix, his multidimensional load-deflection formulations for ball and
roller type bearings formed a basis to define a fully populated bearing stiffness matrix for
single row bearings. In 1989, Lim and Singh [1.15] developed a five dimensional,
symmetric bearing stiffness matrix (which is in fact a six dimensional matrix with last
column and row being all zeros corresponding to the torsional motions) for a single row
spherical contact ball-type and cylindrical contact roller-type bearings. The main
advantage of Lim and Singh’s stiffness model [1.15] over the previous models was the
introduction of flexural and out-of-plane type motions which provided a better
understanding of vibration transmission through rolling element bearings. Lim and Singh
showed the validity of their model in their series of papers [1.15-1.19] through parametric
studies and comparisons with previous analytical and experimental results [1.8-1.9].
Royston and Basdogan [1.4] used Lim and Singh’s model [1.15] to study self-aligning
(spherical) bearings where they proposed a stiffness matrix for such bearings. As the
moment stiffness of self-aligning bearings are negligible, Royston and Basdogan [1.4]
did not consider angular displacement and tilting moment effects which normally brings
the most of the complexity to systems with multiple row bearings; thus, their three
dimensional stiffness matrix was, in fact, a simplified version of Lim and Singh’s five
dimensional model [1.15], where the last two rows and columns of moment stiffness
terms are neglected, but three dimensional translational motions of each rolling element
of both rows are included. Cermelj and Boltezar [1.20] also used Lim and Singh’s
3
stiffness model to investigate the dynamics of a structure containing ball bearings.
Several other authors have also proposed similar Kb formulations to describe the
multidimensional stiffness characteristics of rolling element bearings. For instance, De
Mul et al. [1.21] outlined a general theory for the numerical calculation of the stiffness
matrices of loaded bearings by extending Jones’ [1.14] equations. Hernot et al. [1.22]
calculated the stiffness matrix of angular contact ball bearings using integration methods.
Experimental techniques have also been utilized to measure radial and axial bearing
stiffness coefficients of rolling element bearings [1.7-1.9, 1.23-1.28]. The details of these
studies will be given as we move forward through this dissertation.
1.2.2 Double row bearings
Although single row bearings have been well and extensively studied, only few
publications specific to double row bearings exist. Bercea et al. [1.29] 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 study was limited to a formulation of bearing deflections and
did not include any stiffness formulation or dynamic analysis. Then Nelias and Bercea
[1.30] used their double row tapered rolling bearing model for case studies. Cao and Xiao
[1.31] developed a dynamic model for double row spherical roller bearings based on
energy principles. Later Cao [1.32] improved this model by including the effects of
rotational motions and shaft misalignments. Choi and Yoon [1.33] proposed a method for
4
determining the discrete design variables of an automotive wheel assembly that contained
a double row angular contact ball bearing.
1.2.3 Preload issues
The effects of bearing preloads on the performance of shaft-bearing systems and
their dynamic characteristics have been studied by several authors [1.6, 1.34-1.39].
Akturk et al. [1.40] used a three-degree-of-freedom model to study the effects of bearing
preload and number of rolling elements on vibration characteristics of an angular contact
ball bearing. They observed a reduction in the vibration amplitudes associated with the
ball passage frequency with increasing axial preload. Alfares and Elsharkawy [1.41]
made similar observations using a five-degrees-of-freedom model. They also presented a
reduction in peak-to-peak amplitudes (in time domain) with increasing preload. Bai et al.
[1.42] used the same five-degree-of-freedom model to observe the nonlinear dynamic
characteristics of a rolling element bearing. They showed that unstable periodic solution
of a balanced rotor bearing system can be avoided with a sufficient axial preload.
1.3 Problem Formulation
1.3.1 Bearing deflections, loads and stiffnesses
Figure 1.1(a) illustrates a typical shaft-bearing system where the shaft is
supported by a double row angular contact ball bearing, and is subjected to external
forces and moments. These forces and moments are accommodated by the double row
bearing; if the shaft and the bearing casing are rigid, the double row bearing must support
5
the entire external load. Assuming that the dynamic loads and motions are much smaller
than static forces and displacements; and the shaft-bearing system is allowed to rotate
freely about its rotational (z) axis, the resultant bearing load can be represented in terms
of a 5x1 mean bearing load vector fm  {Fxm , Fym , Fzm , M xm , M ym }T , where the forces and
moments are defined about the geometrical center of the bearings. These forces and
moments cause a relative displacement between the inner and outer rings of the bearing
(defined as bearing deflection, bearing displacement or ring approach), which can be
similarly represented in terms of a 5 dimensional mean bearing displacement vector
qm  { xm ,  ym ,  zm ,  xm ,  ym }T . These forces and displacements acting on the double row
bearing are illustrated in Figure 1.1(b).
The relationship between the bearing load and deflection elements is nonlinear as
shown in a typical load-deflection curve in Figure 1.2. The operating point of the bearing
is dictated by its loading state, and F0 and q 0 respectively define the bearing load and
displacement vectors at the operating point. Here, the slope of the tangent line to the
load-deflection curve at the operating point defines the associated bearing stiffness
element; which is only valid around the operating point. As pointed out by Lim and Singh
[1.15]
and
several
other
investigators
[1.21-1.22],
the
bearing
stiffness
a
multidimensional characteristic which must be represented as a stiffness matrix (Kb). The
elements of the fully populated (5x5) bearing stiffness matrix is mathematically described
as:
6
x
(a)
External forces and moments
Double row
angular contact
ball bearing
Fyext
M yext
ext
z
F
z
M xext
Fxext
Shaft
y
Bearing casing
(b)
y
 ym
Fym
x
r
 ym
 xm
Fxm
M ym
xm
M xm
 ij
Fxm
 xm

M xm
x
Fzm  zm
z
 xm
Figure 1.1 Illustration of bearing loads and displacements. (a) A typical shaft-bearing
system where the shaft is supported by a double row angular contact ball bearing, and is
subjected to external forces and moments. (b) Illustration of mean bearing load and
moment components resulting in translational and rotational displacements.
7
Bearing
Load
Operating
point
Bearing
stiffness
F0
Bearing Deflection
q0
Figure 1.2 A typical nonlinear load-deflection relationship of a rolling element bearing.
The slope of the tangent of the load-deflection curve around the operating point defines
the bearing stiffness.
8
 Fxm
 
 xm
 Fym

  xm
 F
zm
Kb  
  xm

 M xm
  xm

 M ym
 
 xm
Fxm
 ym
Fxm
 zm
Fxm
 xm
Fym
Fym
Fym
 ym
 zm
 xm
Fzm
 ym
Fzm
 zm
Fzm
 xm
M xm
 ym
M xm
 zm
M xm
 xm
M ym
M ym
M ym
 ym
 zm
 xm
Fxm 
 ym 

Fym 
 k xx


 ym 
 k yx
Fzm 
 =  k zx

 ym 
k

 x x
M xm 
k
 ym 
 yx

M ym 
 ym 
k xy
k xz
k x x
k yy
k yz
k y x
k zy
k zz
k z x
k x y
k x z
k x x
k y y
k y z
k y x
k x y 

k y y 

k z y  (1.1)
k x y 

k y y 
q0
q0
Although the physical meaning of the symmetric elements of Kb (i.e. k pq and k qp
where p, q  x, y, z , x , x and p  q ) are different, they have the same units and their
values are identical for conservative systems, thus, Kb is always symmetric. The elements
of Kb are functions of bearing loads (and deflections), as well as certain geometric and
kinematic parameters of the bearing such as the number of rolling elements, unloaded
contact angle, radial clearance, axial distance between bearing rows, unloaded distance
between inner and outer ring curvature centers and Hertzian stiffness constant (which is a
function of geometry and material). Also the elements of Kb are highly dependent on the
configuration of the double row angular contact ball bearing. The effects of bearing
configuration on stiffness elements and consequently on the modal characteristics of a
shaft-bearing assembly are analyzed in detail in this dissertation. Note that some of the
bearing literature uses a 6-dimensional stiffness matrix definition with last row and
column being all zeros corresponding to the torsional motions; 6-dimensional Kb can be
simply obtained by augmenting the 5-dimensional Kb with a row and column of zeros:
9
K b 6 x 6



K b 5 x 5





0 0 0 0 0
0
0 
0

0
0

0q
(1.2)
0
1.3.2 Scope, assumptions and objectives
A double row angular contact ball bearing can be defined as two angular contact
ball bearings (which are typically positioned next to each other) that share common inner
and outer rings; though there are exceptions to this generalization. For example, many
double row designs have a solid outer ring but a split inner ring (mainly for preloading),
such as most of the wheel bearing units. Duplex (or paired) bearings are also obtained by
assembling two single row bearings together; however, their functionality is identical to
double row bearings. Double row angular contact ball bearings typically occupy less
axial space than a duplex bearing of the same size due to the shared bearing rings. Double
row angular contact ball bearings may also offer some economic benefits as well as
handling and mounting benefits versus the duplex bearings; however they may offer less
design flexibility in some cases. The Kb formulation in this dissertation is developed for
double row angular contact ball bearings assuming two bearing rows share common
rings. However, as long as the shaft is sufficiently rigid, and the inner rings of the
bearings are rigidly connected to the shaft (i.e. rings of the two rows do not move
independently except for initial preloading), the formulation is valid for duplex (or
10
paired) angular contact ball bearings that show identical behavior to double row angular
contact ball bearings.
Double row and duplex (paired) angular contact ball bearings can be found in
three arrangements according to the configuration of their rolling elements as shown in
Figure 1.3: (i) Back-to-back (DB) or ‘O’ arrangement, 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, where the load lines converge toward the bore
of the bearing; and (iii) tandem (DT) or series arrangement, 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 larger, thus
it has higher moment stiffness terms and a higher moment load carrying capacity.
However, they are sensitive to misalignments; thus, they must be utilized in applications
where the unavoidable misalignments are minimum. The face-to-face arrangement has a
smaller load center; however, it has a larger misalignment angle and allows larger
misalignments; however their moment stiffness is considerably lower compared to backto-back arrangement. Face-to-face and back-to-back arrangements have the same axial
and radial stiffness characteristics under same axial or radial preloads. The tandem
arrangement, on the other hand, can carry heavier axial loads but only in one direction
due to the fact that all rolling elements are organized in the same direction. Tandem
arrangements are typically mounted with an opposing bearing set to achieve axial and
moment stiffness in the opposite direction as well. Note that the vast majority of the
11
Back-to-Back (DB) ‘O’
Arrangement
Face-to-Face (DF) ‘X’
Arrangement
Tandem (DT)
Arrangement
Duplex (paired)
angular contact
bearings
12
Double row
angular contact
ball bearings
Figure 1.3 Illustration of the three configurations of the duplex (paired) angular contact ball bearings and double row angular
contact ball bearings.
12
commercial double row angular contact ball bearings are found in back-to-back
arrangement, however other arrangements are also utilized [1.1].
Main assumptions utilized for the derivation of the new analytical model are as
follows:
1. The vibratory motions are much smaller than the mean bearing deflections (i.e. |qa(t)|
<< qm ) due to high bearing preloads and static loads. Thus, the time varying bearing
stiffness coefficients are neglected, and linear, time-invariant Kb is defined about the
operating point.
2. The outer ring is fixed in the space and the inner ring is displaced under loads as it is
sufficient to consider only the relative displacement between the bearing rings.
3. The structural deformation of the bearing rings is negligible and only the elastic
deformation at the contact points is considered. Load-deflection relation of each rolling
element is defined by the Hertzian contact stress theory.
4. Both rows of the double row bearing are identical in terms of the structural and
kinematic parameters. Also, angular position of each rolling element relative to one
another remains the same due to rigid cages and retainers.
5. The internal friction is negligible compared to the normal loads on the rolling elements,
thus, the friction does not affect the stiffness coefficients.
6. The bearings are not operating at the overcritical speeds, thus, centrifugal and
gyroscopic moments are neglected. The tribological issues are also disregarded since the
lubrication has no significant effect on bearing stiffness elements unless the bearing
operates at overcritical speeds.
13
The specific objectives of this dissertation are listed below along with subobjectives and the organization in terms of Chapters. Refer to Figure 1.4 for a generic
shaft-bearing system and a simple representative vibration model.
Objective 1: Develop a new, comprehensive five-dimensional bearing stiffness matrix
(Kb) for double row angular contact ball bearings in face-to-face, back-to-back, or
tandem arrangement as shown in Figure 1.3 (Chapter 2).
(1a) Derive explicit expressions for diagonal and off-diagonal elements of Kb
from basic principles (Figures 1.1 and 1.2), that can be computed via direct
substitution for a given mean bearing displacement vector (qm).
(1b) Develop a numerical scheme to compute qm and Kb for a given mean bearing
load vector (fm).
(1c) Verify the diagonal elements of the proposed analytical model with a
commercial code for three arrangements.
(1d) Examine the changes of diagonal and off-diagonal stiffness coefficients of
Kb by varying bearing loads, unloaded contact angle, and angular position of the
bearing.
Objective 2: Design and instrument a new modal experiment to investigate the effects of
axial and radial bearing preloads on the modal characteristics of a shaft-bearing system;
as well as to experimentally validate the proposed stiffness model (Chapter 3).
14
(2a) Develop a simple five-degree-of-freedom vibration model that utilizes Kb
(Figure 1.4(b)), to represent the generic vibration problem of a rigid shaft
supported by a double row angular contact ball bearing (Figure 1.4(a)).
(2b) Investigate the effect of system damping on the modal characteristics of a
shaft bearing assembly by utilizing preload-dependent and preload-independent
viscous damping models. Analyze the effects of axial preloads by employing
alternate viscous damping models.
(2c) Design and instrument a new laboratory experiment consisting of an
automotive wheel-hub assembly (with a back-to-back double row angular contact
ball bearing) to investigate the effects of axial and radial bearing preloads on the
system modes, natural frequencies and resonant amplitudes.
(2d) Validate the new stiffness model based on modal measurements by
comparing the predicted natural frequencies and accelerance spectra with
measurements.
Objective 3: Provide further verification to the proposed analytical model using analytical
and computational methods including finite element based techniques (Chapter 4).
(1a) Justify some critical assumptions of the proposed Kb using a finite element
based contact mechanics code.
(1b) Demonstrate the need for the proposed analytical model by comparing the
modeling approaches between a double row bearing vs. two single row bearings
from statics and stiffness perspective.
15
(1c) Comparatively evaluate four calculations methods (two analytical and two
computational methods) by critically examining calculated bearing loads,
deflections and stiffness elements for simple and combined loading cases.
(1d) Compare the predicted natural frequencies obtained by four methods with
modal measurements.
(1e) Conduct preliminary modal experiments with a damaged bearing and
investigate the effects of bearing preloads on the modal characteristics of a faulty
shaft-bearing system to promote future research.
16
x
 xsa
 xsa
Bearing Casing
Double Row Bearing
F xsa
M xsa
Shaft
M
F zsa
 zsa
z
z
ysa
F ysa
 ysa
y
 ysa
x
x
Ms
RigidShaft
shaft
Rigid
z

Double
rowBearing
bearing
Double
Row
Kb
Cb
y
y
Rigid Casing
Figure 1.4 Generic vibration problem of a shaft-bearing system and a simple
representative vibration model. (a) Schematic of the vibration problem of a shaft
supported by a double row angular contact ball bearing. (b) Analytical vibration model
(of dimension 5) of a rigid shaft supported by double row bearing.
17
References for chapter 1
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1.2 J. Brändlein, P. Eschmann, L. Hasbargen, K. Weigand, Ball and Roller Bearings:
Theory, Design and Application, J. Wiley, Chichester, 1999.
1.3 A. Palmgren, Ball and Roller Bearing Engineering, Philadelphia, 1959.
1.4 T. J. Royston, I. Basdogan, Vibration transmission through self-aligning (spherical)
rolling element bearings: theory and experiment, Journal of Sound and Vibration 215
(5) (1998) 997-1014.
1.5 H.Q. Li, Y.C. Shin, Analysis of bearing configuration effects on high speed spindles
using an integrated thermomechanical spindle model, International Journal of
Machine Tools and Manufacture, 44 (2004) 347-364
1.6 F. P. Wardle, S.J. Lacey, S.Y. Poon, Dynamic and static characteristics of a wide
speed range machine tool spindle, Precision Engineering 5 (4) (1983) 175-183.
1.7 E. P. Gargiulo, A simple way to estimate bearing stiffness, Machine Design, 52 (1980)
107-110.
1.8 M. F. White, Rolling element bearing vibration characteristics: effect of stiffness,
Journal of Applied Mechanics 46 (1979) 677-684.
1.9 J. Kraus, J.J. Blech, S.G. Braun, In situ determination of rolling bearing stiffness and
damping by modal analysis, Journal of Vibration, Acoustics, Stress, and Reliability in
18
Design, Transactions of the American Society of Mechanical Engineers 109 (1987)
235-240.
1.10 M. D. Rajab, Modeling of the transmissibility through rolling element bearing under
radial and moment loads, Ph.D. Dissertation, The Ohio State University, 1982.
1.11 K. Ishida, T. Matsuda, M. Fukui, Effect of gearbox on noise reduction of geared
device, Proceedings of the International Symposium on Gearing and Power
Transmissions, Tokyo, (1981) 13-18.
1.12 J.S. Lin, Experimental analysis of dynamic force transmissibility through bearings,
M.S. Thesis, The Ohio State University, (1989).
1.13 Y. Cao, Y. Altintas, A general method for the modeling of spindle-bearing systems,
Transactions of the American Society of Mechanical Engineers, Journal of
Mechanical Design, 126 (6) (2004) 1089-1104
1.14 A. B. Jones, A general theory for elastically constrained ball and radial roller bearings
under arbitrary load and speed conditions, Transactions of the ASME, Journal of
Basic Engineering 82 (1960) 309-320.
1.15 T. C. Lim, R. Singh, Vibration transmission through rolling element bearings, part I:
bearing stiffness formulation, Journal of Sound and Vibration 139 (2) (1990) 179–199.
1.16 T. C. Lim, R. Singh, Vibration transmission through rolling element bearings, part II:
system studies, Journal of Sound and Vibration 139 (2) (1990) 201–225.
1.17 T. C. Lim, R. Singh, Vibration transmission through rolling element bearings, part III:
geared rotor system studies, Journal of Sound and Vibration 151 (1) (1991) 31–54.
19
1.18 T. C. Lim, R. Singh, Vibration transmission through rolling element bearings, part IV:
statistical energy analysis, Journal of Sound and Vibration 153 (1) (1992) 37-50.
1.19 T. C. Lim, R. Singh, Vibration transmission through rolling element bearings, part V:
effect of distributed contact load on roller bearing stiffness matrix, Journal of Sound
and Vibration 169 (4) (1994) 547–553.
1.20 P. Cermelj, M. Boltezar, An indirect approach investigating the dynamics of a
structure containing ball bearings, Journal of Sound and Vibration, 276 (1-2) (2004)
401–417.
1.21 J. M. De Mul, J. M. Vree, D. A. Vaas, Equilibrium and associated load distribution
in ball and roller bearings loaded in five degrees of freedom while neglecting frictions
I: General theory and application to ball bearings, Transactions of the ASME, Journal
of Tribology, 111 (1989) 142-148.
1.22 X. Hernot, M. Sartor, J. Guillot, Calculation of the stiffness matrix of angular contact
ball bearings by using the analytical approach, Transactions of the ASME, Journal of
Mechanical Design, 122 (2000), 83-90.
1.23 T. L. H. Walford, B.J. Stone, The measurement of the radial stiffness of rolling
element bearings under oscillation conditions, Proceedings of the Institution of
Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 22 (4)
(1980) 175-181.
1.24 T. L. H. Walford, B.J. Stone, Some stiffness and damping characteristics of angular
contact bearings under oscillating conditions, Proceedings of the 2nd International
Conference on Vibrations in Rotating Machinery, (1980).
20
1.25 B. J. Stone, The state of art in the measurement of the stiffness and damping of
rolling element bearings, Annals of the CIRP, 31 (2) (1982) 529-538.
1.26 E. R. Marsh, D. S. Yantek, Experimental measurement of precision bearing dynamic
stiffness, Journal of Sound and Vibration, 202(1) (1997) 55-66.
1.27 R. Tiwari, N. S. Vyas, Estimation of non-linear stiffness parameters of rolling
element bearings from random response of rotor-bearing systems, Journal of Sound
and Vibration, 187 (1995) 229-239.
1.28 G. Pinte, S. Devos, B. Stallaert, W. Symens, J. Swevers, P. Sas, A piezo-based
bearing for the active structural acoustic control of rotating machinery, Journal of
Sound and Vibration, 329 (9) (2010) 1235-1253.
1.29 I. Bercea, D. Nelias, G. Cavallaro, A unified and simplified treatment of the nonlinear equilibrium problem of double-row bearings. Part I: rolling bearing model,
Proceedings of the Institution of Mechanical Engineers, Part J: Journal of
Engineering Tribology 217 (3) (2003) 205-212.
1.30 D. Nelias, I. Bercea, A unified and simplified treatment of the non-linear equilibrium
problem of double-row bearings. Part II: application to taper rolling bearings
supporting a flexible shaft, Proceedings of the Institution of Mechanical Engineers,
Part J: Journal of Engineering Tribology 217 (3) (2003) 213-221.
1.31 M. Cao, J. Xiao, A comprehensive dynamic model of double-row spherical roller
bearing – model development and case studies on surface defects, preloads, and radial
clearance, Mechanical Systems and Signal Processing, 22 (2008) 467-489.
21
1.32 M. Cao, A refined double-row spherical roller bearing model and its application in
performance assessment of moving race shaft misalignments, Journal of Vibration
and Control, 13 (2007) 1145-1168.
1.33 D. H. Choi, K. C. Yoon, A design method of an automotive wheel-bearing unit with
discrete design variables using genetic algorithms, Journal of Tribology, 123 (2001)
181-187.
1.34 K. Kim, S.S. Kim, Effect of preload on running accuracy of spindle, International
Journal of Machine Tools and Manufacture, 29 (1) (1989) 99-105.
1.35 S.Y. Lin, C.T. Chung, R.W. Chang, C.K. Chang, Effect of the bearing preload on the
characteristics of the spindle stiffness, Key Engineering Materials, 419 (2010) 9-12.
1.36 B.R. Jorgensen, Y.C. Shin, Dynamics of spindle-bearing systems at high speeds
including cutting load effects, Journal of Manufacturing Science and Engineering,
120 (1998) 387-394.
1.37 M. Tiwari, K. Gupta, O. Prakash, Dynamic response of an unbalanced rotor supported
on ball bearings, Journal of Sound and Vibration, 238 (5) (2000) 757-779
1.38 C.W. Lin, J.F. Tu, J. Kamman, An integrated thermo-mechanical-dynamic model to
characterize motorized machine tool spindles during very high speed rotation,
International Journal of Machine Tools and Manufacture, 43 (10) (2003) 1035-1050
1.39 I.H. Filiz, G. Gorur, Analysis of preloaded bearings under combined axial and radial
loading, International Journal of Machine Tools and Manufacture, 34 (1) (1994) 1-11.
22
1.40 N. Akturk, M. Uneeb, R. Gohar, The effects of number of balls and preload on
vibrations associated with ball bearings, Transactions of the ASME, Journal of
Tribology 119 (1997) 747-752.
1.41 M. Alfares, A.A. Elsharkawy, Effects of axial preloading of angular contact ball
bearings on the dynamics of a grinding machine spindle system, Journal of Materials
Processing Technology 136 (2003) 48-59
1.42 C. Bai, H. Zhang, Q. Xu, Effects of axial preload of ball bearing on the nonlinear
dynamic characteristics of a rotor-bearing system, Nonlinear Dynamics 53 (2008)
173-190.
23
CHAPTER 2
ANALYTICAL DEVELOPMENT OF A NEW STIFFNESS MATRIX FOR
DOUBLE ROW ANGULAR CONTACT BALL BEARINGS
2.1 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 [2.1-2.3]. 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 simply be 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 Figure 2.1: (i) back-to-back (DB) or ‘O’ arrangement (Figure
24
2.1(a)), 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 (Figure
2.1(b) ), where the load lines converge toward the bore of the bearing; and (iii) tandem
(DT) or series arrangement (Figure 2.1(c) ), 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 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 backto-back arrangement, however other arrangements are also utilized [2.1].
A new, theoretical bearing stiffness matrix (Kb) formulation is proposed in this
chapter for double row angular contact ball bearings by extending Lim and Singh’s [2.4]
formulation for single row bearings as well as Royston and Basdogan’s [2.5] study for
self-aligning (spherical) bearings. Unlike self-aligning bearings [2.5], the moment
stiffness terms of double row angular contact ball bearings are significant and highly
dependent on the configuration of the rolling elements [2.6-2.7]. Thus, tilting moments
and angular motions bring a major complexity to the proposed formulation.
25
(a)
(b)
(c)
Figure 2.1 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.
26
2.2 Literature Review
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 [2.1, 2.8-2.16]. 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 [2.13-2.15]. Although Jones [2.17] 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 [2.4] 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 a single row ball-type and roller-type bearings.
The main advantage of Lim and Singh’s [2.4] 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 [2.4,
2.18-2.21] through parametric studies and comparisons with previous analytical and
experimental results [2.12-2.13]. The importance of moment stiffnesses and crosscoupling stiffness terms are clearly pointed out in these studies. De Mul et al. [2.22] also
outlined a general theory for the numerical calculation of the stiffness matrices of loaded
27
bearings by extending Jones’ [2.17] equations. Hernot et al. [2.23] calculated the stiffness
matrix of angular contact ball bearings by integration techniques.
Cermelj and Boltezar [2.24] used Lim and Singh’s [2.4] stiffness model to further
investigate the dynamics of a structure containing ball bearings. Further, Royston and
Basdogan [2.5] studied double row self-aligning (spherical) bearings where they
proposed a stiffness matrix. As the moment stiffness of self-aligning bearings is
negligible, Royston and Basdogan 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 [2.4], 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.
Experimental techniques have also been utilized to measure radial and axial
bearing stiffness coefficients. For example, Walford and Stone [2.25-2.26] used a twodegree-of-freedom model to extract representative stiffness values from measurements.
Stone [2.27] reviewed various efforts to measure the stiffness and damping coefficients
of rolling element bearings with changes in preload, speed, or lubrication. Kraus et al.
[2.13] 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 [2.28] extracted translational bearing stiffness coefficients by measuring the
resulting responses under known excitation forces. Tiwari and Vyas [2.29] suggested a
method for estimating non-linear bearing parameters without making explicit force
measurements. Finally, Pinte et al. [2.30] experimentally minimized coupling between
28
horizontal and vertical directions by using hinges and leaf springs in order to block outof-plane forces when they developed a piezo-based bearing for active noise control.
Although single row bearings have been extensively studied, publications specific
to double row bearings are sparse. For instance, Bercea et al. [2.6] 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-toback arrangement and did not include any stiffness formulation. Then Nelias and Bercea
[2.7] used their double row tapered rolling bearing model for case studies. Cao and Xiao
[2.31] developed a dynamic model for double row spherical roller bearings based on
energy principles. Later Cao [2.32] improved this model by including the effects of
rotational motions and shaft misalignments. Choi and Yoon [2.33] 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.
2.3 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 condition 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
29
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 [2.1]. The second primary difference is that the contact angle (  ) of ball type
bearings changes under a load, whereas it remains relatively constant for roller type
bearings. Thus, a change in  under loaded conditions must be taken into account.
Besides, the moment 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 fm  {Fxm , Fym , Fzm , M xm , M ym}T and the resulting
mean displacement vector qm  { xm ,  ym ,  zm ,  xm ,  ym }T , as illustrated in Figure 2.2. Here
 xm ,  ym ,  zm and Fxm , Fym , Fzm are the mean displacements and loads in x, y, and z
directions, and  xm ,  ym 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
30
also pointed out by previous investigators [2.4, 2.22]. 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) = qm + 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 chapter, 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 [2.1], 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. The tribological issues will be disregarded since the
lubrication have no significant effect on bearing stiffness elements except at high speeds
31
[2.8]. 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 backto-back, face-to-face, and tandem arrangements (which may be extended for other type of
double row bearings in latter studies) to provide a useful tool for static and dynamic
analyses of these bearings; 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 with three
loading 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.
32
y
 ym
Fym
x
r
 ym
 xm
Fxm
M ym
xm
M xm
 ij
Fxm
 xm
x

M xm
Fzm  zm
z
 xm
Figure 2.2 Mean loads and moments on the bearing and resulting translational and
rotational displacements. The coordinate system is also shown.
33
2.4 New Analytical Formulation: Load-Deflection Relations of a Rolling Element
To define the bearing stiffness, the relationship between fm and qm will be first be
derived. Consider the jth rolling element of the ith row of a double row angular contact
ball bearing shown in Figure 2.3. Assuming the outer ring is fixed, the total elastic
deformation  ( ij ) of this rolling element can be calculated by:
 Aij  Ao
0
 ( ij )  
for
 ( ij )  0
for
 ( ij )  0
(2.1)
i
Here  j is the angular distance of the rolling element from the x-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
(( r )ij and ( z )ij ) of the rolling element are first expressed in terms of the elements of
bearing displacement vector q m :
( r )ij   xm  c1 ym e  cos( ij )   ym  c1 xm e  sin( ij )  rL
(2.2a)
( z )ij   zm  R   xm sin( ij )   ym cos( ij ) 
(2.2b)
Here R is the radius of the inner raceway groove curvature center (pitch radius), o is the
unloaded contact angle, rL is the radial clearance, and e is the axial distance between the
geometric center of the bearing and the bearing rows. The effective load center (E),
34
αj
αo
Outer ring
jth rolling element
of the ith row
ai
ao
Inner ring
Aij
Ao
Figure 2.3 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.
35
which controls the tilting stiffness terms of a double row arrangement, can be defined as
E  2 e  c2 R tan(o ) , and coefficients c1 and c2 are given as follows:
1
for i =1 (Left row)
 1
for i =2 (Right row)
c1  
 1

c2  1
 0

(2.3)
back-to-back (DB) arrangement
face-to-face (DF) arrangement
(2.4)
tandem (DT) arrangement
The net (effective) radial ( r* )ij and axial ( z* )ij displacements of the rolling element are
then expressed including the effects of initial displacement due to the unloaded contact
angle o :
( r* )ij  ( r )ij  Ao cos(o )
(2.5a)
( z* )ij  ( z )ij  c3  Ao sin(o )  ( z 0 )i 
(2.5b)
Here ( z 0 )i defines an axial displacement preload on the ith row obtained by bringing the
inner and outer raceways closer together by a distance ( z 0 )i (for instance in the case of
split inner rings [33]). ( z 0 )i can have a positive value only if the radial clearance is
eliminated ( rL  0 ). Here, c3 is a constant dependent on the type of the rolling element
bearing:
 1
for i = 1 (Left row)
 1
for i = 2 (Right row)
1
for i = 1 (Left row)
 1
for i = 2 (Right row)
Back-to-back (DB) arrangement: c3  
Face-to-face (DF) arrangement: c3  
36
(2.6a)
(2.6b)
Tandem (DT) arrangement: c3  1(*)
(Both rows)
(2.6c)
(*): 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 A( ij ) is finally obtained by vectoral addition of the net displacements:
A( ij ) 
 ( )    ( ) 
* i 2
r j
* i 2
z j
(2.7)
i
Then, the elastic deformation  ( j ) of the rolling element can be determined by Eq.
(2.1) to be used in conjunction with Hertzian contact stress theory in order to obtain the
resultant normal load ( Qij ) on the element:
Qij  K n  ( ij ) 
n
(2.8)
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  ij for the same element is
determined by trigonometry
tan( ) 
i
j
( z* )ij
( r* )ij

( z )ij  c3  Ao sin( o )  ( z 0 )i 
( r )ij  Ao cos( o )
where ( z )ij and ( r )ij are given by Eqs. (2a-b).
37
(2.9)
2.5 Stiffness Matrix
2.5.1 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 [2.4]. To
apply the mathematical definition of the stiffness matrix, the mean load vector fm is first
represented through vectoral sums of Qij
 (
i
j
(i  1, 2;
j  1,  , Z ) for each loaded
)  0  rolling element as follows:


cos( ij ) cos( ij )
 Fxm 


i
i
F 
cos( j ) sin( j )


ym

 2 Z
i

i 
sin(

)
j
fm   Fzm    Q j 

 M  i 1 j 1   R sin( ij )  c1e cos( ij )  sin( ij ) 

 xm 
 

 M ym 
   R sin( ij )  c1e cos( ij )  cos( ij ) 



(2.10)
Expressing Qij and  ij in Eq. (2.10) in terms of mean deflections gives the
explicit expressions between fm and qm:
38
 Fxm 
n
2
2


F 
i
i
i
( r ) j  Ao cos( o )   ( z ) j  c3  Ao sin( o )  ( z 0 )   Ao 
 ym 
2 Z  


fm   Fzm   K n 
2
2
i 1 j 1
M 
( r )ij  Ao cos( o )   ( z )ij  c3  Ao sin( o )  ( z 0 )i 

xm


 M ym 


( r )ij  Ao cos( o )  cos( ij )





( r )ij  Ao cos( o )  sin( ij )



i
i


( z ) j  c3  Ao sin( o )  ( z 0 ) 
x



i
i
i
i
 R ( z ) j  c3  Ao sin( o )  ( z 0 )   c1e  ( r ) j  Ao cos( o )  sin( j ) 


   R ( z )ij  c3  Ao sin( o )  ( z 0 )i   c1e  ( r )ij  Ao cos( o )   cos( ij ) 






 





(2.11)
Substituting ( z )ij and ( r )ij terms from Eq. (2.2a-b) into Eq. (2.11), and assuming
qa (t ) << q m , the five dimensional stiffness matrix Kb around the operating point can be
defined:
 Fpm

 qm
Kb  
 M rm

  qm
Fpm 

 sm 
M rm 

 sm 
qm
 k xx k xy k xz

k yy k yz


k zz
=

 symmetric


k x x
k y x
k z x
k x x
k x y 

k y y 

k z y 
k x y 

k y y 
 qm
(2.12)
Here, the subscripts of the partial derivatives are defined as p, q  x, y , z and r , s  x, y .
The explicit expressions of each stiffness term are symbolically calculated, and given in
their simplest form as follows:
39
 nAi  ( * )i 2

j
r j
* i 2

( ) cos ( ) 
(

)
 z j 
 Aij  Ao
2 Z


k xx  K n 
i 3
i 1 j 1
 Aj 
2
i n
j
i
j
(2.13a)
 nAi  ( * )i 2

j
r j
* i 2

( ) sin( ) cos( ) 
(

)
 z j 
 Aij  Ao
Z
2


k xy  K n 
i 3
i 1 j 1
 Aj 
i n
j
i
j
i
j
(2.13b)
 nAi

( ij ) n ( r* )ij ( z* )ij cos( ij )  i j  1
 Aj  Ao 
2 Z


k xz  K n 
i 3
i 1 j 1
 Aj 
(2.13c)


k x x
 nAij
2
R( z* )ij ( r* )ij  c1e  ( r* )ij 
 i
( ij ) n sin( ij ) cos( ij )  Aj  Ao

2
 R( z* )ij ( r* )ij  c1e  ( z* )ij 

2 Z

 K n 
3
i 1 j 1
 Aij 








(2.13d)


k x y

nAij
2
R( z* )ij ( r* )ij  c1e  ( r* )ij 
 i
( ij ) n cos 2 ( ij )  Aj  Ao

2
 R( z* )ij ( r* )ij  c1e  ( z* )ij 

2 Z

 K n 
3
i 1 j 1
 Aij 

 nAi  ( * )i 2

j
r j
* i 2

( ) sin ( ) 
(

)
 z j 
 Aij  Ao
2 Z


k yy  K n 
i 3
i 1 j 1
 Aj 
i n
j
2







(2.13e)
i
j
40
(2.13f)

nAi 
( ij ) n ( r* )ij ( z* )ij sin( ij ) 1  i j 
 Aj  Ao 
2 Z


k yz  K n 
i 3
i 1 j 1
 Aj 


k y x
 nAij
2
R ( z* )ij ( r* )ij  c1e  ( r* )ij 
 i
( ij ) n sin 2 ( ij )  Aj  Ao

2
 R( z* )ij ( r* )ij  c1e  ( z* )ij 

2 Z

 K n 
3
i 1 j 1
 Aij 








(2.13h)


k y y
(2.13g)


nAij
2

R( z* )ij ( r* )ij  c1e  ( r* )ij 
 i


( ij ) n sin( ij ) cos( ij )  A j  Ao


2
 R( z* )ij ( r* )ij  c1e  ( z* )ij  

2 Z


 K n 
i 3
i 1 j 1
 Aj 
(2.13i )


 nAi  ( * )i 2

j
z j
* i 2

( ) 
(

)
 r j 
 Aij  Ao
Z
2


k zz  K n 
i 3
i 1 j 1
 Aj 
i n
j
k z x
(2.13j)
3


* i

( * )i 
 R ( Ai ) 2   R  ( * )i 2  c e  z j    c e ( z ) j ( Ai ) 2

j
z j
j
1
1
* i
* i




( r ) j
( r ) j


( ij ) n sin( ij ) 

i

nA 
2

 R  ( z* )ij   c1e( z* )ij ( r* )ij i j 
2 Z
A j  Ao 

 K n 
3
i 1 j 1
 Aij 
(2.13k )

41

3



( z* )ij 
2
( z* )ij i 2


i 2
* i

  R( Aj )   R ( z ) j  c1e ( * )i   c1e ( * )i ( Aj )

r j 
r j


( ij )n cos( ij ) 




i
2


* i
* i
* i  nA j

  R ( z ) j  c1e( z ) j ( r ) j  i


 Aj  Ao 


2
Z
k z y  K n 

A 
i
j
i 1 j 1
k x x
k x y




3
(2.13l )
* i 3


(

)j 
( z* )ij  

z
i 2
 R 2 ( Ai ) 2  c eR  ( * )i ( * )i 

 ( Aj )
j
r j
z j
1
* i
* i


 
(

)
(

)
r j
r j

 
( ij ) n sin 2 ( ij ) 
i


nA
2
2

  e 2  R 2  ( z* )ij   i j  R ( z* )ij  c1e( r* )ij  
2 Z
A j  Ao


 K n 
3
i 1 j 1
 Aij 
(2.13m)
3

* i 

( * )i
  R 2 ( Ai ) 2  c eR  ( * )i ( * )i   z j   ( Ai ) 2 ( z ) j  
j
j
1

 r j z j
( r* )ij
( r* )ij  
i n
i
i


( j ) sin( j ) cos( j ) 
i


nA
2
2
   e2  R 2  ( z* )ij   i j  R( z* )ij  c1e( r* )ij  
2 Z
Aj  Ao


 K n 
3
i 1 j 1
 Aij 
(2.13n)
k y y
* i 3


(

)j 
( z* )ij  

z
i 2
 R 2 ( Ai ) 2  c eR  ( * )i ( * )i 

 ( Aj )
j
r j
z j
1
* i
* i


 
(

)
(

)
r j
r j

 
( ij ) n cos 2 ( ij ) 
i


nA
2
2

  e 2  R 2  ( z* )ij   i j  R ( z* )ij  c1e( r* )ij  
Z
2
A j  Ao


 K n 
3
i 1 j 1
 Aij 
(2.13o)
42
k yx  k xy , kzx  kxz , k zy  k yz
(2.13 p-r)
k x x  k x x , k x y  k y x , k x z  k z x
(2.13 s-u)
k y x  k x y , k y y  k y y , k y z  k z y , k y x  k x y
(2.13 v-y)
2.5.2 Numerical estimation of Kb
If the mean displacement vector qm is known the stiffness coefficients k p q
( p , q  x , y , z ,  x ,  y ) can be calculated by direct substitution to Eq. 2.13(a-y). However,
in general fm is known, and the resulting displacement vector qm is unknown. In this case,
the coupled non-linear equations as described by Eqs. (2.10) and (2.11) are numerically
solved to determine qm for a given fm. To implement this method, Eq. (2.10) is
rearranged as follows as:


cos( ij ) cos( ij )
G1   Fxm 


i
i
G   F 
cos( j ) sin( j )


ym
2
  
i
 2 Z i 

sin(

)
j
g  G3    Fzm     Q j 
0
i
i
i
1
i
j


1
G   M 
  R sin( j )  c1e cos( j )  sin( j ) 

 4   xm 
 

G5   M ym 
   R sin( ij )  c1e cos( ij )  cos( ij ) 



(2.14)
Here g  G1 G2 G3 G4 G5  are the functions to be minimized. In this study, a NelderT
Mead simplex algorithm as described in Lagarias et al. [2.34] is used to solve these
equations due to its ease in implementation. It minimizes the sum of the squares of each
term of g to converge to the solution (i.e. iterations continue until G12  G 2 2  G3 2 
G4 2  G5 2   is satisfied where  is a very small number). Note that the solution of qm
requires a reasonable initial guess based on the qm especially for the combined loading
43
MODEL INPUTS
KINEMATIC INPUTS
VECTOR INPUT
 Bearing configuration
qm
 Face-to-face (DF)
 Back-to-back (DB)
 Tandem (DT)
 Number of rolling elements
+
+
Direct
substitution
MODEL OUTPUTS
Bearing
stiffness matrix
fm, Kb
 Pitch radius
OR
 Unloaded contact angle
 Axial distance between bearing
center and each row
 Radial clearance
+
 Distance between inner and
outer curvature centers
qm
+
fm
Solve implicit set of
nonlinear equations
 Hertzian stiffness constant
Figure 2.4 Summary of the proposed stiffness model in terms of inputs and outputs
44
case. A summary of the proposed method in terms of model inputs and outputs is
illustrated in Figure 2.4.
2.6 Numerical Verification of Diagonal Stiffness Elements
The proposed model is verified with a commercial code [2.8] which can output
the diagonal elements of Kb without any need for post-processing. 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 2.1. The bearing is initially unpreloaded, and its outer ring is fixed. First,
the shaft is subjected to an axial load ( Fzext ), that is increased from 1 kN to 10 kN with 1
kN increments (refer to Figure 2.5 for a simple schematic). Assuming the shaft is rigid,
the entire axial load is supported by the double row bearing (i.e. Fzext = Fzm). In the
absence of any radial or moment load, the radial stiffness terms ( k xx and k yy ) as well as
the tilting stiffness terms ( k x x and k y y ) of Kb are equal, yielding three independent
diagonal stiffness elements (kxx , kzz and k y y ) that are illustrated in Figure 2.6 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.
45
Next, a radial shaft load Fxext = 1 kN is applied on the shaft as a point load in the
positive x-direction with an axial distance dz = 74 mm away from the bearing center
(according to Figure 2.5), which imposes a moment load on the bearing Mym = -74 kNmm
and results in a mean load vector fm  {1000 N, 0, 0, 0,  74000 Nmm}T . The solution for qm
in this case is very sensitive to the selection of the initial guess (especially for tandem
arrangement) due to the absence of an axial load. Thus, a more stable loading case with
fm  {1000 N, 0, 3000 N, 0, 74000 Nmm}T is also considered. Due to the non-uniform
loading in the radial direction, k xx  k yy and k x x  k y y for both cases, resulting in five
distinct diagonal stiffness terms which are calculated for the example case and presented
in Tables 2.2 and 2.3, respectively. As seen from both tables, the stiffness elements of the
proposed model show a very good match with the commercial code, with minor errors for
both loading cases. In the absence of axial loading (Table 2.2), stiffness elements in
unloaded directions (y, z and  x for this particular case) are unconventionally small,
especially for a tandem arrangement (however both models still exhibit a very good
match). After the application of Fzext  3 kN, the stiffness values become more
representative, and the agreement between the two models is still excellent, though very
minor deviations for several stiffness elements are seen.
46
Symbol
Z
Value
14
Description
Number of rolling elements in one row
rL (mm)
0
Radial clearance
Kn (N/mm1.5)
395,000
Hertzian stiffness constant
R (mm)
34.75
Radius of the inner raceway groove curvature center
(pitch radius)
Ao (mm)
0.52
Unloaded distance between inner and outer raceway
centers
e (mm)
10.0
Axial distance between the geometric center and one
bearing row
αo (deg)
30
Unloaded contact angle
n
1.5
Load-deflection exponent
Table 2.1 Kinematic properties of the example case: Double row angular contact ball
bearing
47
lSHAFT
dz
Outer ring
x
Fzext
Inner ring
z
O
G
y
Fxext
Double row angular contact ball bearing
(DB, DF or DT configuration)
Figure 2.5 The schematic to illustrate the static shaft loads utilized in the numerical
analysis of example case of Table 2.1.
48
Radial Stiffness (kxx )
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
0
5
0
5
0
10
1
1
1
0.8
0.8
0.8
0.6
0.4
0.2
0
DT
0
10
k  y  y [G Nm m /rad]
k x x [M N/m m ]
DB
Tilting Stiffness (k y y )
 
1
k z z [M N/m m ]
DF
Axial Stiffness (k zz)
1
0.6
0.4
0.2
0
5
10
0
0
5
10
0
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
10
0
0
5
Axial Force (kN)
10
0
5
10
0
5
Axial Force (kN)
10
0.2
1
5
Axial Force (kN)
10
0.4
1
0
5
0.6
1
0
0
0
Figure 2.6 Comparison of axial, radial and tilting stiffness coefficients of the proposed
model and the commercial code [2.8] for the example case of Table 2.1 with respect to
axial load. Key: Discrete points, proposed model; (
back-to-back arrangement; (
), face-to-face arrangement; (
), tandem arrangement; solid line, commercial code
49
),
Bearing
Arrangement
Stiffness Element
Face-to-Face (DF)
Back-to-Back (DB)
Tandem (DT)
Commercial Proposed
Code
Model
Commercial Proposed
Code
Model
Commercial Proposed
Code
Model
kxx (kN/mm)
407
401
360
375
297
313
kyy (kN/mm)
302
306
267
279
36
38
kzz (kN/mm)
311
318
226
235
50
51
kθxθx (MNmm/rad)
55
50
264
263
18
18
kθyθy (MNmm/rad)
95
89
354
353
161
159
Table 2.2 Comparison of diagonal stiffness elements of the proposed model and the
commercial code [2.8] for DF, DB and DT arrangements for the example case given
fm  {1000 N, 0, 0, 0, 74000 Nmm}T
50
Bearing
Arrangement
Face-to-Face (DF)
Back-to-Back (DB)
Tandem (DT)
Stiffness Element
Commercial
Code
Proposed
Model
Commercial Proposed Commercial Proposed
Code
Model
Code
Model
kxx (kN/mm)
442
465
402
420
330
345
kyy (kN/mm)
341
357
328
346
273
274
kzz (kN/mm)
342
359
283
297
217
221
kθxθx (MNmm/rad)
66
61
342
345
156
151
kθyθy (MNmm/rad)
100
96
410
410
211
209
Table 2.3 Comparison of diagonal stiffness elements of the proposed model and the
commercial code [2.8] for DF, DB and DT arrangements for the example case given
fm  {1000 N, 0, 3000 N, 0, 74000 Nmm}T
51
2.7 Examination of Stiffness Coefficients
2.7.1 Effect of mean bearing loads on stiffness coefficients
Typical variations in the stiffness elements under various loads are analyzed and
compared with those of the single row bearing with identical kinematic properties as
described by Lim and Singh’s model [2.4]. First, the effect of pure axial load (
 zm  0, Fzm  0 while other elements of qm and fm are zero) on k zz is analyzed. The
relationship between Fzm and  zm is shown in Figure 2.7(a). Here the slope of the tangent
(about an operating point) defines k zz , which is plotted with respect to  zm (Figure 2.7(b))
as well as Fzm (Figure 2.7(c)). As seen from Figure 2.7(b-c), k zz of DT arrangement is
significantly higher than those of the DF and DB arrangements as expected. In fact, DF
and DB arrangements tend to act as a single row bearing, showing a behavior identical to
Lim and Singh’s model [2.4] as only one row of these arrangements is loaded under a
pure axial load. Note that at a given  zm , k zz of 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 Fzm , k zz of the DT is less than twice of 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 (  xm  0, Fxm  0 while all
other elements of qm and fm are zero). The relationship between Fxm and  xm is shown in
Figure 2.8(a), and its slope ( k xx ) is plotted with respect to  xm in Figure 2.8(b). Here, the
load distribution for all configurations are identical; thus, DF, DB, and DT arrangements
52
1
(a)
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
(b)
0.8
Axial Stiffness (MN/mm)
30
0.6
0.4
0.2
0
0
0.01 0.02 0.03 0.04 0.05
Axial Displacement (mm)
(c)
0.8
0.6
0.4
0.2
0
0
5
Axial Load (kN)
10
Figure 2.7 Relationships between the axial load, axial displacement and axial stiffness
coefficients for the example case. Key: (
face arrangement; (
), Back-to-back arrangement; (
), face-to-
), tandem arrangement; ( ), single row bearing. (a) Fzm vs.  zm ;
(b) k zz vs.  zm ; (c) k zz vs. Fzm .
53
show an identical k xx behavior which is exactly twice of that of a single row bearing at a
given  xm (i.e. all double row configurations act like two parallel springs in the radial
direction). Next, a misalignment about the y-axis (  ym ) is applied, and k y y elements are
investigated. Plots of M ym vs.  ym and k y y vs.  ym are given in Figures 8(a) and (b),
respectively. As expected, k y y of DB arrangement is significantly higher than those of
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,
large number of plots similar to Figures 2.7-2.9 may be generated by considering
different loading and stiffness coefficient combinations. In general, it is more difficult to
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 k x y and k y x
(and their symmetric elements), whereas the matrix becomes fully populated if all
elements of qm are nonzero. Figure 2.10(a-c) illustrates sample variations in some crosscoupling terms under various loading cases. Note that these off-diagonal terms are also
highly dependent on the organization of the rolling elements.
54
25
0.7
(a)
(b)
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
Radial Displacement (mm)
0
0.05
0
0.01
0.02
0.03
0.04
Radial Displacement (mm)
0.05
Figure 2.8 Variation of the radial force and the radial stiffness with respect to radial
displacement of the bearing for the example case. Key: (
(
), face-to-face arrangement; (
), Back-to-back arrangement;
), tandem arrangement; ( ), single row bearing. (a)
Fxm vs.  xm ; (b) k xx vs.  xm .
55
4
3.5
(GNmm/rad)
3
2.5
yy
2
1.5
1
1
0.8
0.6
0.4
0.2
0.5
0
(b)
1.2
k
Bending Moment (MNmm)
1.4
(a)
0
1
2
3
4
Angular Displacement (mrad)
0
5
0
1
2
3
4
Angular Displacement (mrad)
5
Figure 2.9 Variation of the bending moment and the tilting stiffness with respect to
angular displacement of the bearing for the example case. Key: (
arrangement; (
), face-to-face arrangement; (
bearing. (a) M ym vs.  ym ; (b) k y y vs.  ym .
56
), Back-to-back
), tandem arrangement; ( ), single row
1.4
25
(a)
25
(b)
(c)
1.2
20
20
0.8
0.6
k z y (MN/rad)

k x y (MN/rad)

k xz (MN/mm)
1
15
10
15
10
0.4
5
5
0.2
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
0.01 0.02 0.03 0.04
Radial Displacement (mm)
0.05
Figure 2.10 Variation of some off-diagonal elements of Kb under various loads for the
example case. Key: (
), Back-to-back arrangement; (
), face-to-face arrangement; (
), tandem arrangement; ( ), single row bearing. (a) kxz vs.  ym ; (b) k x y vs.  zm ; (c) k z y
vs.  xm .
57
2.7.2 Effect of unloaded contact angle (under radial load) on stiffness coefficients
A radial displacement of  xm = 0.05 mm is applied to the bearing of Table 2.1,
and the effect of unloaded contact angle ( 0 ) on the stiffness coefficients are investigated
and shown in Figures 2.11-2.12 for diagonal and off-diagonal elements of Kb,
respectively. These figures are plotted up to 0 = 85° as numerical issues seem to occur
around 0 = 90° in some cases. The translational stiffness coefficients ( k xx , k yy , k zz ) of
all three arrangements are found to be identical under a pure radial load as shown in
Figure 2.11(a). Here, k xx and k yy follow a similar trend, as they have a maximum value
for 0 = 0° (deep groove ball bearing), which nonlinearly converge to zero as 0
approaches to 90°. On the other hand, k zz is minimum when 0 = 0°, reaches its
maximum value around 0 = 65°, and then decreases for higher 0 values. Recall that
bearings with 0  45 ° are often viewed as “thrust bearings”, thus, an increase in k zz for
higher 0 is an expected result. Figure 2.11(b-c) illustrate k x x and k y y coefficients,
which depend on the bearing configuration. As expected k x x and k y y of the DB
arrangement are higher than the other two arrangements at all 0 values. Also k y y is
greater than k x x for all arrangements and 0 values.
The dominant off-diagonal terms of Kb ( k xz , k x y , k y x , and k z y ) for a given  xm
are plotted with respect to 0 in Figure 2.12(a-d). Other off-diagonal terms are negligible
58
Translational Stiffness (MN/mm)
1
(a)
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
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
10
15
20
25
30 35 40 45 50 55
Unloaded Contact Angle (deg)
60
65
70
75
80
85
0.7
(b)
k x x (GNmm/rad)
 
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
0.8
(c)
k y y (GNmm/rad)
 
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
Figure 2.11 Variations in the diagonal stiffness elements of Kb with 0 given  xm = 0.05
mm for the example case. (a) k xx , k yy , and k zz , (b) k x x , (c) k y y . Key: (
back arrangement; (
), face-to-face arrangement; (
59
), tandem arrangement
), Back-to-
0.5
(a)
k xz (MN/mm)
0.4
0.3
0.2
0.1
0
0
5
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
(b)
k x y (MN/rad)

0
-5
-10
-15
-20
0
15
5
(c)
k y x (MN/rad)

10
5
0
-5
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0
(d)
k z y (MN/rad)

-5
-10
-15
-20
-25
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Unloaded Contact Angle (deg)
Figure 2.12 Variations in dominant off-diagonal stiffness elements of Kb with 0 given
 xm = 0.05 mm for the example case. (a) k xz , (b) k z . Key: (
y
arrangement; (
), face-to-face arrangement; (
60
), tandem arrangement
), Back-to-back
under pure radial load. As seen from Figure 2.12(a), k xz is identical for all three
arrangements at all 0 and has a maximum value when 0 = 45°. On the other hand,
k x y , k y x , and k z y are dependent on the bearing arrangement and 0 . One can observe
k x y and k z y have negative values, and k y x  k x y .
2.7.3 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 x-axis (i.e.  1i = 0 rad, i = 1,2). This assumption should be
verified as the load distribution among the rolling elements vary with angular position for
radial and moment loads which might cause considerable stiffness variations.
Again consider the same example case (of Table 1) is subjected to  xm = 0.05
mm. Three diagonal stiffness elements of Kb ( k xx , k zz , k  ) are normalized with
y y
respect to their values at  1i = 0 rad and plotted against the angular position of the
bearing over a complete ball passage period as shown in Figure 2.13(a-c). These results
are illustrated for the DB arrangement, though all arrangements show a similar behavior.
As seen from Figure 2.13(a-c), the stiffness elements vary between minimum and
maximum values over a ball passage period. In particular, k zz shows the maximum
variation (about 6%) over the ball passage period. The variations in k xx and k y y on the
other hand are negligible. These results suggest that the angular position does not have a
61
significant effect on the stiffness elements, as the maximum variation in a stiffness
coefficient is less than 6% even for a fairly large radial displacement (such as  xm = 0.05
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. 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 k xx , k zz , k  for  xm = 0.05 mm. These results are shown in Figures
y y
2.13(d-f). As seen from the figures, k zz now shows a variation of almost 70% over a ball
passage period, whereas the variations of k xx and k y y are now around 18% and 16%,
respectively. These results clearly show that the angular position is an important
parameter, however its effect can be neglected in the presence of 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. Figure 2.14 shows the
variations in k xx , k zz and k y y for  ym = 0.03 rad (for Z = 14). As seen from the figure,
the variations are arrangement specific for a given  ym , however they are all within 2%;
hence, they are negligible.
62
1.0015
1
(a)
1.0015
(b)
(c)
0.99
1.001
1.0005
0.98
k y y /(k y y ) =0
 
  
k zz /(k zz ) =0

k xx /(k xx ) =0

1.001
0.97
0.96
1.0005
1
1
0.95
0.9995
0.94
0
1
0.5
1
Ball Passage Period
0
1.8
(d)
0.98
1
(e)
0.9
0.88
0.86
k y y /(k y y ) =0
 
  
k zz /(k zz ) =0

0.92
0.5
1
Ball Passage Period
0.96
1.6
0.94
0
(f)
0.98
1.7
0.96
k xx /(k xx ) =0

0.9995
0.5
1
Ball Passage Period
1.5
1.4
1.3
1.2
0.84
0.94
0.92
0.9
0.88
0.86
0.84
0.82
1.1
0.8
1
0
0.5
Ball Passage Period
1
0.82
0
0.5
Ball Passage Period
1
0.8
0
0.5
Ball Passage Period
1
Figure 2.13 Variations in normalized k xx , k zz and k y y over a ball passage period for
back-to-back configuration given  xm = 0.05 mm for the example case. (a-c), with Z = 14
balls; (d-f), with Z = 4 rolling elements
63
Face-to-Face Arrangement
1.015
1.01
1.008
1.02
1.01
0
0.5
0.99
1
0
0.5
0.996
0.995
0.5
yy
0.995
0.985
1
0
0.5
1
1.005
/(k
1
1
0.995
0.99
0
1
k
0.997
0.5
1.01
zz
xx
0.998
0
1.015
)
1.005
1.005
0.995
1
yy =0
0.999
zz =0
1.01
k /(k )
xx =0
1
1.01
1
Back-to-Back Arrangement
1.015
1.001
1.015
)
0.995
1
k /(k )
yy =0
k
1.002
/(k
zz
1
yy
zz =0
1.005
k /(k )
1.004
xx
k /(k )
xx =0
1.006
0.998
1.025
0
0.5
0.99
1
Tandem Arrangement
1.007
1.006
1.004
1.005
1.002
1.004
1.003
)
/(k
zz =0
0.998
0.996
k
1.002
yy
zz
1.003
yy =0
1
1.004
k /(k )
xx
k /(k )
xx =0
1.005
1.001
0.994
1
0.999
0
0.5
1
Ball Passage Period
0.992
1.002
1.001
1
0.999
0
0.5
1
Ball Passage Period
0.998
0
0.5
1
Ball Passage Period
Figure 2.14 Variations in normalized k xx , k zz and k y y over a ball passage period given
 ym = 0.03 rad for DF, DB and DT configurations of the example case.
64
2.8 Conclusion
The chief contribution of this chapter 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. Using the proposed
analytical expressions, one can easily determine the diagonal and non-diagonal (crosscoupling) stiffness coefficients of a double row angular contact ball bearing given the
mean displacement vector qm (by direct substitution) or the mean load vector fm (by
numerical solution of the nonlinear system equations). 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. More extensive verification of the model and its validation with a
modal experiment will be given in the following chapters.
Considering the lack of publications on double row bearings, 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 duplex (paired) bearings which behave as an integrated
(double row) unit when the surrounding structural elements (such the shaft and bearing
rings) are sufficiently rigid. Also, 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,
65
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.
66
References for chapter 2
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2.15 J.S. Lin, Experimental analysis of dynamic force transmissibility through bearings,
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68
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and Vibration 169 (4) (1994) 547–553.
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in ball and roller bearings loaded in five degrees of freedom while neglecting frictions
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of Tribology, 111 (1989) 142-148.
2.23 X. Hernot, M. Sartor, J. Guillot, Calculation of the stiffness matrix of angular contact
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2.24 P. Cermelj, M. Boltezar, An indirect approach investigating the dynamics of a
structure containing ball bearings, Journal of Sound and Vibration, 276 (1-2) (2004)
401–417.
69
2.25 T. L. H. Walford, B.J. Stone, The measurement of the radial stiffness of rolling
element bearings under oscillation conditions, Proceedings of the Institution of
Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 22 (4)
(1980) 175-181.
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contact bearings under oscillating conditions, Proceedings of the 2nd International
Conference on Vibrations in Rotating Machinery, (1980).
2.27 B. J. Stone, The state of art in the measurement of the stiffness and damping of
rolling element bearings, Annals of the CIRP, 31 (2) (1982) 529-538.
2.28 E. R. Marsh, D. S. Yantek, Experimental measurement of precision bearing dynamic
stiffness, Journal of Sound and Vibration, 202(1) (1997) 55-66.
2.29 R. Tiwari, N. S. Vyas, Estimation of non-linear stiffness parameters of rolling
element bearings from random response of rotor-bearing systems, Journal of Sound
and Vibration, 187 (1995) 229-239.
2.30 G. Pinte, S. Devos, B. Stallaert, W. Symens, J. Swevers, P. Sas, A piezo-based
bearing for the active structural acoustic control of rotating machinery, Journal of
Sound and Vibration, 329 (9) (2010) 1235-1253.
2.31 M. Cao, J. Xiao, A comprehensive dynamic model of double-row spherical roller
bearing – model development and case studies on surface defects, preloads, and radial
clearance, Mechanical Systems and Signal Processing, 22 (2008) 467-489.
70
2.32 M. Cao, A refined double-row spherical roller bearing model and its application in
performance assessment of moving race shaft misalignments, Journal of Vibration
and Control, 13 (2007) 1145-1168.
2.33 D. H. Choi, K. C. Yoon, A design method of an automotive wheel-bearing unit with
discrete design variables using genetic algorithms, Journal of Tribology, 123 (2001)
181-187.
2.34 Lagarias, J. C., J. A. Reeds, M. H. Wright, and P. E. Wright, Convergence properties
of the Nelder-Mead simplex method in low dimensions, SIAM Journal of
Optimization, 9 (1) (1998) 112–147.
71
CHAPTER 3
EFFECT OF BEARING PRELOADS ON MODAL CHARACTERISTICS OF A
SHAFT-BEARING ASSEMBLY AND VALIDATION OF THE STIFFNESS
MODEL USING A MODAL EXPERIMENT
3.1 Introduction
It is well known that rolling element bearings must be preloaded to eliminate
large deflections due to external loads, enhance fatigue life, decrease noise, prevent
skidding, and reduce rattle due to clearances [3.1-3.2]. However, the effect of bearing
preloads on the modal or vibration characteristics of a shaft-bearing assembly are not
well understood, though several numerical and experimental studies have been reported
[3.3-3.8]. For instance some shifts in the frequency response function measurements at
various preloads have been seen [3.8-3.9], but changes in the resonant responses due to
bearing preload have not been explored. It seems that a controlled experiment to analyze
the nature and extent of bearing preloads on the modal characteristics of shaft-bearing
systems (including their stiffness and damping characteristics) is yet to be conducted.
This study fills this gap with a focus on double row angular bearings and presents new
72
experimental and analytical results (in the frequency domain) in order to provide a better
understanding on the preload effects.
Double row angular contact ball bearings are similar to duplex (paired) bearings
in terms of design and functionality [3.1]. Conventional double row angular contact ball
bearings have solid inner and outer races, however, certain types (such as most
automotive wheel bearings) are designed with a split inner race; this provides ease in
axial preloading with a lock nut or a similar mechanism [3.1-3.2]. To address such
bearings, a new bearing stiffness matrix (Kb) has been developed in chapter 2. This
chapter will employ the new Kb formulation to guide the design of a new shaft-bearing
system experiment to extensively investigate the effect of bearing preloads on system
modal characteristics, as well as to experimentally validate the new Kb formulation.
3.2 Literature Review
Several aspects of the bearing stiffness formulations and resulting vibration
transmission characteristics have been well studied by prior investigators. White [3.5]
analyzed the influence of preloads on the radial bearing stiffness coefficients in a two
degree-of-freedom model and showed that the bearing nonlinear effects are negligible at
higher preloads. Lim and Singh [3.7-3.8] demonstrated a relationship between preloads
and stiffness coefficients and analyzed the modal properties of a linear, time-invariant
system at various preloads. Royston and Basdogan [3.10] presented similar results for
self-aligning bearings by relating axial and radial preloads to the stiffness coefficients.
Cermelj and Boltezar [3.11] used Lim and Singh’s stiffness model [3.7] to compare
73
analytical and experimental frequency response functions at a given preload. Jorgensen
and Shin [3.12] analyzed high speed effects and found a relationship between the radial
loads, rotational speed, and radial stiffnesses. Yet a few other researchers have focused
on the nonlinear dynamic aspects of the shaft-bearing systems. For example, Akturk et al.
[3.6] used a nonlinear three degree-of-freedom model of an angular contact ball bearing
and observed a reduction in the vibration amplitudes (at the ball passage frequency) with
an increase in the axial preload. Alfares and Elsharkawy [3.13] expressed similar
observations using a five degree-of-freedom model as they also found a reduction in the
peak-to-peak amplitudes with an increase in axial preload. Bai et al. [3.14] used the same
five degree-of-freedom model to observe that unstable periodic solutions of a rotor
bearing system could be avoided with a sufficiently high axial preload.
Experimental studies that demonstrate the effects of bearing preloads on the
vibration characteristics of a shaft-bearing system are limited. Kraus et al. [3.4] utilized
experimental modal analysis to estimate the modal parameters of a shaft-bearing system
using a single degree-of-freedom system (in both axial and transverse directions). They
reported an increase in the stiffness (and natural frequency) of the shaft-bearing system in
axial or transverse direction and also an increase in axial damping but a reduction in
transverse damping with an increase in the axial preload. Lim and Singh [3.8] and
Spiewak and Nickel [3.9] presented limited experimental results (on single row bearings
in the frequency domain) for three axial preloads but did not observe a clear trend
between bearing preloads and vibration amplitudes. Several other researchers [3.12,3.153.18] have measured natural frequencies of the shaft-bearing system at various preloads
74
and rotational speeds but did not focus on the vibratory characteristics of the shaftbearing system nor present any frequency domain results.
3.3 Problem Formulation: Scope, Assumptions and Objectives
The generic vibration problem of a shaft-bearing system containing a double row bearing
is illustrated in Figure 3.1. Here the shaft is subjected to an alternating (with subscript a)
load
vector fsa (t )  {Fxsa (t), Fysa (t ), Fzsa (t ), M xsa (t), M ysa (t)}T
which
represents
typical
vibratory excitations, say due to kinematic errors, torque fluctuations, and mass
unbalances [3.8]. The resulting vibratory motions of the shaft (also shown in Figure 3.1)
are similarly represented by a generalized displacement vector as qsa (t )  { xsa (t ),  ysa (t ),
 zsa (t ),  xsa (t ),  ysa (t )}T . The last two terms of both vectors correspond to tilting moments
and angular motions about the x and y axes, respectively. The rotational term about the zaxis is not included as the shaft is allowed to freely rotate about its rotational (z) axis. For
vibration analysis, assume that the amplitudes of vibratory motions transmitted through
the bearing are much smaller than the mean bearing displacements due to preloads and
external mean loads. This yields a linear, time-invariant system with bearing properties
such as Kb about an operating point. The governing equation of the linearized vibration
model is:
a (t )  Cq a (t )  Kqa (t )  fa (t )
Mq
(3.1)
where M , C , and K are system mass, viscous damping, and stiffness matrices, and
fa (t ) and qa (t ) are the generalized alternating force and displacement vectors,
75
x
 xsa
 xsa
Bearing Casing
Double Row Bearing
F xsa
M xsa
Shaft
M
F zsa
 zsa
z
z
ysa
F ysa
 ysa
y
 ysa
Figure 3.1 Schematic of the vibration transmission problem. Here a shaft is subjected to
dynamic loads and moments fsa (t )  {Fxsa (t ), Fysa (t), Fzsa (t ), M xsa (t ), M ysa (t )}T , resulting
in vibratory motions qsa (t )  { xsa (t ),  ysa (t ),  zsa (t ),  xsa (t ),  ysa (t )}T . Here, subscript a
implies alternating or vibratory. x and y are radial directions, and z is the axial direction.
76
respectively. Although high speed effects are not considered in this study, one could
easily incorporate gyroscopic and centrifugal effects into Eq. (3.1) to analyze overcritical
speeds [3.16]. Note that the bearing preloads and external mean loads do not explicitly
appear in the governing equation. However, they dictate the vibration characteristics of
the bearing system via the diagonal and off-diagonal terms of Kb (as well as via elements
of C), thus controlling the natural modes and forced responses.
The specific objectives of this chapter are as follows: 1. Develop a simple
analytical system model of a shaft-bearing system (with a double row angular contact ball
bearing of alternate configurations) and computationally evaluate the effects of preloads
on the natural frequencies and vibration amplitudes by employing two alternate viscous
damping approximations; 2. Design and instrument a new laboratory experiment
consisting of an automotive wheel-hub assembly (with a back-to-back double row
angular contact ball bearing) to investigate the effects of axial and radial bearing preloads
on the system modes, natural frequencies and resonant amplitudes; 3. Validate the new
stiffness model based on modal measurements.
3.4 Analytical Model of a Rigid Shaft Supported by Double Row Bearing
Since the main purpose of this study is to examine the modal characteristics of
double row angular contact ball bearings, both shaft and bearing casing are assumed to be
rigid. This facilitates the development of a simple analytical model (of dimension nd = 5)
as shown in Figure 3.2 to represent the vibration problem of Figure 3.1. Here, a rigid
shaft (MS ) is supported by a double row angular contact ball bearing represented as a
77
x
x
Ms
Rigid shaft
Double row bearing
Kb
z
Cb
y
y
Figure 3.2 Analytical model (of dimension 5) of a rigid shaft supported by double row
bearing
78
five-dimensional bearing stiffness matrix ( K b ) with an appropriate viscous damping
matrix ( Cb ). Both are directly connected to ground. Mass, damping, and stiffness
matrices of the shaft-bearing assembly essentially become M  MS , C  Cb and K = Kb
where:
 ms
0

MS   0

0
0

0
ms
0
0
0
0
0
0
0
ms
0
0
0
I xx
I yx
0
0 
0 .

I xy 
I yy 
(3.2)
In these studies, K b is evaluated by using the analytical expressions as outlined
in chapter 2. A commercial double row angular contact ball bearing whose properties are
given in Table 3.1 will be utilized in the computational (section 3.5) and experimental
studies (section 3.6). These properties are either given by the bearing manufacturer or
determined from the kinematics, except for the Hertzian stiffness coefficient (Kn) which
is given by an empirical relation [3.1]. Neglecting shaft and casing compliances and
assuming the casing is grounded, the shaft-bearing system can be represented by the
lumped parameter model of Figure 3.2. For the example case, the rigid shaft has a mass
of m = 5 kg, and the moment of inertia of the bearing center about the x and y axes are
calculated from geometry as I xx = I yy = 0.015 kgm2. The product of inertia terms ( I xy =
I yx ) of Eq. (3.2) are zero as the shaft is rotationally symmetric (x, y, and z are the
principal axes); thus MS is diagonal.
79
Symbol
Value
Description
Z
13
Number of rolling elements in one row
rL (mm)
0
Radial clearance
Kn (N/mm1.5)
414,000
Hertzian stiffness constant
R (mm)
31.75
Radius of the inner raceway groove curvature center
(pitch radius)
Ao (mm)
0.65
Unloaded distance between inner and outer raceway
centers
e (mm)
18.20
Axial distance between the geometric center and one
bearing row
αo (deg)
35
Unloaded contact angle
n
1.5
Load-deflection exponent
Table 3.1 Kinematic properties of the double row angular contact ball bearing utilized for
computational and experimental studies
80
The viscous damping matrix (Cb ) , however, cannot be easily predicted due to
unknown dissipative characteristics of rolling element bearings [3.2,3.4] and thus must be
estimated experimentally. To facilitate this, consider a non-proportionally damped multidegree-of-freedom system, and rewrite Eq. (3.1) by defining a state vector
xa (t )  (q a (t ), qa (t ))T of dimension 2nd (where nd is the dimension of the analytical
model).
 a (t )   0 K  q a (t )  f a (t ) 
M C  q

 0 M  q (t )   -M 0  q (t )    0 

 a  
 a  

(3.3)
This is of the form Bx a (t )  Ax a (t )  [f a (t ) 0]T where
M C 
 0 K
B
; A


 0 M
-M 0 
(3.4a-b)
For the generalized eigenvalue problem assume a solution of the form xa (t )  v exp(t )
(Re( ) < 0) ,
Av  Bv
(3.5)
The solution results in 2nd eigenvalues ( r ) and eigenvectors ( vr ) occurring in complex
conjugate pairs. The rth eigenvalue ( r ) is written in the form
r   rr  jr 1   r
(3.6)
and the undamped natural frequencies ( r ) and viscous damping ratios (  r ) of the
system are:
r 
 Re(r     Im(r  
2
81
2
(3.7)
r 
Re(r )
 Re(r )    Im(r ) 
2
2
.
(3.8)
In this study, two alternate viscous damping mechanisms are employed. First, a
preload-independent viscous damping is considered. It is described by the following nonproportional, diagonal matrix ( Cb ) of the form

Cb  diag [cxx , c yy , czz , c x x , c y y ]

(3.9)
where cxx  c yy  czz = 4350 Ns/m and c x x  c y y  7.25 Nms/rad. These damping
coefficients are estimated based on experimental results for an intermediate axial preload
(to be discussed later in section 3.7.2) and then utilized at all loads. Second, a preloaddependent viscous damping model is used where Cb is proportional to Kb :
Cb   K b
(3.10)
Here the proportionality constant  (such as 10-7 s) is found empirically. Since Kb is a
function of the bearing preload, the values of Cb are decided by the extent of preload,
unlike the first model described by Eq. (3.9).
Assuming a sinusoidal excitation f (t ) = Fe jt at a frequency  (rad/s), the
accelerance matrix A(  = [ A pq (   f p / qq ] can be expressed as follows:
A(    2 [ 2M  jC  K]1
(3.11)
which is expanded below (note that each element of a symmetric matrix is a function of
 ):
82
 Axx Axy Axz

Ayy Ayz


Azz
A (   

 symmetric


Ax x
Ay x
Az x
A x x
Ax y 

Ay y 

Az y 
A x y 

A y y 
(3.12)
3.5 Computational Study
The modal characteristics of a shaft-bearing system are first analyzed with a
computational exercise to serve the following four objectives: 1. Demonstrate the utility
of Kb with a simple linear vibration model; 2. Investigate two viscous damping models
described by Eqs. (3.9) and (3.10), and observe a change in the vibration amplitudes with
an increase in axial preload; 3. Guide the design of a laboratory experiment setup (section
3.6) to select an appropriate bearing and analyze its sensitivity to preload; 4. Interpret the
experimental results of next section.
For the sake of illustration, the effects of axial preloads ( Fz 0 ) will be examined,
although other loading scenarios could be easily applied. Here, Fz 0 defines a force
quantity that can be expressed in terms of axial displacement preload  ( z 0 )i  by the
following relation [3.19]:
( z 0 )i   ZK n sin( 0 )1 n 
1/ n
( Fz 0 )1/ n
(3.13)
The double row bearing of Table 3.1 is subjected to Fz 0 ranging from 0.25 kN to
5.0 kN with an increment of 0.25 kN. In the absence of any radial or moment loads on the
bearing the only significant off-diagonal terms of Kb are k x y and k y x (and their
83
symmetric elements k y x and k x y ), which couple translational and rotational motions in
two radial directions, x and y. Also, the translational and rotational stiffness coefficients
in the x and y directions are identical ( k xx  k yy and k x x  k y y ) which results in an
overlap of relevant vibration modes under a pure axial load. For this particular example, a
pair of identical modes are obtained at r = 1 and 2 and then at r = 4 and 5. The third mode
(r = 3) that corresponds to axial (translational) vibrations is unique as it is uncoupled
from all other modes because all the z-axis related off-diagonal terms of Kb are negligible
in the absence of radial or moment loads.
First, consider the preload-independent Cb model of Eq. (3.9). The undamped
natural frequency (r / 2 ) map of the shaft-bearing system with respect to Fz 0 is
illustrated in Figure 3.3(a-c) for three bearing arrangements (face-to-face, back-to-back
and tandem). An increase in Fz 0 affects the diagonal and off-diagonal terms of Kb (as
shown in Table 3.2) and raises r values for all arrangements as seen from Figure 3.3,
but considerable variations among the three configurations are found. For example, the r
= 1,2 mode of the face-to-face arrangement occurs at such a low frequency that it does
not exist until Fz 0 is equal to 1.0 kN. Beyond 1 kN, this mode still remains in the lower
frequency regime (below 160 Hz), whereas the same mode occurs between 760 to 870 Hz
for the back-to-back arrangement and in the 500-660 Hz range for tandem arrangement.
This result could be explained by the fact that the back-to-back arrangement, which is
designed to carry substantial moment loads, has higher moment stiffness terms ( k x x =
k y y around 0.56-0.75 GNmm/rad for this case) that considerably raise the natural
84
Face-to-Face (X) Arrangement
Natural Frequency (Hz)
3000
(a)
2000
1000
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3.5
4
4.5
5
3.5
4
4.5
5
Back-to-Back (O) Arrangement
Natural Frequency (Hz)
3000
(b)
2000
1000
0
0
0.5
1
1.5
2
2.5
3
Tandem Arrangement
Natural Frequency (Hz)
3000
(c)
2000
1000
0
0
0.5
1
1.5
2
2.5
3
Axial Preload (kN)
Figure 3.3 Natural frequencies vs. axial preload for the computational example. (a) Faceto-face arrangement, (b) Back-to-back arrangement, (c) Tandem arrangement. Key for
modes: (
), r = 1, 2; (
), r = 3; (
), r = 4, 5
85
Axial Preload (kN)
Face-to-Face
Back-to-Back
Tandem
Stiffness element (units)
0.5
5
0.5
5
0.5
5
kxx = kyy (kN/mm)
329
454
329
454
531
746
kzz (kN/mm)
378
601
378
601
579
898
kθxθx = kθyθy (MNmm/rad)
134
222
558
753
468
700
kxθy = kθyx (MN/rad)
-6.89
-9.77
-15.2
-22.9
-12.1
-17.8
kyθx = kθxy (MN/rad)
6.89
9.77
15.2
22.9
12.1
17.8
Table 3.2 Non-zero elements of the stiffness matrix calculated by the proposed stiffness
model (of chapter 2) with alternate bearing configurations for Fz 0 = 0.5 kN or 5 kN
86
frequencies of angular (tilting) modes. Since the radial modes (r = 1,2) are highly coupled
with the tilting modes (r = 4,5) through the cross-coupling terms (k x y and k y x ) , the
overlapped radial modes of the back-to-back arrangement also occur at a higher
frequency. Conversely, the face-to-face arrangement, which is designed to provide a
better compensation for shaft misalignments (which are closer to the self-aligning
bearings in this sense), has much lower tilting stiffness coefficients (around 0.1-0.25
GNmm/rad) as it has a smaller effective load center. Lower tilting stiffness coefficients
shift the natural frequencies to the lower frequency regime.
The axial vibration mode (r = 3), however, occurs at the exact same frequencies
for the face-to-face and back-to-back arrangements (shifting from 1240 Hz to 1670 Hz
with increasing Fz 0 ), as they show the same axial vibration behavior. This is because their
k zz values are the same, and there is no coupling between the tilting and axial
coefficients (k z x  k z y  0) in the absence of any radial or moment loads. However,
when the rolling elements are arranged in a tandem configuration, this axial mode occurs
at a higher frequency (shifting from 1540 to 2070 Hz) due to its high axial rigidity (of
course it is assumed that the tandem arrangement is loaded in the direction it can carry
the load).
The system transfer functions at various Fz 0 provide more insights. Thus, the
radial accelerance spectra  A xx ( )  is plotted for five Fz 0 values (0.5, 1, 2, 3, 5 kN) in
Figure 3.4(a-c) for face-to-face, back-to-back, and tandem arrangements, respectively.
These spectra clearly illustrate that the natural frequencies move up or the peaks shift
87
towards the right with an increase in Fz 0 . Again, observe that the back-to-back
arrangement has the highest resonant frequencies, followed by tandem and face-to-face
configurations, respectively. If the vibration amplitudes are to be rank-ordered from
maximum to minimum, the same order of bearing configurations applies (which is
especially clear from the first overlapped mode). It is also apparent that these peak
amplitudes grow with increasing Fz 0 when the preload-independent Cb model of Eq.
(3.9) is selected.
Since the axial vibration mode (r = 3) is uncoupled from other modes of vibration,
its resonant peak cannot be seen in the A xx ( ) spectra. In order to clearly observe this
mode, A zz ( ) spectra are plotted in Figure 3.5(a-c). As expected, the face-to-face and
back-to-back arrangements have identical spectra, whereas this mode occurs at a higher
frequency with higher amplitudes for the tandem configuration. The increasing trend of
vibration amplitude with increasing Fz 0 is also seen in the A zz ( ) spectra.
Now, consider the load-dependent Cb model of Eq. (3.10). If the radial and axial
accelerance spectra
A
xx
( ) and A zz ( )  which are shown in Figures 3.6(a-c) and
3.7(a-c) respectively, are viewed, it can be observed that the vibration amplitudes at all
modes decrease with increasing Fz 0 unlike the previous case. In fact, the peak amplitudes
of the curves corresponding to light preloads (such as 0.5 kN) are elevated, but they are
considerably reduced for heavy axial preloads (such as 5 kN). Also, even though the
undamped natural frequencies do not vary significantly, the shape of the resonant peaks
are altered in comparison to the load-independent viscous damping model of Eq. (3.9).
88
Face-to-Face (X) Arrangement
Accelerance (1/kg)
2.5
(a)
Increasing
preload
2
1.5
1
0.5
0
Increasing preload
0
500
1000
1500
2000
2500
2000
2500
2000
2500
Back-to-Back (O) Arrangement
Accelerance (1/kg)
2
(b)
Increasing
preload
1.5
Increasing preload
r = 4, 5
1
r = 1, 2
0.5
0
0
500
1000
1500
Tandem Arrangement
Accelerance (1/kg)
3
(c)
Increasing
preload
2
1
0
Increasing preload
0
500
1000
1500
Frequency (Hz)
Figure 3.4 Effect of axial preload on the radial accelerance magnitude spectra A xx ( ) of
the computational example with preload-independent damping model described by Eq.
(3.9). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c) Tandem
arrangement. Key: Fz 0 ; (
), 0.5 kN; (
), 1 kN; (
89
) 2 kN; (
) 3 kN; (
) 5 kN.
Face-to-Face (X) Arrangement
Accelerance (1/kg)
3
(a)
Increasing
preload
2
1
0
0
500
1000
1500
2000
2500
2000
2500
Back-to-Back (O) Arrangement
Accelerance (1/kg)
3
(b)
Increasing
preload
2
r=3
1
0
0
500
1000
1500
Tandem Arrangement
Accelerance (1/kg)
4
(c)
Increasing preload
3
2
1
0
0
500
1000
1500
Frequency (Hz)
2000
2500
Figure 3.5 Effect of axial preload on the axial accelerance magnitude spectra A zz ( ) of
the computational example with preload-independent damping model described by Eq.
(3.9). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c) Tandem
arrangement. Key: Fz 0 ; (
), 0.5 kN; (
), 1 kN; (
90
) 2 kN; (
) 3 kN; (
) 5 kN.
Face-to-Face (X) Arrangement
Accelerance (1/kg)
3
Increasing preload
(a)
2
1
Increasing preload
0
0
500
1000
1500
2000
2500
2000
2500
2000
2500
Back-to-Back (O) Arrangement
Accelerance (1/kg)
1.5
Increasing
preload
(b)
Increasing
preload
1
r = 1, 2
0.5
0
r = 4, 5
0
500
1000
1500
Tandem Arrangement
Accelerance (1/kg)
2
(c)
1.5
1
Increasing preload
0.5
0
0
500
1000
1500
Frequency (Hz)
Figure 3.6 Effect of axial preload on the radial accelerance magnitude spectra A xx ( ) of
the computational example with preload-dependent damping model described by Eq.
(3.10). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c) Tandem
arrangement. Key: Fz 0 ; (
), 0.5 kN; (
), 1 kN; (
91
) 2 kN; (
) 3 kN; (
) 5 kN.
Face-to-Face (X) Arrangement
Accelerance (1/kg)
2.5
Increasing
preload
(a)
2
1.5
1
0.5
0
0
500
1000
1500
2000
2500
Back-to-Back (O) Arrangement
Accelerance (1/kg)
2.5
(b)
2
Increasing
preload
r=3
1.5
1
0.5
0
0
500
1000
1500
2000
2500
Tandem Arrangement
Accelerance (1/kg)
2
Increasing
preload
(c)
1.5
1
0.5
0
0
500
1000
1500
Frequency (Hz)
2000
2500
Figure 3.7 Effect of axial preload on the axial accelerance magnitude spectra A zz ( ) of
the computational example with preload-dependent damping model described by Eq.
(3.10). (a) Face-to-face arrangement, (b) Back-to-back arrangement, (c) Tandem
arrangement. Key: Fz 0 ; (
), 0.5 kN; (
), 1 kN; (
92
) 2 kN; (
) 3 kN; (
) 5 kN.
Since the damping characteristics of rolling element bearings are not very well
understood, both damping models must be examined to interpret vibration measurements
as discussed in the next section.
3.6 Experimental Study
3.6.1 Development of a modal experiment
A new experiment composed of an automotive wheel-hub assembly, as shown in
Figure 3.8, is designed to assess the effects of bearing preloads on modal characteristics.
A similar wheel-bearing unit has been investigated by Choi and Yoon [3.20] to optimize
its design variables. In this system, the shaft is supported by a double row angular contact
ball bearing in back-to-back arrangement (with kinematic properties of Table 3.1) that
has a split inner ring. The main reason a back-to-back arrangement is selected is its ease
of preloading with a lock nut through a threaded rod [3.2], and this arrangement is found
in many real-life applications (such as vehicle wheel bearings). Also, the analyses of the
previous section suggest that the radial, axial, and tilting modes of the back-to-back
arrangement are well separated when compared with the face-to-face or tandem
arrangement. The bearing housing (or the knuckle for this case) is rigidly clamped to a
pedestal which is rigidly connected to the ground. The experiment is conducted under
non-rotating (zero speed) conditions to avoid any gyroscopic or centrifugal effects that
have been well studied by several authors [3.12,3.16]. The bearing is initially unloaded in
the axial direction. The initial radial preload for such bearings is given in terms of radial
93
Radial
Bearing/Hub Assembly
Axial Impact Locations
f 1x
f2
f3
f5
Accelerometer
Location 2
f4
f 1z
Accelerometer
Location 1
Shaft with Threaded Rod And Nuts
Load Cell
Fixed
Steering Knuckle Accelerometer
Location 3
Hydraulic Jack
Load Washer
Figure 3.8 Experiment with wheel-hub assembly and illustration of preloading
mechanisms utilizing hydraulic jack (for radial direction) and shaft with threaded rod (for
axial preloading).
94
clearance that is between 0 and -30  m (note that this negative clearance defines a
displacement preload), which creates some uncertainty regarding the precise value of
initial radial preload. Additional bearing preloads are applied via the following
mechanisms.
1. Axial preload ( Fz 0 ): Applied through the shaft by tightening the nuts at both ends
of threaded rod inside the shaft that can be seen in Figure 3.8. The applied load is
measured with a washer type load cell (Omega model LC901-5/8-50K, 28.6 mm
outer diameter 16.4 mm inner diameter, 0 to 222.4 kN compression range, 9.39e-3
mV/V/kN) that is placed between one of the nuts and flange on the shaft.
2. Radial load ( Fx 0 ): Applied as an external static shaft load ( Fxm ) using a hydraulic
jack that is placed on a stable base as shown in Figure 3.8 while the bearing is
axially preloaded at 2.67 kN. The applied radial load, which is carried by the
wheel bearing, essentially imposes a moment load ( M ym ) on the bearing as well.
The amount of applied radial load is measured with a miniature compression load
cell (Omega model LC307-3K, 12.7 mm diameter, 9.7 mm thick, 0 to 13.3 kN
compression range, 0.109 mV/V/kN) which is attached at the top of the jack.
An impulse hammer (PCB model 086C02, 0 to 445N range, 11.2 mV/N) test is
conducted at various axial and radial preloads, and cross-point accelerance measurements
are taken for numerous impact ( f1 , f 2 ,, f5 ) and accelerometer locations (1, 2 and 3) as
shown in Figure 3.8. In these measurements the angular position of the shaft (and
essentially the bearing) is kept the same for all preloads. In the following sub-sections,
95
typical results are presented for various locations of a triaxial accelerometer (PCB model
356A15, 100 mV/g sensitivity in each direction, <5% transverse sensitivity) when the
force (f1z) is applied on the left end of shaft (according to Figure 3.8). Other force
locations are incorporated to observe the mode shapes. The frequency range of interest of
this study will be from 500 to 2500 Hz as the relevant vibration modes of the shaftbearing system occur within this region. The natural frequencies that occur below 500 Hz
correspond to the lower order flexural modes of the casing itself, whereas the modes
beyond 2500 Hz represent local motions. Thus, the modes of shaft-bearing assembly are
well separated from the rest of experimental system (or fixture) modes.
3.6.2 Effects of axial preloads
Axial preloads ranging from 0.45 kN to 2.67 kN are applied with an increment of
0.22 kN; these are within  0.05 kN error margin as it is not possible to experimentally
apply a precise amount of preload. Also, there is an uncertainty of  4 Hz in the
frequency measurements based on the frequency resolution of acquired digital data.
Accelerance results are presented for two accelerometer locations (1 and 2) in Figure 3.8.
Accelerance magnitude measurements at location 1 are first plotted in Figure
3.9(a) up to 2500 Hz for all preloads. In this plot, the low-amplitude peak occurring
around 65 Hz (though not very visible) corresponds to a casing mode and is thus beyond
interest as mentioned earlier. The first preload-dependent mode occurs at 660 Hz when
Fz 0 = 0.45 kN, and its natural frequency increases to 810 Hz as the Fz 0 is gradually
increased to 2.67 kN. The peak amplitude at this mode significantly reduces with an
96
increase in Fz 0 , resembling the load-dependent damping case described by Eq. (3.10) as
discussed in the previous section. As a result of these two combined effects, the peaks
display a clear trend towards right and down with increasing Fz 0 . Around 890 Hz, one
can observe yet another peak with lower amplitudes shifting slightly towards right with
increasing Fz 0 . These two modes, when magnified in Figure 3.9(b), predominantly
correspond to the radial motions of the shaft.
The amplitude at another vibration mode, as observed around 1620 Hz in Figure
3.9(a), is significantly attenuated with an increase in Fz 0 , though it does not exhibit a
considerable frequency shift. Further analysis reveals that this mode predominantly
corresponds to the axial motions of the shaft which is consistent with the computational
study of section 3.5. A reduction in the vibration amplitude can be explained by the fact
that a high Fz 0 eliminates the axial clearances within the bearing and consequently does
not permit any axial vibrations of the shaft. Conversely, when Fz 0 is low, higher
amplitude vibrations seem to occur due to a higher clearance. A nonlinear study of the
clearance issue is beyond the scope of this dissertation.
The next resonant peak shifts from 1890 Hz to 2120 Hz when Fz 0 is increased
from 0.45 kN to 2.67 kN. This mode represents a combination of tilting motions of the
shaft and translational (radial) vibrations. Note that at lower preloads, more than one peak
is seen in this frequency range. The amplitude at this mode also attenuates with
increasing Fz 0 and finally diminishes for Fz 0  2.22 kN. Measurements described so far
97
6
Fig. 3.9(c)
(a)
Accelerance (1/kg)
5
4
3
Fig. 3.9(b)
2
Increasing
preload
1
0
0
500
1000
1500
Frequency (Hz)
2000
2500
1.8
1.6
(b)
Accelerance (1/kg)
1.4
Increasing axial
preload
1.2
1
0.8
0.6
0.4
0.2
0
300
400
500
600
700
Frequency (Hz)
800
900
1000
14
Accelerance (1/kg)
12
(c)
Increasing axial
preload
10
8
6
4
2
0
2000
2200
2400
2600
2800
3000
Frequency (Hz)
3200
3400
3600
3800
Figure 3.9 Effect of axial preload on the measured axial accelerance magnitude spectra at
location 1. (a) 1-2500 Hz, (b) 300-1000 Hz, (c) 2000-3800 Hz. Key: Fz 0 ; (
), 0.45 kN;
(
), 0.67 kN; (
), 0.89 kN; (
), 1.11 kN; (
), 1.78 kN;
(
), 2.00 kN; (
), 2.22 kN; (
) 2.45 kN; (
98
), 1.33 kN; (
) 2.67 kN;
), 1.56 kN; (
0.9
0.8
Increasing axial
preload
Accelerance (1/kg)
0.7
0.6
0.5
Increasing
preload
0.4
0.3
0.2
0.1
0
0
500
1000
1500
Frequency (Hz)
2000
2500
Figure 3.10 Effect of axial preload on the measured axial accelerance magnitude spectra
at location 2. Key: Fz 0 ; (
), 0.45 kN; (
), 0.67 kN; (
1.33 kN; (
), 1.78 kN; (
), 2.00 kN; (
), 1.56 kN; (
2.67 kN;
99
), 0.89 kN; (
), 2.22 kN; (
), 1.11 kN; (
),
) 2.45 kN; (
)
seem to suggest an increase in the damping ratios with increasing Fz 0 which is similar to
the load-dependent damping model of Eq. (3.10).
Although spectra up to 2500 Hz are examined, yet another mode is found, as
shown in Figure 3.9(c), under axial force impacts that starts at 2400 Hz for 0.45 kN and
shifts up to 3500 Hz as Fz 0 increases. Unlike the previous modes, the vibration amplitude
at this mode significantly grows with increasing Fz 0 , resembling the behavior of loadindependent damping model as described by Eq. (3.9). This mode reaches its maximum
amplitude at Fz 0 =2.00 kN, and then amplitude starts to decay under higher Fz 0 without a
significant frequency shift. This shows that the accelerance spectra start to show some
deviations from the general trends at higher preloads.
Similar trends in the system modes (up to 2500 Hz) can also be observed from
location 2 measurements, as shown in Figure 3.10, however with different amplitudes. At
this location the mode occurring at 660 Hz at Fz 0 =0.45 kN (r = 1) has the highest
amplitude, and it still exhibits a shift towards right and down with increasing Fz 0 . The
peak occurring around 890 Hz (r = 2) is more visible at this location. Also, the natural
frequencies of axial shaft motion (r = 3) are found to be slightly lower (around 1500 Hz);
thus, its separation from the next resonant peak (r = 4) is more clear.
3.6.3 Effects of combined radial and moment loads
The effect of radial preloads ( Fx 0 ) and imposed bending moments ( M ym ) is
investigated following the same procedure. Before the application of a radial load, the
100
axial preload is set at 2.67 kN, which is also significantly affected by the applied radial
and moment loads. Then Fx 0 ranging from 0.89 kN to 4.44 kN is applied with 0.44 kN
increments using the hydraulic jack. Figure 3.11 illustrates the accelerance spectra for
each Fx 0 at location 1. The magnitude spectra are plotted up to 3000 Hz in Figure
3.11(a), and the phase spectra are given in Figure 3.11(b). Shifts in the transfer functions
are very clear with increasing Fx 0 , however the direction of the shifts in either frequency
or peak amplitude is mode-dependent. Shifts can also be seen in phase spectra.
A unique characteristic of this case is that the resonant curve occurring within
650-720 Hz range (r = 1 mode) shifts towards the left (instead of right), showing a
reduction in the natural frequency with increasing Fx 0 . This unexpected behavior could
be due to a “relief” in Fz 0 when M ym is simultaneously imposed on the bearing. Peaks at
other modes do not display this behavior as they continue to shift towards the right with
increasing Fx 0 due to an increase in the stiffness coefficients as expected. The second
mode, which is not very visible under pure axial preloads, now occurs between 1130 to
1260 Hz with considerable amplitudes that amplify with increasing radial loads, unlike
the trends seen at the first mode. The axial vibration mode (r = 3) now occurs within a
much narrower bandwidth (1500-1540 Hz), and the peak moves towards right and down
with increasing Fx 0 . The peak within 1960-2250 Hz band (r = 4 mode) behaves in a
manner similar to the r = 2 mode as it shifts right and upwards with increasing radial
load, however a shift in the natural frequency of this mode is much more significant with
an increase in Fx 0 .
101
One can observe similar behavior of the system modes from locations 2 and 3
A zz ( ) spectra, which are displayed in Figures 3.12(a) and 3.12(b), respectively. As
also observed from the axial preloading case, the relative amplitude of modes show
significant variations at each location. Note that the shifts in frequency or amplitude with
an increase in Fx 0 (and M ym ) are again uniform though mode-dependent. The only
exception to these monotonic trends can be observed at the r = 2 mode occurring between
1130 to 1260 Hz. Here, the accelerance amplitude seems to attenuate when Fx 0 increases
from 0.89 kN to 1.78 kN. Beyond 1.78 kN however, the peak amplitude begins to
significantly amplify and reaches its highest value at 4.44 kN. This behavior is observed
at all three measurement locations, and it is most likely due to complex dynamic
interactions within the double row bearing itself; it would require further analysis in a
future study.
3.6.4 Summary of experimental studies
For the first mode (occurring between 660 and 810 Hz) the changes in natural
frequencies and accelerance amplitudes (dB re 1 kg-1) with Fz 0 are summarized in Table
3.3 at measurement locations 1 and 2. The ‘change’ column shows the difference from its
value at the nominal axial preload ( Fz 0 = 0.45 kN). Note that changes do not follow a
linear variation; rather some are abrupt, and some are relatively insignificant. For
example, an increase of Fz 0 from 0.45 kN to 0.67 kN increases the natural frequency by
53 Hz. However, an increase from 0.67 kN to 0.89 kN only induces a 4 Hz difference. In
102
0.7
(a)
Increasing
radial preload
Accelerance (1/kg)
0.6
0.5
0.4
Increasing radial
preload
Increasing radial
preload
0.3
0.2
0.1
0
0
500
1000
1500
Frequency (Hz)
2000
2500
3000
2500
3000
4
(b)
3
Increasing radial
preload
Phase Angle (rad)
2
Increasing radial preload
1
0
-1
-2
-3
-4
0
500
1000
1500
Frequency (Hz)
2000
Figure 3.11 Effect of radial preload (and imposed moment load) on the measured axial
accelerance spectra at location 1. (a) Magnitude spectra, (b) Phase spectra. Key: Fx 0 ; (
), 0.89 kN; (
3.55 kN; (
), 1.33 kN; (
) 4.00 kN; (
) 1.78 kN; (
) 2.22 kN; (
) 4.44 kN;
103
) 2.67 kN; (
), 3.11 kN; (
),
0.5
(a)
0.4
Accelerance (1/kg)
Increasing
radial preload
Increasing radial
preload
0.3
Increasing radial
preload
0.2
0.1
0
0
500
1000
1500
Frequency (Hz)
2000
2500
3000
1.4
(b)
Accelerance (1/kg)
1.2
1
Increasing radial
preload
0.8
Increasing radial
preload
0.6
Increasing radial
preload
0.4
0.2
0
0
500
1000
1500
Frequency (Hz)
2000
2500
3000
Figure 3.12 Effect of radial preload (and imposed moment load) on the measured axial
accelerance magnitude spectra. (a) At location 2, (b) At location 3. Key: Fx 0 ; (
), 0.89
kN; (
), 1.33 kN; (
), 3.55
kN; (
) 4.00 kN; (
) 1.78 kN; (
) 2.22 kN; (
) 4.44 kN;
104
) 2.67 kN; (
), 3.11 kN; (
general, the rate of change in frequency or amplitude with respect to Fz 0 reduces with
increasing Fz 0 .
Similarly, the effect of combined Fx 0 and M ym loading on the natural frequencies
and vibration amplitudes of the r = 1 mode (occurring between 720-650 Hz) are
summarized in Table 3.4 for all three measurement locations. Recall that this is the only
mode whose natural frequencies reduce with increasing Fx 0 . Like the axial preload case
one can observe the changes in natural frequencies and vibration amplitudes follow
somewhat an abrupt manner.
The natural frequency maps (from 500 to 2500 Hz) are plotted with respect to Fz 0
and Fx 0 in Figs. 13(a) and 13(b), respectively. A second order linear regression fit is
applied to each mode, which fits the experimental data in a satisfactorily manner. Similar
results can also be generated for vibration amplitudes at various measurement locations.
Also, note that trends between Figure 3.13(a) and Figure 3.3(b) are similar.
105
Accelerance Amplitude (dB re 1 kg-1)
Nat. Freq. (Hz)
Location 1
Location 1
Location 2
Axial Preload
(kN)
Value
0.45
667
0.67
720
+53
2.4
-2.0
-2.8
-1.8
0.89
724
+57
1.9
-2.5
-5.0
-4.0
1.11
733
+66
1.1
-3.3
-5.4
-4.4
1.33
765
+98
0
-4.4
-6.8
-5.8
1.56
773
+106
-1.7
-6.1
-6.9
-5.9
1.78
797
+130
-3
-7.4
-6.7
-5.7
2.00
801
+134
-6.2
-10.6
-8.0
-7.0
2.22
803
+136
-6.5
-10.9
-8.3
-7.3
2.45
806
+139
-7.2
-11.6
-9.0
-8.0
2.67
806
+139
-7.5
-11.9
-9.5
-8.5
Change
Value
Change
4.4
Value
Change
-1.0
Table 3.3 Measured natural frequencies and peak amplitudes of the first vibration mode
at various axial preloads
106
Accelerance Amplitude (dB re 1 kg-1)
Natural Frequency (Hz)
Location 1
Location 1
Location 2
Location 3
Radial Load
(kN)
Value
0.89
716
1.33
709
-7
-8.1
-0.1
-11.8
-0.2
-4.6
-0.2
1.78
700
-16
-9.2
-1.2
-12.8
-1.2
-5.6
-1.2
2.22
686
-30
-11.4
-3.4
-15.2
-3.6
-7.3
-2.9
2.67
667
-49
-11.8
-3.8
-15.8
-4.2
-8.0
-3.6
3.11
666
-50
-11.2
-3.2
-15.2
-3.6
-7.4
-3.0
3.55
665
-50
-11.7
-3.7
-15.7
-4.1
-7.8
-3.4
4.00
662
-53
-11.8
-3.8
-15.8
-4.2
-7.9
-3.5
4.44
659
-53
-11.9
-3.8
-15.8
-4.2
-8.0
-3.5
Change
Value Change Value Change Value Change
-8.0
-11.6
-4.4
Table 3.4 Measured natural frequencies and peak amplitudes of the first vibration mode
at various radial preloads
107
2400
2200
(a)
Natural Frequency (Hz)
2000
r=4
1800
1600
1400
r=3
1200
r=2
1000
800
r=1
600
0.5
1
1.5
2
Axial Preload Measured by Load Washer (kN)
2.5
2400
2200
(b)
Natural Frequency (Hz)
2000
r=4
1800
r=3
1600
1400
1200
r=2
1000
r=1
800
600
0.5
1
1.5
2
2.5
3
3.5
Radial Preload Measured by Load Cell (kN)
4
4.5
Figure 3.13 Effect of preloads on the measured natural frequencies (between 500-2500
Hz). (a) Under axial preload, (b) Under radial preload (while the bearing is also axially
preloaded). Key: Discrete points, measurements at, ( ), r = 1; (
(
), r = 2; (
), r = 4; continuous line, a second order curve fit of measured data.
108
), r = 3;
3.7 Validation of the Bearing Stiffness Matrix
3.7.1 Comparison of natural frequencies
The five degree-of-freedom analytical formulation of Figure 3.2 is utilized once
again with damping coefficients based on measurements to model the experiment of
Figure 3.8. Only the axial preload case is analyzed as the radial preload case with the
hydraulic jack affects the boundary conditions of the shaft-bearing system; also, it could
impose rotational stiffness terms about the z-axis (according to Figure 3.1) which the
simple model employed here is incapable of estimating. Sample results for an
intermediate axial preload ( Fz 0 = 1.56 kN) will be illustrated.
Table 3.5 lists measured (from experiment of Figure 3.8) and predicted (using five
degree-of-freedom model of Figure 3.2) natural frequencies of the system. The
predictions match well with the measurements with small errors. When there is no
external radial or moment load applied in the model, natural frequencies at r = 1 and r =
2, as well as r = 4 and r = 5, are repeated (as mentioned earlier). With an application of a
slight amount of load in the x-direction (Fxm = 0.3 kN), these repeated roots separate as
shown in the fourth column of Table 3.5. Now, the second, third, and fifth natural
frequencies show a better match with measurements, but the estimation of the first natural
frequency deviates further from experiments.
The last column of Table 3.5 shows a case when a diagonal Kb (with zero crosscoupling terms) were to be used in five-degree-of-freedom model. In this case the natural
frequency calculations deviate significantly from measurements. These results clearly
109
Natural Frequencies (Hz)
Modal Index
(r)
Experiment
5 DOF Model
(original)
5 DOF Model
(with radial load)
5 DOF Model
(with diagonal Kb)
1
773
763
711
1422
2
894
763
864
1422
3
1636
1583
1617
1565
4
2049
2046
2059
1657
5
2246
2046
2127
1657
Table 3.5 Measured and predicted natural frequencies of the experiment (of Figure 3.8),
with several formulations of the analytical model of Figure 3.2.
110
highlight the importance of cross-coupling stiffness terms of Kb, and verify a need for
incorporating the bearing stiffness matrix when analyzing the vibration transmission
paths [3.7].
3.7.2 Comparison of radial accelerance spectra
Next, measured A xx ( ) (with impact location f1x and accelerometer location 1)
are compared with predicted A xx ( ) by the five degree-of-freedom model to further
validate the stiffness model at a given preload and to estimate the damping coefficients.
Since r = 1 and 2 as well as r = 4 and 5 overlap in the absence of radial or moment loads,
and r = 3 mode does not affect A xx ( ) , the dimension of the analytical model can be
conveniently reduced from five to two that describes translation along the x-axis (x) and
the rotation about the y-axis (  y ). Accordingly, the system matrices of this two degreeof-freedom model are written as follows:
M
M 2   11
 M 21
M 12   m 0 


M 22   0 I yy 
C
C 2   11
C21
C12   cxx

C22   0
K
K 2   11
 K 21
K12   k xx

K 22   k y x

0 
c y y 

k x y 

k y y 

(3.14a)
(3.14b)
(3.14c)
where m = mS = 5 kg, and I yy = 0.015 kgm2 (same as before). The elements of K 2 and
C2 are identical to their counterparts in the five degree-of-freedom model.
111
The reduced order model facilitates the analytical derivation of the accelerance.
The closed form solution can be helpful in acquiring a better understanding of the effects
of stiffness elements (and consequently the bearing preloads) on the vibration
characteristics of the shaft-bearing system. The accelerance matrix of the two degree-offreedom model is written as:
 A ( ) A12 ( )   A xx ( ) A x y ( ) 
A 2 (    11


 A 21 ( ) A 22 ( )   A y x ( ) A y y ( ) 
(3.15)
The elements of A2 (  are explicitly described below in terms of the system parameters
(where M12 = M 21 =0, C12 = C21 =0, K12 = K21 ):
 4 M 22  j 3C22   2 K 22
A11 ( ) 
D( )
(3.16a)
 2 K12
A12 ( )  A21 () 
D( )
 4 M11  j 3C11   2 K11
A22 ( ) 
D( )
(3.16b)
(3.16c)
Where:
D( )   4 M 11M 22  j 3 ( M 11C22  C11M 22 )   2 ( M 11 K 22  K11M 22  C11C22 ) 
j (C11 K 22  K11C22 )  K11 K 22  K12 2
(3.17)
Note that these accelerance terms are dictated by all four stiffness elements K11 = k xx ,
K12  K21  kx y and K 22 = k y y . Given the above expressions, one can easily symbolically
112
determine the accelerance magnitudes. For instance, the magnitude of radial accelerance
A
xx
( )  is written in the form:
A xx ( )  A11 ( ) 
N11 ( )
D( )
(3.18a)
where
N11 ( )  M 222  8  (2 K 22 M 22  C22 2 ) 6  K 22 2 4
(3.18b)
and
8
D()  (M112 M 222 )  (C112 M 222  C222 M112  2K22 M112 M 22  2K11M11M 222 )6 
(C112C222  2C112 K22 M 22  2C222 K11M112  K112 M 222  4K11K22 M11M 22  K222 M112  2K122 M11M 22 ) 4 
(C112 K222  2K122C11C22  C222 K112  2M 22 K112 K22  2M11K11K222  2K122 M 22 K11  2K122 M11K22 ) 2 
1/2
(K112 K222  K124  2 K11K22 K122 )
(3.18c)
Measured and predicted A xx ( ) are compared in Figure 3.14(a). Observe that
the two degree-of-freedom model accurately estimates the two system resonances (in
terms of both frequencies and amplitudes) by a judicious selection of the damping
coefficients as C11 = cxx  4350 Ns/m and C22 = c y y = 7.25 Nms/rad. In the context of a
two degree-of-freedom model, such damping values correspond to the modal damping
ratios of  1  0.074 and  2  0.025 . The third curve in Figure 3.14(a) is based on the
five degree-of-freedom model that includes components of both the radial and axial
accelerances. It could be empirically obtained by the following relation, where the factor
of  = 0.18 is chosen to better explain the measurements in Figure 3.14(a); this relation
113
2.5
(a)
Accelerance (1/kg)
2
1.5
1
0.5
0
0
2.5
500
1000
1500
Frequency (Hz)
2000
2500
500
1000
1500
Frequency (Hz)
2000
2500
(b)
Accelerance (1/kg)
2
1.5
1
0.5
0
0
Figure 3.14 Comparison of radial accelerance magnitude spectra
kN. Key: (
), Measured at accelerometer location 1; (
of-freedom model; (
A
xx
( )  for Fz 0 =1.56
), predicted by two degree-
), predicted by five degree-of-freedom model including
participation of the r = 3 mode. (a) Prediction with original I yy  0.015 kgm 2 , (b)
Prediction with modified I yy  0.0154 kgm2
114
is used to account for the participation of axial mode when the system is excited by
applying a force on the shaft in the radial direction:
A obs ( )  A xx ( )   A zz ( )
(3.19)
The predicted results can be further improved by eliminating a small discrepancy in
the second natural frequency. If the equivalent moment of inertia of the shaft-bearing
system rotating about the bearing center were to be chosen as I yy = 0.0154 kgm2 instead
of the calculated value ( I yy = 0.015 kgm2), the resonance peaks overlap at 2046 Hz as
shown in Figure 3.14(b). Overall, predictions obtained of the analytical model, which
utilizes Kb, match very well with measurements.
3.8 Conclusion
Three key contributions of analytical and experimental studies emerge. First, the
effects of axial preloads are analytically investigated on a shaft-bearing assembly
containing a double row angular contact ball bearing. In these analyses the bearing is
represented via a new stiffness matrix (Kb). The vibration characteristics of face-to-face,
back-to-back, and tandem arrangements are evaluated on a comparative basis. Analyses
show that the natural frequencies of back-to-back arrangement that are related to radial
and tilting motions occur at higher frequencies due to higher moment stiffness terms.
Similarly, the axial vibration frequency of the tandem arrangement is higher than the
other two arrangements due to high axial stiffness. In these analyses two alternate viscous
damping models are employed. The first one is a preload-independent non-proportional
viscous damping model, whose values are experimentally determined for an intermediate
115
preload and kept constant for all preloads. This model results in higher peak amplitudes
with increased axial preloads. The second model is a preload-dependent, proportional
viscous damping model, which yields lower vibration amplitudes with increased
preloads.
Second, the effects of bearing preloads on the shaft-bearing dynamics are
experimentally investigated for an automotive wheel-hub assembly containing a double
row angular contact ball bearing (with back-to-back arrangement) under two different
preloading mechanisms. The axial preload is applied through the shaft by tightening the
lock nuts, and radial preload applied via a static shaft load using a hydraulic jack which
also imposes moment load on the bearing. Measurements show that each vibration mode
responds somewhat differently to changes in axial or radial preloads. In general, the
natural frequencies of the shaft-bearing system increase with an increase in the bearing
preload due to the increase in the bearing stiffness elements. The first mode of the shaftbearing system, however, does not follow this rule and displays an opposite trend when
the system is radially preloaded. The resonant amplitudes, on the other hand, attenuate at
certain modes, but amplify at other modes with increasing axial or radial preloads. These
results suggest a mode-dependent damping mechanism which requires further
investigation. Although two alternate viscous damping mechanisms (with preloadindependent non-proportional and preload dependent proportional models) are
investigated, measurements show that the bearing damping can not be well represented
by either of these mechanisms alone. More accurate damping models should be
established in future studies.
116
Third, the Kb formulation is experimentally validated at an intermediate axial
preload ( Fz 0 = 1.56 kN). The calculated natural frequencies of the analytical model
utilizing Kb match well with measurements. Also, the predicted and measured radial
accelerance spectra show a very good correlation when damping coefficients from
experimental data are utilized. However, the damping characteristics at several modes are
yet to be fully understood. Future work should resolve this issue, by employing amplitude
and/or frequency dependent damping model, or by using nonlinear contact damping
formulations. Finally, experimental methods of this chapter could be extended to study
other bearing types.
117
References for chapter 3
3.1 T. A. Harris, Rolling Bearing Analysis, J. Wiley, New York, 2001.
3.2 F.P. Wardle, S.J. Lacey, S.Y. Poon, Dynamic and static characteristics of a wide
speed range machine tool spindle, Precision Engineering 5 (4) (1983) 175-183.
3.3 B.J. Stone, The state of art in the measurement of the stiffness and damping of
rolling element bearings, Annals of the CIRP, 31 (2) (1982) 529-538.
3.4 J. Kraus, J.J. Blech, S.G. Braun, In situ determination of rolling bearing stiffness and
damping by modal analysis, Transactions of the American Society of Mechanical
Engineers, Journal of Vibration, Acoustics, Stress, and Reliability in Design, 109
(1987) 235-240.
3.5 M. F. White, Rolling element bearing vibration characteristics: effect of stiffness,
Transactions of the American Society of Mechanical Engineers, Journal of Applied
Mechanics 46 (1979) 677-684.
3.6 N. Akturk, M. Uneeb, R. Gohar, The effects of number of balls and preload on
vibrations associated with ball bearings, Journal of Tribology 119 (1997) 747-752.
3.7 T.C. Lim, R. Singh, Vibration transmission through rolling element bearings, part I:
bearing stiffness formulation, Journal of Sound and Vibration 139 (2) (1990) 179–
199.
118
3.8 T.C. Lim, R. Singh, Vibration transmission through rolling element bearings, part II:
system studies, Journal of Sound and Vibration 139 (2) (1990) 201–225.
3.9 S.A. Spiewak, T. Nickel, Vibration based preload estimation in machine tool spindles,
International Journal of Machine Tools & Manufacture 41 (2001) 567-588
3.10 T.J. Royston, I. Basdogan, Vibration transmission through self-aligning (spherical)
rolling element bearings: theory and experiment, Journal of Sound and Vibration, 215
(5) (1998) 997-1014
3.11 P. Cermelj, M. Boltezar, An indirect approach investigating the dynamics of a
structure containing ball bearings, Journal of Sound and Vibration 276 (2004) 401–
417.
3.12 B.R. Jorgensen, Y.C. Shin, Dynamics of spindle-bearing systems at high speeds
including cutting load effects, Journal of Manufacturing Science and Engineering,
120 (1998) 387-394
3.13 M. Alfares, A.A. Elsharkawy, Effects of axial preloading of angular contact ball
bearings on the dynamics of a grinding machine spindle system, Journal of Materials
Processing Technology 136 (2003) 48-59
3.14 C. Bai, H. Zhang, Q. Xu, Effects of axial preload of ball bearing on the nonlinear
dynamic characteristics of a rotor-bearing system, Nonlinear Dynamics 53 (2008)
173-190
3.15 J.L. Stein, J.F. Tu, A state-space model for monitoring thermally induced preload in
anti-friction spindle bearings of high-speed machine tools, Transactions of the
119
American Society of Mechanical Engineers Journal of Dynamic Systems,
Measurement and Control, 116 (1994) 372-386
3.16 C.W. Lin, J.F. Tu, J. Kamman, An integrated thermo-mechanical-dynamic model to
characterize motorized machine tool spindles during very high speed rotation,
International Journal of Machine Tools and Manufacture, 43 (10) (2003) 1035-1050
3.17 Y. Cao, Y. Altintas, A general method for the modeling of spindle-bearing systems,
Transactions of the American Society of Mechanical Engineers, Journal of
Mechanical Design, 126 (6) (2004) 1089-1104
3.18 H.Q. Li, Y.C. Shin, Analysis of bearing configuration effects on high speed spindles
using an integrated thermomechanical spindle model, International Journal of
Machine Tools and Manufacture, 44 (2004) 347-364
3.19 X. Hernot, M. Sartor, J. Guillot, Calculation of the stiffness matrix of angular contact
ball bearings by using the analytical approach, Transactions of the ASME, Journal of
Mechanical Design, 122 (2000), 83-90.
3.20 D.H. Choi, K.C. Yoon, A design method of an automotive wheel-bearing unit with
discrete design variables using genetic algorithms, Journal of Tribology, 123 (1)
(2001) 181-189
120
CHAPTER 4
CRITICAL EXAMINATION OF STIFFNESS CALCULATIONS USING
ANALYTICAL AND COMPUTATIONAL METHODS
4.1 Introduction
Finite element methods have been utilized in the analysis of shaft-bearing systems
for several decades [4.1-4.12]. In most of these studies, bearings are represented as
translational nonlinear stiffnesses to represent the nonlinear nature of load-deflection
relations of rolling element bearings. El-Saeidy [4.11], for instance, incorporated a finite
element code to investigate a shaft-bearing system and focused on bearing clearances and
nonlinearities. Cermelj and Boltezar [4.12] utilized finite element analysis to investigate
vibration transmission through ball bearings. Several researchers focused on the problem
of contacting bodies within the context of finite element analysis [4.13-4.17], though
Vijayakar’s previous work [4.18-4.20], which has been implemented into a commercial
and widely used code [4.21], will be a main reference for this study. Finite element based
contact mechanics models offer an alternative to Hertzian contact based analytical
approaches (which relates the load carried by a rolling element Qij with its resultant
121
elastic deformation  ij as described by Eq. (4.1), n = 1.5 for ball bearings) in the analysis
of rolling element bearings.
Qij  K n  ij 
n
(4.1)
In addition to finite element based methods, several computational codes provide
powerful tools in the analysis of rolling element bearings. The code utilized in chapter 2
[4.22] for the verification of the stiffness model is one of the most well-known of such
codes. Widely used finite element codes do not directly calculate Kb elements; and even
more bearing-specific codes typically do not output all elements of Kb.
4.2 Problem Formulation
The main goal of this study is to comparatively evaluate the stiffness coefficients
using analytical and computational tools and critically examine some of the assumptions
of the new analytical formulation. Two analytical and two computational methods, as
discussed in detail in the next section, are utilized: (i) Method A: Kb formulation for
double row angular contact ball bearings (as outlined in chapter 2); (ii) Method B: Kb
formulation for single row bearings developed by Lim and Singh [4.23]; (iii) Method C:
A finite element/contact mechanics based commercial code (Calyx) developed by
Vijayakar [4.21] that is specialized for the analysis of gearing/bearing (2D or 3D) multibody systems; and (iv) Method D: A commercial code (RomaxDesigner) which is also
especially useful for the design and analysis of bearing/gearing systems [4.22].
122
The problem of an externally loaded rigid shaft supported by two or more rolling
element bearings is statically indeterminate by its nature, which is difficult (if not
impossible) to solve by hand, especially if the loading is multidimensional.
Computational tools (such as Methods C and D) easily provide the solution for the
statically indeterminate problem by calculating bearing loads and deflections which may
be used to calculate Kb by using analytical methods. Methods C and D (especially
Method C) also allow detailed bearing modeling in comparison to analytical models.
Computational methods, however, also have some limitations. For instance although
Method C has double row modeling capabilities for tapered roller bearings, its current
version does not allow modeling of double row angular contact ball bearings as an
integrated unit when the bearing rows are aligned in opposite directions (i.e. face-to-face
or back-to-back); thus, two single row bearings must be modeled. Also, bearing stiffness
coefficients may not be directly calculated with Method C, thus, indirect methods such as
finite difference approximation must be used. Method D, on the other hand, only outputs
the diagonal elements of Kb.
The specific objectives are as follows: 1. Justify some assumptions of the
proposed stiffness model (Method A) using a finite element based contact mechanics
code (Method C); 2. Conduct a study on the stiffness modeling of double row (or duplex)
angular contact ball bearings in a shaft bearing system, and comparatively evaluate the
modeling approaches between a double row bearing vs. two single row bearings from
statics and bearing stiffness perspective; such studies have not been conducted by
previous investigators to the best of our knowledge; 3. Investigate the finite difference
123
approximation used to calculate the bearing stiffness elements; 4. Comparatively evaluate
the four calculation methods (A, B, C, and D) by critically examining bearing loads,
deflections, and stiffness elements for various loading cases utilizing double row and two
single row modeling approaches; 5. Evaluate the results obtained by the four calculation
methods with respect to modal measurements of chapter 3; 6. Conduct preliminary
experiments with a faulty bearing to evaluate the effect of bearing damage on the modal
characteristics of a shaft-bearing system and to investigate the effect of bearing preloads
on a comparative basis with a healthy system.
4.3 Analytical and Computational Methods
4.3.1 Method A: Gunduz and Singh’s model
The stiffness matrix model for double row angular contact ball bearings derived in
chapter 2 is designated as Method A.
4.3.2 Method B: Lim and Singh’s model
The stiffness matrix model developed by Lim and Singh [4.23] in 1989 for single
row bearings is designated as Method B. (i.e. Method A is an extension of Method B to
double row angular contact ball bearings). This method is more commonly known as the
‘REBM Method’ in industry (REBM stands for Rolling Element Bearing Matrix), which
is a code [4.24] that implements the analytical expressions outlined in [4.23] to a user
friendly environment through a graphical user interface. It prompts the user whether the
bearing is a ball or roller type, and whether the input vector is fm or qm. If qm is the input,
124
the program calculates and outputs the Kb and fm by direct substitution. If fm is the input,
the program first calculates qm by solving the nonlinear system equations (a reasonable
initial guess for qm is necessary), then calculates and outputs Kb via direct substitution.
However, self-written MATLAB© codes are utilized in this dissertation instead
of using REBM in the implementation of Method B for three reasons: First, the REBM
does not permit a negative value of the contact angle. A negative contact angle provides
significant ease for systems containing two (or more) single row bearings aligned in the
opposite directions (i.e. face-to-face and back-to-back arrangements). Second, the
nonlinear system of equations may be solved with a simplex algorithm as described by
Lagarias et al. [4.25] rather than using a Newton-Raphson method as utilized by REBM
[4.24]. It is found that the prior algorithm might provide better results for certain
combined loading cases. Third, a better control on solution tolerances can be achieved.
4.3.3 Method C: Finite element/contact mechanics based commercial code (Calyx)
The unique algorithm utilized by Method C [4.18-4.21] combines a semianalytical finite element approach with detailed contact modeling of the contact regions.
Method C has certain advantages which have not been offered by other methods available
in the field. First, it is excellent for finite element modeling and analysis of
gearing/bearing systems due to its unique contact modeling capabilities. Second, it does
not make assumptions such as Hertzian contact, rigid ring, and rigid shaft that are made
by most of the theoretical models. Also, the program has excellent post-processing and
3D visualization capabilities which are not offered by other available commercial codes.
125
Figure 4.1 shows a screenshot from the 3D visualizer of the code that illustrates a shaft
supported by two single row angular bearings found in back-to-back arrangement
subjected to a combined load.
Besides its numerous advantages, the program also has certain drawbacks from
the perspective of stiffness modeling. First, the current version does not contain a ball
bearing menu, thus, the curvature of the rollers of roller-type bearings must be adjusted to
represent balls and obtain point-type contacts, which might be difficult and timeconsuming for inexperienced users. Second, the contact angles of multiple row ball-type
bearings may not be controlled separately, thus, two separate single row bearings have to
be modeled when the bearing rows are aligned in opposite directions. Also, Method C
does not directly calculate elements of Kb, therefore one has to utilize indirect calculation
methods to determine stiffness elements.
The contact modeling between the rolling elements and bearing races is an
iterative/evolving process as there are many variables to control. Figure 4.2(a) illustrates
a typical loaded rolling element of an angular contact ball bearing that forms an elliptical
contact zone with bearing races (defined as contact ellipse with radii ‘ao’ and ‘bo’). A
parabolic load distribution along both dimensions of the ellipse is typical as shown in
Figure 4.2(b-c). To achieve such desired contact characteristics the 3D visualizer of the
program, as shown in Figure 4.3, can be effectively used as a part of the iterative process.
To achieve proper contact modeling, geometric parameters of the bearing (some
of which are not so well-defined, such as the curvature and axial position of the bearing
races), and 4 important contact grid parameters must be properly controlled,
126
Fz=3 kN
Fx=1 kN
Figure 4.1 A screenshot from the 3D visualizer of Calyx [4.21] that illustrates a shaft
supported by two single row angular bearings found in a back-to-back arrangement
subjected to a combined load as shown on the figure. The stress distribution on the rolling
elements and the load lines that pass through the contact points may be seen. The outer
rings of the bearings are not shown for illustrative purposes.
127
which are defined as follows: 1. Separation tolerance (Stol): If the unloaded separation
between two points of contacting bodies (as shown in Figure 4.2) is greater than Stol, then
the entire contact grid is eliminated from further consideration; 2. The number of contact
grids in the axial direction (2N+1): The contact surface is divided into 2N+1 grid cells in
the axial direction about a symmetry axis where N is a user-defined input; 3. The number
of contact grids in the profile direction (2M +1): Similarly, the contact surface is divided
into 2M +1 grids in the profile direction (that is normal to axial direction) about an axis of
symmetry where M is a user-defined input; and 4. The dimension of the grid cells in the
profile direction ( s ): This controls the dimension of the grid cells such that the width of
the grid is (2M + 1)Δs. Figure 4.4 schematically shows the definitions of M, N, and s
with an example case with N = 3 and M = 1. Note that choosing a correct width for grid
cells is crucial for obtaining the correct contact pressures. Using a too wide grid for a
fixed M may result in loss of resolution, whereas a too narrow grid (as shown in Figure
4.4(b)) may cause the contact zone to become truncated resulting in artificially high
contact pressures.
4.3.4 Method D: A commercial code for gearing/bearing systems (RomaxDesigner)
Method D [4.22] has been utilized earlier in chapter 2 for the preliminary
verification of the diagonal elements of proposed Kb. The program is a blackbox tool
from the user point of view (as the algorithms are not published), however it is extremely
useful from the perspective of bearing stiffness modeling due to the following reasons:
First, it contains a bearing library which has a large collection of commercial single and
128
(a) Contact ellipse (b)
Stol
2b0
2a0
2b0
2a0
Figure 4.2 Illustration of the contact of a rolling element of an angular contact ball
bearing. (a) Formation of the contact ellipse under loaded conditions. (b) Parabolic stress
distribution along both dimensions of the contact surface.
129
(a)
(b)
Figure 4.3 Illustration of the parabolic contact pressure distribution on the rolling
elements with Calyx© [4.21]. (a) Several rolling elements shown within the inner and
outer races. (b) A zoomed view to one rolling element.
130
(a)
(2 M  1) s
s is sufficient
(b)
(2 M  1) s
s is not sufficient
Contact zone is truncated
Figure 4.4 Illustration of contact grid variables (M, N, and s ) with an example case with
N=3 and M=1, resulting in 7 grids in axial and 3 grids in profile direction. Blue dots
represent contact grids, and elliptical area represents the contact zone. (a) Grid width is
sufficiently large. (b) Grid width is too small, contact zone is truncated.
131
Figure 4.5 Several screenshots from the 3D visualizer of RomaxDesigner© [4.22] that
illustrates a shaft supported by a double row angular bearing.
132
double row bearings from the catalogs of major bearing manufacturers. The custom
bearing design option is also available. Second, it has both single row and double
row/duplex (face-to-face, back-to-back, or tandem) bearing modeling capabilities. Third,
it calculates and outputs the diagonal stiffness elements of the bearings, possibly using
some analytical method that utilizes Hertzian theory. Figure 4.5 illustrates some
screenshots from the 2D and 3D visualizer of the code.
4.4 Justification of Some Assumptions of Analytical Model (Method A) using Finite
Element Model (Method C)
4.4.1 Justification of rigid ring assumption
To justify the rigid ring assumption of the proposed K b , a model of a simple shaft
supported by two angular contact ball bearings arranged back-to-back whose outer races
are connected to the ground is created with Method C. The properties of the bearing are
chosen according to the double row bearing analyzed in chapter 2 as given in Table 2.1.
A combined load which imposes axial, radial and moment load on both bearings is
applied. The magnitude of deformations of rolling elements and bearing races around the
contact zones ( cr ) are compared with those of that are away from the contact zones
( ucs ) using the 3D visualizer of the code (r and s are the indices of finite element grids).
The locations and magnitudes of the elastic deformations on the rolling elements and
bearing rings are shown in Figure 4.6 in colors. As seen from the figure the elastic
deformations only occur at the contact points, while the sections away from the contact
133
(a)
(c)
(b)
Deformations occur only
at the contact points
Deformations occur only at
the contact points
(d)
Figure 4.6 Locations and magnitudes of elastic deformations (in colors) of rolling
elements and bearing rings for a combined load case. (a-b) Elastic deformation of rolling
elements; (c) deformation of outer rings; and (d) deformation of inner rings.
134
points remain undeformed, such that  cr   ucs  0 . This justifies that the structural
deformations of the bearing rings may be neglected.
Note that one should not expect to obtain the same results when a virtually rigid
material (with a very high elastic modulus ‘Emod’) is selected for the bearing rings. Using
a very large Emod not only makes bearing rings undeformable, but also changes the
characteristics of the contact, which corresponds to a higher Kn value. Rigid shaft
assumption, however, can be justified in this manner as discussed in the next subsection.
4.4.2 Justification of rigid shaft assumption
Using the same shaft-bearing system, the deformation zones of the system are
investigated using Method C. As seen in Figure 4.7(a), no deformation zones have been
observed on the shaft, indicating the rigid shaft assumption is realistic. The maximum
shear stresses also occur around the contacts between the rolling elements and bearing
rings as seen in Figure 4.7(b).
An alternative way of verifying this assumption can be achieved by increasing the
elastic modulus (Emod) of the shaft ten-fold and comparing the bearing deflections with
those of the original case. By assuming the shaft rotates with 1000 rpm, radial deflections
2
1
2
,  xm
) and mutual angular displacement of both bearings (  1ym   ym
)
of each bearing (  xm
are calculated between 0-10 ms with 1 ms increments by static analysis. Here, time axis
only corresponds to different angular positions of the bearing since the analysis is static.
The results illustrated in Figure 4.8(a-c) shows that the rigid body deflections of the
bearings does not change if a rigid shaft is used instead of a steel (elastic) shaft.
135
(a) Fz=3 kN
Fx=1 kN
No displacement zone on the shaft
(b)
No shear stresses on the shaft
Figure 4.7 Elastic deformation and maximum shear stresses distribution along the shaftbearing system for a combined load case including the shaft. (a) Total elastic
deformations, and (b) maximum shear stress distribution.
136
-6
(a)
xm (m)
-6.05
1
-6.1
-6.15
-6.2
-16.7
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
Time (ms)
7
8
9
10
(b)
xm (m)
-16.8
2
-16.9
-17
-17.1
0
(c)
-0.535
1,2
ym (mrad)
-0.53
-0.54
-0.545
0
Figure 4.8 Rigid body deflections of two bearings in radial and angular directions with
1
rigid and elastic shafts with respect to time assuming a shaft speed of 1000 rpm. (a)  xm
2
2
vs. t; (b)  xm
vs. t; and (c)  1ym   ym
vs. t. Key: (
MPa); (
), rigid shaft (Emod = 2050 MPa).
137
), elastic (steel) shaft (Emod = 205
4.5 Analysis of a Double Row bearing vs. Two Single Row Bearings
Two approaches may be practiced in modeling of double row/duplex angular
contact ball bearings: Modeling two single row bearings vs. modeling an integrated
double row bearing as shown in Figure 4.9. The first approach essentially yields 2
i
i
i
i
i
i
i
i
i
,  ym
,  zm
,  xm
, Fym
, Fzmi , M xm
separate qim  { xm
,  ym
}T and fmi  {Fxm
, M ym
}T vectors and 2
separate K ib matrices (i.e. one for each individual bearing) where:
 k xxi k xyi k xzi

k yyi k yzi


k zzi
K ib  

 symmetric


k xi x
k yi  x
k zi x
i
k x x
k xi y 

k yi  y 

k zi y 

ki x y 
ki y y 

 i  1, 2 
(4.2)
D
D
D
D
D T
,  ym
,  zm
,  xm
The second approach, on the other hand, results in a single qmD  { xm
,  ym
}
D
D T
and fmD  {FxmD , FymD , FzmD , M xm
, M ym
} vector and a K bD matrix for the double row bearing
where:
 k xxD k xyD k xzD

k yyD k yzD


k zzD
K bD  

 symmetric


138
k xD x
k yD x
k zD x
D
k x x
k xD y 

k yD y 

k zD y 

kDx y 
kDy y 
(4.3)
(a)
TWO SINGLE ROW BEARINGS
Applicable methods: B, C and D
lSHAFT
zG
x
BRG2
BRG1
ext
Fz
O
y
z
G
ext
Fx
zBRG1
zBRG2
(b)
ONE DOUBLE ROW BEARING
Applicable methods: A and D
lSHAFT
zG
DOUBLE
ROW BRG
x
ext
Fz
O
y
z
G
ext
Fx
Figure 4.9 Illustration of two approaches that can be practiced in modeling of double
row/duplex angular contact ball bearings: (a) Modeling two single row bearings, and (b)
modeling an integrated double row bearing. Here, the rigid shaft is subjected to radial
( Fxext ) and axial ( Fzext ) external loads at point O.
139
By definition, two bearing rows positioned next to each other that share common
inner and outer rings correspond to a double row bearing, whereas split rings define two
single row bearings; however, there are always exceptions to this generalization. For
example, many double row designs have a solid outer ring but a split inner ring (for
preloading), such as most of the wheel bearing units. Duplex bearings are also obtained
by assembling two single row bearings together; however, their functionality is identical
to double row bearings (for instance, Method D treats duplex bearings as a double row
bearing). As long as the shaft is sufficiently rigid, and the inner rings of the bearings are
rigidly connected to the shaft (i.e. rings of the two bearings do not move independently
except for initial preloading), both modeling approaches should yield equivalent results
independent of their inner and outer rings being solid or split. However, as far as bearing
loads deflections and stiffnesses are concerned, two modeling approaches (with double
row vs. two single rows) would yield completely different values. This issue brings up
the question whether a two single row bearing model can be used to represent a double
row model, and if so, under what conditions are integrated double row models necessary
or more helpful. This section investigates these questions by analyzing bearing loads,
deflections, and stiffnesses. Method D is extensively utilized as it has both single and
double row modeling capabilities.
140
4.5.1 Examination of bearing deflections
4.5.1.1 Two single row bearings
Consider an externally loaded shaft supported by two single row angular contact
ball bearings which may organize in face-to-face, back-to-back, or tandem arrangements.
Analysis with Method C reveals that the axial deflection and angular displacements of
2
1
2
1
2
,  xm
,  1ym   ym
) independent of the
both bearings are identical (i.e.  zm
  zm
  xm
external load or bearing arrangement. The radial displacements, however, are not equal
2
1
2
and  1ym   ym
) unless the external loading forms a special case. Pure axial
(i.e.  xm
  xm
2
1
2
 0 or symmetrical radial loading (say in xloading where  xm
  xm
 0 and  1ym   ym
1
2
2
direction) without an imposed moment where  xm
  xm
 0 and  1ym   ym
 0 are some
examples of such special cases. The same analysis with Method D yields similar but
slightly different results such that the axial deflections and angular displacements of two
1
2
1
2
,  xm
,
bearings are almost equal but not identical, unlike Method C (i.e.  zm
  zm
  xm
2
1
2
 1ym   ym
). The radial displacements, however, are again not equal (i.e.  xm
and
  xm
2
 1ym   ym
) except in special cases.
4.5.1.2 Double row bearing
If two single row bearings are replaced with a double row bearing with its
geometric center being at the middle of those of the two rows (i.e. zG  ( z BRG1  z BRG 2 ) / 2
), analysis with Method D shows the radial displacements of the double row bearing is
141
equal to the arithmetic average of those of the two single row bearings. Axial and angular
displacements of the double row bearing are identical to those of the two single row
bearings:
D
 xm


D
ym

1
2
 xm
  xm
2
2
 1ym   ym
2
(4.4a)
(4.4b)
D
1
2
 zm
  zm
  zm
(4.4c)
D
1
2
 xm
  xm
  xm
(4.4d)
D
1
2
 ym
  ym
  ym
(4.4e)
The radial deflections of the double row bearing may also be expressed in terms
of mean deflections of one of the two single row bearings (or vice versa) as shown by Eq.
4.5(a-d). Note that Eqs. 4.4(a-b) can be easily obtained from Eq. 4.5(a-d).
D
1
1
 xm
  xm
 e sin(  ym
);
D
1
1
 ym
  ym
 e sin(  xm
);
D
2
2
 xm
  xm
 e sin(  ym
)
D
2
2
 ym
  ym
 e sin(  xm
)
(4.5a-b)
(4.5c-d)
where e  zBRG 2  zG  zG  z BRG1 according to Figure 4.9 ( zBRG 2  zG  z BRG1 ).
4.5.1.3 Example case
Consider a 148 mm long, uniform solid shaft with a diameter of 50 mm, z-axis
being its rotational axis according to the global coordinate system attached at one end of
the shaft at point ‘O’ as shown in Figure 4.9(a-b). First, two single row bearings are
142
placed on the shaft (in all three arrangements) with z-coordinates z BRG1 = 64 mm, zBRG 2 =
84 mm. These two bearings are then replaced with a double row bearing with its
geometric center located at zG = 74 mm ( zG  ( z BRG1  z BRG 2 ) / 2 with e = 10 mm). The
kinematic properties of the bearing(s) are given in Table 2.1. The external load is applied
at point O with magnitudes Fxext  1 kN and Fzext  3 kN. Fxext imposes a bending moment
i
( M ym
) on both bearings. A combined asymmetrical loading case ensures that the analysis
is general and does not form a special case.
The bearing deflections occurring in x (radial), z (axial), and  y (tilting)
directions (  xm ,  zm , and  ym ) are calculated by Method D for back-to-back, face-to-face,
and tandem arrangements. As seen from Table 4.1(a-c) the deflections of the double row
bearing are almost identical to the arithmetic average of the deflections of two single row
bearings verifying Eqs. 4.4(a-e), and 4.5(a-d). One should also notice the significant
difference between the deflections among the three arrangements. For instance,  ym of
DB arrangement is much lower than the other two arrangements due to its higher moment
rigidity while  zm of DT arrangement are significantly smaller than the other two
arrangements as expected.
143
(a)
Back-to-Back Arrangement
Deflection
Bearing 1
Bearing 2
Average*
Double Row
δxm (μm)
-4.32
-13.06
-8.69
-8.63
δzm (μm)
21.26
21.28
21.27
21.27
βym (mrad)
-0.444
-0.439
-0.441
-0.440
(b)
Face-to-Face Arrangement
Deflection
Bearing 1
Bearing 2
Average*
Double Row
δxm (μm)
5.93
-27.05
-10.56
-10.72
δzm (μm)
19.7
19.49
19.60
19.54
βym (mrad)
-1.64
-1.623
-1.632
-1.616
(c)
Tandem Arrangement
Deflection
Bearing 1
Bearing 2
Average*
Double Row
δxm (μm)
-9.42
-33.34
-21.38
-21.26
δzm (μm)
5.07
5.03
5.05
4.98
βym (mrad)
-1.204
-1.208
-1.206
-1.201
*: Arithmetic average of the deflection components of bearings 1 and 2.
Table 4.1 Single row versus double row bearing displacement analyses with Method D
for the example case of section 4.5.1.3. Bearings are organized in (a) back-toarrangement; (b) face-to-face arrangement; and (c) tandem arrangement
144
4.5.2 Examination of bearing forces and moments
4.5.2.1 Two single row bearings
When a shaft supported by two single row bearings is subjected to an external
load, the load is shared among the two bearings. In the absence of initial preloads and
assuming the shaft is rigid, the sum of the forces carried by both bearings is equal to the
net external forces.
1
Fxm
 Fxm2  Fxext
(4.6a)
1
2
Fym
 Fym
 Fyext
(4.6b)
1
Fzm
 Fzm2  Fzext
(4.6c)
i
(q = x,y,z; i = 1,2), such expressions
Although these relations are valid for Fqm
i
i
that relate Fqext and M pext (q = x, y, z and p = x, y) to M xm
and M ym
(i = 1,2), may not be
easily found; these relations would depend on geometric parameters and organization of
the rolling elements; arithmetic sum of bearing moments has no meaning. The elements
i
i
of fmi  {Fxmi , Fymi , Fzmi , M xm
, M ym
}T of each individual bearing can only be determined
through the solution of the multidimensional statically indeterminate problem when two
single row bearing model is used.
4.5.2.2 Double row bearing
When the shaft is rigid, a double row bearing carries the entire external load (thus
the problem is statically determinate), and the elements of fmD can be simply calculated
using the following relations:
145
1
FxmD  Fxext  Fxm
 Fxm2
(4.7a)
1
2
FymD  Fyext  Fym
 Fym
(4.7b)
1
FzmD  Fzext  Fzm
 Fzm2
(4.7c)
D
M xm
 M xext  zG / Fy Fyext
(4.7d)
D
M ym
 M yext  zG / Fx Fxext
(4.7e)
Here zG / Fx and zG / Fy are the axial distances between the bearing’s geometric
center (point G) and the application point (or equivalent center) of Fx and Fy,
respectively.
4.5.2.3 Example case (cont’d)
Consider the same example case described in section 4.5.1.3 and shown in Figure 4.9.
Nonzero elements of f mi ( Fxm , Fzm , and M ym ) are calculated using Method D and
1
outlined in Table 4.2(a-c). As expected, FqmD  Fqext  Fqm
 Fqm2 (q = x, z) is satisfied,
i
whereas such quantifiable relationships are not seen for M ym
. A critical observation here
1
1
is that sgn( Fxm
)  sgn( Fxm2 ) for all configurations, and sgn( Fzm
)  sgn( Fzm2 ) for face-to-
face and back-to-back configurations as a result of the imposed moment. In other words,
to share the moment load imposed by Fxext , the two bearings are loaded in the opposite x
and z directions, which causes one bearing to carry loads greater than the external loads
(call this a primary bearing), while the other one (call this a secondary bearing) is loaded
in the negative direction. Note that the primary bearing and the secondary bearing are
dictated by the configuration of the two bearings.
146
(a)
Back-to-Back Arrangement
Load
Bearing 1
Bearing 2
Sum*
Double Row
Fxm (kN)
1.609
-0.6093
1.00
1.00
Fzm (kN)
3.369
-0.3695
3.00
3.00
Mym (kNmm)
-39.894
-11.937
-51.83
-74.00
(b)
Face-to-Face Arrangement
Load
Bearing 1
Bearing 2
Sum*
Double Row
Fxm (kN)
-1.589
2.589
1.00
1.00
Fzm (kN)
-1.179
4.179
3.00
3.00
Mym (kNmm)
-37.53
-78.26
-115.79
-74.00
(c)
Tandem Arrangement
Load
Bearing 1
Bearing 2
Sum*
Double Row
Fxm (kN)
2.441
-1.441
1.00
1.00
Fzm (kN)
2.091
0.909
3.00
3.00
Mym (kNmm)
-59.51
24.41
-35.10
-74.00
*: Arithmetic sum of the load components of bearings 1 and 2.
Table 4.2 Single row versus double row bearing load analyses with Method D for the
example case of section 4.5.1.3. Bearings are organized in (a) back-to-arrangement; (b)
face-to-face arrangement; and (c) tandem arrangement
147
4.5.3 Examination of multi-dimensional stiffness coefficients
4.5.3.1 Two single row bearings vs. a double row bearing
In
the
previous
subsections
it
is
shown
that
D
1
2
 zm
  zm
  zm
and
1
FzmD  Fzm
 Fzm2  Fzext . Thus, bearings 1 and 2 exhibit the characteristics of two parallel
springs in an axial direction; thus k zz of a double row bearing, which replaces the two
single row bearings, may simply be obtained by arithmetic summation of k 1zz and k zz2 .
Similarly, the two bearings are also in parallel connection in radial directions.
1
2
2
This is not as obvious, since  xm
and  1ym   ym
due to the angular displacements in
  xm
the presence of moment loads. However, when such multidimensional effects are
D
1
2
disregarded (recall k xx and k yy are 1-dimensional definitions),  xm
and
  xm
  xm
D
1
2
can be obtained as evident from Eqs. 4.5(a-d). Given FxmD  Fxext
 ym
  ym
  ym
D
1
2
1
 Fym
 Fym
 Fyext (Eqs. 4.7a-b), the two bearings behave as two
 Fxm
 Fxm2 and Fym
2
D
parallel springs in radial directions; thus, the arithmetic sum of k 1pp and k pp
gives k pp
(p
= x,y). Therefore, the following simple relations can be written for translational stiffness
coefficients:
1
k xxD  k xx
 k xx2
(4.8a)
k yyD  k 1yy  k yy2
(4.8b)
k zzD  k zz1  k zz2
(4.8c)
148
However, k rr (r   x ,  y ) of two single row bearings and a double row bearing
D
1
2
D
1
2
and  ym
are
may not simply be correlated. Although  xm
  xm
  xm
  ym
  ym
D
1
2
D
2
and M ym
, thus two bearings are not in parallel
satisfied, M xm
 M xm
 M xm
 M 1ym  M ym
connection in rotational dimensions, thus krrD  krr1  krr2 (r   x ,  y ) . In fact, there is
simply no way of obtaining k rrD from k rri (i =1,2), since k rrD depend on geometric
parameters (such as ‘e’) and the organization of rolling elements, thus, they must be
derived from load-deflection relations. Note that the overall tilting characteristics of a
shaft-bearing system are dictated by k D and kD terms and k i  and ki  terms have no
y y
x x
x x
y y
meaning individually.
So far off-diagonal terms of K b have not been considered as Method D does not
output such values. However, the results obtained for diagonal elements should be easily
extendable to off-diagonal elements, such that the translational off diagonal-stiffness
elements ( k pq ) (p,q = x,y,z; p  q ) of K ib (i = 1,2), such as kxy, or kxz, may be summed to
obtain k pq of K bD , whereas rotational related off-diagonal terms such as kxθy, or kθxθy
should require a double row formulation.
4.5.3.2 Example case (cont’d)
The example case of the previous subsections is continued here for stiffness elements.
Five diagonal elements of Kb have been calculated for all three arrangements and given
in Tables 4.3(a-c) for the described combined load. As seen from the tables, the
149
(a)
Back-to-Back Arrangement
Bearing 1 Bearing 2
Sum*
Double Row
kxx (kN/mm)
290.5
110.6
401.9
401.1
kyy (kN/mm)
293.7
33.9
327.6
327.6
kzz (kN/mm)
238.4
44.2
282.6
282.6
kθxθx (MNmm/rad)
153.5
13.1
166.6
342.2
kθyθy (MNmm/rad)
154.2
43.9
198.1
410.1
(b)
Face-to-Face Arrangement
Bearing 1 Bearing 2
Sum*
Double Row
kxx (kN/mm)
144.2
297.8
441.9
442.3
kyy (kN/mm)
53.6
287.7
341.3
341.7
kzz (kN/mm)
86.1
256.2
342.3
342.2
kθxθx (MNmm/rad)
28.3
155.1
183.4
65.7
kθyθy (MNmm/rad)
82.9
175.6
258.5
100.6
(c)
Tandem Arrangement
Bearing 1 Bearing 2
Sum*
Double Row
kxx (kN/mm)
166.2
162.6
328.8
329.0
kyy (kN/mm)
128.8
144.5
273.3
272.8
kzz (kN/mm)
134.7
82.5
217.2
216.9
kθxθx (MNmm/rad)
70.2
56.2
126.4
156.1
kθyθy (MNmm/rad)
103.7
50.3
153.9
211.4
*: Arithmetic sum of the stiffness elements of bearings 1 and 2
Table 4.3 Single row versus double row bearing stiffness analyses with Method D for the
example case of section 4.5.1.3. Bearings are organized in (a) back-to-arrangement; (b)
face-to-face arrangement; and (c) tandem arrangement
150
arithmetic sum of translational stiffness coefficients of two single row bearings are
almost identical to those of the double row bearing for all arrangements (i.e.
D
2
k pp
 k 1pp  k pp
; p = x,y,z), satisfying Eqs. 4.8(a-c). However k rrD cannot be calculated
from k rri ( r   x ,  y ; i = 1,2).
4.6. Methods for Calculating Bearing Stiffness Coefficients
4.6.1 Analytical methods
Numerous Kb formulations have been proposed by several authors to describe the
multidimensional stiffness characteristics of rolling element bearings. Lim and Singh’s
[4.23] formulation (Method B) for single row ball and roller type bearings, Royston and
Basdogan’s [4.26] formulation for spherical bearings, and the proposed formulation for
double row angular contact ball bearings as described in chapter 2 (Method A) are some
of them. Kb formulations defined by De Mul et al. [4.27] and Hernot et al. [4.28] are
other examples. As mentioned earlier, some computational tools such as Method D [4.22]
are also capable of calculating some elements of Kb analytically.
4.6.2 Finite difference approximation
Finite difference approximation methods can be used in conjunction with Method
C to calculate the Kb matrix, or with Method D to determine the unrevealed off-diagonal
terms. Since the slope of the load-deflection curve around the operating point defines the
stiffness, elements of
 F / q q
m
should be approximated using finite difference
151
Bearing
Load
Operating
point
F(q0+Δq)
Bearing
stiffness
F(q0)
F(q0-Δq)
q0-Δq
Bearing Deflection
q0
q0+Δq
Figure 4.10 The typical load-deflection relationship of a rolling element bearing
illustrating the operating point and bearing stiffness. Multiple points around the operating
point can be analyzed to approximate bearing stiffness by finite difference
approximation.
152
relations, which require the static analysis of multiple points around the operating point
as shown in Figure 4.10. The lowest order relations, which require the lowest number of
calculation points, are given by Eqs. 4.9(a-c):
Forward difference:
F(q 0  q)  F(q 0 )
F

 O(q)
q q=q0
q
(4.9a)
Central difference:
F (q 0  q)  F (q 0  q)
F

 O( q) 2
q q=q0
2 q
(4.9b)
F(q 0 )  F(q 0  q)
F

 O(q)
q q=q0
q
(4.9c)
Backward difference:
The error in a method's solution is defined as the difference between its
approximation and the exact solution. The two sources of errors of finite difference
methods are round-off errors, the loss of precision due to rounding of decimal quantities,
and truncation errors, caused by the truncation of the higher order terms of Taylor series.
The last term in above equations, which are given in terms of order of magnitude ( O ) of
q , stands for truncation errors. Forward and backward difference methods are single-
sided approximations, and their truncation errors are in the order of q whereas the
central difference method is a double-sided method and its truncation error is in the order
of (q)2 . Note that Eqs. 4.9(a-c) all require two calculation points. Higher order
approximations of  F / q q come with the expense of a higher number of calculation
m
points such as given below:
153
Forward difference:
F (q 0  2q)  6F (q 0  q)  3F (q 0 )  2F (q 0  q)
F

 O ( q ) 3
q q=q0
6 q
(4.10a)
Central difference:
F (q 0  2q)  8F (q 0  q)  8F(q 0  q)  F (q 0  2q)
F

 O ( q ) 4
q q=q0
12q
(4.10b)
Backward difference:
2F(q 0  q)  3F(q 0 )  6F(q 0  q)  F(q 0  2q)
F

 O (q)3
q q=q0
6q
(4.10c)
In this study, first order forward (Eq. 4.9(a)) and second order central difference (Eq.
4.9(b)) methods will be used; higher order (third and above) approximations will not be
utilized to avoid a high number of calculation points which could result in excessive
computing times with Method C. For instance k zz  (Fzm /  zm )qm may be approximated
using Eq. 4.9(a) as:
k zz 
Fzm (q 0   zme z )  Fzm (q 0 )
 zm
(4.11)
where q 0 is the bearing deflection vector at the operating point, and ez is the unit vector
in z direction. The cross–coupling stiffness terms may be calculated in a similar way. For
instance, the k x y term, which is highly significant in the presence of axial load, may be
approximated with Eq. 4.9(a) as:
154
k x y 
Fxm (q 0   yme y )  Fxm (q 0 )
 ym
(4.12)
Note that the calculation of the fully populated (5x5) Kb around an operating point using
a finite difference approximation has a high computational cost. Normally, a lower order
method Eq. 4.9(a-c) requires a minimum of 50 (25x2) simulations around the operating
point. However, since the stiffness matrix is symmetric, the number of calculations may
be reduced to 30 (15x2) points, still a very high number for the calculation of a single Kb.
4.7. Investigation of Bearing Loads, Deflections and Stiffness Elements with Four
Calculation Methods
4.7.1 Pure axial load
Four calculation methods are evaluated by comparing bearing loads, deflections
and stiffnesses for the example case described in section 4.5.1.3. First, the shaft is
subjected to a pure axial load of 1 kN (i.e. Fzext = 1 kN and Fxext = 0 according to Figure
4.9), and the load distribution among the two bearings are calculated with Methods C and
D for all three configurations and shown in Table 4.4(a). As expected, bearing 2 for the
face-to-face, and bearing 1 for back-to-back arrangement carries the entire thrust load,
whereas Fzext is almost equally distributed between two bearings in the tandem
arrangement. Method C gives almost the ideal results; however, the secondary bearing of
DF and DB arrangements are found to support some negative loads when Method D is
used. This load is about -1% of Fzext and might be due to some complex interactions
155
within the bearing. Accordingly, the load carried by the primary bearing is about 101% of
Fzext .
i
Second,  zm
(i =1,2) are calculated using four calculation methods and shown in
Table 4.4(b). Methods A and B give identical results since they utilize the same theory.
Note that only  zm of the loaded bearing can be calculated with Method B, the unloaded
bearing must exhibit the same  zm due to system rigidity.  zm obtained by Method D is
very close (less than half a micron) to the value obtained by Methods A and B, which
once again suggests that the theory used by Method D is similar to that of Method A.
Method C, however, gives a slightly higher  zm value when compared with other
methods. As seen from Table 4.4(b),  zm obtained by Method C is about 4 μm greater
than those obtained by Methods A or B.
Finally, k zz of both bearings are calculated with four methods. Here, k zz of
Method C are calculated by plugging  zm values obtained by Method C into analytical
expressions of Method B. The results, as illustrated in Table 4.4(c), show that k zz of
Methods A and B are very close to that of Method D. k zz obtained by using  zm of
Method C in Method B are higher because a larger  zm of Method C correspond to a
different operating point on the load-deflection curve of Method B that is relatively
farther away from the origin. However, k zz of Method C is expected to be lower than
those of other methods as it allows higher  zm under the same Fzext . In fact, Method C
156
(a)
Load
Fzm (N)
Arrangement
Method C
Bearing 1 Bearing 2
Method D
Bearing 1 Bearing 2
Face-to-face
0
1000
-9.5
1009.5
Back-to-back
1000
0
1009.5
-9.5
Tandem
500.029
499.971
501.5
498.6
(b)
Axial
Arrang.
Deflection
δzm (μm)
Method A
Method B
Method C (in B)
Method D
Double Row
Brg 1
Brg 2
Brg 1
Brg 2
Brg 1
Brg 2
DF
9.83
-
9.83
14.23
14.23
10.29
10.29
DB
9.83
9.83
-
14.23
14.23
10.29
10.29
DT
6.26
6.26
6.26
10.51
10.51
6.53
6.51
(c)
Method A
Axial
Arrang.
Stiffness
Double Row
kzz
(kN/mm)
Method B
Method C (in B)
Method D
Brg 1
Brg 2
Brg 1
Brg 2
Brg 1
Brg 2
DF
157.4
-
157.4
0
195.9
0.048
152.1
DB
157.4
157.4
-
195.9
0
152.1
0.047
DT
244.4
122.2
122.2
163.6
163.6
117.6
117.4
Table 4.4 Comparative evaluation of the four calculation methods for a shaft-bearing
system of three different bearing configurations that is under pure axial load. (a) Fzm
calculated Methods C and D; (b) δzm calculated by four calculation methods; and (c) kzz
calculated by four calculation methods.
157
has a different load-deflection curve than Method B; thus, utilizing  zm obtained by
Method C in Method B reduces the accuracy of stiffness calculations.
Instead, the finite difference approximation, as described in section 4.6.2 is
ext
i
utilized to calculate k zz with Method C. Thus,  zm
(i =1,2) are calculated for Fz
= 990,
999, 1000, 1001, and 1010 N, and sample results are presented for the back-to-back
arrangement. Selected calculation points permit the approximation of k 1zz with Eq. 4.9(a)
(first order method) or Eq. 4.9(b) (second order method), using two different step sizes
(ΔFz
ext
1
= ΔFzm = 1 N, 10 N). Calculated  zm and k zz are given in Tables 4.5(a) and
4.5(b), respectively. As seen from Table 4.5(b), k 1zz values calculated with two step sizes
and two approximation orders result in the same exact value of k 1zz = 164.00 kN/mm ( k zz2
is obviously zero), which indicates the truncation errors and nonlinear effects have no
impact on the calculated results. Note that k 1zz = 164.00 kN/mm calculated by Eq. 4.9(ab) matches fairly well with other methods, however, it is still slightly higher, although
k zz of Method C was expected to be slightly lower. Such slight differences between the
methods were expected since Method C does not employ Hertzian contact theory, thus its
load-deflection relations at the contact points is not described by a simple relation such as
Eq. (4.1).
158
.Method C (Back-to-back arrangement)
(a)
Fz
ext
1
(N)
1000
1001
1010
999
990
2
1
2
δzm (μm)
δzm (μm)
Fzm (N)
Fzm (N)
14.70592
14.71201
14.76689
14.69982
14.64494
14.70592
14.71201
14.76689
14.69982
14.64494
1000
1001
1010
999
990
0
0
0
0
0
Method C (Back-to-back Arrangement)
(b)
ΔFz
ext
(N)
Bearing 1
Bearing 2
1st Order 2nd Order 1st Order 2nd Order
kzz
1
164.00
164.00
0
0
(kN/mm)
10
164.00
164.00
0
0
Table 4.5 Calculation of axial deflections, forces, and stiffness elements around the
i
operating point with Method C for the back-to-back arrangement. (a) Calculation of  zm
and Fzmi (i=1,2) around the operating point. (b) Calculation of k zz with finite difference
approximation using two step sizes and accuracy orders.
159
4.7.2 Combined load: Back-to-back arrangement
Next, a combined load case, with Fzext  3 kN and Fxext  1 kN according to Figure
4.9 is analyzed for back-to-back arrangement; this loading case imposes a combination of
i
i
), and bending moment ( M ym
) on both bearings. The load
axial ( Fzmi ), radial ( Fxm
distribution among the two bearings calculated by Methods C and D are shown in Table
1
)  sgn( Fqm2 )
4.6(a). One can notice that two methods give similar results and sgn( Fqm
1
 Fqext  0  Fqm2 ( q  x, z ), this was shown earlier in section 4.5.
causing Fqm
i
(i =
Then,  xm ,  zm , and  ym terms are evaluated. As seen from Table 4.6(b),  xm
1
varies from -4.32 μm
1,2) show some variations among the four calculation methods.  xm
2
(Method D) to -6.94 μm (Method B), while the variation of  xm
is greater and occurs
between -9.43 μm (Method B) to -15.69 μm (Method C). Note that the methods may not
be rank ordered from according to deflections as such orders seem to be relatively
1
2
and minimum  xm
at the same
random. For instance, Method B results in maximum  xm
time.  ym values of Method B are also found to be greater than the other methods. The
reason for this discrepancy is that Method B has to input the forces calculated by Method
C or D; thus, Method B calculations describe the deflections of two separate bearings and
1
2
2
not a system behavior. This also explains why  zm
and  1ym   ym
occurs for
  zm
2
1
2
and  1ym   ym
in Methods C and D. On the other hand,  zm
Method B, whereas  zm
  zm
and  ym of Method A for a double row bearing match very well with those of Methods C
160
(a)
Method C
Bearing 1 Bearing 2
Load
Method D
Bearing 1
Bearing 2
Fxm (kN)
1.580
-0.580
1.609
-0.609
Fzm (kN)
3.325
-0.325
3.370
-0.370
Mym (kNmm)
-41.06
-11.31
-39.89
-11.94
(b)
Deflection
Method B
Method A
Method C
Method D
Double Row
Brg 1
Brg 2
Brg 1
Brg 2
Brg 1
Brg 2
δxm (μm)
-8.33
-6.94
-9.43
-6.06
-15.69
-4.32
-13.06
δzm (μm)
21.29
20.24
21.63
24.39
24.39
21.26
21.28
βym (mrad)
-0.440
-0.573
-0.634
-0.482
-0.482
-0.444
-0.439
(c)
Stiffness
Method A
Double Row
Method B
Brg 1 Brg 2
Method C (in B)
Brg 1 Brg 2
kxx (kN/mm)
420.3
302.9
112.8
333.6
127.0
290.5
110.6
kzz (kN/mm)
297.6
247.4
47.1
278.5
49.6
238.4
44.2
kθyθy(MNmm/rad)
410.1
150.0
44.2
168.9
46.1
154.2
43.9
Method D
Brg 1
Brg 2
Table 4.6 Comparative evaluation of the four calculation methods of a shaft-bearing
system of back-to-back configuration that is under combined load. (a) Bearing loads
calculated by Methods C and D; (b) Bearing deflections calculated by four calculation
methods; and (c) Bearing stiffness elements calculated by four calculation methods
161
and D. These results demonstrate a significant advantage of Method A over Method B
since one can simply input fmD in Method A ( fmD  {1000 0 3000 0 -74000}T for this case)
and obtain accurate results independent from the complexity of the external load.
Finally kxx, kzz, and kθyθy are calculated using the four methods and shown in Table
4.6(c). As seen from the table, the three methods for single row bearings (B, C, and D)
give fairly close stiffness values, with Method C being slightly greater than the others.
Here, the stiffness elements of Method C are again calculated using the deflections of
Method C in explicit expressions of Method B. Observe from Table 4.6(c) that
D
2
k pp
 k 1pp  k pp
(p = x,y,z), whereas k rrD  krr1  k rr2 (r   x ,  y ) as discussed in section
D
4.4; where k pp
and k rrD correspond to stiffness terms calculated by Method A, and k ipp
and k rri (i = 1,2) are the stiffness elements calculated by methods B, C or D, which verify
the results obtained by Method A.
Bearing stiffness coefficients may again attempted to be calculated using finite
difference approximation. When a shaft is supported by two (or more) bearings, it is not
possible to apply a disturbance on one bearing without affecting the statics of the
other(s). Here, this disturbance is applied as a part of the external load just like in the
previous subsection. For the calculation of k xx and k y y , Fxext = 1 kN; and for the
calculation of k zz , Fzext = 3 kN is varied around the operating point; though the variation
i
i
i
of Fxext is mainly investigated here.  xm
,  zm
, and  ym
are calculated with Method C for
ext
Fx
= 990, 999, 999.99, 1000, 1000.01, 1001, and 1010 N, and are tabulated in Table
162
(a)
Method C (Back-to-back arrangement)
Fxext (N)
1000
δxm1 (μm)
-5.394
δxm2 (μm)
-14.553
δzm1 (μm)
23.108
δzm2 (μm)
23.108
βym1 (mrad)
-0.458
βym2 (mrad)
-0.458
1000.01
-5.394
-14.553
23.108
23.108
-0.458
-0.458
1001
-5.393
-14.556
23.108
23.108
-0.458
-0.458
1010
-5.386
-14.586
23.109
23.109
-0.460
-0.460
999.99
-5.394
-14.553
23.108
23.108
-0.458
-0.458
999
-5.395
-14.550
23.107
23.107
-0.458
-0.458
990
-5.402
-14.521
23.106
23.106
-0.456
-0.456
(b)
Method C (Back-to-back arrangement)
Fxext (N)
1000
Fxm1 (N)
1586.98
Fxm2 (N)
-586.98
Fzm1 (N)
3327.52
Fzm2 (N)
-327.52
Mym1 (Nmm)
-40818.60
Mym2 (Nmm)
-11418.57
1000.01
1586.99
-586.98
3327.53
-327.53
-40818.99
-11418.69
1001
1588.61
-587.61
3327.88
-327.88
-40857.34
-11431.07
1010
1603.35
-593.35
3331.13
-331.13
-41206.00
-11543.60
999.99
1586.96
-586.97
3327.52
-327.52
-40818.21
-11418.44
999
1585.34
-586.34
3327.16
-327.16
-40779.86
-11406.06
990
1570.60
-580.60
3323.91
-323.91
-40431.20
-11293.53
Table 4.7 Calculation of bearing deflections and loads at multiple points around the
i
i
i
,  zm
and  ym
, and (b) calculation of Fxmi , Fzmi and
operating point. (a) Calculation of  xm
i
M ym
(i = 1,2) around the operating point.
163
4.7(a); force and moment distribution among the two bearings are similarly given in
Table 4.7(b). Selected calculation points allow the approximation of k xx and k y y with a
first (Eq. 4.9(a) or (c)) or a second order method (Eq. 4.9(b)), with three step sizes (i.e.
ext
ΔFx
= 0.01, 1, 10 N).
ext
Coefficients k xx and k y y are first calculated using Eq. 4.9(a) with ΔFx
k zz is similarly calculated using ΔFz
ext
= 10 N;
= 10 N. The results, as shown in Table 4.8(a), do
not match with analytical results (of Table 4.6(c)), such that k xx of both bearings
(especially k 1xx ) are excessively large, and k zz2 is too small. k 1zz and k i  (i = 1,2) are
y y
somehow calculated in reasonable orders, and they are fairly consistent with the
analytical results. To explore the reason for the inconsistencies, three analyses have been
carried out. First, the finite difference approximation method is used with Method D, the
same problems have been observed with slightly different values as seen from the results
on the right hand side of Table 4.8(a), which indicate the problem is not related with
Method C, but about the application of finite difference approximation. Next, several step
ext
sizes (i.e. ΔFx
= 0.01, 1, 10 N) and accuracy orders are used to approximate kxx and
kθyθy with Method C, however almost same exact results are obtained at all cases as seen
from the results of Table 4.8(b). Finally, a second order curve fit has been applied to the
collected data, as illustrated in Figure 4.11(a-b) for bearings 1 and 2 respectively, and a
numerical differentiation  Fxm /  xm  is applied on the curve fit to approximate kxx. As
seen from Figure 4.11(c-d), kxx values obtained with numerical differentiation (shown
164
with solid line) are in the same order of magnitude with finite difference approximation
(shown with discrete markers). This occurs simply because numerical differentiation
utilizes finite difference approximation techniques to differentiate.
The analyses show the large errors are not caused by truncation errors as the
results do not show any improvement with a change in step size ( q ) or use of a higher
order approximation. They are obviously not due to round-off errors either, as a sufficient
number of decimal digits is incorporated into the analyses. In fact, the problem is more
fundamental. When a disturbance is applied in terms of an external force, it not only
creates a disturbance in the desired dimension, but also in other dimensions. For instance,
ext
when a disturbance is applied in terms of ΔFx , it not only imposes a  xm term, but also
terms such as  zm and  ym , which ultimately causes an improper use of equations
such as Eq. 9(a-c). Unfortunately, it is not possible to quantify the individual contribution
of these multidimensional disturbances, which have more drastic effects when the
bearings are already under an asymmetrical combined load. That is why even k zz
calculations are corrupted by these multidimensional disturbances although a disturbance
in the Fz
ext
is not expected to create a considerable change in  xm or  ym .
A proper application of the finite difference approximation for such a shaftbearing system under combined load case could be achieved with the following steps: (i)
Use the actual system model to solve the static problem and determine the operating
points of both bearings; (ii) Create a separate, simple shaft-bearing model for each
bearing (consider one bearing at a time); (iii) Apply bearing forces (or deflections) and
165
(a)
Stiffness
Method C
Bearing 1 Bearing 2
kxx (kN/mm)
1489.80
kzz (kN/mm)
230.53
kθyθy(MNmm/rad) 216.91
(b)
kxx (kN/mm)
219.30
4.94
70.63
Method D
Bearing 1 Bearing 2
1366.67
223.85
199.44
258.33
3.08
68.66
Method C (Back-to-back Arrangement)
Bearing 1
Bearing 2
ΔFxext (N)
1st Order 2nd Order 1st Order 2nd Order
0.01
1489.832 1489.832
219.313
219.313
1
1489.805 1489.806
219.308
219.308
10
1489.805 1489.805
219.308
219.308
kθyθy
(MNmm/rad)
0.01
1
10
216.904
216.916
216.916
216.905
216.917
216.916
70.634
70.629
70.629
70.631
70.629
70.629
Table 4.8 Calculated bearing stiffness elements for the combined load case with finite
difference approximation. (a) Calculation of kxx, kzz and kθyθy using Methods C and D, and
(b) Calculation of kxx and kθyθy by Method C using three different step sizes and two
different accuracy orders.
166
Bearing 1
1700
Bearing 2
-500
(a)
-550
F2xm (N)
F1xm (N)
1600
(b)
1500
1400
-6.12
-6.1
-6.08 -6.06 -6.04 -6.02
-6
-600
-650
-15.85 -15.8 -15.75 -15.7 -15.65 -15.6 -15.55 -15.5
-5.98
1
2
xm (m)
250
(c)
dF1xm/dx (kN/mm)
dF1xm/dx (kN/mm)
1600
xm (m)
1500
1400
1300
-6.12
-6.1
-6.08 -6.06 -6.04 -6.02
-6
-5.98
(d)
240
230
220
210
200
-15.85 -15.8 -15.75 -15.7 -15.65 -15.6 -15.55 -15.5
1
xm (m)
2
xm (m)
Figure 4.11 Application of a second order curve fit to the data collected by Method C and
1
1
2
vs.  xm
; (b) Fxm2 vs.  xm
, and (c)
numerical differentiation to approximate kxx. (a) Fxm
1
Fxm
x
Fxm2
vs.  ; (d)
x
2
vs.  xm
. Key: (a-b) ( ), actual data; (
1
xm
x  1xm
), second order
2
x  xm
curve fit. (c-d) ( ), finite difference approximation; (
167
), numerical differentiation.
bring each bearing to its operating point; (iv) Apply proper force/deflection disturbances
about the geometric center of the bearing to eliminate (or minimize) the multidimensional
disturbances; (v) Use one or more finite difference approximation techniques to estimate
stiffness elements. Obviously this procedure is time costly and tedious to carry out. These
results, once again, demonstrate the merits of Method A, which can be especially useful
for such combined loading cases, in which indirect methods (such as finite difference
approximation) could be impractical to carry out.
4.8. Comparison of Predicted Natural Frequencies with Experimental Results
Next, predicted natural frequencies of a shaft-bearing system obtained by four
methods are compared with respect to the modal measurements of chapter 3. Only axial
preload is analyzed as the radial preload with hydraulic jack affects the boundary
conditions of the shaft-bearing system. Recall from section 3.6 that the axial preload
applied on the double row bearing was a compressive force on the split inner rings that
loads each row by Fz 0 as shown in Figure 4.12(a). The previous simulations of this
chapter, however, assume the bearings are not preloaded and the axial load is applied as
an external load ( Fzext as shown in Figure 4.12(b)), which loads only the first (left) row of
the back-to-back arrangement leaving the second (right) row unloaded. Thus, the
previous simulations (of this chapter) may not be used to model the experimental setup.
The modeling of Fz 0 with Methods C and D, however, is not very simple, since Method
C does not contain a separate preload menu; Method D, on the other hand, contains a
preload menu (which can be applied in terms of force or displacement to the inner or
168
outer rings), however, an application of the preload does not seem to make any difference
on the calculated bearing deflection and stiffness values (although it should); thus, the
preload menu of the code is not useful for our purposes at this point. Due to such
difficulties, the system symmetry is made use of here, and half of the shaft-bearing
system with a single row bearing (as shown in Figure 4.12(c)) is modeled instead of
modeling the whole system. For the symmetric system, Fz 0 is applied as Fzext , and  z 0
value for both rows is obtained easily.
For a sample preload ( Fz 0 = 1.56 kN as analyzed in chapter 3),  z 0 has been
calculated by methods A, C, and D, and the natural frequencies of the system are
determined using the five-degree-of-freedom model (of Figure 3.2) by incorporating Kb.
 z 0 values obtained by Methods A and B are under pure axial load. The comparison of
natural frequencies with the measurements is shown in Table 4.9. As seen from the table,
the natural frequencies obtained by Methods C and D match well with the measurements
as well as with the results obtained by Method A. It seems Method A establishes a lower
bound, whereas Method C, which deviates slightly higher than other methods, forms an
upper bound on the natural frequencies. Higher deviations of Method C were expected as
it yields higher k zz values as evident from Table 4.4(c). Also, recall that in the absence of
radial or moment loads, natural frequencies at r = 1 and r = 2, as well as r = 4 and r = 5,
are repeated; thus, each method yields a pair of overlapped modes. Recall that these
modes could be split with an application of slight amount of radial load as discussed in
chapter 3.
169
Next, the predicted radial accelerance spectra  A xx ( )  of three methods (A, C,
and D) obtained by the five-degree-of-freedom model are compared with the measured
A xx ( ) (with impact location f1x and accelerometer location 1, according to Figure 3.8).
Recall from chapter 3 that since r = 1 and 2 as well as r = 4 and 5 overlap, and r = 3
mode does not affect A xx ( ) , the dimension of the analytical model could be
conveniently reduced from five to two (that are translation along the x-axis (x) and the
rotation about the y-axis (  y )); refer to section 3.8 for the details. The comparison of the
measured and predicted A xx ( ) with three methods are shown in Figure 4.13. Here, the
non-proportional, preload-independent damping model as described in section 3.5 is
utilized for all methods with ct  4350 Ns/m and cr = 7.25 Nms/rad, corresponding to
translational and rotational damping values of a diagonal damping matrix respectively
(i.e. Cb  diag ([ct ct ct cr cr ]) ). As seen from Figure 4.13, three methods match well
with the experimental results (while Method C shows slightly higher deviations).
170
(a) Fz 0
Simulations of the previous section
(with pure axial load)
Fzm
Fz 0
 z0
(b)
Experimentally applied axial preload
Fzext
 z0
Both rows
are loaded
Loaded
row
Assume
symmetry
(c)
 zm
Unloaded
row
Fz 0
 z0
Modeling of the experimental preloading
with methods C and D
Figure 4.12 Modeling of experimental preload utilizing system symmetry. (a) Axial
preload ( Fz 0 ) applied in experiment; (b) External axial load utilized in the simulations of
section 4.4-4.7; (c) Modeling of experimental preload with Methods C and D making use
of the system symmetry.
171
Natural Frequencies (Hz)
Modal Index (r)
Experiment
Method A
Method C
Method D
1
773
763
841
801
2
894
763
841
801
3
1636
1583
1649
1598
4
2049
2046
2224
2090
5
2246
2046
2224
2090
Table 4.9 A comparison of measured and predicted natural frequencies obtained by three
calculation methods (A, C and D) using the five degree-of-freedom model.
172
1.8
1.6
Accelerance (1/kg)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000
1500
Frequency (Hz)
Figure 4.13 Comparison of radial accelerance magnitude spectra
kN with three methods of calculation. Key: (
(
), predicted by Method A; (
2000
A
xx
( )  for Fz 0 =1.56
), Measured from the modal experiment;
), predicted by Method C; (
Method D.
173
2500
) predicted by
4.9 Preliminary Modal Experiments with a Faulty Bearing
The wheel bearing used in the experiments of chapter 3 was a healthy bearing that
was free of any major defects and did not show any problem in preloading. From a health
monitoring point of view, the effect of bearing damage on the modal characteristics of a
shaft-bearing assembly under different preloads can be questioned. In order to investigate
this problem experimentally, a wheel bearing which is of the same type analyzed in
chapter 3 is manually damaged such that the bearing consequently exhibited an excessive
endplay or internal axial clearance (so that one could hold the shaft and shake it to hear
the rattling noise while faulty bearing is mounted in the assembly). As a result, the
bearing was unable to hold axial preload properly, although some force measurements
could be still read from the load cell, which does not reflect the actual preload carried by
the bearing.
The experimental setup analyzed in chapter 3, as shown in Figure 4.14, is utilized
once again to investigate the effect of bearing damage and effects of Fz 0 on the faulty
system. In comparison to the previous setup, a new (SCADAS) system has been used for
data acquisition; the new system allows 20 channel analysis, which makes it possible to
use 6 triaxial and 1 uniaxial accelerometers with an impact hammer. The locations of the
uniaxial (#0) and triaxial (# 1, 2, 3, 4, 5 and 6) accelerometers are illustrated in Figure
4.14. Incorporation of 6 triaxial and 1 uniaxial accelerometers (equivalent to 19 uniaxial
accelerometers) provides better investigation of the system mode shapes while yielding a
large amount of data (e.g. number of FRF data = 19 x number of impact locations x
number of preloads x 2 (for a healthy and a faulty bearing)). The new data acquisition
174
system is also integrated with LMS Testlab [4.29], which saves significant time in data
acquisition and provides very useful on-line modal analysis capabilities.
Figure 4.15 shows sample cross-point accelerance magnitude measurements (
A0 z / F6 z ( ) according to Figure 4.14) for faulty and healthy systems at 6 preload
levels. As seen from the figure, measurements from faulty and healthy systems show
dramatic differences, such that the system with the faulty bearing does not exhibit any of
the rigid body modes of the healthy bearing that are found in the 500-1500 Hz range.
Instead, it behaves as a single-degree-of-freedom system with a resonance occurring
around 160 Hz, which does not show any frequency shifts when Fz 0 is changed.
However, the amplitude of the resonance shows some reduction, indicating the damping
of the system is still somehow affected. In a higher frequency regime, the spectra almost
has a constant value for all Fz 0 ; no other peaks are seen until relatively high frequencies
(say 1400 Hz). In the context of single-degree-of-freedom vibration theory this region
corresponds to a mass controlled region, with a magnitude of A0 z / F6 z ( ) mass
controlled
 0.02
g/N. Conversion of this value to SI units yields 0.1962 kg 1  0.2 kg 1 corresponding to
an effective mass of 5 kg, which is nothing but the mass of the shaft (the actual value is
slightly greater than 5 kg, which may account for the mass of the inner rings). This
indicates that the system resonance corresponds to translational (axial) vibrations of the
shaft as a result of excessive internal clearance within the bearing. This prediction can be
easily verified by animating the mode shape (using LMS Testlab [4.29]) using the whole
data set collected by all accelerometers. Note that this mode is not observed in a healthy
175
system, as the lowest modes of the healthy system (occur above 500 Hz and correspond
to coupled radial and tilting motions of the shaft.
Accordingly, the faulty system is conveniently modeled as a single-degree-offreedom system in the axial (z) dimension with a mass of mS  5 kg. The (axial) stiffness
of the model can be easily extracted from the modal data as k zz  n2 mS 
5.05*106 N/m  5.05 kN/mm . Note that the calculated k zz is too low compared to the
representative stiffness values of a rolling element bearing; this is obviously due to the
bearing damage which also causes the resonance to occur at such a low frequency. A
viscous damping constant ( czz ) for the single-degree-of-freedom model can be similarly
assessed from the modal data. As mentioned earlier, the damping value would be lightly
dependent on the bearing preload, so one can determine czz for a given Fz 0 .
176
Fixed
Steering
Knuckle
(STX knuckle)
Radial (+Y)
Axial (+Z)
3
Radial (+X)
6
5
1
0
Impact Location
Shaft with Threaded
Rod And Nuts
Load Cell
(uniaxial)
2
4
Load Washer
Figure 4.14 Experiment with wheel-hub assembly using faulty and healthy bearings.
Locations of the accelerometers, sensors, and impact are illustrated.
177
0.08
1.00
F
F
F
F
F
F
F
F
F
F
F
F
F
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
FRF bearing:0:-Z/bearing:6:+Z
Faulty Bearing:
Amplitude
Healthy Bearing:
Amplitude
Axial
Preload:
g/N
Function
Function
Function
Function
Function
Function
Function
Function
Function
Function
Function
Function
Function
0.02
0.00
0.00
0.00
Hz
1800.00
Figure 4.15 Comparison of measured cross-point accelerance magnitude spectra
 A0 z

( )  for faulty and healthy bearings at various Fz 0 . Key: Fz 0 , faulty bearing: (

 F6 z

0.45 kN; (
), 0.67 kN; (
healthy bearing: (
kN; (
), 0.89 kN; (
), 0.45 kN; (
), 1.33 kN; (
), 0.67 kN; (
), 2.22 kN.
178
), 1.78 kN; (
), 0.89 kN; (
),
), 2.22 kN;
), 1.33 kN; (
), 1.78
4.10 Conclusion
The major contribution of this chapter is the verification of the new stiffness
model (of chapter 2) by using computational and analytical methods. Bearing loads,
deflections and stiffness elements are compared using four calculation methods; a
comparison of natural frequencies and radial accelerance spectra with respect to the
experimental measurements (of chapter 3) is also provided. More specific contributions
are as follows: First, the finite element based contact mechanics tool [4.21] has been
incorporated to verify some assumptions utilized in proposed Kb model. Second,
modeling approaches between a double row bearing vs. two single row bearings are
comparatively evaluated from statics and stiffness perspective. Third, four methods are
comparatively evaluated for axial and combined loading cases. This study reveals the
differences between Hertzian contact-based analytical methods and a finite element based
approach. Also, an error committed by the finite difference approximation to calculate the
bearing stiffness elements under a combined load case has been investigated.
Finally, the modal characteristics of a shaft-bearing system with a faulty bearing,
using a similar setup analyzed in chapter 3 are analyzed on a preliminary basis. The
initial experiments show that the system with a faulty bearing exhibits a single-degree-offreedom behavior, unlike the system with a healthy bearing, and the effect of axial
preloads on the faulty system is found to be far less significant. As a part of future work,
more extensive study should be conducted, and the effect of radial preloads on modal
characteristics should also be examined.
179
References for chapter 4
4.1 J.S. Archer, Consistent mass matrix for distributed mass systems, Journal of
Structural Division, Proceedings of the ASCE, 89 (1963) 161-178.
4.2 J.S. Archer, Consistent matrix formulations for structural analysis using finiteelement techniques, AIAA Journal, 3 (10) (1965) 1910-1918.
4.3 R.L. Ruhl, J.F. Booker, A finite element model for distributed parameter turborotor
systems, ASME Journal of Engineering for Industry, 94 (1972) 126-132.
4.4 H.D. Nelson, J.M. McVaugh, The dynamics of rotor-bearing systems using finite
elements, ASME Journal of Engineering for Industry, 98 (1976) 593-600.
4.5 R. Gash, Vibration of large turbo-rotors in fluid-film bearings on an elastic
foundation, Journal of Sound and Vibration, 47 (1) (1976) 53-73.
4.6 E.J. Gunter, The influence of internal friction on the stability of high speed rotors,
ASME Journal of Engineering for Industry, 89 (4) (1967) 683-688.
4.7 E.S. Zorzi, H.D. Nelson, Finite element simulation of rotor-bearing systems with
internal damping, ASME Journal of Engineering for Power, 99 (1) (1977) 71-76.
4.8 H.D. Nelson, A finite rotating shaft element using Timoshenko beam theory, ASME
Journal of Mechanical Design, 102 (1980) 793-803.
180
4.9 T.S. Sankar, E. Hashish, Finite element and modal analysis of rotor-bearing systems
under stochastic loading conditions, ASME Journal of Vibration, Acoustics, Stress
and Reliability in Design, 106 (1984) 80-89.
4.10 Z. Abduljabbar, M.M. ElMadany, A.A. AlAbdulwahab, Active vibration control of
a
flexible
rotor,
Computers
and
Structures,
58
(3)
(1996)
499-511.
4.11 F.M.A. El-Saeidy, Finite element modeling of a rotor shaft rolling bearings system
with consideration of bearing nonlinearities, Journal of Vibration and Control, 4 (5)
(1998) 541–602.
4.12 P. Cermelj, M. Boltezar, An indirect approach investigating the dynamics of a
structure containing ball bearings, Journal of Sound and Vibration, 276 (1-2) (2004)
401–417.
4.13 K.J. Bathe, A. Chowdhury, A solution method for planar and axisymmetric contact
problems, International Journal for Numerical Methods in Engineering, 21 (1)
(1985) 65-88.
4.14 A. Chowdhury, K.J. Bathe, A solution method for static and dynamic analysis of
three-dimensional contact problems with friction, Computers and Structures, 24 (6)
(1986) 855-873.
4.15 W.H. Chen, P. Tsai, Finite element analysis of elastodynamic sliding contact
problems with friction, Computers and Structures, 22 (6) (1986) 925-938.
4.16 B.R. Torstenfelt, Contact problems with friction in general-purpose finite element
computer programs, Computers and Structures, 16 (1983) 487-493.
181
4.17 B.R. Torstenfelt, An automatic incrementation technique for contact problems with
friction, Computers and Structures, 19 (3) (1984) 393-400.
4.18 S.M. Vijayakar, H.R. Busby, D.R. Houser, Linearization of multibody frictional
contact problems, Computers and Structures, 29 (4) (1988) 569-576.
4.19 S.M. Vijayakar, H.R. Busby, L. Wilcox, Finite element analysis of threedimensional conformal contact with friction, Computers and Structures, 33 (1) (1989)
49-61.
4.20 S.M. Vijayakar, A combined surface integral and finite element solution for a threedimensional contact problem, International Journal for Numerical Methods in
Engineering, 31 (3) (1991) 525-545.
4.21 Advanced Numerical Solutions, Calyx© User’s Manual, www.ansol.us (2011).
4.22 Romax Technology Limited, RomaxDesigner©, www.romaxtech.com (2011).
4.23 T. C. Lim, R. Singh, Vibration transmission through rolling element bearings, part I:
bearing stiffness formulation, Journal of Sound and Vibration 139 (2) (1990) 179–
199.
4.24 T.C. Lim, Rolling Element Bearing Matrix Calculation Program (REBM)© (2011).
4.25 Lagarias, J. C., J. A. Reeds, M. H. Wright, and P. E. Wright, Convergence properties
of the Nelder-Mead simplex method in low dimensions, SIAM Journal of
Optimization, 9 (1) (1998) 112–147.
4.26 T. J. Royston, I. Basdogan, Vibration transmission through self-aligning (spherical)
rolling element bearings: theory and experiment, Journal of Sound and Vibration 215
(5) (1998) 997-1014.
182
4.27 J. M. De Mul, J. M. Vree, D. A. Vaas, Equilibrium and associated load distribution
in ball and roller bearings loaded in five degrees of freedom while neglecting
frictions I: General theory and application to ball bearings, Transactions of the
ASME, Journal of Tribology, 111 (1989) 142-148.
4.28 X. Hernot, M. Sartor, J. Guillot, Calculation of the stiffness matrix of angular
contact ball bearings by using the analytical approach, Transactions of the ASME,
Journal of Mechanical Design, 122 (2000), 83-90.
4.29 LMS International, LMS Test Lab© Manual, www.lmsintl.com, 2011.
183
CHAPTER 5
CONCLUSION
5.1 Summary
A new stiffness model for the double row angular contact ball bearings is
proposed to provide a better understanding of the multidimensional stiffness
characteristics of these components, and for the analysis of mechanical systems that
might contain double row (or duplex) angular contact ball bearings. Prior researchers had
not focused on the double row bearings except Royston and Basdogan [5.1], who studied
self-aligning bearings and neglected all angular displacement and tilting moment effects
due to unique properties of self-aligning bearings. The angular displacements and
moment stiffnesses of double row angular contact ball bearings however, are significant,
and the stiffness coefficients are highly dependent on the configuration of the rolling
elements. Also, the contact angle (  ) of ball-type bearings change under a static load,
unlike roller-type bearings where the contact angle remain relatively constant.
In Chapter 2, a systematic, analytical approach to determine the five-dimensional
bearing stiffness matrix (Kb) for double row angular contact ball bearings of face-to-face,
back-to-back, and tandem arrangements is developed. This work is an extension of Lim
184
and Singh’s stiffness model for single row bearings [5.2], however the proposed
formulation is more complex due to the presence of two rows that can organize in
different configurations. Using the proposed explicit expressions, one can determine the
diagonal and off-diagonal (cross-coupling) stiffness coefficients of a double row angular
contact ball bearing given the mean displacement vector qm (by direct substitution) or the
mean load vector fm (by numerical solution of the nonlinear system equations). The
diagonal elements of the stiffness matrix are verified with a commercial code [5.3]
through a comparison of the stiffness elements for an example case under three loading
scenarios. 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 more insight. The proposed Kb formulation is also valid for duplex (paired)
bearings which behave as an integrated double row unit, especially when the surrounding
structural elements (such the shaft and bearing rings) are sufficiently rigid.
In chapter 3, the role of bearing preloads on the modal characteristics of a shaftbearing assembly with a double row angular contact ball bearing is investigated. First, a
five-degree-of-freedom analytical model which includes the proposed Kb, is developed to
comparatively evaluate the preload effects on the modal responses of three double row
configurations. Changes in the resonant amplitudes with bearing preloads are examined
for preload-independent and preload-dependent viscous damping models. It has been
shown that the bearing preloads significantly affect the vibration characteristics of the
shaft-bearing assembly due to major changes in both diagonal and off-diagonal elements
of the stiffness matrix and such effects depend on the bearing configuration. Second, a
185
new experiment consisting of a vehicle wheel bearing assembly with a double row
angular contact ball bearing in a back-to-back arrangement has been designed. The
bearing is subjected to axial or radial preloads in a controlled manner. Experiments with
two preloading mechanisms show that the nature and extent of bearing preloads
considerably affect the natural frequencies and resonant amplitudes, thus influencing the
vibration behavior of the bearing assembly. Finally, the proposed stiffness model is
validated by comparing the predicted natural frequencies and accelerance spectra of the
five-degree-of-freedom model with modal measurements.
In chapter 4, the new stiffness model (developed in chapter 2) has been verified
using analytical and computational methods [5.2-5.4]. First, a finite element based
contact mechanics code [5.4] has been incorporated to verify some assumptions utilized
in proposed Kb model. Second, modeling approaches between a double row bearing vs.
two single row bearings are comparatively evaluated from statics and stiffness
perspective. Third, four calculation methods are comparatively evaluated through a
critical examination of bearing loads, deflections and stiffness elements. Fourth,
predicted natural frequencies and accelerance magnitude spectra using four methods are
compared with the measurements of chapter 3. In all of these studies, the advantages and
the benefits of the proposed model have been illustrated. The differences between
Hertzian contact-based analytical methods and finite element based methods have also
been revealed. Finally, the modal characteristics of a shaft-bearing system with a faulty
bearing, using a similar setup analyzed in chapter 3, is analyzed on a preliminary basis;
186
and the effects of bearing preloads on a healthy and a faulty system are comparatively
evaluated to promote future research.
5.2 Contributions
In this dissertation, several inter-related contributions (as related to the stiffness
characteristics of double row angular contact ball bearings, and their effects on the modal
characteristics of a shaft-bearing system) are evident:
1. A new, comprehensive five-dimensional stiffness matrix has been proposed for double
row (and duplex) angular contact ball bearings in face-to-face, back-to-back, or tandem
arrangement. The diagonal elements of the new analytical model have been verified using
a commercial code; and the effects of critical geometric and kinematic parameters on the
stiffness elements are investigated to better understand the underlying physics.
2. The proposed stiffness matrix is validated using a new modal experiment on an
automotive wheel-hub assembly (with a back-to-back double row angular contact ball
bearing) using the natural frequency and sinusoidal transfer function results of a five
degree-of-freedom model. The effects of axial and radial bearing preloads on the natural
frequencies, system modes and resonant amplitudes of a shaft-bearing system are
extensively investigated using controlled experiments and analytical methods. The effect
of system damping is examined using alternate preload-independent and preloaddependent viscous damping models.
187
3. The new stiffness model is further verified using analytical and computational methods
(including finite element based techniques). The need for the new formulation is
quantitatively demonstrated by comparing modeling approaches between a double row
bearing vs. two single row bearings. Good agreement between the bearing load,
deflection and stiffness calculations with four calculation methods is obtained, and
predicted natural frequencies and accelerance spectra are successfully compared with
modal measurements. Preliminary modal experiments with a faulty bearing are also
included to promote future research.
5.3 Future Work
Several areas of potential research based on the present study include the
following:
1.
Extend the proposed stiffness formulation for double row (and duplex) angular
contact ball bearings to different bearing types such as double row cylindrical and
tapered roller bearings.
2.
Extend the load-dependent and load-independent viscous damping models (utilized
in chapter 3), and develop a comprehensive, five-dimensional bearing damping
matrix ( Cb ). The experimental results of chapter 3 suggest a preload and modedependent dissipative mechanism, thus, the effects of preload and  (frequency)
should be included in the Cb model. Hertzian contact theory should be extended to
188
characterize the contact (or Hertzian) damping between the rolling elements and
bearing races.
3.
Develop a vibration based method to estimate bearing preloads and in-situ stiffnesses
by making use of the frequency and amplitude shifts of the resonant peaks with
preload measured from the shaft-bearing system; this study would ultimately lead to
on-line bearing stiffness and preload monitoring. A methodology chart for the
development of this method is suggested in Figure 5.1.
4.
Examine the effects of bearing damage on the modal characteristics of a shaftbearing assembly on a more extensive basis and characterize the effects of bearing
preloads on such faulty systems. Analyze the effects of radial and moment loads on
the modal characteristics of a faulty system experimentally. Also, develop advanced
fault diagnostic and on-line monitoring techniques using the proposed stiffness
matrix.
189
Structural Dynamic Model
Compare
the forced
response

{x}

{x}
[M] [C] [K]
Known
Frequency
response
{x}
Vibration
measurements
Unknown
Iterative
loop
Kinematic bearing
parameters/constants
ˆ

r
uˆ r
[K]
[K]b
Global
stiffness
matrix
Bearing
stiffness
matrix
ˆr
[C]
Estimated
modal
properties
Global
damping
matrix
++
Bearing
Preloads
Solve Implicit set of
Nonlinear Equations
Figure 5.1 Proposed methodology for vibration based preload and stiffness estimation
190
References for chapter 5
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rolling element bearings: theory and experiment, Journal of Sound and Vibration
215 (5) (1998) 997-1014
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bearing stiffness formulation, Journal of Sound and Vibration 139 (2) (1990) 179–
199.
5.3 Advanced Numerical Solutions, Calyx© User’s Manual, www.ansol.us (2011).
5.4 Romax Technology Limited, RomaxDesigner©, www.romaxtech.com (2011).
191
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194
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