multi-dimensional stiffness characteristics of double row angular contact ball bearings and their

224 Pages·2012·1.66 MB·English

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 cross-
coupling 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.
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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 shaft-
bearing 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.

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DEDICATION

Dedicated to my mom and dad

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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.
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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

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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
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2.5.2 Numerical estimation of K ..................................................................... 43
b
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
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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
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