Advanced Dynamics of Rolling Elements

306 Pages·1984·5.767 MB·English

by  Pradeep K. Gupta (auth.)

Advanced Dynamics
of Rolling Elements
Pradeep K. Gupta
Advanced Dynamics
of Rolling Elements
With 97 Figures
Springer-Verlag
Berlin Heidelberg New York Tokyo
Pradeep K. Gupta
PKG Incorporated
117 Southbury Road
Clifton Park, New York 12065
U.S.A.
Library of Congress Cataloging in Publication Data
Gupta, Pradeep K.
Advanced dynamics of rolling elements.
Bibliography: p.
Includes indexes.
I. Rolling contact. 2. Ball-bearings. 3. Roller bearings. I. Title.
TJI83.5.G87 1984 621.8'22 84-10567
© 1984 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1984
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York,
New York 10010, U.S.A.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
Printed and bound by R. R. Donnelley and Sons, Harrisonburg, Virginia.
Printed in the United States of America.
9 8 7 6 5 4 3 2 I
TSBN-13: 978-1-4612-9767-3 e-TSBN-13: 978-1-4612-5276-4
DOl: 10.1007/978-1-4612-5276-4
to Shelly
Preface
In any rotating machinery system, the bearing has traditionally been a crit
ical member of the entire system, since it is the component that permits the
relative motion between the stationary and moving parts. Depending on
the application, a number of different bearing types have been used, such as
oil-lubricated hydrodynamic bearings, gas bearings, magnetic suspensions,
rolling element bearings, etc. Hydrodynamic bearings can provide any
desired load support, but they are limited in stiffness and the associated
power loss may be quite large. Gas bearings are used for high-precision
applications where the supported loads are relatively light, bearing power
losses are very low, and the rotating speeds generally high. For super
precision components where no frictional dissipation or bearing power loss
can be tolerated, magnetic suspensions are employed; again, the load support
requirements are very low. Rolling element bearings have been widely used
for those applications that require greater bearing versatility, due to the
requirements for high-load and high-stiffness characteristics, while allowing
moderate power loss and permitting variable speeds. A study of the dynamic
interaction of rolling elements is, therefore, the subject of this text.
Texts covering the analysis and design methodology of rolling elements
are very limited. Notable works include Analysis of Stresses and Deflections
(Jones, 1946, Vols. I and II), Ball and Roller Bearings, Their Theory, Design
and Application (Eschmann, Hasbargen, and Weigand, 1958), Ball and Roller
Bearing Engineering (Palmgren, 1959, 3rd ed.), Advanced Bearing Technology
(Bisson and Anderson, 1965), and Rolling Bearing Analysis (Harris, 1966).
Most of these texts were published at a time when the available computa-
VII
viii Preface
tional means were very limited. Thus, all of this work was restricted to solving
an equilibrium problem, which is a first approximation to the analysis of
rolling bearings.
During the last decade, the operating speeds and temperatures that
bearings must survive have increased greatly, due to the demand for higher
efficien0'. This situation has created considerable interest in the advance
ment of rolling bearing technology. With the advent of modem high-speed
computers, the simple quasi-static equilibrium models of the 1960s have
been replaced by advanced dynamic models capable of providing real-time
performance simulations of rolling bearings. To a rolling bearing engineer,
such an advancement offers design tools for improved designs against
skidding, skewing, cage instabilities, lubricant traction behavior, and time
varying operating environments, as compared to the simple fatigue life and
stiffness computations provided by the static models.
The principal motivation of the text is to document the vastly improved
state of the art from an engineer's viewpoint. The text is a result of the
research I have carried out under both government and industrial sponsor
ship over the past several years. The objectives are to
1. present a generalized formulation of a dynamic model to simulate
rolling bearing performance under arbitrary operating conditions,
and
2. provide the reader with a computer code to implement the analytical
model in actual practice.
The dynamic model basically consists of the formulation of the coordinate
frames and equations of motion of the bearing elements, and the geometric
interactions which provide the foundation for modeling the applied force
and moment vectors. These subjects are covered in Chapters 2 and 3, follow
ing the basic background presented in the first chapter. Specialized subjects
particularly applicable to oil-lubricated rolling bearings are included in
Chapters 4 and 5. Chapter 6 is devoted to numerical analysis, which logically
precedes the presentation of the computer program framework and struc
ture in Chapter 7. Typical results of the computer program and dynamic
simulations of rolling bearings are presented in Chapter 8. In view of the
very limited experimental data, a rigorous experimental validation of the
analytical models has been difficult; however, I have attempted to present
some comparisons between the analytical results and the recently available
experimental data in Chapter 9. I expect that experimental validations and
subsequent improvements of the analytical models will continue as more
experimental data become available. Finally, in Chapter 10 I have briefly
presented some general design guidelines from an engineer's viewpoint.
I have named the computer program developed in this text ADORE
(Advanced Dynamics Of Rolling Elements). In view of the complexity and
length of the code, I have made ADORE operational on some of the world
wide computer networks, such as the CYBERNET Services. * The intent is
*CYBERNET is a registered trademark of Control Data Corporation.
Preface ix
to provide the reader immediate access to the code. As may be expected,
sophisticated computer codes are constantly updated as the technology be
hind the codes advances and the understanding of the fundamental model
grows. Because ADORE is operational on a widely available computer net
work, I will be able to effectively maintain and support the code, and the user
will have the benefit of the latest version of the program which may be sub
stantially improved in comparison to the original program presented in this
text. I greatly encourage interested readers to use ADORE on the network
and, if possible, to supply me with comments and suggestions which will be
crucial to its future improvement. In order to provide adequate instructions
to the user while running ADORE on the network, I have included the source
listing of the main program in one of the appendices. I hope this listing, which
is strictly for reference purposes, will help the reader understand some of the
finer nuances of ADORE and that, through example, it will inspire sophis
ticated readers to develop their own codes, which will eventually help
advance the general technology behind ADORE.
The text is directed principally toward users and designers of advanced
rolling bearing systems; in particular, engine manufacturers, manufacturers
of inertial guidance systems, and the rolling bearing industry. It will also be
very helpful to research staff both in government and in industrial labora
tories.
It is my greatest pleasure to acknowledge Mechanical Technology Incor
porated for the research opportunities and environment provided to me
during my tenure there as a Senior Scientist (1971-82). In particular, I would
like to mention the encouragement and support provided by Dr. Donald F.
Wilcock, Dr. Jed A. Walowit, Mr. Oscar Pinkus, Mr. Wilbur Shapiro, and
Dr. Jeffrey A. Asher. I am very grateful to Mrs. Terri Brandt and Mrs. Pat
Marx for the outstanding typing of the manuscript. My thanks are also
due to Ms. Donna Graham for her ingenious editing, and Miss Rose Ann
Coons who drew all of the illustrations in the book. Finally, the encourage
ment I received from my wife Shelly and my daughters Neha and Priya has
been crucial to the preparation of this text. The extensive time I spent on
the computer, particularly during nights and holidays, has been a great
sacrifice on their part and I am extremely thankful for their support.
Clifton Park, New York Pradeep K. Gupta
Contents
Chapter 1. Introduction
1.1 Rolling Bearing Elements and Basic Interactions 2
1.2 Types of Analytical Models 4
1.2.1 Quasi-Static Model 4
1.2.2 Dynamic Model 6
1.3 Nomenclature 7
1. 3.1 Coordinate Frames 8
1.3.2 Vector Transformations 8
1.3.3 List of Symbols 9
1.4 Summary 11
Chapter 2. Equations of Motion and Coordinate Transformations 12
2.1 Coordinate Frames and Transformations 12
2.2 Equations of Motion 16
2.2.1 Mass Center Motion 16
2.2.2 Rotational Motion 16
2.3 Moving Coordinate Frames 18
2.4 General Motion Simulation 19
2.5 Summary 21
Chapter 3. Geometric Interactions in Rolling Bearings 22
3.1 Rolling Element/Race Interactions 24
3.1.1 Ball/Race Interactions 24
Xl
XII Contents
3.1.2 Roller/Race Interactions 34
3.1.3 Roller/Race-Flange Interactions 41
3.2 Rolling Element/Cage Interactions 46
3.2.1 Geometric and Kinematic Considerations 47
3.2.2 Hydrodynamic Models 52
3.2.3 Dry Contact Models 59
3.3 Race/Cage Interactions 59
3.3.1 Geometric and Kinematic Considerations 59
3.3.2 Hydrodynamic Models 61
3.3.3 Dry Contact Models 61
3.4 Interactions Between Rolling Elements 61
3.4.1 Ball Bearings 62
3.4.2 Roller Bearings 63
3.5 External System Interactions and Constraints 64
3.5.1 Equilibrium Constraint for Ball Bearings 64
3.5.2 Equilibrium Constraint for Roller Bearings 72
3.6 Summary 75
Chapter 4. Elastohydrodynamic Lubrication 76
4.1 General Consideration in Lubricant Traction Modeling 77
4.1.1 Rolling Element/Race Contact Zone 77
4.1.2 Lubricant Rheology 78
4.1.3 Typical Traction-Slip Behavior 79
4.2 An E1astohydrodynamic Traction Model 80
4.2.1 Film Thickness Computation 81
4.2.2 Computation of Traction 85
4.2.3 Estimation of Lubricant Constitutive Equation 88
4.3 Traction Behavior of Some Lubricants 91
4.3.1 U.S. Specification MIL-L-23699 92
4.3.2 U.S. Specification MIL-L-7808 94
4.3.3 Traction Fluid Santotrac 30 94
4.3.4 Polyphenyl Ether 95
4.3.5 SAE-30-Type Oil 96
4.4 Summary 97
Chapter 5. Churning and Drag Losses 100
5.1 Estimation of Drag Forces 100
5.2 Estimation of Churning Moments 102
5.2.1 Loss on the Cylindrical Surface 103
5.2.2 Loss on the End Surface 104
5.3 Effective Lubricant Viscosity and Density 104
5.4 Summary 105
Contents Xlii
Chapter 6. Numerical Integration of the Equations of Motion 106
6.1 Dimensional Organization 107
6.2 Explicit Algorithms 109
6.2.1 Step-Changing Criterion 111
6.3 Implicit Algorithms 115
6.3.1 Predictor Formula 115
6.3.2 Corrector Formula 117
6.3.3 Step-Changing Criterion 118
6.3.4 Change of Order 119
6.3.5 Computational Considerations 119
6.4 Selection of a Method 119
6.5 External Constraints 120
6.5.1 Equilibrium Constraints 120
6.5.2 Fictitious Damping 121
6.6 Summary 121
Chapter 7. The Computer Program ADORE 123
7.1 Program Overview 126
7.2 Structure of ADORE 126
7.3 ADORE Capabilities 128
7.3.1 Bearing Types 128
7.3.2 Types of Cages 128
7.3.3 Operating Conditions 129
7.3.4 External Constraints 129
7.3.5 Radial Preloads 129
7.3.6 Material Properties 130
7.3.7 Lubricant Traction 130
7.3.8 Churning and Drag 130
7.3.9 Roller Skew 130
7.3.10 Rolling Element Skid 131
7.3.11 Cage Instability 131
7.3.12 Bearing Power Loss 131
7.3.13 Wear 131
7.3.14 Geometric Imperfections 131
7.3.15 Bearing Noise 132
7.3.16 Bearing Life for Arbitrary Load and Speed Cycles 132
7.3.17 Flexibility in Units 133
7.3.18 Graphic Output 133
7.3.19 Integrating Algorithms 133
7.3.20 Restart Capabilities 133
7.4 Input/Output Data 133
7.4.1 Input Data 134
7.4.2 Print Output 136
7.4.3 Plot Output 140