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DYNAMIC STUDIES OF ROLLING ELEMENT
BEARINGS WITH WAVINESS AS A
DISTRIBUTED DEFECT
by
CHETTU KANNA BABU
INDUSTRIAL TRIBOLOGY MACHINE DYNAMICS AND
MAINTENANCE ENGINEERING CENTER
Submitted
in fulfillment of the requirements of the degree of
Doctor of Philosophy
to the
INDIAN INSTITUTE OF TECHNOLOGY DELHI
MARCH-2013
TO MY FATHER
Certificate
This is to certify that the thesis entitled “Dynamic Studies of Rolling Element
Bearings with Waviness as a Distributed Defect” being submitted by Mr. Chettu
Kanna Babu (2006ITZ8049) to the Indian Institute of Technology Delhi, New Delhi,
India, for the award of the degree of Doctor of Philosophy is a record of bonafide
research work carried out by him under our guidance and the candidate has fulfilled the
requirements for the submission of this thesis. The thesis, in our opinion has attained a
standard required for a Ph.D. degree of this institute. As per our awareness the results
contained in this thesis have not been submitted in part or in full to any other university
or institute for award of any degree or diploma.
Prof. N. Tandon
Dr. R. K. Pandey
(Supervisor)
(Supervisor)
Professor
Associate Professor
Industrial Tribology, Machine Dynamics and
Department of Mechanical
Maintenance Engineering Centre (ITMMEC)
Engineering
I.I.T. Delhi, New Delhi-110016, India
I.I.T. Delhi, New Delhi-110016,
India
Date:
Place: IIT Delhi, New Delhi
i
Acknowledgements
I wish to express my deep sense of gratitude and indebtedness to my thesis
supervisors, Prof. N. Tandon and Dr. R. K. Pandey, for their constant support,
inspiration and encouragement during the course of this endeavor. Their comments and
suggestions at various stages have immensely improved my understanding of the subject
and paved the way from time to time for solutions of governing equations and
experimental studies reported in this thesis.
I acknowledge all the supports provided by Prof. O. P. Gandhi (Head, Industrial
Tribology, Machine Dynamics and Maintenance Engineering Centre) in completion of
this thesis. I am highly thankful to Dr. A. K. Darpe (External expert in SRC),
Department of Mechanical Engineering, Indian Institute of Technology Delhi, for his
valuable suggestions and vital feedbacks during my presentations. I express my sincere
gratitude to internal SRC members Prof. O. P. Gandhi and Prof. V. K. Agarwal for
their kind technical suggestions and help time to time. I would also like to acknowledge
the directly or indirectly help provided by other faculty members of IIT Delhi during
perusal of this study.
Vital helps rendered by Mr. Ashok Kumar, Mr. Avatar Singh and Mr. P. N.
Jha in Machine Dynamics Laboratory (ITMMEC) during my experimental studies are
gratefully acknowledged. I would also like to thankfully record the necessary helps
extended by other staff members of the centre. I must mention and record my special
thanks to my co-researchers Dr. Vinod Patel, Ms. Sidra Khanam and Mr. U. Sudeep at
ii
ITMMEC, for their moral and academic interaction supports during pursue of this
endeavor.
I most respectfully record my gratitude to the management of Hindustan
Aeronautics Limited, Bangalore, India, for granting needful permission and sponsoring
me to pursue my Ph. D. programme at ITMMEC, IIT Delhi. My sincerely thank with
great regards to Dr. V. Sridhara, Mr. P. V. Mustafa, Prof. B. Satyanarayana, Mr. M.
N. Manjunatha, Mr. R. K. Bharati, Mr. P.R. Nagasubramanian, Mr. G.
Harikrishnan, Mr. Girish K Degonkar and Prof. K. Ramji for their helping hands
during pursuing of my research activities. I also thank Mr. R. P. Singh, Mr. Asit Behra,
Mr. Viswanath, Dr. Ramkumar and Dr. G. V. Rao for their kind helps during pursuing
of this research programme.
I express my deep felt love and gratitude to my family members, wife
(Umamaheswari), daughter (Shanmukha priya), and all nears and dears for having
patiently put up with me and constantly encouraging me throughout this endeavor at the
cost of their own sufferings.
(Chettu Kanna Babu)
Date:
Place: IIT Delhi, New Delhi
iii
Abstract
Rolling
bearings
find
wide
applications
in
industrial
and
domestic
machines/appliances. In spite of high precision manufacturing of the components of the
rolling bearings, geometrical imperfections happen to arise in the bearings at the
beginning (in new piece itself) due to manufacturing/assembly constraints or during the
use of the bearings. The geometrical imperfections invite vibrations in the bearings
during operation, which highly influence the rolling bearing’s performance parameters
such as temperature rise, wear and reliable life. Therefore, it is worth exploring the
combined effects of the geometrical imperfections and frictional moments on the
vibration behaviors of the rolling bearings. In this thesis attempts have been made to
understand the role of the geometrical imperfection (distributed defect) on the vibrations
of rolling bearings using theoretical and experimental approaches. The main objective of
the study reported in this thesis is to develop a general and realistic non-linear 6-DOF
dynamic model of a rotor-bearings system for investigations of vibrations of the rotorbearings system. The waviness on races and balls, frictional moments, shaft deflection,
centrifugal force and gyroscopic moment of the balls has been accounted in this proposed
analytical/numerical model for studies of the vibrations. However, the additional
objective of this thesis is to investigate the combined influence of the lubricant starvation
and waviness on the vibrations of the bearing-rotor system.
Non-linear vibration analysis of angular contact ball bearings has been simulated
herein considering load dependent and load independent components of frictional
iv
moments in the bearing. Six degrees of freedom of rigid rotor is considered in the
dynamic modeling of the rotor-bearing system. Moreover, surface waviness on inner
race, outer race and ball is considered and has been incorporated in the model by
representing it as sinusoidal function. The proposed model is validated with the
experiments and published results of researchers by incorporating needful changes in
degrees of freedom in the proposed model. Based on the computed results, it is observed
that the load independent component of frictional moment significantly reduces the
amplitudes of vibrations. The influence of inner race waviness is relatively more on the
vibration in comparison to waviness of outer race and ball. Moreover, at large amplitude
of waviness and increase in the order of waviness, vibration enhances considerably.
Non-linear vibration model of the angular contact ball bearings supported rigid
rotor has been simulated herein considering the combined influences of the lubricant
starvation and waviness on the balls and races of the bearings. Based on the results of the
proposed model, it is observed that for the same order of the waviness on the bearing
components, the vibrations in the transverse direction of the rotational axis enhances
considerably in case of starved contacts in comparison to fully flooded contacts of the
bearings. Moreover, the influence of inner race waviness is relatively more on the
vibrations in comparison to same orders and amplitudes of waviness present on outer race
and balls. At large amplitudes of waviness and increase in the orders of waviness,
vibrations enhance significantly. It is necessary to mention here that due to severity of
lubricant’s starvation; the amplitudes of vibrations reach the same order as due to
increase in the orders of the waviness on the races and balls. Based on the proposed
v
vibration model, authors found that the effects of starvation on vibrations are significant
and it must be accounted for in the dynamic models of bearings for achieving accurate
results.
The model has also been used for vibration analysis of high speed angular contact
ball bearings supporting the rotor bearing system under the effect of shaft deflection,
centrifugal force and gyroscopic moment of the ball. Concepts pertaining to inner ring
centrifugal displacement, frictional moments and waviness of balls and races have also
been discussed. Three dimensional Timoshenko beam element with six degrees of
freedom for each node has been adapted for computation of shaft deflection. Newton
Raphson with centered finite divided difference formulae have been used to solve the
deformation consistent equations. Nonlinear equations have been solved using the
Runge-Kutta 4th order method. Results have been validated through comparison with the
research results established by predecessors. Based on the computed results, it is
observed that the influence of outer race waviness on vibration is more as compared to
the influence of inner race waviness and ball. Due to the centrifugal force and gyroscopic
moment, side bands are observed around defect frequency and operating frequency in the
spectrum. It is also important to note that the vibration enhances considerably owing to
increase in the order and/or amplitude of waviness. Other major reasons attributing to
change in vibration frequencies are seen as shaft deflection, centrifugal force and
gyroscopic moment of the ball
Experimental investigations of vibration response of deep groove ball bearings in
presence of waviness on either of bearings’ races with fully flooded and starved condition
vi
are attempted. Waviness on inner and outer races of bearings is created by electric
discharge machining. Experimentations for vibration response of bearings with fully
flooded and starved conditions have been conducted on a test rig using both healthy and
defective (with waviness) bearings. The vibration signals have been acquired using
accelerometer, signal conditioner and FFT. The acquired signals have been processed in
time and frequency domains. Characteristic defect frequencies and its harmonics are
broadly investigated in the frequency spectra. Comparisons of vibration responses of
healthy and defective (with waviness on either of races) with fully flooded and starved
bearings are provided and discussed the effect of starvation on vibrations.
**************
vii
Contents
Certificate
i
Acknowledgements
ii
Abstract
iv
Contents
viii
List of Figures
xii
List of Tables
xxvii
Nomenclature
xxx
Chapter-1
Introduction
1.1
Defects in Rolling Bearings
4
1.2
Techniques of Defect Detection
7
1.3
Scope of Study
11
1.4
Organization of Thesis
11
Literature Review
13
2.1
Dynamic Models of Rolling Bearings/Rotor Systems with
Distributed Defects
13
2.1.1
Models considering Lubricated Contacts
14
2.1.2
Models neglecting Lubricated Contacts
23
2.2
Dynamic Models of Rolling Bearings with Nonlinearity Effects
28
2.3
Lubricated Rolling/Sliding Concentrated Contacts
32
Chapter-2
2.4
Conclusions of Literature Review
viii
33
2.5
Objectives of Studies in the Thesis
34
Dynamic Model of Rotor Ball Bearing System with 6 DOF
33
3.1
Rigid Rotor Supported on Lubricated Ball Bearing
33
3.2
Deformed Rotor Supported on Lubricated Ball Bearings
41
3.3
Waviness Models for Ball and Races
47
3.4
Frictional Moments at the Contacts of Balls and Races
50
3.5
Dynamic Coefficients between Bearing Elements
52
3.6
Computational Procedure for Rigid Rotor Supported on
Lubricated Ball Bearing
54
3.7
Computational Procedure for Deformed Rotor Supported on
Lubricated Ball Bearing
56
Numerical Results of Dynamic Model for 6 DOF
Rigid Rotor Ball Bearing System
58
Chapter-3
Chapter-4
4.1
Numerical Results of Dynamic Model for Fully Flooded Condition 58
4.1.1
Validation with Theoretical Models and Experimental Results
59
4.1.2
Influence of Frictional Moments
61
4.1.3
Influence of Frictional Moments with Inner race Waviness
64
4.1.4
Influence of Frictional Moments with Outer race Waviness
83
4.1.5
Influence of Frictional Moments with Ball Waviness
96
4.1.6
Concluding Remarks
100
4.2
Numerical Results of Dynamic Model for Starved
4.2.1
Starvation
Condition
101
101
ix
4.2.2
Damping Coefficients and Starved Film Thickness at Contacts
103
4.2.3
Computational Procedure
104
4.2.4
Inner race Waviness
104
4.2.5
Outer race Waviness
111
4.2.6
Ball Waviness
117
4.2.7
Concluding Remarks
120
Numerical Results of Dynamic Model for 6 DOF
Deformed Rotor Ball Bearing System
141
5.1
Numerical Results of Dynamic Model with Shaft deflection
141
5.1.1
Validation of the Proposed Model
142
5.1.2
Inner race Waviness
144
5.1.3
Outer race Waviness
149
5.1.4
Ball Waviness
154
5.1.5
Conclusions
158
5.2
Influence of Centrifugal Force and Gyroscopic Moment of the Ball 160
and Inner ring Centrifugal Displacement on Vibration
5.2.1
Inner ring Centrifugal Displacement of Angular Contact
Ball Bearing
160
5.2.2
Rolling Element Deflection and Load Analysis
161
5.2.3
Calculation of Frictional Moment
165
5.2.4
Bearing Force and Moments acting on the Shaft
167
5.2.5
Computational Procedure
168
5.2.6
Validation of the Proposed Model
169
5.2.7
Outer race Waviness
171
Chapter-5
x
5.2.8
Inner race Waviness
177
5.2.9
Ball Waviness
181
5.2.10 Conclusions
Chapter-6
181
Experimentation
183
6.1
Experimental Setup and Instrumentation
183
6.2
Experimental Results and Discussions
197
6.2.1
Theoretical Model validation with Experimental Results of
Fully Flooded Condition
197
6.2.2
Theoretical Model validation with Experimental Results of
Starved Condition
219
6.3
Concluding Remarks
237
Conclusions and Scope for Future Work
238
7.1
Vibration at Low Operating Speed of Bearings
238
7.2
Vibration at High Operating Speed of Bearings
239
7.3
Scope for Future Work
240
Chapter-7
References
241
Appendix-A
xi