DYNAMICS OF CRACKED ROTORS
by
ASHISH K DARPE
DEPARTMENT OF MECHANICAL ENGINEERING
Submitted
in fulfillment of the requirements of the degree of DOCTOR OF PHILOSOPHY
to the
INDIAN INSTITUTE OF TECHNOLOGY, DELHI NEW DELHI - 110016, INDIA
JUNE, 2002
I. T. DELHI.
LI99ARY Notfal.1=?1:15:-.
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© Indian Institute of Technology, New Delhi 2002
All rights reserved
CERTIFICATE
This is to certify that the thesis entitled "Dynamics of Cracked Rotors" being submitted by Ashish K Darpe to the Indian Institute of Technology, Delhi for the award of the degree of Doctor of Philosophy is a record of bona fide research work carried out by him under our supervision and guidance. The thesis work, in our opinion has reached the requisite standard fulfilling the requirement for the degree of Doctor of Philosophy.
The results contained in this thesis have not been submitted in part or in full, to any other University or Institute for the award of any Degree or Diploma.
(Dr. K Gupta)
Professor Associate Professor
Department of Mechanical Engineering Indian Institute of Technology, Delhi
New Delhi, 110016
ACKNOWLEDGEMENTS
I would like to express my sincere thanks and deep sense of gratitude towards my thesis supervisors, Professor K Gupta and Dr. A Chawla for their constant encouragement and inspiring guidance. Their critical reviews and constructive comments improved my grasp of the subject and steered me to the fruitful completion of the work.
Prof. Gupta, despite a very hectic schedule made it a point to spare time for valuable discussions and sharing wealth of his three decades of knowledge and experience on the subject. He has been very kind in his approach and cheered me all through during the course of this work. Dr. Chawla provided excellent help and understanding in every step of my work right from day one. My gratitude towards him for allowing me to exploit the most updated computing facility at the A.I. and Simulation Lab. I must also acknowledge his prompt response to every request of having discussions with him.
I also take this opportunity to express my heartfelt thanks to the SRC members Prof B. Seth, Dr. Om Prakash, Dr. S.P.Singh and also to Prof J. S. Rao and Dr. Atul Bhaskar (former SRC members) for their useful suggestions. Particular mention should be made of Dr.S.P.Singh for his ever willing support and many lively discussions.
During the course of this research work, I had the privilege of working in various research projects in related areas under the leadership of Prof B.C.Nakra, Prof J. S.Rao, Prof K Gupta and Dr. A Chawla. I extend warm respects to them for their continuous support and acknowledge their forbearance while I tried to divide time between PhD and project work.
I would like to express my heartfelt thanks to Prof C. W. Lee (Korea Advanced Institute of Science and Technology, Korea) for the useful discussions during his visit to I.I.T. Delhi and to Prof R. H. Plaut (Virginia Polytechnic Institute and State University, U.S.), Dr. A.S. Sekhar (I.I.T. Kharagpur) for their prompt replies to my queries / reprints.
Mr. K. N. Madhu has been both, a guide in the laboratory work and an elder brother for me. All the worries and frustration used to end on his table as he cheered and nursed the quest of every research scholar in the Vibration Research Laboratory.
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When things went out of control during thesis compilation, Mr. T. S. Arvindan lent a much-needed helping hand. Thanks for his willing help. I am also thankful to Mr.
Gamdoor Singh, Mr. Sivakumar, Mr. Ayodhya Prasad and Mr. Jaitley for their kind support and co-operation and to my friends Amit and Sunny for their joyful company.
I acknowledge the strong moral support of Prof. Gupta, Dr. Chawla, Prof. J.S.Rao and their families as they stood behind me during the periods of crisis. I will remain indebted for the cozy stay of my family to the unflinching support of the families of Prof.
S. G. Deshmukh and Mr. K. N. Madhu. My special regards to Prof. Deshmukh, Mrs.
Rugmani Sundaran, Dr. S. Mukherjee, Prof Bijwe and Dr. Mrs. Bijwe for their good wishes and encouragement and to my sisters for their prayers and love. I also express my affection to all my friends and relatives through this work.
I owe every bit of this work to my family. Every member of the family has been a source of strength and inspiration for me. My parents made supreme sacrifices for me throughout my career and particularly during this endeavor. No words can express my feelings towards them as they willingly gave up a more pleasant and comfortable life at Nagpur and accompanied me here to provide strong support. My wife has been a cheerleader to the core and has an uncanny knack of shooting down occasional bursts of sadness. Many thanks to her for showing me silver lining in every dark cloud. And how can I forget my daughter, who with her smile and innocence showered love that wiped all the traces of tiredness day after day.
Ashish Kamalakar Darpe New Delhi India
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ABSTRACT
Dynamic response analysis of a cracked rotor is attempted. A response dependent breathing crack model accounts for the breathing of crack. Transient response and breathing behaviour of an accelerating / decelerating cracked rotor is analysed. The effects of various parameters on the peak response variation and breathing behaviour are investigated. The response of the accelerating cracked rotor through subharmonic resonances is studied using time domain, frequency domain and orbit plots.
The coupled vibrations of a cracked rotor are investigated using a response dependent breathing crack model. Initially a simple Jeffcott rotor is analysed considering the longitudinal and lateral vibrations. Additional external excitation in the form of periodic axial impulses is applied and the lateral vibration response is analysed. Later, to study all the coupling mechanisms together, the stiffness matrix of a Timoshenko beam element is modified accounting for bending-longitudinal-torsional vibration coupling mechanisms of crack. Using the breathing crack model and the FE model of the cracked rotor, the response of the cracked rotor is analysed applying external excitation in the form of periodic axial impulses and torsional harmonic excitation. The excitation frequency in one mode is tuned to the natural frequency of vibration in the other mode, so as to excite resonance conditions. The interaction of the excitation frequencies with the rotational frequency and its harmonics is the focus of study.
A mathematical model for a two-crack rotor is developed. The stiffness of the two- crack rotor is derived and then the dynamic response analysis is carried out to find the influence of crack orientation angle on the breathing behaviour of the cracks and on the response of the rotor. The dynamic response of an asymmetric rotor is then attempted to investigate the influence of an additional crack and its orientation on the rotor response.
The effect of bow is incorporated in the equations of motion of the cracked rotor and its influence on the response of the rotor is also studied.
Finally, experimental investigations of some of the key findings of the analytical work are attempted. Peak response variation, time domain, frequency domain and orbit plot during the passage through subharmonic resonances and coupled longitudinal and bending vibration response of cracked rotor are verified.
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CONTENTS
CERTIFICATE
ACKNOWLEDGEMENTS ii
ABSTARACT iv
CONTENTS
List of figures viii
List of photographs xvii
List of tables xviii
List of symbols xix
Chapter 1 INTRODUCTION 1
Chapter 2 LITERATURE REVIEW 12
2.1 INTRODUCTION 13
2.2 LITERATURE REVIEW 16
2.2.1 Crack models 17
2.2.2 Stiffness estimation 24
2.2.3 Vibrations of a cracked rotor 25
2.2.4 Field experience 39
2.2.5 Experimental work 41
2.2.6 Other aspects 43
2.3 CONCLUSIONS 48
Chapter 3 TRANSIENT RESPONSE AND BREATHING BEHAVIOUR 53
3 1 INTRODUCTION 54
3.2 EQUATIONS OF MOTION AND CRACK MODELS 57
3.2.1 Breathing crack model 58
3.2.2 Switching crack model 67
3.2.3 Open crack model 70
3.3 RESPONSE DURING PASSAGE THROUGH FIRST CRITICAL SPEED 70 3.3.1 Breathing behaviour of accelerating and decelerating rotor 70 3.3.2 Effect of crack depth, unbalance phase and unbalance on peak
response 80
3.3.3 Effect of unbalance phase, crack depth and damping on breathing
behaviour 87
3.4 RESPONSE DURING PASSAGE THROUGH SUBCRITICAL
RESONANCES 93
3.4.1 Time and frequency domain response during subharmonic resonances 93 3.4.2 Orbital analysis of the cracked rotor near subharmonic resonances 98
3.5 CONCLUSIONS 108
Chapter 4 RESPONSE OF A CRACKED JEFFCOTT ROTOR TO AXIAL
EXCITATION 111
4.1 INTRODUCTION 111
4.2 EQUATIONS OF MOTION ACCOUNTING FOR COUPLING BETWEEN
LATERAL AND LONGITUDINAL MOTION 113
4.3 RESPONSE OF THE ROTOR WITHOUT EXTERNAL EXCITATION 121 4.3.1 Unbalance response at half the bending critical speed 122 4.3.2 Unbalance response at 1/10th the bending critical speed 124 4.4 RESPONSE OF THE ROTOR TO AXIAL IMPULSES 126 4.4.1 Response to a single impulse applied only once 128 4.4.2 Response to periodic impulses (single impulse per rotation) 130 4.4.3 Response to periodic impulses (four impulses per rotation) 132 4.4.4 Response to periodic impulses (ten impulses per rotation) 133 4.4.5 Effect of crack depth on the response 136 4.4.6 Response at a speed that is not an integer fraction of bending critical
speed 137
4.5 CONCLUSIONS 141
Chapter 5 COUPLED BENDING, LONGITUDINAL AND TORSIONAL
VIBRATIONS 143
5.1 INTRODUCTION 143
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5.2 FINITE ELEMENT MODEL OF A CRACKED ROTOR SEGMENT 145 5.3 MODELING OF BREATHING BEHAVIOUR OF CRACK 156 5.4 COUPLING OF BENDING AND TORSIONAL VIBRATIONS 166 5.5 COUPLING OF LONGITUDINAL AND TORSIONAL VIBRATIONS 176
5.6 CONCLUSIONS 182
Chapter 6 EXPERIMENTATION 186
6.1 INTRODUCTION 186
6.2 PREPARATION OF TEST SPECIMEN 188
6.3 EXPERIMENTAL SETUP AND INSTRUMENTATION 193 6.4 PEAK RESPONSE VARIATION WITH UNBALANCE PHASE 197 6.5 CHANGES IN THE ORBIT ORIENTATION DURING PASSAGE
THROUGH SUBCRITICAL RESONANCES 203
6.6 IN RESPONSE NEAR SUBHARMONIC RESONANCES 207 6.7 COUPLED LONGITUDINAL AND BENDING VIBRATIONS 212
6.7.1 Non-rotating Shaft 215
6.7.2 Rotating Shaft 222
6.8 CONCLUSIONS 232
Chapter 7 EFFECTS OF ASYMMETRY AND BOW 236
7.1 DYNAMICS OF A TWO CRACK ROTOR 237
7.1.1 Introduction 237
7.1.2 Stiffness of a two crack rotor 239
7.1.3 Rotor with two breathing cracks 252
7.1.4 Rotor with one open and one breathing crack 265 7.2 DYNAMICS OF A BOWED ROTOR WITH A CRACK 275 7.2.1 Equations of motion for the cracked rotor with bow 276 7.2.2 Unbalance response of the cracked bowed rotor 278
7.3 CONCLUSIONS 293
Chapter 8 CONCLUSIONS AND SCOPE OF FUTURE WORK 296
REFERENCES 305
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