Non-linear dynamics and chaos of flexible rotor systems

NON-LINEAR DYNAMICS AND CHAOS OF FLEXIBLE ROTOR SYSTEMS

TIKAM CHAND GUPTA

DEPARTMENT OF APPLIED MECHANICS INDIAN INSTITUTE OF TECHNOLOGY DELHI

SEPTEMBER, 2011

©Indian Institute of Technology Delhi (IITD), New Delhi, 2011

NON- LINEAR DYNAMICS AND CHAOS OF FLEXIBLE ROTOR SYSTEMS

by

TIKAM CHAND GUPTA Department of Applied Mechanics

Submitted

in fulfillment of the requirements of Degree of

DOCTOR OF PHILOSOPHY

to the

INDIAN INSTITUTE OF TECHNOLOGY DELHI

SEPTEMBER, 2011

CERTIFICATE

This is to certify that the thesis entitled `NON-LINEAR DYNAMICS AND CHAOS OF FLEXIBLE ROTOR SYSTEMS' being submitted by Tikam chand Gupta for the award of the degree of Doctor of Philosophy has been prepared under our supervision in conformity with the rules and regulations of the Indian Institute of Technology, Delhi. We further certify that the thesis has attained 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 full, to any other University for any degree or diploma.

(Prof. D. K. Sehgal) Professor,

Department of Applied Mechanics, Indian Institute of Technology, New Delhi —110016

(Prof. K. Gupta) Professor,

Department of Mechanical Engineering, Indian Institute of Technology,

New Delhi - 110016

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ACKNOWLEDGEMENTS

Gratitude to my supervisors Prof. K. Gupta, Professor of Mechanical Engineering and Prof. D. K. Sehgal, Professor of Applied Mechanics, whose motivation, guidance and kind help throughout the course of investigations led to the completion of this work. The continual revision of the my research papers by Prof. K. Gupta definitely improved my research capabilities.

The financial support provided by `Quality Improvement Programme' is gratefully acknowledged, without which it was not possible to pursue Ph.D. programe at IIT, Delhi.

I sincerely acknowledge the encouragement and freedom offered by Prof. B. P.

Patel of Applied Mechanics; to discuss anytime on any topic related to engineering mechanics, nonlinear vibrations, computational methods and spirituality.

I express my thanks to Mr. Vikram Singh Rawat (Computational Lab., Department of Applied Mechanics) and Mr. K. N. Madhu (Vibrations Research Lab., Department of Mechanical Engineering), for their prompt help and co-operation.

I wish to thank all my colleagues and friends in the Computational Lab. and Vibration Research Lab., who gave me full co-operation whenever needed.

Finally, I would like to express my gratitude to my parents, wife, children and other family members for their moral support, encouragement and understanding that made it possible for me to complete this research work.

Date Tikam Chand Gupta

Place: New Delhi

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ABSTRACT

The present thesis recommends the use of flexible rotor modeling as against rigid rotor modeling, for the purpose of investigation of the effect of ball bearing nonlinearities and varying compliance of bearing on the dynamic response of realistic rotor ball bearing systems. The nonlinearity in the dynamic response is induced by the nonlinear Hertzian deformation force between the balls and races, and the internal radial clearance. The superiority of the flexible rotor modeling as against the rigid rotor modeling has been established successfully by performing detailed dynamic analysis on the following two rotor systems: (i) a flexible horizontal rotor supported on two deep groove ball bearings with radial clearance and (ii) a flexible vertical rotor supported by an angular contact ball bearing mounted at the lower end and a deep groove ball bearing mounted at the upper end. Only theoretical simulation is performed for dynamic response characterization of the horizontal rotor but vertical rotor is studied both theoretically and experimentally. A generalized Timoshenko beam FE formulation is used to model the flexible rotor. The nonlinear dynamic response is quantitatively characterized by various tools like Poincare maps, Floquet multipliers, time histories, phase plots and Lyapunov exponents.

First, the nonlinear dynamic response of an unbalanced flexible rotor supported on deep groove ball bearing with clearance, is numerically computed using Newmark-(3 implicit time integration scheme. A significant difference between dynamic responses of rotor at centrally mounted disc location and bearing location is observed, which establishes the need to follow the flexible rotor formulation instead of rigid rotor formulation. Then, a parametric analysis has been carried out to investigate the effect of

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different system parameters, such as rotating unbalance, rotational speed, rotor shaft flexibility and bearing radial clearance, on the nonlinear dynamic response of horizontal flexible rotor supported on deep groove ball bearings.

Accounting for the nonlinear dynamic response of a horizontal balanced flexible rotor ball bearing system, a parametric study is theoretically performed to investigate the time- varying nonlinear dynamic stiffness of ball bearing as a function of various time-invariant parameters like flexibility of shaft, radial clearance and, rotational speed.

To capture unstable solutions as well and to further characterize the nonlinear dynamic response for instability, the existing time domain shooting technique has been modified by coupling it with the fixed point algorithm (FPA) and has been used to derive quasi- periodic solutions and Floquet multipliers. The maximum value of Lyapunov exponent is determined using Wolf's algorithm with Gram-Schimdt orgthogonalization. The maximum values of Floquet multipliers and Lyapunov exponents are taken as signatures to determine the instability and chaotic nature of the dynamical system. The instability and chaos in the non-linear dynamic response is found to depend explicitly upon stiffness ratio (shaft stiffness/Hertzian load deflection factor) of rotor ball bearing system, radial clearance of ball bearing, unbalance force and rotational speed. A range of parameters has been identified for which the flexible rotor system is susceptible to instability and/or chaos.

Experiments have been performed on a vertical rotor to capture the effects of self- excitation due to varying compliance of bearing and ball bearing nonlinearities, in the dynamic response. The measured nonlinear dynamic response is analyzed using various analytical tools like orbits, phase plots, displacement-time plots, FFT and Poincare maps.

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A FE model is then formulated to theoretically simulate the dynamic response of the vertical rotor having the same dimensions and specifications as of experimental rig. In an attempt to study instability of the dynamic response of vertical flexible rotor supported on ball bearings, it has been found that the nonlinear dynamic response of the vertical rotor ball bearing system can not be derived successfully by modified shooting method.

Therefore, the Newmark-(3 implicit time integration technique has been used to derive nonlinear dynamic response of vertical rotor.

From the experimental as well as theoretical studies on vertical rotor ball bearing system, it has been found that the orbit of the centrally mounted disc and the rotor centre in upper ball bearing are not concentric with the bearing' axis line and with the increase in rotational speed, the centre of the disc orbit approaches the bearings' axis line. The Poincare maps and FFT constructed based experimentally measured dynamic response reveal the motion as chaotic. This observation is found to be in contradiction to the conclusion reached by the theoretical analysis.

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CONTENTS

Pages

CERTIFICATE i

ACKNOWLEDGEMENTS ii

ABSTRACT iii

CONTENTS vi

LIST OF FIGURES xi

LIST OF TABLES xvi

NOMENCLATURE xvii

CHAPTER 1

1 INTRODUCTION 1

2 ORGANIZATION OF THE THESIS 3

CHAPTER 2 LITERAURE REVIEW

2.1 INTRODUCTION 7

2.2 HISTORICAL BACKGROUND 8

2.3 INTRODUCTION OF BALL BEARINGS SYSTEMS 8

2.4 RESEARCH TRENDS IN ROTOR BALL BEARING SYSTEMS 10

2.5 LITERATURE REVIEW 13

2.5.1 Varying Compliance 13

2.5.2 Clearance Nonlinearity 15

2.5.3 Ball Bearing stiffness 18

2.5.4 Nonlinear Dynamics of Flexible rotors with fluid film bearing 20 2.5.5 Nonlinear Dynamics of flexible rotor with ball bearing 22

2.5.6 Experiments in rotor ball bearing system 23 2.5.7 Computational Methods for Non-linear Dynamic Response of Rotor

Bearing Systems 26

2.5.8 Lyapunov Exponents 28

2.6 MOTIVATION 29

CHAPTER 3 FLEXIBLE ROTOR SYSTEMS: THEORY AND FORMULATION

3.1

3.2

3.3 DEEP GROOVE BALL BEARING 41

3.3.1 Dynamic Stiffness of Ball Bearing 43

3.3.2 Ball Bearing Damping 44

3.4 ANGULAR CONTACT BALL BEARING 45

3.4.1 Kinematical Equations of Angular Contact Ball Bearing 45 3.4.2 Stiffness Matrix of Angular Contact Ball Bearing 46

Formulation 48

3.5 RIGID DISC 53

3.6 EXCITATION 54

3.7 METHODS OF SOLUTION AND ANALYSIS 55

3.7.1 Whirling Frequencies 55

3.7.2 Newmark-(3 and Newton-Raphson Method 57

3.7.3 Modified Shooting Method 59

Non-Autonomous Shooting Technique 62

Fixed Point Algorthim for Quasi-Periodic Solutions 66

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Modified Non-Autonomous Shooting Method 67 3.7.4 Method of Computing Lyapunov Exponents 69

Wolfs Algorithm 71

3.8 SUMMARY 72

CHAPTER 4 NON-LINEAR VIBRATION ANALYSIS OF AN UNBALANCED FLEXIBLE ROTOR

4.1 INTRODUCTION 75

4.2 VALIDATION 77

4.2.1 Mass and Stiffness Matrices 78

4.2.2 Gyroscopic Matrix 79

4.2.3 Non-Linear Dynamic Response 80

4.3 RESULTS AND DISCUSSION 83

4.3.1 Natural Frequencies of Rotor Bearing System 85

4.3.2 Dynamic Response 87

Motion of the Rotor Centre at the Disc Location 87 Motion of the Rotor Centre at the Bearing Location 91

4.4 CONCLUSIONS 99

CHAPTER 5 DYNAMIC STIFFNESS OF BALL BEARING AND WHIRL FREQUENCIES

5.1 INTRODUCTION 101

5.2 METHOD OF PARAMETRIC ANALYSIS 104

5.2.1 Flow Chart 106

5.3 RESULTS AND DISCUSSION 109

viii

5.3.1 Dynamic Stiffness of Bearing 109

5.3.2 Whirl Frequencies 119

5.4 CONCLUSIONS 126

CHAPTER 6 INSTABILITY AND CHAOS OF A FLEXIBLE ROTOR BALL BEARING SYSTEM

6.1 INTRODUCTION 129

6.2 VALIDATION OF MODIFIED SHOOTING METHOD 131

6.3 RESULTS AND DISCUSSION 135

6.4 CONCLUSIONS 151

CHAPTER 7 NON-LINEAR DYNAMIC ANALYSIS OF VERTICAL FLEXIBLE ROTOR SYSTEM

7.1 INTRODUCTION 153

7.2 EXPERIMENTAL STUDIES ON VERTICAL ROTOR 156

7.2.1 Experimental Test Rig 156

7.2.2 Instrumentation 159

7.2.3 Estimation of Ball Bearing Stiffness 161 7.2.4 Analysis of Results and Discussion 162 7.2.5 Conclusions of Experimental Studies on Vertical Rotor 182

7.3 THEORETICAL SIMULATION OF VERTICAL ROTOR 184

7.3.1 Introduction 184

7.3.2 Results and Discussion of Theoretical Simulation 186 7.3.3 Conclusions of Theoretical Simulation on Vertical Rotor 205

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CHAPTER 8 CONCLUSIONS 206 8.1 HORIZONTAL FLEXIBLE ROTOR BALL BEARING SYSTEM 207 8.2 VERTICAL FLEXIBLE ROTOR ANGULAR CONTACT BALL BEARING

SYSTEM 211

8.3 SCOPE FOR THE FUTURE WORK 213

REFERENCES 215

Appendix A 224

Appendix B 226

Appendix C 227

Appendix D 232

About the Author 235

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