ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS

A Text Book

Developed under the Curriculum Development Scheme of

Quality Improvement Programme at IIT Guwahati

Sponsored by

All India Council of Technical Education

Dr Rajiv Tiwari Department of Mechanical Engineering

Indian Institute of Technology Guwahati 781039

April 2010

PREFACE

The present book materials is the outcome of an elective course on Rotor Dynamics offered by me to undergraduate, graduate and post-graduate students at IIT Guwahati over last ten years. Moreover, it contains materials of some of the research works done by me with my graduate students.

The modeling and analysis of rotor-bearing dynamics are now reached a mature state. In broad sense this area covers several categories namely modeling, analysis, identification and condition monitoring of rotor-bearing systems. The finite element (FE) method has been used extensively for modeling and analyses of rotors. Till today, the condition monitoring of rotor-bearing systems based on vibrations mainly concerned with the feature based fault detection and diagnostics. As a result of this the methods available so far are not reliable and fail-safe up to the expectation of fellow engineers working in the fields. For model-based condition monitoring of the rotor-bearing systems, identification methods for system parameters are under development. For the identification of rotor system parameters the literature available is not so rich and a lot of possibilities have been appeared in the literature. The very purpose of this course material is to give a basic understanding of the rotor dynamics phenomena with the help of simple rotor models and subsequently the modern analysis methods for real life rotor systems. This background will be helpful in the identification of rotor-bearing system parameters and its use in futuristic model-based condition monitoring and fault diagnostic and prognostics.

The present course material compiles review of some of the available literatures in a systematic and lucid form so as to boost research in the developing area of the rotor dynamics. Throughout the book materials are supplemented by numerical examples both by the hand calculation and computation extensive methods. It is expected that with this course material, students will get sufficient exposure and motivation for applying FEM in rotor dynamics and allied areas. We sincerely acknowledge the Quality Improvement Programme at IIT Guwahati sponsored by AICTE, New Delhi for funding towards the development of the course. Our heartfelt thanks to the help offered by the graduate students, research scholars and project, technical and office staffs at IIT Guwahati. This work is dedicated to my daughter Ranu, son Antariksha, and wife Vibha for their patience.

(R. Tiwari)

CONTENTS

Preface ii Content iv Nomenclature viii

Chapter 1 A Brief History of Rotor Dynamics 1-23

1.1 From the Rakine to Jeffcott Rotor Models 2 1.2 Rotor Dynamics Phenomena Studies from Stodola to Lund 5 1.3 Development of Rotor Dynamics Analysis Tools 11 1.4 Softwares for Rotor Dynamics Analysis 12 1.5 Dynamic Balancing of Rotors 12 1.6 Condition Monitoring of Rotating Machineries 13 Concluding Remarks 15 Exercise Problems 16 References 16

Chapter 2 Transverse Vibrations-I: Simple Rotor Systems 24-84

2.1 Single-DOF Undamped Rotor Model 25 2.1.1 Unbalance force model 26 2.1.2 Equations of motion 28 2.1.3 Free vibrations 28 2.1.4 Forced vibrations 28 2.1.5 Attenuation of vibrations 31 2.2 A Single-DOF Damped Rotor Model 35 2.3 Rankine Rotor Model 39 2.4 Jeffcott Rotor Model 40 2.4.1 Steady state response (Approach 1) 43 2.4.2 Steady state response (Approach 2) 44 2.4.3 Steady state response (Approach 3) 48 2.5 A Jeffcott Rotor Model with an Offset Disc 51 2.5.1 Calculation of natural frequencies 56 2.5.2 Unbalance forced response 56 2.5.3 Bearing reaction forces 57 2.6 Suppression of Critical Speeds 69 Concluding Remarks 75 Exercise Problems 76 References 83

Chapter 3 Bearing and Seal Systems 84-161

3.1 Rolling Element Bearings 86 3.1.1 Linear radial stiffness of rolling bearing 91 3.1.2 Nonlinear stiffness of rolling bearings 100 3.1.3 Linear stiffness determination under combined loadings 113 3.1.4 Radial stiffness at high-speed conditions 115

3.2 Hydrodynamic Oil-Lubricated Journal Bearings 117 3.2.1 Types of hydrodynamic bearings 119 3.2.2 Reynolds equation and its basic assumptions 121 3.2.3 Basic concepts and assumptions of oil-film bearing models 123 3.2.4 Short and long hydrodynamic radial bearings 126 3.2.5 Numerical analysis procedure of finite radial bearings 131 3.2.6 Friction force and lubricant flow rate 134 3.2.8 Dynamic characteristics of fluid-film bearings 136 3.3 Dynamic Seals 138

6.3.1 Classification of Seals 139 6.3.2 Theoretical estimation of dynamic coefficients of seals 142 6.3.3 Fluid-film dynamic force equations 151 3.4 Squeeze-film Dampers 152 Concluding Remarks 157 Exercise Problems 158 References 159

Chapter 4 Transverse Vibrations-II: Simple Rotor-Bearing-Foundation Systems 162-205

4.1 Symmetrical Long Rigid Shaft in Flexible Anisotropic Bearings 162 4.2 A Symmetrical Long Rigid Shaft on Anisotropic Bearings 173 4.2.1 Unbalance response 175 4.2.2 Bearing forces 181 4.3 A Symmetrical Flexible Shaft on Anisotropic Bearings 183 4.4 A Rotor on Flexible Bearings and Foundations 190 Concluding Remarks 195 Exercise Problems 196 References 204

Chapter 5 Transverse Vibrations-III: Simple Rotor Systems with Gyroscopic Effects 206-269

5.1 Angular Momentum 207 5.2 Gyroscopic Moments in Rotating Systems 209 5.2.1 Motion of a rotor mounted on two bearings 209 5.2.2 Gyroscopic moments though Coriolis component of accelerations 210 5.2.3 Gyroscopic moments in a rotating thin blade 212 5.2.4 Gyroscopic moments in a multi-bladed propeller 214 5.3 Synchronous Motion 216 5.3.1 A cantilever rotor with a thin disc 216 5.2.2 A cantilever rotor with a long stick 221 5.4 Asynchronous Rotational Motion 228 5.5 Asynchronous General Motion 234 5.6 Gyroscopic Effects by the Dynamics Approach 251 5.7 Analysis of Gyroscopic effects with Energy Methods 256 Concluding Remarks 262 Exercise Problems 263 References 269

Chapter 6 Torsional Vibrations of Rotors-I: The Direct and Transfer Matrix Methods 271-360

6.1 A Simple Rotor System with a Single Disc Mass 272 6.2 A Two-Disc Torsional Rotor System 275 6.3 A Two-Disc Rotor System with a Stepped Shaft 280 6.4 A Three-Disc Rotor System 285 6.4.1 A direct approach 285 6.4.2 An indirect approach 291 6.5 Transfer Matrix Methods 296 6.5.1 A point matrix 297 6.5.2 A field matrix 299 6.6 Geared Systems 320 6.7 Branched Systems 330 6.8 TMM for Damped Torsional Vibrations 341 6.9 Modelling of Reciprocating Machine Systems 344

6.9.1 An equivalent polar moment of inertia 345 6.9.2 An equivalent torsional stiffness of crack shafts 347 6.9.3 Torque variations in a reciprocating machinery 348

Concluding Remarks 351 Exercise Problems 352 References 360

Chapter 7 Torsional Vibrations of Rotors-II: The Continuous System and FE Methods 361-419

7.1 Torsional Vibrations of Continuous Shaft Systems 361 7.1.1 The Hamiltons principle 362 7.1.2 Lagranges equation 365 7.1.3 Governing differential equations 367

7.2 Applications of Finite Element Methods 377 7.2.1 Galerkin method 378

7.2.2 Rayleigh-Ritz Method 384 7.2.3 Assembled System Equations 386 7.2.4 Application of Boundary Conditions 388 7.2.5 Free Torsional Vibrations 389

7.3 Development of Geared Element for Branched Systems 402 Concluding Remarks 413 Exercise Problems 414 References 419

Chapter 8 Transverse Vibrations-IV: Multi-DOF Rotors 420-504

8.1 Influence Coefficient Method 420 8.1.1 A static case 421 8.1.2 A dynamic case 423 8.2 Transfer Matrix Methods 441 8.2.1 A field matrix 441 8.2.2 A point matrix 444 8.2.3 An overall transfer matrix 446 8.2.4 Free vibrations 448 8.2.5 Forced response 450 8.2.6 Gyroscopic effects 451

8.3 Dunkerleys Formula 485 Concluding Remarks 490 Appendix 8.1 Load deflection relations for various boundary conditions of the shaft 491 Exercise Problems 495 References 504

Chapter 9 The Continuous and Finite Element Transverse Vibration Analyses of Simple Rotor Systems 505-591

9.1 Governing Equations in Continuous Systems 505 9.2 Natural Frequencies and Mode Shapes 511 9.3 Forced Vibrations 516 9.4 A Brief Review on Application of FEM in Rotor-Bearing Systems 519 9.5 A Finite Element Formulation 521 9.5.1 FE Formulation in a weak form 522 9.5.2 Derivations of Shape functions 523

9.5.3 Satisfaction of the compatibility and completeness conditions 526 9.5.4 Finite element governing equations 528 9.5.5 The consistent load matrix 531 9.5.6 System equations of motion 535 9.5.7 Eigen value problems 538 9.6 Proportional Damping 567 9.7 The Static and Dynamic Condensations 569 9.7.1 Static (Guyan) reduction 570

9.7.2 The dynamic reduction 575 Concluding Remarks 580 Exercise Problems 581 References 588

Chapter 10 Transverse Vibrations-VI: Finite Element Analysis of Rotors with Gyroscopic Effects 592-652

10.1 Rotor Systems with a Single Disc 592 10.2 Timoshenko Beam Theory 606 10.3 Finite Element formulations of the Timoshenko beam 614 10.3.1 Weak formulations of the Timoshenko beam element for the static case 615 10.3.2 Derivation of shape functions 617

10.3.3 Weak formulation of the Timoshenko beam element for the dynamic case 623 10.4 Whirling of Timoshenko Shafts 629 10.4.1 Equations of motion of a spinning Timoshenko shaft 629

10.4.2 Finite element formulation 635 10.4.3 The weak form finite element formulations 636 10.4.4 Rigid disc element 637 10.4.5 System equations of motion 638 10.4.6 Eigen value problem 638 Appendix 10.1 Timoshenko beam model 643 Appendix 10.2 Rotating Timoshenko beam model Concluding Remarks 648 Exercise Problems 648 References 652

Chapter 11 Instability in Rotating Machines 653-739

11.1 Oil Whirl 653 11.2 Stability Analysis using Linearized Stiffness and Damping Coefficients 656 11.3 Stability Analysis with Fluid-Film Non-Linearity 662 11.4 Resonant Whip 664 11.5 Internal Friction 666 11.6 Effect of Rotor Polar Asymmetry 677 11.7 An Asymmetric Rotor with Uniformly Distributed Mass 681 11.8 Self Excited Vibrations 700 11.9 System with Variable or Nonlinear Characteristics 703

11.9.1 A Rotor with Variable Elasticity 705 11.9.2 Physical analysis of a horizontal symmetric shaft with gravity effects 707 11.9.3 Analytical Solution of the Equation of Motion of asymmetric rotor 708

11.10 Sub-Critical Vibrations of a Jeffcott Rotor 714 11.10.1 Equations of Motion 714 11.10.2 Unbalance Response 716 11.11 Stream Whirl 720 11.12 Instability due to Seals 724 11.13 Non-linear Equations of Motion of the Jeffcott Rotor (Run-up and run-down) 726 Concluding Remarks 732 Exercise Problems 733 References 735 Appendix 11A Newmark Method 737

Chapter 12 Instability of Multi-DOF Rotors Mounted On Flexible Bearings 740-767

12.1 Rotors Mounted on Flexible Bearings 740 12.1.1 Fluid Film Bearing Characteristics for Short Bearing Approximation 741 12.1.2 FEM Formulation for Bearings 744 Concluding Remarks 762 References 762 Exercise Problems 763

Chapter 13 Dynamic Balancing of Rotors 766-810

13.1 Unbalances in the Rigid and Flexible Rotors 767 13.2 Principle of Rigid Rotor Balancing 769 13.2.1 Static Balancing (Single plane balancing) 769

13.2.2 Static Balancing (Two plane balancing) 769 13.2.3 Couple unbalance 770 13.2.4 Dynamic unbalance 771 13.2.5 Various expressions of unbalance 771 13.3 Balancing of Practical Rigid Rotor 777 13.3.1 Single plane balancing 777 13.3.2 Two plane balancing (Cradle balancing machines) 778 13.3.3 Two plane balancing (The influence coefficient method) 785 13.4 Balancing of Flexible Rotors 790

13.4.1 Modal Balancing Method 792 13.4.2 Influence Coefficient Methods 801 Concluding Remarks 805 Exercise Problems 806 References 810

Chapter 14 Experimental Estimation of Dynamic Parameters of Bearings Dampers and Seals 811-897

14.1 Previous Literature Reviews and Surveys 813 14.2 Basic Concepts and Assumptions of Bearing Models 816 14.3 Abstract definition of the Identification 820 14.4 Static force method 821 14.5 Methods Using Dynamic Loads 827 14.6 Development of a General Identification Algorithm in Rotor-Bearing Systems 831 14.7 Use of Electromagnetic Vibrators 833

14.7.1 Complex Receptance Method 833 14.7.2 Estimation using the Direct Complex Impedance 839 14.7.3 Estimation by Multi-Frequency Testing 843

14.8 Use of Centrifugal Forces 846 14.9 Transient methods 855 14.10 Methods using Unknown Excitation 860 14.11 Special Methods of Estimation of Dynamic Parameter of Seals 862 14.11.1 Experimental Estimation Procedures 863

14.11.2 Previous Seal Rotor Dynamic Data and Resources 865 14.12 Accuracy of Estimated Bearing Coefficients 866 14.13 General Remarks and Future Directions on Bearings 867 14.13.1 General Remarks on Bearings 868 14.13.2 Future Directions on Bearings 870 14.14 General Remarks and Future Directions on Seals 873 14.15 Conditioning of Regression Matrices for Simultaneous Estimation of the Residual Unbalance and Bearing Dynamic Parameters 875 14.16 Simultaneous Identification of Residual Unbalances and Bearing Dynamic Parameters from Impulse Responses of Flexible Rotor-Bearing Systems 887 Concluding Remarks 890 Exercise Problems 891 References 894

Chapter 15 Measurements in Rotating Machineries 898-944

15.1 Specifications of Measuring Instruments 899 15.2 Uncertainty Analysis of Estimated Parameters 904 15.3 Transducers 910 15.3.1 Displacement Sensors 911 15.3.2 Accelerometers 921 15.4 Signal Conditioning & Analysis Equipments 928 15.4.1 Filters 928 15.4.2 Measurement amplifier 929 15.4.3 Oscilloscope, Spectrum analyzer and Data Acquisition System 929 15.5 Vibration Exciter Systems 930 15.5.1 Electromagnetic Systems 931 15.5.2 Mechanical-Type Exciters 932

15.5.3 Hydraulic and Pneumatic Systems 932 15.5.4 Impact hammer 933 15.6 Sound Measurements 936 Concluding Remarks 941 Exercise Problems 941 References 944

Chapter 16 Signal Processing in Rotating Machineries 945-1006

16.1 Display of Vibration Measurements 945 16.2 Accuracy of Vibration Measurements 951 16.3 Fourier Series 961 16.4 Fourier Transform and Fourier Integral 967 16.5 Discrete Fourier Transform 970 16.6 Fast Fourier Transform 978 16.7 Leakage Error and Countermeasures 979 16.8 Applications of FFT to Rotor Vibrations 986 16.9 Properties of Random Discrete Signals 988 16.9.1 Probability, Probability Distribution Function, and Probability Density Function 988 16.9.2 Random Process, Ensemble, and Sample Function 992 16.9.3 Stationary and Ergodic Process 992 16.9.4 Probability Distribution and Probability Distribution Function 994 16.9.5 Ensemble Average, Temporal Average, Mean, Variance 995 16.9.6 Auto-correlation Function and Covariance 997 16.9.7 Cross-correlation Function 998 16.9.8 Coherence Function 999 16.10 Protection against spurious signals 999 16.10.1 Electrical noise 999 16.10.2 Runout 1000 16.10.3 Removing runout from a vibration signal 1001 16.10.4 Electronic differentiation and integration 1002 Concluding Remarks 1003 References 1004 Exercise Problems 1005

Chapter 17 Vibration Based Condition Monitoring in Rotating Machineries 1007-1053

17.1 Rotor unbalance 1008 17.2 Shaft Bow or Thermal Bow 1012 17.3 Misalignment 1015 17.4 Rubs 1019 17.5 Mechanical Looseness of Components 1024 17.6 Shaft Cracks 1027 17.7 Rolling Element Bearing Faults 1031 17.8 Faults in Gears 1040 Concluding Remarks 1044 Exercise Problems 1044 Appendix: 17.A Discrete Wavelet Transform 1045 References 1046

Chapter 18 Active Magnetic Bearings In Rotors 1054-1144

18.1 Introduction 1054 18.1.1 Classification and Properties of Active Magnetic Bearings 1055 18.1.2 Advantages and Limitations of Active Magnetic Bearings 1059 18.2 Literature Survey on Design and Analysis of AMBs 1059 18.3 Basics of Active Magnetic Bearings 1067 18.3.1 Operating Principle of an AMB 1068 18.3.2 Fundamental Relations 1069 18.3.3 Dynamics of Active Magnetic Bearings 1070 18.3.4 Differential driving mode 1073 18.3.5 PID Controller 1076 18.4 Block Diagrams and Transfer Functions 1077 18.4.1 Block diagrams and transfer functions of magnetic bearing systems 1077 18.4.2 The PID Controller and Its Transfer function 1078 18.4.3 Transfer function and block diagram of the overall active magnetic bearing system 1079 18.5 Tuning of the Controller Parameters 1082 18.5.1 Stability Criteria 1083 18.5.2 PID Controller 1083 18.5.3 PD Controller 1085 18.6 A Single-DOF Rotor System 1088 18.7 Two-DOF Rotor Systems 1097 18.8 Four-DOFs Rigid-Rotor Flexible-Bearing Systems 1105 18.9 Flexible Rotor-Bearing Systems 1117 18.9.1 Flexible Rotor-bearing System and Formulations 1117 18.9.2 Modeling of Rotor-bearings system 1117 18.9.3 Shaft model 1118 18.9.4 Rigid disc model 1119 18.9.5 Bearing model 1119 18.9.6 Active magnetic bearing model 1120 18.9.7 Equations of motion of the rotor substructure 1120 18.9.8 Natural whirl frequency of the system 1122 18.9.9 Plots of unbalance response obtained for flexible rotor-bearing system 1126 18.9.10 Campbell Diagram 1129 Concluding Remarks 1130 References 1132

Nomenclature

c damping coefficient [C] damping matrix d diameter e eccentricity E Youngs modulus f(t) time dependent force F amplitude of the force [F] field matrix for TMM G modulus of rigidity [G] gyroscopic matrix I diametral second moment of area (for the circular shaft = pi d4/64) Id diametral mass moment of inertia Ip polar mass moment of inertia j 1 , imaginary number J polar second moment of area (for the circular shaft = pi d4/32) k stiffness [K] stiffness matrix l length m mass of disc M(t) moment [M] mass matrix n gear ratio [P] point matrix for TMM S shear force {S} state vector for TMM U strain energy [U] transfer matrix for TMM x linear displacement in the horizontal direction {x} displacement matrix X amplitude of the linear displacement in the horizontal direction y linear displacement in the vertical direction Y amplitude of the linear displacement in the vertical direction z linear displacement in the axial direction Z mechanical impedance T kinetic energy T(t) toque [T] overall transfer matrix for TMM phase between the force and the displacement x(t) angular displacement about x-axis x amplitude of x y(t) angular displacement about y-axis y amplitude of y z(t) angular displacement about z-axis z amplitude of z whirl frequency density angular position of the unbalance with x-axis spin (or angular) speed of the rotor damping ratio

Subscripts:

b bearing c critical d diametral E external i imaginary L linear nf natural frequency NL non-linear p polar r rotor/real f foundation

Back subscript

R right L left

Superscripts: i imaginary r real

Units:

cm centimeter G giga (109) Hz cycle per sec (Hertz) m meter mm millimeter M mega (106) N Newton Pa Pascal (N/m2) rad radian rpm revolution per minute sec second m micrometer (10-6m)

Abbreviations:

AMB active magnetic bearing DOF degree of freedom EOM equation of motion FEM finite element method SHM simple harmonic motion TMM transfer matrix method