ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS A Lecture Notes 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 September 2005
CONTENT Content Preface ii vi Chapter 1 Analysis of Simple Rotor Systems 1-70 1.1 Single DOF Rotor Model 1 1.2 Rankine Rotor Model 9 1.3 Jeffcott Rotor Model 10 1.4 Symmetrical Rigid Shaft in Flexible Anisotropic Bearings 41 1.5 Symmetrical Rigid Shaft in Flexible Anisotropic Bearings with Damping and Cross Coupling 51 1.6 Asymmetrical Flexible Shaft in Flexible Anistropic Bearings 59 with Damping and Cross Coupling 1.7 Effects of Flexible Foundations 67 Chapter 2 Torsional Vibrations 71-130 2.1 Simple System with Single Rotor Mass 71 2.2 A Two-Disc Torsional System 72 2.3 System with a Stepped Shaft 76 2.4 MDOF Systems 77 2.4.1 Transfer matrix method 78 2.5 Geared Systems 89 2.6 Branched Systems 99 2.7 Damping in Torsional Systems 103 2.8 Torsional Vibration for Continuous Systems 107 2.8.1 Hamilton s Principle 107 2.8.2 Lagrange s Equation 107 2.8.3 Governing Differential Equations 108 2.8.4 Finite Element Formulation 111 2.8.5 Assembled System Equations 117 2.8.6 Application of Boundary Conditions 118 2.8.7 Free Torsional Vibration 119 2.8.8 Geared element for branched systems 122 2.9 Modelling of reciprocating machine systems 127 Chapter 3 Transverse Vibrations of Multi-DOF Rotors 131-187 3.1 Method of Influence Coefficients 131 3.2 Transfer Matrix Method: (Myklestand & Prohl method) 142 3.3 Mechanical Impedance (and Receptance) Method 159 3.4 Dynamic Stiffness Matrix Method 172 3.5 Dunkerley s Formula 185 Chapter 4 Finite Element Analysis of Simple Rotor Systems 188-220 4.1 Literature Review 188 4.2 Euler-Bernoulli Beam Theory 190 4.3 Finite Element Formulation 192 ii
4.3.1 Weak Form 194 4.4 System Equations of Motion 195 4.5 Eigen Value Problem 195 4.6 Forced Vibration Analysis (The consistence load matrix) 211 Chapter 5 Gyroscopic Effects in Rotors 221-270 5.1 Synchronous Motion 221 5.2 Asynchronous Motion (Rotational Motion only) 228 5.3 Asynchronous Motion (General motion) 230 5.4 Gyroscopic Effect (General Approach) 243 5.5 Gyroscopic Effect (Energy Method) 247 5.6 Finite Element Analysis of Rotors 255 5.6.1 Finite Element Formulation with Timoshenko Beam Model 255 5.6.2 Weak Form FEM Formulation 258 5.6.3 Rigid Disc Element 259 5.6.4 System Equations of Motion 260 5.6.5 Eigen Value Problem 261 Appendix 5.1 Rotating Timoshenko Beam Model 267 Chapter 6 Bearing and Seal Systems 271-319 6.1 Rolling Element Bearings 272 6.1.1 Bearing Elastic Deformation 276 6.2 Hydrodynamic Oil-Lubricated Journal Bearings 288 6.2.1 Basic Concept and Assumptions of Bearing Models 290 6.2.2 Reynolds Equation and Approximate Solutions 291 6.2.3 Finite Bearings 294 6.2.4 Friction 298 6.2.5 Lubricant Flow Rate 299 6.2.6 Dynamic Characteristics 300 6.3 Dynamic Seals 303 6.3.1 Classification of Seals 303 6.3.2 Theoretical Estimation of Dynamic Coefficients of Seals 306 6.3.3 Fluid-Film Dynamic Force Equations 317 Chapter 7 Instability in Rotating Machines 320-359 7.1 Oil Whirl 320 7.2 Stability Analysis using Linearized Stiffness and Damping Coefficients 321 7.3 Stability Analysis Allowing for Oil-Film Non-Linearity 324 7.4 Resonant Whip 325 7.5 Internal Friction 326 7.6 Effect of Rotor Polar Asymmetry 330 7.7 Free Vibration and Stability of Motion of a Rotor with Uniformly Distributed Mass 333 7.8 Self Excited Vibrations 345 7.9 System with Variable or Nonlinear Characteristics 348 7.10 Examples of Systems with Variable Elasticity 350 7.11 With Gravity (Horizontal Shaft) 352 iii
7.12 Solution of the Equation 353 Chapter 8 Rotors Mounted on Flexible Bearings 360-376 8.1 Fluid Film Bearing Characterstics for Short Bearing Approximation 360 8.2 FEM Formulation for Bearings 364 8.3 Natural Whirl Frequency and Stability Analysis 364 8.4 Numerical Examples and Discussions 369 Appendix 8.1 Rotor mounted on flexible bearings 374 Chapter 9 Dynamic Balancing of Rotors 377-397 9.1 Balancing of Rigid Rotor 377 9.1.1 Cradle balancing machine 377 9.1.2 The Influence Coefficient Method 384 9.2 Balancing of Flexible Rotors 390 9.2.1 Modal Balancing Method 390 9.7.2 Influence Coefficient Methods 394 Chapter 10 Experimental Estimation of Dynamic Parameters of Bearings 398-430 10.1 Static force method 399 10.2 Use of Electromagnetic Vibrator 405 10.2.1 Complex Receptance Method 406 10.2.2 Direct Complex Impedance Derivation 416 10.2.3 Multi Frequency Testing 417 10.3 Use of Centrifugal Forces 420 10.4 Transient methods 426 Chapter 11 Measurements and Diagnostics in Rotors 431-470 11.1 Signal Measurement and Display 431 11.2 Shaft Imbalance 434 11.3 Misalignment, Pre-Loaded Shaft 434 11.4 Rubs 436 11.5 Loose Components 437 11.6 Shaft Cracks 437 11.7 Rolling Element Bearing Faults 438 1.8 Faults in Gears 440 11.9 Protection Against Spurious Signals 441 11.9.1 Electrical noise 441 11.9.2 Runout 441 11.9.3 Removing runout from a vibration signal 442 11.9.4 Electronic differentiation and integration 443 11.10 Vibration Measuring Instrument 443 11.11 Seismometer Instruments with Low Natural Frequency (Velometer) 444 11.12 Accelerometer-Instrument with High Natural Frequency 445 11.13 Measurement and Signal Processing 447 iv
11.13.1 Measurement and Sampling Problem 448 11.13.2 Fourier Series 450 11.13.3 Fourier Transform 452 11.13.4 Discrete Fourier Transform 454 11.13.5 First Fourier Transform 459 11.13.6 Leakage Error and Countermeasures 459 11.13.7 Applications of FFT to Rotor Vibrations 461 11.14 Description of the Rotor Test Rig at IIT Guwahati 463 11.15 Description of the Instruments 465 11.15.1 Impact hammer 465 11.15.2 Measurement amplifier 467 11.15.3 Proximity probe transducer 467 11.15.4 Pulse analyzer (Data acquisition system) 467 11.16 Measurements and Analysis of the Test Rig Data 468 Chapter 12 An Introduction to MATLAB 472-492 12.1 The Software Package MATLAB 472 12.2 Matrices and Matrix Operations in MATLAB 472 12.3 Using the MATLAB Operator for Matrix Division 473 12.4 Manipulating the Elements of a Matrix 474 12.5 Transposing Matrices 475 12.6 Special Matrices 476 12.7 Generating Matrices with Specified Element Values 477 12.8 Some Special Matrix Operations 477 12.9 Element-by-Element Operations 477 12.10 Input and Output in MATLAB 478 12.11 MATLAB Graphics 480 12.12 Scripting in MATLAB 483 12.13 Functions in MATLAB 487 12.14 User-Defined Functions 489 12.15 Some Pitfalls in MATLAB 491 v
PREFACE The present course 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 eight years. Moreover it contains materials of some of the project works done by 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 rotorbearing system parameters and its use in futuristic model based condition monitoring and fault diagnostic and prognostics. The present course material compiles some of the available literature in a systematic and lucid form so as to boost research in the developing area of the rotor dynamics. Lecturer materials are supplemented by numerical examples. 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 wife, Vibha, and my son, Antariksh. (R. Tiwari) vi