ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS

Transcription

1 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 September 2005

2 CONTENT Content Preface ii vi Chapter 1 Analysis of Simple Rotor Systems Single DOF Rotor Model Rankine Rotor Model Jeffcott Rotor Model Symmetrical Rigid Shaft in Flexible Anisotropic Bearings Symmetrical Rigid Shaft in Flexible Anisotropic Bearings with Damping and Cross Coupling Asymmetrical Flexible Shaft in Flexible Anistropic Bearings 59 with Damping and Cross Coupling 1.7 Effects of Flexible Foundations 67 Chapter 2 Torsional Vibrations Simple System with Single Rotor Mass A Two-Disc Torsional System System with a Stepped Shaft MDOF Systems Transfer matrix method Geared Systems Branched Systems Damping in Torsional Systems Torsional Vibration for Continuous Systems Hamilton s Principle Lagrange s Equation Governing Differential Equations Finite Element Formulation Assembled System Equations Application of Boundary Conditions Free Torsional Vibration Geared element for branched systems Modelling of reciprocating machine systems 127 Chapter 3 Transverse Vibrations of Multi-DOF Rotors Method of Influence Coefficients Transfer Matrix Method: (Myklestand & Prohl method) Mechanical Impedance (and Receptance) Method Dynamic Stiffness Matrix Method Dunkerley s Formula 185 Chapter 4 Finite Element Analysis of Simple Rotor Systems Literature Review Euler-Bernoulli Beam Theory Finite Element Formulation 192 ii

3 4.3.1 Weak Form System Equations of Motion Eigen Value Problem Forced Vibration Analysis (The consistence load matrix) 211 Chapter 5 Gyroscopic Effects in Rotors Synchronous Motion Asynchronous Motion (Rotational Motion only) Asynchronous Motion (General motion) Gyroscopic Effect (General Approach) Gyroscopic Effect (Energy Method) Finite Element Analysis of Rotors Finite Element Formulation with Timoshenko Beam Model Weak Form FEM Formulation Rigid Disc Element System Equations of Motion Eigen Value Problem 261 Appendix 5.1 Rotating Timoshenko Beam Model 267 Chapter 6 Bearing and Seal Systems Rolling Element Bearings Bearing Elastic Deformation Hydrodynamic Oil-Lubricated Journal Bearings Basic Concept and Assumptions of Bearing Models Reynolds Equation and Approximate Solutions Finite Bearings Friction Lubricant Flow Rate Dynamic Characteristics Dynamic Seals Classification of Seals Theoretical Estimation of Dynamic Coefficients of Seals Fluid-Film Dynamic Force Equations 317 Chapter 7 Instability in Rotating Machines Oil Whirl Stability Analysis using Linearized Stiffness and Damping Coefficients Stability Analysis Allowing for Oil-Film Non-Linearity Resonant Whip Internal Friction Effect of Rotor Polar Asymmetry Free Vibration and Stability of Motion of a Rotor with Uniformly Distributed Mass Self Excited Vibrations System with Variable or Nonlinear Characteristics Examples of Systems with Variable Elasticity With Gravity (Horizontal Shaft) 352 iii

4 7.12 Solution of the Equation 353 Chapter 8 Rotors Mounted on Flexible Bearings Fluid Film Bearing Characterstics for Short Bearing Approximation FEM Formulation for Bearings Natural Whirl Frequency and Stability Analysis Numerical Examples and Discussions 369 Appendix 8.1 Rotor mounted on flexible bearings 374 Chapter 9 Dynamic Balancing of Rotors Balancing of Rigid Rotor Cradle balancing machine The Influence Coefficient Method Balancing of Flexible Rotors Modal Balancing Method Influence Coefficient Methods 394 Chapter 10 Experimental Estimation of Dynamic Parameters of Bearings Static force method Use of Electromagnetic Vibrator Complex Receptance Method Direct Complex Impedance Derivation Multi Frequency Testing Use of Centrifugal Forces Transient methods 426 Chapter 11 Measurements and Diagnostics in Rotors Signal Measurement and Display Shaft Imbalance Misalignment, Pre-Loaded Shaft Rubs Loose Components Shaft Cracks Rolling Element Bearing Faults Faults in Gears Protection Against Spurious Signals Electrical noise Runout Removing runout from a vibration signal Electronic differentiation and integration Vibration Measuring Instrument Seismometer Instruments with Low Natural Frequency (Velometer) Accelerometer-Instrument with High Natural Frequency Measurement and Signal Processing 447 iv

5 Measurement and Sampling Problem Fourier Series Fourier Transform Discrete Fourier Transform First Fourier Transform Leakage Error and Countermeasures Applications of FFT to Rotor Vibrations Description of the Rotor Test Rig at IIT Guwahati Description of the Instruments Impact hammer Measurement amplifier Proximity probe transducer Pulse analyzer (Data acquisition system) Measurements and Analysis of the Test Rig Data 468 Chapter 12 An Introduction to MATLAB The Software Package MATLAB Matrices and Matrix Operations in MATLAB Using the MATLAB Operator for Matrix Division Manipulating the Elements of a Matrix Transposing Matrices Special Matrices Generating Matrices with Specified Element Values Some Special Matrix Operations Element-by-Element Operations Input and Output in MATLAB MATLAB Graphics Scripting in MATLAB Functions in MATLAB User-Defined Functions Some Pitfalls in MATLAB 491 v

6 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