Modal Analysis Theory and Testing (W. Heylen) Front

MODAL ANALYSIS THEORY AND TESTING

Ward Heylen Stefan Lammens Paul Sas Division of Production Engineering, Machine Design and Automation Katholieke Universiteit Leuven Belgium

CONTENTS

PART A. THEORY

A.O. Introduction A.I. Analytical and experimental modal analysis A.1.0. Introduction A.1.1. Single degree of freedom systems A.l.1.1. System equation, transfer function A.I.1.2. System poles, naturalfrequencies,damping ratios A. 1.1.3. Residues A.I.1.4. Transfer function plots A.I.1.5. Frequency response function, impulse response function A.I.1.6. Influence of mass, damping and stiffness changes A. 1.2. Multiple degree of freedom systems A.l.2.1. System equation, transfer function A.l.2.2. System poles, naturalfrequencies,damping factors A.l.2.3. Modal vectors, residues A.l.2.4. Modal participation factors A. 1.2.5. Frequency response function matrix, impulse response function matrix A.I.2.6. Undamped and proportionally damped systems A.l.2.7. Orthogonality, modal coordinates A.l.2.8. Modal vector scaling A.l.2.9. Analytical and experimental approach A.1.3. Single degree of freedom system: example A.1.4. Multiple degree of freedom system: example A.l.4.1. General viscous damping A.l.4.2. Proportional viscous damping A.l.4.3. No damping A. 1.5. Conclusions A.2. (Digital) signal processing: basic theory A.2.0. Introduction A.2.1. Fourier transform for different signal types A.2.1.1. Periodic signals A.2.1.2. Non-periodic functions A.2.1.3. Sampled time functions A.2.1.4. Sampled time andfrequencytransform A.2.2. Some analysis parameters A.2.3. Properties and relations A.2.3.1. Time scaling A.2.3.2. Time shift A.2.3.3. Frequency shift and zoom transform A.2.3.4. Energy relationships

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A.2.3.5. Integration and differentiation A.2.3.6. Convolution A.2.4. Errors and windows A.2.4.1. Aliasing A.2.4.2. Leakage A.2.4.3. Windows A.2.5. Other transforms A.2.5.1. Laplace transform A.2.5.2. Z-transform A.2.6. Time and frequency functions and applications A.2.6.1. Autopower spectrum and autocorrelation function A.2.6.2. Crosspower spectrum and crosscorrelation function A.2.6.3. Averaging A.2.6.4. Frequency response functions and coherence functions A.2.7. Conclusions A.3. Modal parameter estimation A.3.0. Introduction A.3.1. Basic modal model equations A.3.2. Basic concepts A.3.2.1. Single versus multiple degree of freedom methods A.3.2.2. Local versus global parameter estimates A.3.2.3. Single versus multiple input A.3.2.4. Modal model versus direct model A.3.2.5. Low order complete versus high order incomplete A.3.2.6. Real mode shapes versus complex mode shapes A.3.2.7. Time domain versusfrequencydomain implementation A.3.2.8. Classification A.3.3. Single degree of freedom methods A.3.3.1. Peak picking A.3.3.2. Mode picking A.3.3.3. Circle fitting A.3.4. Multiple degree of freedom time domain methods A.3.4.1. Ibrahim time domain method (ITD) A.3.4.2. Polyreference least squares complex exponential method (LSCE) A.3.4.3. Eigensystem realisation algorithm (ERA) A.3.4.4. Time domain direct parameter identification (TDPI) A.3.5. Multiple degree offreedomfrequency domain methods A.3.5.1. (Non linear) least squares frequency domain method (LSFD) A.3.5.2. Identification of structural system parameters (ISSPA) A.3.5.3. Orthogonal polynomial method (OP) A.3.5.4. Frequency domain direct parameter identification (FDPI) A.3.5.5. Complex mode indicator function (CMIF) A.3.6. Conclusions A.4. Model validation A.4.0. Introduction A.4.1. Modal scale factor (MSF) and modal assurance criterion (MAC) A.4.2. Mode participation A.4.3. Reciprocity A.4.4. Mode complexity A.4.5. Modal phase collinearity and mean phase deviation A.4.6. Modal confidence factor A.4.7. Synthesis offrequencyresponse functions A.4.8. Conclusion A.5. Use of modal parameters A.6. Model updating A.6.1. Introduction

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A.6.2. General scheme and related topics A.6.2.1. General scheme A.6.2.2. Inherent incompatibilities between analytical and experimental data A.6.2.3. Example A.6.2.4. Conclusion A.6.3. Model matching A.6.3.1. Introduction A.6.3.2. Reduction techniques A.6.3.2.1. General remarks A.6.3.2.2. Dynamic reduction A.6.3.2.3. System Equivalent Reduction and Expansion Process A.6.3.2.4. Improved Reduced System A.6.3.3. Expansion techniques A.6.3.3.1. Introduction A.6.3.3.2. Eigenvector Mixing A.6.3.3.3. Modal Co-ordinate Method A.6.3.3.4. Interpolation A.6.3.3.5. MAC expansion A.6.3.4. Example A.6.3.5. Conclusion A.6.4. Correlation techniques A.6.4.1. Introduction A.6.4.2. Resonance frequency differences A.6.4.3. Visual comparison of mode shapes A.6.4.4. Modal Assurance Criterion A.6.4.5. Co-Ordinate Modal Assurance Criterion A.6.4.6. Visual comparison offrequencyresponse functions A.6.4.7. Cross and mixed orthogonalities A.6.4.8. Global mass A.6.4.9. Influence of the mesh incompatibility A.6.4.10. Example A.6.4.11. Conclusion A.6.5. Selection of updating parameters A.6.5.1. Introduction A.6.5.2. The different sorts of updating parameters A.6.5.3. Validity and reliability of the updated model A.6.5.4. Error localisation A.6.5.4.1. Co-Ordinate Modal Assurance Criterion A.6.5.4.2. Error Matrix Method A.6.5.4.3. Force Balance Method A.6.5.4.4. Sensitivity based methods A.6.5.4.5. Example A.6.5.5. Conclusion A.6.6. Correction methods A.6.6.1. Introduction A.6.6.2. Direct correction methods A.6.6.2.1. Lagrange multipliers A.6.6.2.2. Orthogonality based methods A.6.6.2.3. Matrix perturbation method A.6.6.3. Iterative, sensitivity based methods A.6.6.3.1. The components of the residuals vector A.6.6.3.1.1. The (relative) difference of the resonance frequencies A.6.6.3.1.2. The difference between the analytical and experimental mode shapes A.6.6.3.1.3. The off-diagonal elements of the mixed orthogonalities A.6.6.3.1.4. The force balance

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A.6.6.3.1.5. The difference of analytical and experimental frequency response functions A.6.6.3.1.6. The difference of analytical and experimental total mass A.6.6.3.1.7. The parameter changes A.6.6.3.2. The minimisation problem A.6.6.4. Example A.6.6.5. Conclusion A.6.7. Conclusion Appendices AA.2.1. Applications of several time andfrequencyfunctions AA.2.1.1. Autopower spectrum and autocorrelation function AA.2.1.2. Crosspower spectrum and crosscorrelation function AA.2.1.3. Cepstrum analysis AA.2.1.4. Conclusions AA.3.1 Least squares complex exponential method: example References Symbols and Notation

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