About the authors
Preface
Acknowledgments
List of Abbreviations 21
List of Symbols 23
1 Introduction 1
1.1 Noise and Vibration 1
1.2 Noise and Vibration Analysis 2
1.3 Application Areas 3
1.4 Analysis of Noise and Vibrations 4
1.4.1 Experimental Analysis 5
1.5 Standards 5
1.6 Becoming a Noise and Vibration Analysis Expert 5
1.6.1 The Virtue of Simulation 6
1.6.2 Learning Tools and the Format of This Book 6
2 Dynamic Signals and Systems 9
2.1 Introduction 9
2.2 Periodic Signals 11
2.2.1 Sine Waves 11
2.2.2 Complex Sines 11
2.2.3 Interacting Sines 13
2.2.4 Orthogonality of Sines 15
2.3 Random Signals 16
2.4 Transient Signals 17
2.5 RMS Value and Power 18
2.6 Linear Systems 19
2.6.1 The Laplace Transform 20
2.6.2 The Transfer Function 24
2.6.3 The Impulse Response 25
2.6.4 Convolution 26
2.7 The Continuous Fourier Transform 29
2.7.1 Characteristics of the Fourier Transform 32
2.7.2 The Frequency Response 34
2.7.3 Relationship Between the Laplace and Frequency Domains 35
2.7.4 Transient Versus Steady-State Response 35
2.8 Chapter Summary 37
2.9 Problems 38
References 39
3 Time Data Analysis 41
3.1 Introduction to Discrete Signals 41
3.1.1 Discrete Convolution 42
3.2 The Sampling Theorem 42
3.2.1 Aliasing 44
3.2.2 Discrete Representation of Analog Signals 45
3.2.3 Interpolation and Resampling 46
3.3 Filters 50
3.3.1 Analog Filters 51
3.3.2 Digital Filters 53
3.3.3 Smoothing Filters 55
3.3.4 Acoustic Octave Filters 55
3.3.5 Analog RMS Integration 57
3.3.6 Frequency Weighting Filters 58
3.4 Time Series Analysis 59
3.4.1 Min- and Max-analysis 60
3.4.2 Time Data Integration 60
3.4.3 Time Data Differentiation 65
3.4.4 FFT-based Processing 68
3.5 Chapter Summary 68
3.6 Problems 70
References 71
4 Statistics and Random Processes 73
4.1 Introduction to the Use of Statistics 73
4.1.1 Ensemble and Time Averages 74
4.1.2 Stationarity and Ergodicity 74
4.2 Random Theory 75
4.2.1 Expected Value 75
4.2.2 Errors in Estimates 75
4.2.3 Probability Distribution 76
4.2.4 Probability Density 77
4.2.5 Histogram 77
4.2.6 Sample Probability Density Estimate 78
4.2.7 Average Value and Variance 78
4.2.8 Central Moments 80
4.2.9 Skewness 80
4.2.10 Kurtosis 81
4.2.11 Crest Factor 81
4.2.12 Correlation Functions 82
4.2.13 The Gaussian Probability Distribution 83
4.3 Statistical Methods 85
4.3.1 Hypothesis Tests 85
4.3.2 Test of Normality 88
4.3.3 Test of Stationarity 89
Frame statistics 89
The reverse arrangements test 90
The runs test 93
4.4 Quality Assessment of Measured Signals 94
4.5 Chapter Summary 96
4.6 Problems 98
References 98
5 Fundamental Mechanics 99
5.1 Newton’s Laws 99
5.2 The Single Degree-of-freedom System (SDOF) 100
5.2.1 The Transfer Function 101
5.2.2 The Impulse Response 102
5.2.3 The Frequency Response 104
5.2.4 The Q-factor 107
5.2.5 SDOF Forced Response 108
5.3 Alternative Quantities for Describing Motion 108
5.4 Frequency Response Plot Formats 109
5.4.1 Magnitude and Phase 111
5.4.2 Real and Imaginary Parts 114
5.4.3 The Nyquist Plot – Imaginary vs. Real Part 114
5.5 Determining Natural Frequency and Damping Ratio 117
5.5.1 Peak in the Magnitude of FRF 117
5.5.2 Peak in the Imaginary Part of FRF 117
5.5.3 Resonance Bandwidth (3 dB Bandwidth) 118
5.5.4 Circle in the Nyquist Plot 118
5.6 Rotating Mass 119
5.7 Some Comments on Damping 120
5.7.1 Hysteretic Damping 121
5.8 Models Based on SDOF Approximations 121
5.8.1 Vibration Isolation 122
5.8.2 Resonance Frequency and Stiffness Approximations 124
5.9 The Two-degree-of-freedom System (2DOF) 125
5.10 The Tuned Damper 128
5.11 Chapter Summary 129
5.12 Problems 131
References 132
6 Modal Analysis Theory 133
6.1 Waves on a String 133
6.2 Matrix Formulations 135
6.2.1 Degree-of-freedom 135
6.3 Eigenvalues and Eigenvectors 136
6.3.1 Undamped System 136
6.3.2 Mode Shape Orthogonality 140
6.3.3 Modal Coordinates 141
6.3.4 Proportional Damping 143
6.3.5 General Damping 145
6.4 Frequency Response of MDOF Systems 149
6.4.1 Frequency Response from [M], [C], [K] 149
6.4.2 Frequency Response from Modal Parameters 150
6.4.3 Frequency Response from [M], [K], and _ – Modal Damping 155
6.4.4 Mode Shape Scaling 155
6.4.5 The Effect of Node Lines on FRFs 157
6.4.6 Antiresonance 158
6.4.7 Impulse Response of MDOF Systems 158
6.5 Free Decays 158
6.6 Chapter Summary 159
6.7 Problems 161
References 162
7 Transducers for Noise and Vibration Analysis 163
7.1 The Piezoelectric Effect 163
7.2 The Charge Amplifier 164
7.3 Transducers with Built-In Impedance Converters, ‘IEPE’ 165
7.3.1 Low-frequency Characteristics 167
7.3.2 High-frequency Characteristics 168
7.3.3 Transducer Electronic Data Sheet, TEDS 168
7.4 The Piezoelectric Accelerometer 169
7.4.1 Frequency Characteristics 170
7.4.2 Mounting Accelerometers 172
7.4.3 Electrical Noise 172
7.4.4 Choosing an Accelerometer 173
7.5 The Piezoelectric Force Transducer 174
7.6 The Impedance Head 176
7.7 The Impulse Hammer 177
7.8 Accelerometer Calibration 177
7.9 Measurement Microphones 178
7.10 Microphone Calibration 180
7.11 The Geophone 180
7.12 MEMS-Based Sensors 181
7.13 Shakers for Structure Excitation 181
7.14 Some Comments on Measurement Procedures 183
7.15 Problems 184
References 185
8 Frequency Analysis Theory 187
8.1 Periodic Signals – The Fourier Series 187
8.2 Spectra of Periodic Signals 189
8.2.1 Frequency and Time 190
8.3 Random Processes 190
8.3.1 Spectra of Random Processes 191
8.4 Transient Signals 193
8.5 Interpretation of spectra 194
8.6 Chapter Summary 196
8.7 Problems 197
References 197
9 Experimental Frequency Analysis 199
9.1 Frequency Analysis Principles 199
9.1.1 Nonparametric Frequency Analysis 200
9.2 Octave and Third-octave Band Spectra 201
9.2.1 Time Constants 201
9.2.2 Real-time Versus Serial Measurements 202
9.3 The Discrete Fourier Transform (DFT) 202
9.3.1 The Fast Fourier Transform, FFT 204
9.3.2 The DFT in Short 205
9.3.3 The Basis of the DFT 205
9.3.4 Periodicity of the DFT 207
9.3.5 Properties of the DFT 209
9.3.6 Relation Between DFT and Continuous Spectrum 210
9.3.7 Leakage 211
9.3.8 The Picket-fence Effect 214
9.3.9 Time Windows for Periodic Signals 215
Amplitude correction of window effects 217
Power correction of window effects 217
Comparison of common windows 219
Frequency resolution 223
9.3.10 Time Windows for Random Signals 223
9.3.11 Oversampling in FFT Analysis 224
9.3.12 Circular Convolution and Aliasing 225
9.3.13 Zero Padding 226
9.3.14 Frequency Domain Processing 227
9.3.15 Zoom FFT 228
9.4 Chapter Summary 229
9.5 Problems 230
References 231
10 Spectrum and Correlation Estimates Using the DFT 233
10.1 Averaging 233
10.2 Spectrum Estimators for Periodic Signals 235
10.2.1 The Autopower Spectrum 235
10.2.2 Linear Spectrum 236
10.2.3 Phase Spectrum 237
10.3 Estimators for PSD and CSD 237
10.3.1 The Periodogram 238
10.3.2 Welch’s Method 239
10.3.3 Window Correction for Welch Estimates 240
10.3.4 Bias Error in Welch Estimates 241
10.3.5 Random Error in Welch Estimates 246
10.3.6 The Smoothed Periodogram Estimator 252
10.3.7 Bias Error in Smoothed Periodogram Estimates 254
10.3.8 Random Error in Smoothed Periodogram Estimates 254
10.4 Estimators for Correlation Functions 255
10.4.1 Correlation Estimator By Long FFT 256
10.4.2 Correlation Estimator By Welch’s Method 258
10.4.3 Variance of the Correlation Estimator 259
10.4.4 Effect of Measurement Noise on Correlation Function Estimates 261
10.5 Estimators for Transient Signals 263
10.5.1 Windows for Transient Signals 265
10.6 A Signal Processing Framework for Spectrum and Correlation Estimation 266
10.7 Spectrum Estimation in Practice 267
10.7.1 Linear Spectrum Versus PSD 268
10.7.2 Example of a Spectrum of a Periodic Signal 270
10.7.3 Practical PSD Estimation 271
10.7.4 Spectrum of Mixed Property Signal 272
10.7.5 Calculating RMS Values in Practice 274
10.7.6 RMS From Linear Spectrum of Periodic Signal 274
10.7.7 RMS from PSD 276
10.7.8 Weighted RMS Values 277
10.7.9 Integration and Differentiation in the Frequency Domain 278
10.8 Multi-channel Spectral and Correlation Analysis 279
10.8.1 Matrix Notation for MIMO Spectral Analysis 280
10.8.2 Arranging Spectral Matrices in MATLAB/Octave 281
10.8.3 Multi-channel Correlation Functions 282
10.9 Chapter Summary 282
10.10Problems 283
References 284
11 Measurement and Analysis Systems 287
11.1 Principal Design 288
11.2 Hardware for Noise and Vibration Analysis 289
11.2.1 Signal Conditioning 289
11.2.2 Analog-to-Digital Conversion, ADC 290
Quantization and Dynamic Range 290
Setting the Measurement Range 291
Sampling Accuracy 293
Anti-Alias Filters 294
Sigma–Delta ADCs 295
11.2.3 Practical Issues 297
11.2.4 Hardware Specifications 298
Absolute Amplitude Accuracy 299
Anti-Alias Protection 299
Simultaneous Sampling 299
Cross-Channel Match 299
Dynamic Range 300
Cross-Channel Talk 301
11.2.5 Transient (Shock) Recording 301
11.3 FFT Analysis Software 301
11.3.1 Block Processing 302
11.3.2 Data Scaling 303
11.3.3 Triggering 303
11.3.4 Averaging 304
11.3.5 FFT Setup Parameters 306
11.4 Chapter Summary 306
11.5 Problems 306
References 307
12 Rotating Machinery Analysis 309
12.1 Vibrations in Rotating Machines 309
12.2 Understanding Time–Frequency Analysis 310
12.3 Rotational Speed Signals (Tachometer Signals) 312
12.4 RPM Maps 314
12.4.1 The Waterfall Plot 315
12.4.2 The Color Map Plot 316
12.5 Smearing 316
12.6 Order Tracks 318
12.7 Synchronous Sampling 319
12.7.1 DFT Parameters after Resampling 323
12.8 Averaging Rotation-speed-dependent Signals 323
12.9 Adding Change in RMS with Time 325
12.10Parametric Methods 329
12.11Chapter Summary 330
12.12Problems 331
References 331
13 Single-input Frequency Response Measurements 333
13.1 Linear Systems 334
13.2 Determining Frequency Response Experimentally 334
13.2.1 Method 1 – the H1 Estimator 335
13.2.2 Method 2 – the H2 Estimator 337
13.2.3 Method 3 – the Hc Estimator 338
13.3 Important Relationships for Linear Systems 339
13.4 The Coherence Function 340
13.5 Errors in Determining the Frequency Response 341
13.5.1 Bias Error in FRF Estimates 341
13.5.2 Random Error in FRF Estimates 343
13.5.3 Bias and Random Error Trade-offs 345
13.6 Coherent Output Power 345
13.7 The Coherence Function in Practice 346
13.7.1 Non-random Excitation 348
13.8 Impact Excitation 348
13.8.1 The Force Signal 349
13.8.2 The Response Signal and Exponential Window 352
13.8.3 Impact Testing Software 352
13.8.4 Compensating for the Influence of the Exponential Window 354
13.8.5 Sources of Error 356
13.8.6 Improving Impact Testing by Alternative Processing 357
13.9 Shaker Excitation 358
13.9.1 Signal-to-noise Ratio Comparison 359
13.9.2 Pure Random Noise 359
13.9.3 Burst Random Noise 361
13.9.4 Pseudo-random Noise 362
13.9.5 Periodic Chirp 363
13.9.6 Stepped-sine Excitation 363
13.10Examples of FRF Estimation – No Extraneous Noise 364
13.10.1 Pure Random Excitation 364
13.10.2 Burst Random Excitation 365
13.10.3 Periodic Excitation 367
13.11Example of FRF Estimation – with Output Noise 367
13.12Examples of FRF Estimation – with Input and Output Noise 369
13.12.1 Sources of Error during Shaker Excitation 371
13.12.2 Checking the Shaker Attachment 371
13.12.3 Other Sources of Error 372
13.13Chapter Summary 373
13.14Problems 374
References 375
14 Multiple-input Frequency Response Measurement 377
14.1 Multiple-input Systems 377
14.1.1 The 2-input/1-output System 378
14.1.2 The 2-input/1-output System – matrix notation 379
14.1.3 The H1 Estimator for MIMO 380
14.1.4 Multiple Coherence 382
14.1.5 Computation Considerations for Multiple-input System 384
14.1.6 The Hv Estimator 384
14.1.7 Other MIMO FRF Estimators 385
14.2 Conditioned Input Signals 386
14.2.1 Conditioned Output Signals 388
14.2.2 Partial Coherence 389
14.2.3 Ordering Signals Prior to Conditioning 390
14.2.4 Partial Coherent Output Power Spectra 391
14.2.5 Backtracking the H-systems 391
14.2.6 General Conditioned Systems 391
14.3 Bias and Random Errors for Multiple-input Systems 392
14.4 Excitation Signals for MIMO Analysis 393
14.4.1 Pure Random Noise 394
14.4.2 Burst Random Noise 394
14.4.3 Periodic Random Noise 395
14.4.4 The Multiphase Stepped-sine Method (MPSS) 395
14.5 Data Synthesis and Simulation Examples 396
14.5.1 Burst Random – Output Noise 396
14.5.2 Burst and Periodic Random – Input Noise 399
14.5.3 Periodic Random – Input and Output Noise 399
14.6 Real MIMO Data Case 403
14.7 Chapter Summary 406
14.8 Problems 407
References 408
15 Orthogonalization of Signals 409
15.1 Principal Components 409
15.1.1 Principal Components Used to Find Number of Sources 411
15.1.2 Data Reduction 413
15.2 Virtual Signals 416
15.2.1 Virtual Input Coherence 419
15.2.2 Virtual Input/Output Coherence 421
15.2.3 Virtual Coherent Output Power 422
15.3 Noise Source Identification (NSI) 426
15.3.1 Multiple Source Example 426
15.3.2 Automotive Example 429
15.4 Chapter Summary 429
15.5 Problems 432
References 432
16 Experimental Modal Analysis 433
16.1 Introduction to Experimental Modal Analysis 433
16.1.1 Main Steps in EMA 434
16.2 Experimental Setup 435
16.2.1 Points and DOFs 436
16.2.2 Selecting Measurement DOFs 436
16.2.3 Measurement System 437
16.2.4 Sensor Considerations 438
16.2.5 Data Acquisition Strategies 438
16.2.6 Suspension 439
16.2.7 Measurement Checks 440
16.2.8 Calibration 442
16.2.9 Data Acquisition 442
16.2.10 Mode Indicator Functions 442
16.2.11 Data Quality Assessment 445
16.2.12 Checklist 445
16.3 Introduction to Modal Parameter Extraction 445
16.4 SDOF Parameter Extraction 448
16.4.1 The Least Squares Local Method 448
16.4.2 The Least Squares Global Method 449
16.4.3 The Least Squares (Local) Polynomial Method 450
16.5 The Unified Matrix Polynomial Approach, UMPA 451
16.5.1 Mathematical Framework 451
16.5.2 Choosing Model Order 454
16.5.3 Matrix Coefficient Normalization 455
16.5.4 Data Compression 457
16.6 Time Versus Frequency Domain Parameter Extraction for EMA 459
16.7 Time Domain Parameter Extraction Methods 462
16.7.1 Converting Bandpass Filtered FRFs Into IRFs 463
16.7.2 The Ibrahim Time Domain Method 464
16.7.3 The Multiple-reference Ibrahim Time Domain Method (MITD) 467
16.7.4 Prony’s Method 471
16.7.5 The Least Squares Complex Exponential Method 472
16.7.6 Polyreference Time Domain 473
16.7.7 The Modified Multiple-reference Ibrahim Time Domain Method
(MMITD) 477
16.8 Frequency Domain Parameter Extraction Methods 479
16.8.1 The Least squares complex frequency domain method 480
16.8.2 The Frequency Domain Direct Parameter Identification Method (FDPI)483
16.8.3 The Frequency Z-Domain Direct Parameter Method, FDPIz 487
16.8.4 The Complex Mode indicator Function Method, CMIF 487
16.9 Methods for mode shape estimation and scaling 489
16.9.1 Least Squares Frequency Domain – Single Reference Case 489
16.9.2 Least Squares Frequency Domain – Multiple Reference Case 491
16.9.3 Least Squares Frequency Domain - Multiple Reference Without MPFs 493
16.9.4 Least Squares Time Domain 494
16.9.5 Scaling Modal Model When Poles and Mode Shapes are Known 495
16.10Evaluating the extracted parameters 495
16.10.1 Synthesized FRFs 496
16.10.2 The MAC matrix 496
16.11Chapter Summary 498
16.12Problems 499
References 500
17 Operational Modal Analysis (OMA) 503
17.1 Principles for OMA 504
17.2 Data Acquisition Principles 505
17.3 OMA Modal Parameter Extraction for OMA 506
17.3.1 Spectral Functions for OMA Parameter Extraction 506
17.3.2 Correlation Functions for OMA Parameter Extraction 510
17.3.3 Half spectra 512
17.3.4 Time versus Frequency Domain Parameter Extraction for OMA 513
17.3.5 Modal Parameter Estimation Methods for OMA 513
17.3.6 Least Squares Frequency Domain, OMA Versions 514
17.4 Scaling OMA modal models 516
17.4.1 Scaling an OMA Model Using the Mass Matrix 517
17.4.2 The OMAH method 517
17.5 Chapter Summary 520
17.6 Problems 521
References 522
18 Advanced Analysis Methods 525
18.1 Shock Response Spectrum 525
18.2 The Hilbert Transform 528
18.2.1 Computation of the Hilbert Transform 529
18.2.2 Envelope Detection by the Hilbert Transform 530
18.2.3 Relating Real and Imaginary Parts of Frequency Response Functions 531
18.3 Cepstrum Analysis 535
18.3.1 Power Cepstrum 536
18.3.2 Complex Cepstrum 537
18.3.3 The Real Cepstrum 539
18.3.4 Inverse Cepstrum 539
18.4 The Envelope Spectrum 539
18.5 Creating Random Signals with Known Spectral Density 542
18.6 Identifying Harmonics In Noise 543
18.6.1 The Three-parameter Sine Fit Method 544
18.6.2 Periodogram Ratio Detection, PRD 545
18.7 Harmonic Removal 548
18.7.1 Frequency Domain Editing, FDE 548
18.7.2 Cepstrum Based Harmonic Removal Methods 549
18.8 Chapter Summary 550
18.9 Problems 552
References 552
19 Practical Vibration Measurements and Analysis 555
19.1 Introduction to a Plexiglas Plate 555
19.2 Forced Response Simulation 556
19.2.1 Frequency Domain Forced Response for Periodic Inputs 557
19.2.2 Frequency Domain Forced Response for Random Inputs 559
19.2.3 Time Domain Computation of Forced Response for Any Inputs 559
Time Domain Response By Frequency Domain Computation 559
Time Domain Response By Digital Filters 560
19.2.4 Plexiglas Plate Forced Response Example 563
19.3 Spectra of periodic signals 564
19.4 Spectra of random signals 565
19.5 Data With Random and Periodic Content 568
19.5.1 Car Idling Sound 569
19.5.2 Container Ship Measurement 573
19.6 Operational Deflection Shapes – ODS 574
19.6.1 Plexiglas Plate ODS Example – Single Reference 577
19.6.2 Plexiglas Plate ODS Example – Multiple-Reference 578
19.7 Impact Excitation and FRF Estimation 581
19.8 Plexiglas EMA Example 585
19.8.1 FRF Quality Assessment 585
19.8.2 EMA Modal Parameter Extraction, MPE 590
19.9 Methods for EMA Modal Parameter Estimation, MPE 595
19.9.1 Time Domain Variable Settings 595
19.9.2 High Order Methods for EMA MPE 598
19.9.3 Low Order methods for EMA MPE 600
19.9.4 The Complex Mode Indicator Function, CMIF 604
19.9.5 Calculating Scaled Mode Shapes 604
19.10Conclusions of EMA MPE 609
19.11OMA examples 610
19.11.1 OMA Using Synthesized Data for Plexiglas Plate 610
19.11.2 OMA on Measured Data of Plexiglas Plate 618
19.11.3 OMA of a Supension Bridge 622
19.11.4 OMA On Container Ship 628
References 632
A Appendix A: Complex Numbers 635
B Appendix B: Logarithmic Diagrams 639
C Appendix C: Decibels 643
D Appendix D: Some Elementary Matrix Algebra 645
E Appendix E: Eigenvalues and the SVD 649
E.1 Eigenvalues and Complex Matrices 649
E.2 The Singular Value Decomposition (SVD) 650
F Appendix F: Organizations and Resources 653
G Appendix G: Checklist for Experimental Modal Analysis Testing 655
Reference 657
Index 665
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