About the Author ............................................... xv
Preface ...................................................... xvii
Acknowledgements .............................................. xxi
List of Abbreviations ....................................... xxiii
Notation ...................................................... xxv
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 ........................ 3
1.4.1 Experimental Analysis ............................ 4
1.5 Standards ............................................... 4
1.6 Becoming a Noise and Vibration Analysis Expert .......... 4
1.6.1 The Virtue of Simulation ......................... 4
1.6.2 Learning Tools and the Format of this Book ....... 5
2 Dynamic Signals and Systems .................................. 7
2.1 Introduction ............................................ 7
2.2 Periodic Signals ........................................ 8
2.2.1 Sine Waves ....................................... 8
2.2.2 Complex Sines ................................... 10
2.2.5 Interacting Sines ............................... 11
2.2.4 Orthogonality of Sines .......................... 12
2.3 Random Signals ......................................... 13
2.4 Transient Signals ...................................... 14
2.5 RMS Value and Power .................................... 15
2.6 Linear Systems ......................................... 16
2.6.7 The Laplace Transform ........................... 17
2.6.2 The Transfer Function ........................... 20
2.6.5 The Impulse Response ............................ 21
2.6.4 Convolution ..................................... 22
2.7 The Continuous Fourier Transform ....................... 25
2.7.1 Characteristics of the Fourier Transform ........ 27
2.7.2 The Frequency Response .......................... 29
2.7.3 Relationship between the Laplace and Frequency
Domains ......................................... 29
2.7.4 Transient versus Steady-state Response .......... 30
2.8 Chapter Summary ........................................ 31
2.9 Problems ............................................... 32
References .................................................. 33
3 Time Data Analysis .......................................... 35
3.1 Introduction to Discrete Signals ....................... 35
3.2 The Sampling Theorem ................................... 35
3.2.1 Aliasing ........................................ 37
3.2.2 Discrete Representation of Analog Signals ....... 38
3.2.3 Interpolation and Resampling .................... 40
3.3 Filters ................................................ 42
3.3.1 Analog Filters .................................. 43
3.3.2 Digital Filters ................................. 45
3.3.3 Smoothing Filters ............................... 46
3.3.4 Acoustic Octave Filters ......................... 47
3.3.5 Analog RMS Integration .......................... 49
3.3.6 Frequency Weighting Filters ..................... 49
3.4 Time Series Analysis ................................... 51
3.4.1 Min-and Max-analysis ............................ 51
3.4.2 Time Data Integration ........................... 51
3.4.3 Time Data Differentiation ....................... 55
3.4.4 FFT-based Processing ............................ 58
3.5 Chapter Summary ........................................ 58
3.6 Problems ............................................... 59
References .................................................. 60
4 Statistics and Random Processes ............................. 63
4.1 Introduction to the Use of Statistics .................. 63
4.1.1 Ensemble and Time Averages ...................... 64
4.1.2 Stationarity and Ergodicity ..................... 64
4.2 Random Theory .......................................... 65
4.2.1 Expected Value .................................. 65
4.2.2 Errors in Estimates ............................. 65
4.2.3 Probability Distribution ........................ 66
4.2.4 Probability Density ............................. 66
4.2.5 Histogram ....................................... 67
4.2.6 Sample Probability Density Estimate ............. 68
4.2.7 Average Value and Variance ...................... 68
4.2.8 Central Moments ................................. 70
4.2.9 Skewness ........................................ 70
4.2.10 Kurtosis ........................................ 70
4.2.11 Crest Factor .................................... 71
4.2.12 Correlation Functions ........................... 71
4.2.13 The Gaussian Probability Distribution ........... 72
4.3 Statistical Methods .................................... 74
4.3.1 Hypothesis Tests ................................ 74
4.3.2 Test of Normality ............................... 11
4.3.3 Test of Stationarity ............................ 11
4.4 Quality Assessment of Measured Signals ................. 81
4.5 Chapter Summary ........................................ 84
4.6 Problems ............................................... 85
References .................................................. 86
5 Fundamental Mechanics ....................................... 87
5.1 Newton's Laws .......................................... 87
5.2 The Single Degree-of-freedom System (SDOF) ............. 88
5.2.1 The Transfer Function ........................... 88
5.2.2 The Impulse Response ............................ 89
5.2.3 The Frequency Response .......................... 91
5.2.4 The Q-factor .................................... 94
5.2.5 SDOF Forced Response ............................ 95
5.3 Alternative Quantities for Describing Motion ........... 95
5.4 Frequency Response Plot Formats ........................ 97
5.4.1 Magnitude and Phase ............................. 97
5.4.2 Real and Imaginary Parts ....................... 100
5.4.3 The Nyquist Plot - Imaginary vs. Real Part ..... 100
5.5 Determining Natural Frequency and Damping ............. 103
5.5.1 Peak in the Magnitude of FRF ................... 103
5.5.2 Peak in the Imaginary Part of FRF .............. 103
5.5.3 Resonance Bandwidth (3 dB Bandwidth) ........... 104
5.5.4 Circle in the Nyquist Plot ..................... 104
5.6 Rotating Mass ......................................... 104
5.7 Some Comments on Damping .............................. 106
5.7.7 Hysteretic Damping ............................. 106
5.8 Models Based on SDOF Approximations ................... 107
5.8.1 Vibration Isolation ............................ 107
5.8.2 Resonance Frequency and Stiffness
Approximations ................................. 110
5.9 The Two-degree-of-freedom System (2DOF) ............... 110
5.10 The Tuned Damper ...................................... 113
5.11 Chapter Summary ....................................... 115
5.12 Problems .............................................. 115
References ................................................. 116
6 Modal Analysis Theory ...................................... 119
6.1 Waves on a String ..................................... 119
6.2 Matrix Formulations ................................... 120
6.2.1 Degree-of-freedom .............................. 121
6.3 Eigenvalues and Eigenvectors .......................... 122
6.3.1 Undamped System ................................ 122
6.3.2 Mode Shape Orthogonality ....................... 125
6.3.3 Modal Coordinates .............................. 127
6.3.4 Proportional Damping ........................... 128
6.3.5 General Damping ................................ 130
6.4 Frequency Response of MDOF Systems .................... 133
6.4.1 Frequency Response from [M], [С], [K] .......... 133
6.4.2 Frequency Response from Modal Parameters ....... 134
6.4.3 Frequency Response from [M], [K], and ζ -
Modal Damping .................................. 138
6.4.4 Mode Shape Scaling ............................. 138
6.4.5 The Effect of Node Lines on FRFs ............... 139
6.4.6 Antiresonance .................................. 140
6.4.7 Impulse Response of MDOF Systems ............... 141
6.5 Time Domain Simulation of Forced Response ............. 141
6.6 Chapter Summary ....................................... 143
6.7 Problems .............................................. 144
References ................................................. 145
7 Transducers for Noise and Vibration Analysis ............... 147
7.1 The Piezoelectric Effect .............................. 147
7.2 The Charge Amplifier .................................. 148
7.3 Transducers with Built-in Impedance Converters,
'IEPE' ................................................ 149
7.3.1 Low-frequency Characteristics .................. 150
7.3.2 High-frequency Characteristics ................. 151
7.3.3 Transducer Electronic Data Sheet, TEDS ......... 152
7.4 The Piezoelectric Accelerometer ....................... 152
7.4.1 Frequency Characteristics ...................... 153
7.4.2 Mounting Accelerometers ........................ 155
7.4.3 Electrical Noise ............................... 155
7.4.4 Choosing an Accelerometer ...................... 155
7.5 The Piezoelectric Force Transducer .................... 157
7.6 The Impedance Head .................................... 158
7.7 The Impulse Hammer .................................... 159
7.8 Accelerometer Calibration ............................. 159
7.9 Measurement Microphones ............................... 161
7.10 Microphone Calibration ................................ 162
7.11 Shakers for Structure Excitation ...................... 162
7.12 Some Comments on Measurement Procedures ............... 163
7.13 Problems .............................................. 164
References ................................................. 165
8 Frequency Analysis Theory .................................. 167
8.1 Periodic Signals - The Fourier Series ................. 167
8.2 Spectra of Periodic Signals ........................... 169
8.2.1 Frequency and Time ............................. 170
8.3 Random Processes ...................................... 170
8.3.1 Spectra of Random Processes .................... 171
8.4 Transient Signals ..................................... 173
8.5 Interpretation of spectra ............................. 173
8.6 Chapter Summary ....................................... 175
8.7 Problems .............................................. 175
References ................................................. 176
9 Experimental Frequency Analysis ............................ 177
9.1 Frequency Analysis Principles ......................... 177
9.1.1 Nonparametric Frequency Analysis ............... 178
9.2 Octave and Third-octave Band Spectra .................. 179
9.2.1 Time Constants ................................. 179
9.2.2 Real-time versus Serial Measurements ........... 179
9.3 The Discrete Fourier Transform (DFT) .................. 180
9.3.1 The Fast Fourier Transform, FFT ................ 181
9.3.2 The DFT in Short ............................... 182
9.3.3 The Basis of the DFT ........................... 183
9.3.4 Periodicity of the DFT ......................... 183
9.3.5 Properties of the DFT .......................... 186
9.3.6 Relation between DFT and Continuous Spectrum ... 186
9.3.7 Leakage ........................................ 187
9.3.8 The Picket-fence Effect ........................ 189
9.3.9 Time Windows for Periodic Signals .............. 191
9.3.10 Time Windows for Random Signals ................ 198
9.3.11 Oversampling in FFT Analysis ................... 199
9.3.12 Circular Convolution and Aliasing .............. 199
9.3.13 Zero Padding ................................... 200
9.3.14 Zoom FFT ....................................... 201
9.4 Chapter Summary ....................................... 202
9.5 Problems .............................................. 203
References ................................................. 204
10 Spectrum and Correlation Estimates Using the DFT ........... 205
10.1 Averaging ............................................. 205
10.2 Spectrum Estimators for Periodic Signals .............. 206
10.2.1 The Autopower Spectrum ......................... 207
10.2.2 Linear Spectrum ................................ 208
10.2.3 Phase Spectrum ................................. 208
10.3 Estimators for PSD and CSD ............................ 209
10.3.1 The Periodogram ................................ 209
10.3.2 Welch's Method ................................. 211
10.3.3 Window Correction for Welch Estimates .......... 211
10.3.4 Bias Error in Welch Estimates .................. 212
10.3.5 Random Error in Welch Estimates ................ 217
10.3.6 The Smoothed Periodogram Estimator ............. 221
10.3.7 Bias Error in Smoothed Periodogram Estimates ... 223
10.3.8 Random Error in Smoothed Periodogram
Estimates ...................................... 224
10.4 Estimator for Correlation Functions ................... 224
10.5 Estimators for Transient Signals ...................... 226
10.5.1 Windows for Transient Signals .................. 227
10.6 Spectrum Estimation in Practice ....................... 228
10.6.1 Linear Spectrum Versus PSD ..................... 228
10.6.2 Example of a Spectrum of a Periodic Signal ..... 229
10.6.3 Practical PSD Estimation ....................... 231
10.6.4 Spectrum of Mixed Property Signal .............. 233
10.6.5 Calculating RMS Values in Practice ............. 234
10.6.6 RMS From Linear Spectrum of Periodic Signal .... 234
10.6.7 RMS from PSD ................................... 236
10.6.8 Weighted RMS Values ............................ 236
10.6.9 Integration and Differentiation in the
Frequency Domain ............................... 238
10.7 Multi-channel Spectral Analysis ....................... 238
10.7.1 Matrix Notation for MIMO Spectral Analysis ..... 239
10.7.2 Arranging Spectral Matrices in MATLAB/Octave ... 240
10.8 Chapter Summary ....................................... 240
10.9 Problems .............................................. 241
References ................................................. 242
11 Measurement and Analysis Systems ........................... 245
11.1 Principal Design ...................................... 246
11.2 Hardware for Noise and Vibration Analysis ............. 246
11.2.1 Signal Conditioning ............................ 247
11.2.2 Analog-to-digital Conversion, ADC .............. 247
11.2.3 Practical Issues ............................... 253
11.2.4 Hardware Specifications ........................ 255
11.2.5 Transient (Shock) Recording .................... 257
11.3 FFT Analysis Software ................................. 257
11.3.1 Block Processing ............................... 258
11.3.2 Data Scaling ................................... 259
11.3.3 Triggering ..................................... 259
11.3.4 Averaging ...................................... 260
11.3.5 FFT Setup Parameters ........................... 261
11.4 Chapter Summary ....................................... 261
11.5 Problems .............................................. 261
References ................................................. 262
12 Rotating Machinery Analysis ................................ 263
12.1 Vibrations in Rotating Machines ....................... 263
12.2 Understanding Time-Frequency Analysis ................. 264
12.3 Rotational Speed Signals (Tachometer Signals) ......... 265
12.4 RPM Maps .............................................. 267
12.4.1 The Waterfall Plot ............................. 268
12.4.2 The Color Map Plot ............................. 268
12.5 Smearing .............................................. 269
12.6 Order Tracks .......................................... 272
12.7 Synchronous Sampling .................................. 272
12.7.1 DFTParameters after Resampling ................. 276
12.8 Averaging Rotation-speed-dependent Signals ............ 276
12.9 Adding Change in RMS with Time ........................ 277
12.10 Parametric Methods ................................... 281
12.11 Chapter Summary ...................................... 282
12.12 Problems ............................................. 282
References ................................................. 283
13 Single-input Frequency Response Measurements ............... 285
13.1 Linear Systems ........................................ 286
13.2 Determining Frequency Response Experimentally ......... 286
13.2.1 Method 1 - the Ht Estimator .................... 286
13.2.2 Method 2 - the H2 Estimator .................... 288
13.2.3 Method 3 - the Hc Estimator .................... 289
13.3 Important Relationships for Linear Systems ............ 290
13.4 The Coherence Function ................................ 291
13.5 Errors in Determining the Frequency Response .......... 291
13.5.1 Bias Error in FRF Estimates .................. 292
13.5.2 Random Error in FRF Estimates .................. 293
13.5.3 Bias and Random Error Trade-offs ............... 295
13.6 Coherent Output Power ................................. 295
13.7 The Coherence Function in Practice .................... 296
13.7.1 Non-random Excitation .......................... 297
13.8 Impact Excitation ..................................... 297
13.8.1 The Force Signal ............................... 298
13.8.2 The Response Signal and Exponential Window ..... 300
13.8.3 Impact Testing Software ........................ 300
13.8.4 Compensating for the Influence of the
Exponential Window ............................. 303
13.8.5 Sources of Error ............................... 305
13.8.6 Improving Impact Testing by Alternative
Processing ..................................... 306
13.9 Shaker Excitation ..................................... 306
13.9.1 Signal-to-noise Ratio Comparison ............... 307
13.9.2 Pure Random Noise .............................. 308
13.9.3 Burst Random Noise ............................. 310
13.9.4 Pseudo-random Noise ............................ 310
13.9.5 Periodic Chirp ................................. 311
13.9.6 Stepped-sine Excitation ........................ 311
13.10 Examples of FRF Estimation - No Extraneous Noise ..... 312
13.10.1 Pure Random Excitation ........................ 312
13.10.2 Burst Random Excitation ....................... 312
13.10.3 Periodic Excitation ........................... 314
13.11 Example of FRF Estimation - with Output Noise ........ 315
13.12 Examples of FRF Estimation - with Input and Output
Noise ................................................ 316
13.12.1 Sources of Error during Shaker Excitation ..... 318
13.12.2 Checking the Shaker Attachment ................ 318
13.12.3 Other Sources of Error ........................ 319
13.13 Chapter Summary ...................................... 319
13.14 Problems ............................................. 321
References ................................................. 321
14 Multiple-input Frequency Response Measurement ............. 323
14.1 Multiple-input Systems ................................ 323
14.1.1 The 2-input/I-output System .................... 324
14.1.2 The 2-input/I-output System - matrix notation .. 325
14.1.3 The H1 Estimator for MIMO ...................... 326
14.1.4 Multiple Coherence ............................. 327
14.1.5 Computation Considerations for Multiple-input
System ......................................... 329
14.1.6 The Hv Estimator ............................... 329
14.1.7 Other MIMO FRF Estimators ...................... 330
14.2 Conditioned Input Signals ............................. 331
14.2.1 Conditioned Output Signals ..................... 333
14.2.2 Partial Coherence .............................. 333
14.2.3 Ordering Signals Prior to Conditioning ......... 334
14.2.4 Partial Coherent Output Power Spectra .......... 334
14.2.5 Backtracking the H-systems ..................... 335
14.2.6 General Conditioned Systems .................... 336
14.3 Bias and Random Errors for Multiple-input Systems ..... 336
14.4 Excitation Signals for MIMO Analysis .................. 337
14.4.1 Pure Random Noise .............................. 337
14.4.2 Burst Random Noise ............................. 338
14.4.3 Periodic Random Noise .......................... 338
14.4.4 The Multiphase Stepped-sine Method (MPSS) ...... 338
14.5 Data Synthesis and Simulation Examples ................ 339
14.5.1 Burst Random - Output Noise .................... 339
14.5.2 Burst and Periodic Random-Input Noise .......... 342
14.5.3 Periodic Random - Input and Output Noise ....... 342
14.6 Real MIMO Data Case ................................... 345
14.7 Chapter Summary ....................................... 348
14.8 Problems .............................................. 349
References ................................................. 350
15 Orthogonalization of Signals ............................... 351
15.1 Principal Components .................................. 351
15.1.1 Principal Components Used to Find Number of
Sources ........................................ 353
15.1.2 Principal Components Used for Data Reduction ... 355
15.2 Virtual Signals ....................................... 360
15.2.1 Virtual Input Coherence ........................ 361
15.2.2 Virtual Input/Output Coherence ................. 364
15.2.3 Virtual Coherent Output Power .................. 364
15.3 Noise Source Identification (NSI) ..................... 367
15.3.1 Multiple Source Example ........................ 367
15.3.2 Automotive Example ............................. 370
15.4 Chapter Summary ....................................... 372
15.5 Problems .............................................. 373
References ................................................. 373
16 Advanced Analysis Methods .................................. 375
16.1 Shock Response Spectrum ............................... 375
16.2 The Hilbert Transform ................................. 378
16.2.1 Computation of the Hilbert Transform ........... 379
16.2.2 Envelope Detection by the Hilbert Transform .... 379
16.2.3 Relating Real and Imaginary Parts of
Frequency Response Functions ................... 380
16.3 Cepstrum Analysis ..................................... 384
16.3.1 Power Cepstrum ................................. 385
16.3.2 Complex Cepstrum ............................... 387
16.3.3 Inverse Cepstrum ............................... 387
16.4 The Envelope Spectrum ................................. 388
16.5 Creating Random Signals with Known Spectral Density ... 390
16.6 Operational Deflection Shapes - ODS ................... 391
16.6.1 Multiple-Reference ODS ......................... 392
16.7 Introduction to Experimental Modal Analysis ........... 393
16.7.1 Main Steps in EMA .............................. 393
76.7.1 Data Checks .................................... 394
16.7.1 Mode Indicator Functions ....................... 395
16.7.2 The MAC Matrix ................................. 397
16.7.3 Modal Parameter Extraction ..................... 398
16.8 Chapter Summary ....................................... 399
16.9 Problems .............................................. 400
References ................................................. 400
Appendix A Complex Numbers .................................... 403
Appendix В Logarithmic Diagrams ............................... 407
Appendix С Decibels ........................................... 411
Appendix D Some Elementary Matrix Algebra ..................... 413
Reference .................................................. 415
Appendix E Eigenvalues and the SVD ............................ 417
E.1 Eigenvalues and Complex Matrices ...................... 417
E.2 The Singular Value Decomposition (SVD) ................ 418
Reference .................................................. 419
Appendix F Organizations and Resources ........................ 421
Bibliography .................................................. 423
Index ......................................................... 429
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