About the Author |
|
xv | |
Preface |
|
xvii | |
Acknowledgements |
|
xxi | |
|
|
xxiii | |
Notation |
|
xxv | |
|
|
1 | (6) |
|
|
1 | (1) |
|
1.2 Noise and Vibration Analysis |
|
|
2 | (1) |
|
|
3 | (1) |
|
1.4 Analysis of Noise and Vibrations |
|
|
3 | (1) |
|
1.4.1 Experimental Analysis |
|
|
4 | (1) |
|
|
4 | (1) |
|
1.6 Becoming a Noise and Vibration Analysis Expert |
|
|
4 | (3) |
|
1.6.1 The Virtue of Simulation |
|
|
4 | (1) |
|
1.6.2 Learning Tools and the Format of this Book |
|
|
5 | (2) |
|
2 Dynamic Signals and Systems |
|
|
7 | (28) |
|
|
7 | (1) |
|
|
8 | (5) |
|
|
8 | (2) |
|
|
10 | (1) |
|
|
11 | (1) |
|
2.2.4 Orthogonality of Sines |
|
|
12 | (1) |
|
|
13 | (1) |
|
|
14 | (1) |
|
|
15 | (1) |
|
|
16 | (9) |
|
2.6.1 The Laplace Transform |
|
|
17 | (3) |
|
2.6.2 The Transfer Function |
|
|
20 | (1) |
|
2.6.3 The Impulse Response |
|
|
21 | (1) |
|
|
22 | (3) |
|
2.7 The Continuous Fourier Transform |
|
|
25 | (6) |
|
2.7.1 Characteristics of the Fourier Transform |
|
|
27 | (2) |
|
2.7.2 The Frequency Response |
|
|
29 | (1) |
|
2.7.3 Relationship between the Laplace and Frequency Domains |
|
|
29 | (1) |
|
2.7.4 Transient versus Steady-state Response |
|
|
30 | (1) |
|
|
31 | (1) |
|
|
32 | (3) |
|
|
33 | (2) |
|
|
35 | (28) |
|
3.1 Introduction to Discrete Signals |
|
|
35 | (1) |
|
|
35 | (7) |
|
|
37 | (1) |
|
3.2.2 Discrete Representation of Analog Signals |
|
|
38 | (2) |
|
3.2.3 Interpolation and Resampling |
|
|
40 | (2) |
|
|
42 | (9) |
|
|
43 | (2) |
|
|
45 | (1) |
|
|
46 | (1) |
|
3.3.4 Acoustic Octave Filters |
|
|
47 | (2) |
|
3.3.5 Analog RMS Integration |
|
|
49 | (1) |
|
3.3.6 Frequency Weighting Filters |
|
|
49 | (2) |
|
|
51 | (7) |
|
3.4.1 Min- and Max-analysis |
|
|
51 | (1) |
|
3.4.2 Time Data Integration |
|
|
51 | (4) |
|
3.4.3 Time Data Differentiation |
|
|
55 | (3) |
|
3.4.4 FFT-based Processing |
|
|
58 | (1) |
|
|
58 | (1) |
|
|
59 | (4) |
|
|
60 | (3) |
|
4 Statistics and Random Processes |
|
|
63 | (24) |
|
4.1 Introduction to the Use of Statistics |
|
|
63 | (2) |
|
4.1.1 Ensemble and Time Averages |
|
|
64 | (1) |
|
4.1.2 Stationarity and Ergodicity |
|
|
64 | (1) |
|
|
65 | (9) |
|
|
65 | (1) |
|
4.2.2 Errors in Estimates |
|
|
65 | (1) |
|
4.2.3 Probability Distribution |
|
|
66 | (1) |
|
4.2.4 Probability Density |
|
|
66 | (1) |
|
|
67 | (1) |
|
4.2.6 Sample Probability Density Estimate |
|
|
68 | (1) |
|
4.2.7 Average Value and Variance |
|
|
68 | (2) |
|
|
70 | (1) |
|
|
70 | (1) |
|
|
70 | (1) |
|
|
71 | (1) |
|
4.2.12 Correlation Functions |
|
|
71 | (1) |
|
4.2.13 The Gaussian Probability Distribution |
|
|
72 | (2) |
|
|
74 | (7) |
|
|
74 | (3) |
|
|
77 | (1) |
|
4.3.3 Test of Stationarity |
|
|
77 | (4) |
|
4.4 Quality Assessment of Measured Signals |
|
|
81 | (3) |
|
|
84 | (1) |
|
|
85 | (2) |
|
|
86 | (1) |
|
|
87 | (32) |
|
|
87 | (1) |
|
5.2 The Single Degree-of-freedom System (SDOF) |
|
|
88 | (7) |
|
5.2.1 The Transfer Function |
|
|
88 | (1) |
|
5.2.2 The Impulse Response |
|
|
89 | (2) |
|
5.2.3 The Frequency Response |
|
|
91 | (3) |
|
|
94 | (1) |
|
5.2.5 SDOF Forced Response |
|
|
95 | (1) |
|
5.3 Alternative Quantities for Describing Motion |
|
|
95 | (2) |
|
5.4 Frequency Response Plot Formats |
|
|
97 | (6) |
|
5.4.1 Magnitude and Phase |
|
|
97 | (3) |
|
5.4.2 Real and Imaginary Parts |
|
|
100 | (1) |
|
5.4.3 The Nyquist Plot - Imaginary vs. Real Part |
|
|
100 | (3) |
|
5.5 Determining Natural Frequency and Damping |
|
|
103 | (1) |
|
5.5.1 Peak in the Magnitude of FRF |
|
|
103 | (1) |
|
5.5.2 Peak in the Imaginary Part of FRF |
|
|
103 | (1) |
|
5.5.3 Resonance Bandwidth (3 dB Bandwidth) |
|
|
104 | (1) |
|
5.5.4 Circle in the Nyquist Plot |
|
|
104 | (1) |
|
|
104 | (2) |
|
5.7 Some Comments on Damping |
|
|
106 | (1) |
|
|
106 | (1) |
|
5.8 Models Based on SDOF Approximations |
|
|
107 | (3) |
|
5.8.1 Vibration Isolation |
|
|
107 | (3) |
|
5.8.2 Resonance Frequency and Stiffness Approximations |
|
|
110 | (1) |
|
5.9 The Two-degree-of-freedom System (2DOF) |
|
|
110 | (3) |
|
|
113 | (2) |
|
|
115 | (1) |
|
|
115 | (4) |
|
|
116 | (3) |
|
|
119 | (28) |
|
|
119 | (1) |
|
|
120 | (2) |
|
|
121 | (1) |
|
6.3 Eigenvalues and Eigenvectors |
|
|
122 | (11) |
|
|
122 | (3) |
|
6.3.2 Mode Shape Orthogonality |
|
|
125 | (2) |
|
|
127 | (1) |
|
6.3.4 Proportional Damping |
|
|
128 | (2) |
|
|
130 | (3) |
|
6.4 Frequency Response of MDOF Systems |
|
|
133 | (8) |
|
6.4.1 Frequency Response from [ M], [ C], [ K] |
|
|
133 | (1) |
|
6.4.2 Frequency Response from Modal Parameters |
|
|
134 | (4) |
|
6.4.3 Frequency Response from [ M], [ K], and ε - Modal Damping |
|
|
138 | (1) |
|
|
138 | (1) |
|
6.4.5 The Effect of Node Lines on FRFs |
|
|
139 | (1) |
|
|
140 | (1) |
|
6.4.7 Impulse Response of MDOF Systems |
|
|
141 | (1) |
|
6.5 Time Domain Simulation of Forced Response |
|
|
141 | (2) |
|
|
143 | (1) |
|
|
144 | (3) |
|
|
145 | (2) |
|
7 Transducers for Noise and Vibration Analysis |
|
|
147 | (20) |
|
7.1 The Piezoelectric Effect |
|
|
147 | (1) |
|
|
148 | (1) |
|
7.3 Transducers with Built-In Impedance Converters, `IEPE' |
|
|
149 | (3) |
|
7.3.1 Low-frequency Characteristics |
|
|
150 | (1) |
|
7.3.2 High-frequency Characteristics |
|
|
151 | (1) |
|
7.3.3 Transducer Electronic Data Sheet, TEDS |
|
|
152 | (1) |
|
7.4 The Piezoelectric Accelerometer |
|
|
152 | (5) |
|
7.4.1 Frequency Characteristics |
|
|
153 | (2) |
|
7.4.2 Mounting Accelerometers |
|
|
155 | (1) |
|
|
155 | (1) |
|
7.4.4 Choosing an Accelerometer |
|
|
155 | (2) |
|
7.5 The Piezoelectric Force Transducer |
|
|
157 | (1) |
|
|
158 | (1) |
|
|
159 | (1) |
|
7.8 Accelerometer Calibration |
|
|
159 | (2) |
|
7.9 Measurement Microphones |
|
|
161 | (1) |
|
7.10 Microphone Calibration |
|
|
162 | (1) |
|
7.11 Shakers for Structure Excitation |
|
|
162 | (1) |
|
7.12 Some Comments on Measurement Procedures |
|
|
163 | (1) |
|
|
164 | (3) |
|
|
165 | (2) |
|
8 Frequency Analysis Theory |
|
|
167 | (10) |
|
8.1 Periodic Signals - The Fourier Series |
|
|
167 | (2) |
|
8.2 Spectra of Periodic Signals |
|
|
169 | (1) |
|
|
170 | (1) |
|
|
170 | (3) |
|
8.3.1 Spectra of Random Processes |
|
|
171 | (2) |
|
|
173 | (1) |
|
8.5 Interpretation of spectra |
|
|
173 | (2) |
|
|
175 | (1) |
|
|
175 | (2) |
|
|
176 | (1) |
|
9 Experimental Frequency Analysis |
|
|
177 | (28) |
|
9.1 Frequency Analysis Principles |
|
|
177 | (2) |
|
9.1.1 Nonparametric Frequency Analysis |
|
|
178 | (1) |
|
9.2 Octave and Third-octave Band Spectra |
|
|
179 | (1) |
|
|
179 | (1) |
|
9.2.2 Real-time versus Serial Measurements |
|
|
179 | (1) |
|
9.3 The Discrete Fourier Transform (DFT) |
|
|
180 | (22) |
|
9.3.1 The Fast Fourier Transform, FFT |
|
|
181 | (1) |
|
|
182 | (1) |
|
9.3.3 The Basis of the DFT |
|
|
183 | (1) |
|
9.3.4 Periodicity of the DFT |
|
|
183 | (3) |
|
9.3.5 Properties of the DFT |
|
|
186 | (1) |
|
9.3.6 Relation between DFT and Continuous Spectrum |
|
|
186 | (1) |
|
|
187 | (2) |
|
9.3.8 The Picket-fence Effect |
|
|
189 | (2) |
|
9.3.9 Time Windows for Periodic Signals |
|
|
191 | (7) |
|
9.3.10 Time Windows for Random Signals |
|
|
198 | (1) |
|
9.3.11 Oversampling in FFT Analysis |
|
|
199 | (1) |
|
9.3.12 Circular Convolution and Aliasing |
|
|
199 | (1) |
|
|
200 | (1) |
|
|
201 | (1) |
|
|
202 | (1) |
|
|
203 | (2) |
|
|
204 | (1) |
|
10 Spectrum and Correlation Estimates Using the DFT |
|
|
205 | (40) |
|
|
205 | (1) |
|
10.2 Spectrum Estimators for Periodic Signals |
|
|
206 | (3) |
|
10.2.1 The Autopower Spectrum |
|
|
207 | (1) |
|
|
208 | (1) |
|
|
208 | (1) |
|
10.3 Estimators for PSD and CSD |
|
|
209 | (15) |
|
|
209 | (2) |
|
|
211 | (1) |
|
10.3.3 Window Correction for Welch Estimates |
|
|
211 | (1) |
|
10.3.4 Bias Error in Welch Estimates |
|
|
212 | (5) |
|
10.3.5 Random Error in Welch Estimates |
|
|
217 | (4) |
|
10.3.6 The Smoothed Periodogram Estimator |
|
|
221 | (2) |
|
10.3.7 Bias Error in Smoothed Periodogram Estimates |
|
|
223 | (1) |
|
10.3.8 Random Error in Smoothed Periodogram Estimates |
|
|
224 | (1) |
|
10.4 Estimator for Correlation Functions |
|
|
224 | (2) |
|
10.5 Estimator for Transient Signals |
|
|
226 | (2) |
|
10.5.1 Windows for Transient Signals |
|
|
227 | (1) |
|
10.6 Spectrum Estimation in Practice |
|
|
228 | (10) |
|
10.6.1 Linear Spectrum Versus PSD |
|
|
228 | (1) |
|
10.6.2 Example of a Spectrum of a Periodic Signal |
|
|
229 | (2) |
|
10.6.3 Practical PSD Estimation |
|
|
231 | (2) |
|
10.6.4 Spectrum of Mixed Property Signal |
|
|
233 | (1) |
|
10.6.5 Calculating RMS Values in Practice |
|
|
234 | (1) |
|
10.6.6 RMS From Linear Spectrum of Periodic Signal |
|
|
234 | (2) |
|
|
236 | (1) |
|
10.6.8 Weighted RMS Values |
|
|
236 | (2) |
|
10.6.9 Integration and Differentiation in the Frequency Domain |
|
|
238 | (1) |
|
10.7 Multi-channel Spectral Analysis |
|
|
238 | (2) |
|
10.7.1 Matrix Notation for MIMO Spectral Analysis |
|
|
239 | (1) |
|
10.7.2 Arranging Spectral Matrices in MATLAB/Octav |
|
|
240 | (1) |
|
|
240 | (1) |
|
|
241 | (4) |
|
|
242 | (3) |
|
11 Measurement and Analysis Systems |
|
|
245 | (18) |
|
|
246 | (1) |
|
11.2 Hardware for Noise and Vibration Analysis |
|
|
246 | (11) |
|
11.2.1 Signal Conditioning |
|
|
247 | (1) |
|
11.2.2 Analog-to-digital Conversion, ADC |
|
|
247 | (6) |
|
|
253 | (2) |
|
11.2.4 Hardware Specifications |
|
|
255 | (2) |
|
11.2.5 Transient (Shock) Recording |
|
|
257 | (1) |
|
11.3 FFT Analysis Software |
|
|
257 | (4) |
|
|
258 | (1) |
|
|
259 | (1) |
|
|
259 | (1) |
|
|
260 | (1) |
|
11.3.5 FFT Setup Parameters |
|
|
261 | (1) |
|
|
261 | (1) |
|
|
261 | (2) |
|
|
262 | (1) |
|
12 Rotating Machinery Analysis |
|
|
263 | (22) |
|
12.1 Vibrations in Rotating Machines |
|
|
263 | (1) |
|
12.2 Understanding Time-Frequency Analysis |
|
|
264 | (1) |
|
12.3 Rotational Speed Signals (Tachometer Signals) |
|
|
265 | (2) |
|
|
267 | (2) |
|
12.4.1 The Waterfall Plot |
|
|
268 | (1) |
|
12.4.2 The Color Map Plot |
|
|
268 | (1) |
|
|
269 | (3) |
|
|
272 | (1) |
|
12.7 Synchronous Sampling |
|
|
272 | (4) |
|
12.7.1 DFT Parameters after Resampling |
|
|
276 | (1) |
|
12.8 Averaging Rotation-speed-dependent Signals |
|
|
276 | (1) |
|
12.9 Adding Change in RMS with Time |
|
|
277 | (4) |
|
|
281 | (1) |
|
|
282 | (1) |
|
|
282 | (3) |
|
|
283 | (2) |
|
13 Single-input Frequency Response Measurements |
|
|
285 | (38) |
|
|
286 | (1) |
|
13.2 Determining Frequency Response Experimentally |
|
|
286 | (4) |
|
13.2.1 Method 1 - the H1 Estimator |
|
|
286 | (2) |
|
13.2.2 Method 2 - the H2 Estimator |
|
|
288 | (1) |
|
13.2.3 Method 3 - the Hc Estimator |
|
|
289 | (1) |
|
13.3 Important Relationships for Linear Systems |
|
|
290 | (1) |
|
13.4 The Coherence Function |
|
|
291 | (1) |
|
13.5 Errors in Determining the Frequency Response |
|
|
291 | (4) |
|
13.5.1 Bias Error in FRF Estimates |
|
|
292 | (1) |
|
13.5.2 Random Error in FRF Estimates |
|
|
293 | (2) |
|
13.5.3 Bias and Random Error Trade-offs |
|
|
295 | (1) |
|
13.6 Coherent Output Power |
|
|
295 | (1) |
|
13.7 The Coherence Function in Practice |
|
|
296 | (1) |
|
13.7.1 Non-random Excitation |
|
|
297 | (1) |
|
|
297 | (9) |
|
|
298 | (2) |
|
13.8.2 The Response Signal and Exponential Window |
|
|
300 | (1) |
|
13.8.3 Impact Testing Software |
|
|
300 | (3) |
|
13.8.4 Compensating for the Influence of the Exponential Window |
|
|
303 | (2) |
|
|
305 | (1) |
|
13.8.6 Improving Impact Testing by Alternative Processing |
|
|
306 | (1) |
|
|
306 | (6) |
|
13.9.1 Signal-to-noise Ratio Comparison |
|
|
307 | (1) |
|
|
308 | (2) |
|
13.9.3 Burst Random Noise |
|
|
310 | (1) |
|
13.9.4 Pseudo-random Noise |
|
|
310 | (1) |
|
|
311 | (1) |
|
13.9.6 Stepped-sine Excitation |
|
|
311 | (1) |
|
13.10 Examples of FRF Estimation - No Extraneous Noise |
|
|
312 | (3) |
|
13.10.1 Pure Random Excitation |
|
|
312 | (1) |
|
13.10.2 Burst Random Excitation |
|
|
312 | (2) |
|
13.10.3 Periodic Excitation |
|
|
314 | (1) |
|
13.11 Example of FRF Estimation - with Output Noise |
|
|
315 | (1) |
|
13.12 Examples of FRF Estimation - with Input and Output Noise |
|
|
316 | (3) |
|
13.12.1 Sources of Error during Shaker Excitation |
|
|
318 | (1) |
|
13.12.2 Checking the Shaker Attachment |
|
|
318 | (1) |
|
13.12.3 Other Sources of Error |
|
|
319 | (1) |
|
|
319 | (2) |
|
|
321 | (2) |
|
|
321 | (2) |
|
14 Multiple-input Frequency Response Measurement |
|
|
323 | (28) |
|
14.1 Multiple-input Systems |
|
|
323 | (8) |
|
14.1.1 The 2-input/1-output System |
|
|
324 | (1) |
|
14.1.2 The 2-input/1-output System - matrix notation |
|
|
325 | (1) |
|
14.1.3 The H1 Estimator for MIMO |
|
|
326 | (1) |
|
14.1.4 Multiple Coherence |
|
|
327 | (2) |
|
14.1.5 Computation Considerations for Multiple-input System |
|
|
329 | (1) |
|
|
329 | (1) |
|
14.1.7 Other MIMO FRF Estimators |
|
|
330 | (1) |
|
14.2 Conditioned Input Signals |
|
|
331 | (5) |
|
14.2.1 Conditioned Output Signals |
|
|
333 | (1) |
|
|
333 | (1) |
|
14.2.3 Ordering Signals Prior to Conditioning |
|
|
334 | (1) |
|
14.2.4 Partial Coherent Output Power Spectra |
|
|
334 | (1) |
|
14.2.5 Backtracking the H-systems |
|
|
335 | (1) |
|
14.2.6 General Conditioned Systems |
|
|
336 | (1) |
|
14.3 Bias and Random Errors for Multiple-input Systems |
|
|
336 | (1) |
|
14.4 Excitation Signals for MIMO Analysis |
|
|
337 | (2) |
|
|
337 | (1) |
|
14.4.2 Burst Random Noise |
|
|
338 | (1) |
|
14.4.3 Periodic Random Noise |
|
|
338 | (1) |
|
14.4.4 The Multiphase Stepped-sine Method (MPSS) |
|
|
338 | (1) |
|
14.5 Data Synthesis and Simulation Examples |
|
|
339 | (6) |
|
14.5.1 Burst Random - Output Noise |
|
|
339 | (3) |
|
14.5.2 Burst and Periodic Random - Input Noise |
|
|
342 | (1) |
|
14.5.3 Periodic Random - Input and Output Noise |
|
|
342 | (3) |
|
|
345 | (3) |
|
|
348 | (1) |
|
|
349 | (2) |
|
|
350 | (1) |
|
15 Orthogonalization of Signals |
|
|
351 | (24) |
|
15.1 Principal Components |
|
|
351 | (9) |
|
15.1.1 Principal Components Used to Find Number of Sources |
|
|
353 | (2) |
|
15.1.2 Principal Components Used for Data Reduction |
|
|
355 | (5) |
|
|
360 | (7) |
|
15.2.1 Virtual Input Coherence |
|
|
361 | (3) |
|
15.2.2 Virtual Input/Output Coherence |
|
|
364 | (1) |
|
15.2.3 Virtual Coherent Output Power |
|
|
364 | (3) |
|
15.3 Noise Source Identification (NSI) |
|
|
367 | (5) |
|
15.3.1 Multiple Source Example |
|
|
367 | (3) |
|
15.3.2 Automotive Example |
|
|
370 | (2) |
|
|
372 | (1) |
|
|
373 | (2) |
|
|
373 | (2) |
|
16 Advanced Analysis Methods |
|
|
375 | (48) |
|
16.1 Shock Response Spectrum |
|
|
375 | (3) |
|
16.2 The Hilbert Transform |
|
|
378 | (6) |
|
16.2.1 Computation of the Hilbert Transform |
|
|
379 | (1) |
|
16.2.2 Envelope Detection by the Hilbert Transform |
|
|
379 | (1) |
|
16.2.3 Relating Real and Imaginary Parts of Frequency Response Functions |
|
|
380 | (4) |
|
|
384 | (4) |
|
|
385 | (2) |
|
|
387 | (1) |
|
|
387 | (1) |
|
16.4 The Envelope Spectrum |
|
|
388 | (2) |
|
16.5 Creating Random Signals with Known Spectral Density |
|
|
390 | (1) |
|
16.6 Operational Deflection Shapes - ODS |
|
|
391 | (2) |
|
16.6.1 Multiple Reference ODS |
|
|
392 | (1) |
|
16.7 Introduction to Experimental Modal Analysis |
|
|
393 | (6) |
|
|
393 | (1) |
|
|
394 | (1) |
|
16.7.3 Mode Indicator Functions |
|
|
395 | (2) |
|
|
397 | (1) |
|
16.7.5 Modal Parameter Extraction |
|
|
398 | (1) |
|
|
399 | (1) |
|
|
400 | (3) |
|
|
400 | (3) |
|
Appendix A Complex Numbers |
|
|
403 | (4) |
|
Appendix B Logarithmic Diagrams |
|
|
407 | (4) |
|
|
411 | (2) |
|
Appendix D Some Elementary Matrix Algebra |
|
|
413 | (4) |
|
|
415 | (2) |
|
Appendix E Eigenvalues and the SVD |
|
|
417 | (4) |
|
E.1 Eigenvalues and Complex Matrices |
|
|
417 | (1) |
|
E.2 The Singular Value Decomposition (SVD) |
|
|
418 | (1) |
|
|
419 | (2) |
|
Appendix F Organizations and Resources |
|
|
421 | (2) |
Bibliography |
|
423 | (6) |
Index |
|
429 | |