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Mechanical Vibration and Shock Analysis, Volume 3, Random Vibration, 3rd Edition

Mechanical Vibration and Shock Analysis, Volume 3, Random Vibration, 3rd Edition

Christian Lalanne

ISBN: 978-1-118-93117-2

Apr 2014, Wiley-ISTE

595 pages

$165.99

Description

The vast majority of vibrations encountered in the real environment are random in nature. Such vibrations are intrinsically complicated and this volume describes the process that enables us to simplify the required analysis, along with the analysis of the signal in the frequency domain.
The power spectrum density is also defined, together with the requisite precautions to be taken in its calculations as well as the processes (windowing, overlapping) necessary to obtain improved results.
An additional complementary method – the analysis of statistical properties of the time signal – is also described. This enables the distribution law of the maxima of a random Gaussian signal to be determined and simplifies the calculation of fatigue damage by avoiding direct peak counting.

Foreword to Series xiii

Introduction   xvii

List of Symbols   xix

Chapter 1. Statistical Properties of a Random Process 1

1.1. Definitions  1

1.1.1. Random variable 1

1.1.2. Random process 2

1.2. Random vibration in real environments    2

1.3. Random vibration in laboratory tests     3

1.4. Methods of random vibration analysis 3

1.5. Distribution of instantaneous values     5

1.5.1. Probability density 5

1.5.2. Distribution function6

1.6. Gaussian random process    7

1.7. Rayleigh distribution 12

1.8. Ensemble averages: through the process   12

1.8.1. n order average 12

1.8.2. Centered moments 14

1.8.3. Variance  14

1.8.4. Standard deviation 15

1.8.5. Autocorrelation function   16

1.8.6. Cross-correlation function 16

1.8.7. Autocovariance 17

1.8.8. Covariance 17

1.8.9. Stationarity 17

1.9. Temporal averages: along the process 23

1.9.1. Mean   23

1.9.2. Quadratic mean – rms value 25

1.9.3. Moments of order n 27

1.9.4. Variance – standard deviation  28

1.9.5. Skewness 29

1.9.6. Kurtosis 30

1.9.7. Crest Factor 33

1.9.8. Temporal autocorrelation function 33

1.9.9. Properties of the autocorrelation function  39

1.9.10. Correlation duration 41

1.9.11. Cross-correlation 47

1.9.12. Cross-correlation coefficient  50

1.9.13. Ergodicity 50

1.10. Significance of the statistical analysis (ensemble or temporal) 52

1.11. Stationary and pseudo-stationary signals 52

1.12. Summary chart of main definitions     53

1.13. Sliding mean 54

1.14. Test of stationarity 58

1.14.1. The reverse arrangements test (RAT)   58

1.14.2. The runs test 61

1.15 Identification of shocks and/or signal problems 65

1.16. Breakdown of vibratory signal into “events”: choice of signal samples 68

1.17. Interpretation and taking into account of environment variation   75

Chapter 2. Random Vibration Properties in the Frequency Domain   79

2.1. Fourier transform 79

2.2. Power spectral density 81

2.2.1. Need  81

2.2.2. Definition 82

2.3. Amplitude Spectral Density    89

2.4. Cross-power spectral density 89

2.5. Power spectral density of a random process  90

2.6. Cross-power spectral density of two processes  91

2.7. Relationship between the PSD and correlation function of a process  93

2.8. Quadspectrum – cospectrum   93

2.9. Definitions  94

2.9.1. Broadband process 94

2.9.2. White noise 95

2.9.3. Band-limited white noise 95

2.9.4. Narrow band process 96

2.9.5. Colors of noise 97

2.10. Autocorrelation function of white noise 98

2.11. Autocorrelation function of band-limited white noise 99

2.12. Peak factor  101

2.13. Effects of truncation of peaks of acceleration signal on the PSD 101

2.14. Standardized PSD/density of probability analogy 105

2.15. Spectral density as a function of time 106

2.16. Sum of two random processes   106

2.17. Relationship between the PSD of the excitation and the response of a linear system 108

2.18. Relationship between the PSD of the excitation and the cross-power spectral density of the response of a linear system    111

2.19. Coherence function 112

2.20. Transfer function calculation from random vibration measurements 114

2.20.1. Theoretical relations 114

2.20.2. Presence of noise on the input 116

2.20.3. Presence of noise on the response    118

2.20.4. Presence of noise on the input and response 120

2.20.5. Choice of transfer function 121

Chapter 3. Rms Value of Random Vibration    127

3.1. Rms value of a signal as a function of its PSD  127

3.2. Relationships between the PSD of acceleration, velocity and displacement 131

3.3. Graphical representation of the PSD     133

3.4. Practical calculation of acceleration, velocity and displacement rms values 135

3.4.1. General expressions 135

3.4.2. Constant PSD in frequency interval    135

3.4.3. PSD comprising several horizontal straight line segments   137

3.4.4. PSD defined by a linear segment of arbitrary slope 137

3.4.5. PSD comprising several segments of arbitrary slopes   147

3.5. Rms value according to the frequency 147

3.6. Case of periodic signals 149

3.7. Case of a periodic signal superimposed onto random noise  151

Chapter 4. Practical Calculation of the Power Spectral Density  153

4.1. Sampling of signal 153

4.2. PSD calculation methods    158

4.2.1. Use of the autocorrelation function    158

4.2.2. Calculation of the PSD from the rms value of a filtered signal 158

4.2.3. Calculation of PSD starting from a Fourier transform     159

4.3. PSD calculation steps 160

4.3.1. Maximum frequency 160

4.3.2. Extraction of sample of duration T 160

4.3.3. Averaging 167

4.3.4. Addition of zeros 170

4.4. FFT  175

4.5. Particular case of a periodic excitation 177

4.6. Statistical error 178

4.6.1. Origin 178

4.6.2. Definition 180

4.7. Statistical error calculation 180

4.7.1. Distribution of the measured PSD 180

4.7.2. Variance of the measured PSD  183

4.7.3. Statistical error 183

4.7.4. Relationship between number of degrees of freedom, duration and bandwidth of analysis 184

4.7.5. Confidence interval 190

4.7.6. Expression for statistical error in decibels  202

4.7.7. Statistical error calculation from digitized signal 204

4.8. Influence of duration and frequency step on the PSD 212

4.8.1. Influence of duration 212

4.8.2. Influence of the frequency step 213

4.8.3. Influence of duration and of constant statistical error frequency step 214

4.9. Overlapping 216

4.9.1. Utility   216

4.9.2. Influence on the number of degrees of freedom 217

4.9.3. Influence on statistical error  218

4.9.4. Choice of overlapping rate   221

4.10. Information to provide with a PSD     222

4.11. Difference between rms values calculated from a signal according to time and from its PSD   222

4.12. Calculation of a PSD from a Fourier transform 223

4.13. Amplitude based on frequency: relationship with the PSD  227

4.14. Calculation of the PSD for given statistical error 228

4.14.1. Case study: digitization of a signal is to be carried out  228

4.14.2. Case study: only one sample of an already digitized signal is available 230

4.15. Choice of filter bandwidth  231

4.15.1. Rules  231

4.15.2. Bias error 233

4.15.3. Maximum statistical error   238

4.15.4. Optimum bandwidth 240

4.16. Probability that the measured PSD lies between ± one standard deviation 243

4.17. Statistical error: other quantities 245

4.18. Peak hold spectrum 250

4.19. Generation of random signal of given PSD 252

4.19.1. Random phase sinusoid sum method 252

4.19.2. Inverse Fourier transform method    255

4.20. Using a window during the creation of a random signal from a PSD   256

Chapter 5. Statistical Properties of Random Vibration in the Time Domain 259

5.1. Distribution of instantaneous values     259

5.2. Properties of derivative process   260

5.3. Number of threshold crossings per unit time 264

5.4. Average frequency 269

5.5. Threshold level crossing curves   272

5.6. Moments  279

5.7. Average frequency of PSD defined by straight line segments    282

5.7.1. Linear-linear scales 282

5.7.2. Linear-logarithmic scales 284

5.7.3. Logarithmic-linear scales 285

5.7.4. Logarithmic-logarithmic scales 286

5.8. Fourth moment of PSD defined by straight line segments     288

5.8.1. Linear-linear scales 288

5.8.2. Linear-logarithmic scales 289

5.8.3. Logarithmic-linear scales 290

5.8.4. Logarithmic-logarithmic scales 291

5.9. Generalization: moment of order n  292

5.9.1. Linear-linear scales 292

5.9.2. Linear-logarithmic scales 292

5.9.3. Logarithmic-linear scales 292

5.9.4. Logarithmic-logarithmic scales 293

Chapter 6. Probability Distribution of Maxima of Random Vibration  295

6.1. Probability density of maxima 295

6.2. Moments of the maxima probability distribution 303

6.3. Expected number of maxima per unit time   304

6.4. Average time interval between two successive maxima     307

6.5. Average correlation between two successive maxima   308

6.6. Properties of the irregularity factor  309

6.6.1. Variation interval 309

6.6.2. Calculation of irregularity factor for band-limited white noise  313

6.6.3. Calculation of irregularity factor for noise of form G = Const f b 316

6.6.4. Case study: variations of irregularity factor for two narrowband signals 320

6.7. Error related to the use of Rayleigh’s law instead of a complete probability density function 321

6.8. Peak distribution function 323

6.8.1. General case. 323

6.8.2. Particular case of narrowband Gaussian process 325

6.9. Mean number of maxima greater than the given threshold (by unit time) 328

6.10. Mean number of maxima above given threshold between two times  331

6.11. Mean time interval between two successive maxima 331

6.12. Mean number of maxima above given level reached by signal excursion above this threshold 332

6.13. Time during which the signal is above a given value 335

6.14. Probability that a maximum is positive or negative 337

6.15. Probability density of the positive maxima 337

6.16. Probability that the positive maxima is lower than a given threshold 338

6.17. Average number of positive maxima per unit of time 338

6.18. Average amplitude jump between two successive extrema  339

6.19. Average number of inflection points per unit of time 341

Chapter 7. Statistics of Extreme Values  343

7.1. Probability density of maxima greater than a given value   343

7.2. Return period 344

7.3. Peak ƒÜp expected among p N peaks  344

7.4. Logarithmic rise 345

7.5. Average maximum of p N peaks  346

7.6. Variance of maximum 346

7.7. Mode (most probable maximum value)    346

7.8. Maximum value exceeded with risk α 346

7.9. Application to the case of a centered narrowband normal process   346

7.9.1. Distribution function of largest peaks over duration T     346

7.9.2. Probability that one peak at least exceeds a given threshold   349

7.9.3. Probability density of the largest maxima over duration T   350

7.9.4. Average of highest peaks 353

7.9.5. Mean value probability    355

7.9.6. Standard deviation of highest peaks    356

7.9.7. Variation coefficient 357

7.9.8. Most probable value 358

7.9.9. Median  358

7.9.10. Value of density at mode 360

7.9.11. Value of distribution function at mode 361

7.9.12. Expected maximum    361

7.9.13. Maximum exceeded with given risk α  361

7.10. Wideband centered normal process     363

7.10.1. Average of largest peaks 363

7.10.2. Variance of the largest peaks  366

7.10.3. Variation coefficient 367

7.11. Asymptotic laws 368

7.11.1. Gumbel asymptote 368

7.11.2. Case study: Rayleigh peak distribution 369

7.11.3. Expressions for large values of p N 370

7.12. Choice of type of analysis    371

7.13. Study of the envelope of a narrowband process 374

7.13.1. Probability density of the maxima of the envelope    374

7.13.2. Distribution of maxima of envelope 379

7.13.3. Average frequency of envelope of narrowband noise     381

Chapter 8. Response of a One-Degree-of-Freedom Linear System to Random Vibration 385

8.1. Average value of the response of a linear system 385

8.2. Response of perfect bandpass filter to random vibration   386

8.3. The PSD of the response of a one-dof linear system 388

8.4. Rms value of response to white noise 389

8.5. Rms value of response of a linear one-degree of freedom system subjected to bands of random noise   395

8.5.1. Case where the excitation is a PSD defined by a straight line segment in logarithmic scales    395

8.5.2. Case where the vibration has a PSD defined by a straight line segment of arbitrary slope in linear scales    401

8.5.3. Case where the vibration has a constant PSD between two frequencies  404

8.5.4. Excitation defined by an absolute displacement 409

8.5.5. Case where the excitation is defined by PSD comprising n straight line segments 411

8.6. Rms value of the absolute acceleration of the response 414

8.7. Transitory response of a dynamic system under stationary random excitation 415

8.8. Transitory response of a dynamic system under amplitude modulated white noise excitation   423

Chapter 9. Characteristics of the Response of a One-Degree-of-Freedom Linear System to Random Vibration   427

9.1. Moments of response of a one-degree-of-freedom linear system: irregularity factor of response 427

9.1.1. Moments 427

9.1.2. Irregularity factor of response to noise of a constant PSD    431

9.1.3. Characteristics of irregularity factor of response 433

9.1.4. Case of a band-limited noise 444

9.2. Autocorrelation function of response displacement 445

9.3. Average numbers of maxima and minima per second 446

9.4. Equivalence between the transfer functions of a bandpass filter and a one-degree-of-freedom linear system    449

9.4.1. Equivalence suggested by D.M. Aspinwall 449

9.4.2. Equivalence suggested by K.W. Smith   451

9.4.3. Rms value of signal filtered by the equivalent bandpass filter  453

Chapter 10. First Passage at a Given Level of Response of a One-Degree-of-Freedom Linear System to a Random Vibration 455

10.1. Assumptions 455

10.2. Definitions  459

10.3. Statistically independent threshold crossings  460

10.4. Statistically independent response maxima 468

10.5. Independent threshold crossings by the envelope of maxima   472

10.6. Independent envelope peaks 476

10.6.1. S.H. Crandall method 476

10.6.2. D.M. Aspinwall method 479

10.7. Markov process assumption 486

10.7.1. W.D. Mark assumption   486

10.7.2. J.N. Yang and M. Shinozuka approximation 493

10.8. E.H. Vanmarcke model 494

10.8.1. Assumption of a two state Markov process 494

10.8.2. Approximation based on the mean clump size 500

Appendix    511

Bibliography  571

Index 591

Summary of Other Volumes in the Series  597