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 function 6 1.6 Gaussian random process7 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 function16 1.8.6 Cross-correlation function 16 1.8.7 Autocovariance 17 1.8.
8 Covariance 17 1.8.9 Stationarity17 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 Kurtosis30 1.
9.7 Crest Factor 33 1.9.8 Temporal autocorrelation function33 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 signals52 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 - cospectrum93 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 time106 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 input116 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 T160 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 method252 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 scales286 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 scales291 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 scales293 Chapter 6 Probability Distribution of Maxima of Random Vibration 295 6.1 Probability density of maxima 295 6.2 Moments of the maxima probability distribution303 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.