Cover -- Title Page -- Copyright -- Contents -- Foreword to Series -- Introduction -- List of Symbols -- Chapter 1. Statistical Properties of a Random Process -- 1.1. Definitions -- 1.1.1. Random variable -- 1.1.2. Random process -- 1.2. Random vibration in real environments -- 1.3. Random vibration in laboratory tests -- 1.4. Methods of random vibration analysis -- 1.5. Distribution of instantaneous values -- 1.5.1. Probability density -- 1.5.2. Distribution function -- 1.6. Gaussian random process -- 1.7. Rayleigh distribution -- 1.8. Ensemble averages: through the process -- 1.8.1. n order average -- 1.8.2. Centered moments -- 1.8.3. Variance -- 1.8.4. Standard deviation -- 1.8.5. Autocorrelation function -- 1.8.6. Cross-correlation function -- 1.8.7. Autocovariance -- 1.8.8. Covariance -- 1.8.9. Stationarity -- 1.9. Temporal averages: along the process -- 1.9.1. Mean -- 1.9.2. Quadratic mean - rms value -- 1.9.3. Moments of order n -- 1.9.4. Variance - standard deviation -- 1.9.5. Skewness -- 1.9.6. Kurtosis -- 1.9.7. Crest Factor -- 1.9.8. Temporal autocorrelation function -- 1.9.9. Properties of the autocorrelation function -- 1.9.10. Correlation duration -- 1.9.11. Cross-correlation -- 1.9.12. Cross-correlation coefficient -- 1.9.13. Ergodicity -- 1.10. Significance of the statistical analysis (ensemble or temporal) -- 1.11. Stationary and pseudo-stationary signals -- 1.13. Sliding mean -- 1.14. Test of stationarity -- 1.14.1. The reverse arrangements test (RAT) -- 1.14.2. The runs test -- 1.15 Identification of shocks and/or signal problems -- 1.16. Breakdown of vibratory signal into "events": choice of signal samples -- 1.17. Interpretation and taking into account of environment variation -- Chapter 2. Random Vibration Properties in the Frequency Domain -- 2.1. Fourier transform -- 2.2. Power spectral density -- 2.2.1. Need.

2.2.2. Definition -- 2.3. Amplitude Spectral Density -- 2.4. Cross-power spectral density -- 2.5. Power spectral density of a random process -- 2.6. Cross-power spectral density of two processes -- 2.7. Relationship between the PSD and correlation function of a process -- 2.8. Quadspectrum - cospectrum -- 2.9. Definitions -- 2.9.1. Broadband process -- 2.9.2. White noise -- 2.9.3. Band-limited white noise -- 2.9.4. Narrow band process -- 2.9.5. Colors of noise -- 2.10. Autocorrelation function of white noise -- 2.11. Autocorrelation function of band-limited white noise -- 2.12. Peak factor -- 2.13. Effects of truncation of peaks of acceleration signal on the PSD -- 2.14. Standardized PSD/density of probability analogy -- 2.15. Spectral density as a function of time -- 2.16. Sum of two random processes -- 2.17. Relationship between the PSD of the excitation and the response of a linear system -- 2.18. Relationship between the PSD of the excitation and the cross-power spectral density of the response of a linear system -- 2.19. Coherence function -- 2.20. Transfer function calculation from random vibration measurements -- 2.20.1. Theoretical relations -- 2.20.2. Presence of noise on the input -- 2.20.3. Presence of noise on the response -- 2.20.4. Presence of noise on the input and response -- 2.20.5. Choice of transfer function -- Chapter 3. Rms Value of Random Vibration -- 3.1. Rms value of a signal as a function of its PSD -- 3.2. Relationships between the PSD of acceleration, velocity and displacement -- 3.3. Graphical representation of the PSD -- 3.4. Practical calculation of acceleration, velocity and displacement rms values -- 3.4.1. General expressions -- 3.4.2. Constant PSD in frequency interval -- 3.4.3. PSD comprising several horizontal straight line segments -- 3.4.4. PSD defined by a linear segment of arbitrary slope.

3.4.5. PSD comprising several segments of arbitrary slopes -- 3.5. Rms value according to the frequency -- 3.6. Case of periodic signals -- 3.7. Case of a periodic signal superimposed onto random noise -- Chapter 4. Practical Calculation of the Power Spectral Density -- 4.1. Sampling of signal -- 4.2. PSD calculation methods -- 4.2.1. Use of the autocorrelation function -- 4.2.2. Calculation of the PSD from the rms value of a filtered signal -- 4.2.3. Calculation of PSD starting from a Fourier transform -- 4.3. PSD calculation steps -- 4.3.1. Maximum frequency -- 4.3.2. Extraction of sample of duration T -- 4.3.3. Averaging -- 4.3.4. Addition of zeros -- 4.4. FFT -- 4.5. Particular case of a periodic excitation -- 4.6. Statistical error -- 4.6.1. Origin -- 4.6.2. Definition -- 4.7. Statistical error calculation -- 4.7.1. Distribution of the measured PSD -- 4.7.2. Variance of the measured PSD -- 4.7.3. Statistical error -- 4.7.4. Relationship between number of degrees of freedom, duration and bandwidth of analysis -- 4.7.5. Confidence interval -- 4.7.6. Expression for statistical error in decibels -- 4.7.7. Statistical error calculation from digitized signal -- 4.8. Influence of duration and frequency step on the PSD -- 4.8.1. Influence of duration -- 4.8.2. Influence of the frequency step -- 4.8.3. Influence of duration and of constant statistical error frequency step -- 4.9. Overlapping -- 4.9.1. Utility -- 4.9.2. Influence on the number of degrees of freedom -- 4.9.3. Influence on statistical error -- 4.9.4. Choice of overlapping rate -- 4.10. Information to provide with a PSD -- 4.11. Difference between rms values calculated from a signal according to time and from its PSD -- 4.12. Calculation of a PSD from a Fourier transform -- 4.13. Amplitude based on frequency: relationship with the PSD.

4.14. Calculation of the PSD for given statistical error -- 4.14.1. Case study: digitization of a signal is to be carried out -- 4.14.2. Case study: only one sample of an already digitized signal is available -- 4.15. Choice of filter bandwidth -- 4.15.1. Rules -- 4.15.2. Bias error -- 4.15.3. Maximum statistical error -- 4.15.4. Optimum bandwidth -- 4.16. Probability that the measured PSD lies between ± one standard deviation -- 4.17. Statistical error: other quantities -- 4.18. Peak hold spectrum -- 4.19. Generation of random signal of given PSD -- 4.19.1. Random phase sinusoid sum method -- 4.19.2. Inverse Fourier transform method -- 4.20. Using a window during the creation of a random signal from a PSD -- Chapter 5. Statistical Properties of Random Vibration in the Time Domain -- 5.1. Distribution of instantaneous values -- 5.2. Properties of derivative process -- 5.3. Number of threshold crossings per unit time -- 5.4. Average frequency -- 5.5. Threshold level crossing curves -- 5.6. Moments -- 5.7. Average frequency of PSD defined by straight line segments -- 5.7.1. Linear-linear scales -- 5.7.2. Linear-logarithmic scales -- 5.7.3. Logarithmic-linear scales -- 5.7.4. Logarithmic-logarithmic scales -- 5.8. Fourth moment of PSD defined by straight line segments -- 5.8.1. Linear-linear scales -- 5.8.2. Linear-logarithmic scales -- 5.8.3. Logarithmic-linear scales -- 5.8.4. Logarithmic-logarithmic scales -- 5.9. Generalization: moment of order n -- 5.9.1. Linear-linear scales -- 5.9.2. Linear-logarithmic scales -- 5.9.3. Logarithmic-linear scales -- 5.9.4. Logarithmic-logarithmic scales -- Chapter 6. Probability Distribution of Maxima of Random Vibration -- 6.1. Probability density of maxima -- 6.2. Moments of the maxima probability distribution -- 6.3. Expected number of maxima per unit time.

6.4. Average time interval between two successive maxima -- 6.5. Average correlation between two successive maxima -- 6.6. Properties of the irregularity factor -- 6.6.1. Variation interval -- 6.6.2. Calculation of irregularity factor for band-limited white noise -- 6.6.3. Calculation of irregularity factor for noise of form G = Const. f b -- 6.6.4. Case study: variations of irregularity factor for two narrowband signals -- 6.7. Error related to the use of Rayleigh's law instead of a complete probability density function -- 6.8. Peak distribution function -- 6.8.1. General case -- 6.8.2. Particular case of narrowband Gaussian process -- 6.9. Mean number of maxima greater than the given threshold (by unit time) -- 6.10. Mean number of maxima above given threshold between two times -- 6.11. Mean time interval between two successive maxima -- 6.12. Mean number of maxima above given level reached by signal excursion above this threshold -- 6.13. Time during which the signal is above a given value -- 6.14. Probability that a maximum is positive or negative -- 6.15. Probability density of the positive maxima -- 6.16. Probability that the positive maxima is lower than a given threshold -- 6.17. Average number of positive maxima per unit of time -- 6.18. Average amplitude jump between two successive extrema -- 6.19. Average number of inflection points per unit of time -- Chapter 7. Statistics of Extreme Values -- 7.1. Probability density of maxima greater than a given value -- 7.2. Return period -- 7.3. Peak lp expected among Np peaks -- 7.4. Logarithmic rise -- 7.5. Average maximum of Np peaks -- 7.6. Variance of maximum -- 7.7. Mode (most probable maximum value) -- 7.8. Maximum value exceeded with risk a -- 7.9. Application to the case of a centered narrowband normal process -- 7.9.1. Distribution function of largest peaks over duration T.

7.9.2. Probability that one peak at least exceeds a given thr.