Mechanical Vibration and Shock Analysis, Random Vibration

Christian Lalanne is a Consultant Engineer who previously worked as an expert at the French Atomic Energy Authority and who has specialized in the study of vibration and shock for more than 40 years. He has been associated with the new methods of drafting testing specifications and associated informatic tools.

• Foreword to Series xiiiIntroduction xviiList of Symbols xixChapter 1 Statistical Properties of a Random Process 11.1 Definitions 11.1.1 Random variable 11.1.2 Random process 21.2 Random vibration in real environments 21.3 Random vibration in laboratory tests 31.4 Methods of random vibration analysis 31.5 Distribution of instantaneous values 51.5.1 Probability density 51.5.2 Distribution function 61.6 Gaussian random process 71.7 Rayleigh distribution 121.8 Ensemble averages: through the process 121.8.1 n order average 121.8.2 Centered moments 141.8.3 Variance 141.8.4 Standard deviation 151.8.5 Autocorrelation function 161.8.6 Cross-correlation function 161.8.7 Autocovariance 171.8.8 Covariance 171.8.9 Stationarity 171.9 Temporal averages: along the process 231.9.1 Mean 231.9.2 Quadratic mean – rms value 251.9.3 Moments of order n 271.9.4 Variance – standard deviation 281.9.5 Skewness 291.9.6 Kurtosis 301.9.7 Crest Factor 331.9.8 Temporal autocorrelation function 331.9.9 Properties of the autocorrelation function 391.9.10 Correlation duration 411.9.11 Cross-correlation 471.9.12 Cross-correlation coefficient 501.9.13 Ergodicity 501.10 Significance of the statistical analysis (ensemble or temporal) 521.11 Stationary and pseudo-stationary signals 521.12 Summary chart of main definitions 531.13 Sliding mean 541.14 Test of stationarity 581.14.1 The reverse arrangements test (RAT) 581.14.2 The runs test 611.15 Identification of shocks and/or signal problems 651.16 Breakdown of vibratory signal into “events”: choice of signal samples 681.17 Interpretation and taking into account of environment variation 75Chapter 2 Random Vibration Properties in the Frequency Domain 792.1 Fourier transform 792.2 Power spectral density 812.2.1 Need 812.2.2 Definition 822.3 Amplitude Spectral Density 892.4 Cross-power spectral density 892.5 Power spectral density of a random process 902.6 Cross-power spectral density of two processes 912.7 Relationship between the PSD and correlation function of a process 932.8 Quadspectrum – cospectrum 932.9 Definitions 942.9.1 Broadband process 942.9.2 White noise 952.9.3 Band-limited white noise 952.9.4 Narrow band process 962.9.5 Colors of noise 972.10 Autocorrelation function of white noise 982.11 Autocorrelation function of band-limited white noise 992.12 Peak factor 1012.13 Effects of truncation of peaks of acceleration signal on the PSD 1012.14 Standardized PSD/density of probability analogy 1052.15 Spectral density as a function of time1062.16 Sum of two random processes 1062.17 Relationship between the PSD of the excitation and the response of a linear system 1082.18 Relationship between the PSD of the excitation and the cross-power spectral density of the response of a linear system 1112.19 Coherence function 1122.20 Transfer function calculation from random vibration measurements 1142.20.1 Theoretical relations 1142.20.2 Presence of noise on the input 1162.20.3 Presence of noise on the response 1182.20.4 Presence of noise on the input and response 1202.20.5 Choice of transfer function 121Chapter 3 Rms Value of Random Vibration 1273.1 Rms value of a signal as a function of its PSD 1273.2 Relationships between the PSD of acceleration, velocity and displacement 1313.3 Graphical representation of the PSD 1333.4 Practical calculation of acceleration, velocity and displacement rms values 1353.4.1 General expressions 1353.4.2 Constant PSD in frequency interval 1353.4.3 PSD comprising several horizontal straight line segments 1373.4.4 PSD defined by a linear segment of arbitrary slope 1373.4.5 PSD comprising several segments of arbitrary slopes 1473.5 Rms value according to the frequency 1473.6 Case of periodic signals 1493.7 Case of a periodic signal superimposed onto random noise 151Chapter 4 Practical Calculation of the Power Spectral Density 1534.1 Sampling of signal 1534.2 PSD calculation methods 1584.2.1 Use of the autocorrelation function 1584.2.2 Calculation of the PSD from the rms value of a filtered signal 1584.2.3 Calculation of PSD starting from a Fourier transform 1594.3 PSD calculation steps 1604.3.1 Maximum frequency 1604.3.2 Extraction of sample of duration T1604.3.3 Averaging 1674.3.4 Addition of zeros 1704.4 FFT 1754.5 Particular case of a periodic excitation 1774.6 Statistical error 1784.6.1 Origin 1784.6.2 Definition 1804.7 Statistical error calculation 1804.7.1 Distribution of the measured PSD 1804.7.2 Variance of the measured PSD 1834.7.3 Statistical error 1834.7.4 Relationship between number of degrees of freedom, duration and bandwidth of analysis 1844.7.5 Confidence interval 1904.7.6 Expression for statistical error in decibels 2024.7.7 Statistical error calculation from digitized signal 2044.8 Influence of duration and frequency step on the PSD 2124.8.1 Influence of duration 2124.8.2 Influence of the frequency step 2134.8.3 Influence of duration and of constant statistical error frequency step 2144.9 Overlapping 2164.9.1 Utility 2164.9.2 Influence on the number of degrees of freedom 2174.9.3 Influence on statistical error 2184.9.4 Choice of overlapping rate 2214.10 Information to provide with a PSD 2224.11 Difference between rms values calculated from a signal according to time and from its PSD 2224.12 Calculation of a PSD from a Fourier transform 2234.13 Amplitude based on frequency: relationship with the PSD 2274.14 Calculation of the PSD for given statistical error 2284.14.1 Case study: digitization of a signal is to be carried out 2284.14.2 Case study: only one sample of an already digitized signal is available 2304.15 Choice of filter bandwidth 2314.15.1 Rules 2314.15.2 Bias error 2334.15.3 Maximum statistical error 2384.15.4 Optimum bandwidth 2404.16 Probability that the measured PSD lies between ± one standard deviation 2434.17 Statistical error: other quantities 2454.18 Peak hold spectrum 2504.19 Generation of random signal of given PSD 2524.19.1 Random phase sinusoid sum method 2524.19.2 Inverse Fourier transform method 2554.20 Using a window during the creation of a random signal from a PSD 256Chapter 5 Statistical Properties of Random Vibration in the Time Domain 2595.1 Distribution of instantaneous values 2595.2 Properties of derivative process 2605.3 Number of threshold crossings per unit time 2645.4 Average frequency 2695.5 Threshold level crossing curves 2725.6 Moments 2795.7 Average frequency of PSD defined by straight line segments 2825.7.1 Linear-linear scales 2825.7.2 Linear-logarithmic scales 2845.7.3 Logarithmic-linear scales 2855.7.4 Logarithmic-logarithmic scales 2865.8 Fourth moment of PSD defined by straight line segments 2885.8.1 Linear-linear scales 2885.8.2 Linear-logarithmic scales 2895.8.3 Logarithmic-linear scales 2905.8.4 Logarithmic-logarithmic scales 2915.9 Generalization: moment of order n 2925.9.1 Linear-linear scales 2925.9.2 Linear-logarithmic scales 2925.9.3 Logarithmic-linear scales 2925.9.4 Logarithmic-logarithmic scales 293Chapter 6 Probability Distribution of Maxima of Random Vibration 2956.1 Probability density of maxima 2956.2 Moments of the maxima probability distribution 3036.3 Expected number of maxima per unit time 3046.4 Average time interval between two successive maxima 3076.5 Average correlation between two successive maxima 3086.6 Properties of the irregularity factor 3096.6.1 Variation interval 3096.6.2 Calculation of irregularity factor for band-limited white noise 3136.6.3 Calculation of irregularity factor for noise of form G = Const.f b 3166.6.4 Case study: variations of irregularity factor for two narrowband signals 3206.7 Error related to the use of Rayleigh’s law instead of a complete probability density function 3216.8 Peak distribution function 3236.8.1 General case 3236.8.2 Particular case of narrowband Gaussian process 3256.9 Mean number of maxima greater than the given threshold (by unit time) 3286.10 Mean number of maxima above given threshold between two times 3316.11 Mean time interval between two successive maxima 3316.12 Mean number of maxima above given level reached by signal excursion above this threshold 3326.13 Time during which the signal is above a given value 3356.14 Probability that a maximum is positive or negative 3376.15 Probability density of the positive maxima 3376.16 Probability that the positive maxima is lower than a given threshold 3386.17 Average number of positive maxima per unit of time 3386.18 Average amplitude jump between two successive extrema 3396.19 Average number of inflection points per unit of time 341Chapter 7 Statistics of Extreme Values 3437.1 Probability density of maxima greater than a given value 3437.2 Return period 3447.3 Peak lp expected among Np peaks 3447.4 Logarithmic rise 3457.5 Average maximum of Np peaks 3467.6 Variance of maximum 3467.7 Mode (most probable maximum value) 3467.8 Maximum value exceeded with risk α 3467.9 Application to the case of a centered narrowband normal process 3467.9.1 Distribution function of largest peaks over duration T 3467.9.2 Probability that one peak at least exceeds a given threshold 3497.9.3 Probability density of the largest maxima over duration T 3507.9.4 Average of highest peaks 3537.9.5 Mean value probability 3557.9.6 Standard deviation of highest peaks 3567.9.7 Variation coefficient 3577.9.8 Most probable value 3587.9.9 Median 3587.9.10 Value of density at mode 3607.9.11 Value of distribution function at mode 3617.9.12 Expected maximum 3617.9.13 Maximum exceeded with given risk α 3617.10 Wideband centered normal process 3637.10.1 Average of largest peaks 3637.10.2 Variance of the largest peaks 3667.10.3 Variation coefficient 3677.11 Asymptotic laws 3687.11.1 Gumbel asymptote 3687.11.2 Case study: Rayleigh peak distribution 3697.11.3 Expressions for large values of Np 3707.12 Choice of type of analysis 3717.13 Study of the envelope of a narrowband process 3747.13.1 Probability density of the maxima of the envelope 3747.13.2 Distribution of maxima of envelope 3797.13.3 Average frequency of envelope of narrowband noise 381Chapter 8 Response of a One-Degree-of-Freedom Linear System to Random Vibration 3858.1 Average value of the response of a linear system 3858.2 Response of perfect bandpass filter to random vibration 3868.3 The PSD of the response of a one-dof linear system 3888.4 Rms value of response to white noise 3898.5 Rms value of response of a linear one-degree of freedom system subjected to bands of random noise 3958.5.1 Case where the excitation is a PSD defined by a straight line segment in logarithmic scales 3958.5.2 Case where the vibration has a PSD defined by a straight line segment of arbitrary slope in linear scales 4018.5.3 Case where the vibration has a constant PSD between two frequencies 4048.5.4 Excitation defined by an absolute displacement 4098.5.5 Case where the excitation is defined by PSD comprising n straight line segments 4118.6 Rms value of the absolute acceleration of the response 4148.7 Transitory response of a dynamic system under stationary random excitation 4158.8 Transitory response of a dynamic system under amplitude modulated white noise excitation 423Chapter 9 Characteristics of the Response of a One-Degree-of-Freedom Linear System to Random Vibration 4279.1 Moments of response of a one-degree-of-freedom linear system: irregularity factor of response 4279.1.1 Moments 4279.1.2 Irregularity factor of response to noise of a constant PSD 4319.1.3 Characteristics of irregularity factor of response 4339.1.4 Case of a band-limited noise 4449.2 Autocorrelation function of response displacement 4459.3 Average numbers of maxima and minima per second 4469.4 Equivalence between the transfer functions of a bandpass filter and a one-degree-of-freedom linear system 4499.4.1 Equivalence suggested by D.M Aspinwall 4499.4.2 Equivalence suggested by K.W Smith 4519.4.3 Rms value of signal filtered by the equivalent bandpass filter 453Chapter 10 First Passage at a Given Level of Response of a One-Degree-of-Freedom Linear System to a Random Vibration 45510.1 Assumptions 45510.2 Definitions 45910.3 Statistically independent threshold crossings 46010.4 Statistically independent response maxima 46810.5 Independent threshold crossings by the envelope of maxima 47210.6 Independent envelope peaks 47610.6.1 S.H Crandall method 47610.6.2 D.M Aspinwall method 47910.7 Markov process assumption 48610.7.1 W.D Mark assumption 48610.7.2 J.N Yang and M Shinozuka approximation 49310.8 E.H Vanmarcke model 49410.8.1 Assumption of a two state Markov process 49410.8.2 Approximation based on the mean clump size 500Appendix 511Bibliography 571Index 591Summary of Other Volumes in the Series 597