Lalanne C. Mechanical Vibration and Shock Analysis. Random Vibration (Volume 3)

2nd Edition. — Wiley–ISTE, 2009. — 610 p. — ISBN: 978-1848211247.The vast majority of vibrations encountered in a real–world environment are random in nature. Such vibrations are intrinsically complicated, but this volume describes a process enabling the simplification of the analysis required, and the analysis of the signal in the frequency domain. Power spectrum density is also defined, with the requisite precautions to be taken in its calculation described together with the processes (windowing, overlapping) necessary for improved results. A further complementary method, the analysis of statistical properties of the time signal, is described. This enables the distribution law of the MAXIMA of a random Gaussian signal to be determined and simplifies calculation of fatigue damage to be made by the avoidance of the direct counting of peaks. The Mechanical Vibration and Shock Analysis five–volume series has been written with both the professional engineer and the academic in mind. Christian Lalanne explores every aspect of vibration and shock, two fundamental and extremely significant areas of mechanical engineering, from both a theoretical and practical point of view. The five volumes cover all the necessary issues in this area of mechanical engineering. The theoretical analyses are placed in the context of both the real world and the laboratory, which is essential for the development of specifications.ForewordList of SymbolsStatistical Properties of a Random Process
Definitions
Random variable
Random process
Random vibration in real environments
Random vibration in laboratory tests
Methods of random vibration analysis
Distribution of instantaneous values
Probability density
Distribution function
Gaussian random process
Rayleigh distribution
Ensemble averages: through the process
In order average
Centered moments
Variance
Standard deviation
Autocorrelation function
Cross-correlation function
Autocovariance
Covariance
Stationarity
Temporal averages: along the process
Mean
Quadratic mean – rms value
Moments of order n
Variance – standard deviation
Skewness
Kurtosis
Temporal autocorrelation function
Properties of the autocorrelation function
Correlation duration
Cross-correlation
Cross-correlation coefficient
Ergodicity
Significance of the statistical analysis (ensemble or temporal)
Stationary and pseudo-stationary signals
Summary chart of main definitions
Sliding mean
Identification of shocks and/or signal problems
Breakdown of vibratory signal into events: choice of signal samples
Interpretation and taking into account of environment variationRandom Vibration Properties in the Frequency Domain
Fourier transform
Power spectral density
Need
Definition
Cross-power spectral density
Power spectral density of a random process
Cross-power spectral density of two processes
Relationship between the PSD and correlation function of a process
Quadspectrum – cospectrum
Definitions
Broad band process
White noise
Band-limited white noise
Narrow band process
Pink noise
Autocorrelation function of white noise
Autocorrelation function of band-limited white noise
Peak factor
Effects of truncation of peaks of acceleration signal on the PSD
Standardized PSD/density of probability analogy
Spectral density as a function of time
Relationship between the PSD of the excitation and the response of a linear system
Relationship between the PSD of the excitation and the cross-power spectral density of the response of a linear system
Coherence function
Transfer function calculation from random vibration measurements
Theoretical relations
Presence of noise on the input
Presence of noise on the response
Presence of noise on the input and response
Choice of transfer functionRms Value of Random Vibration
Rms value of a signal as a function of its PSD
Relationships between the PSD of acceleration, velocity and displacement
Graphical representation of the PSD
Practical calculation of acceleration, velocity and displacement rms values
General expressions
Constant PSD in frequency interval
PSD comprising several horizontal straight line segments
PSD defined by a linear segment of arbitrary slope
PSD comprising several segments of arbitrary slopes
Rms value according to the frequency
Case of periodic signals
Case of a periodic signal superimposed onto random noisePractical Calculation of the Power Spectral Density
Sampling of signal
PSD calculation methods
Use of the autocorrelation function
Calculation of the PSD from the rms value of a filtered signal
Calculation of the PSD starting from a Fourier transform
PSD calculation steps
Maximum frequency
Extraction of sample of duration T
Averaging
Addition of zeros
FFT
Particular case of a periodic excitation
Statistical error
Origin
Definition
Statistical error calculation
Distribution of the measured PSD
Variance of the measured PSD
Statistical error
Relationship between number of degrees of freedom, duration and bandwidth of analysis
Confidence interval
Expression for statistical error in decibels
Statistical error calculation from digitized signal
Influence of duration and frequency step on the PSD
Influence of duration
Influence of the frequency step
Influence of duration and of constant statistical error frequency step
Overlapping
Utility
Influence on the number of dofs
Influence on statistical error
Choice of overlapping rate
Information to provide with a PSD
Difference between rms values calculated from a signal according to time and from its PSD
Calculation of a PSD from a Fourier transform
Amplitude based on frequency: relationship with the PSD
Calculation of the PSD for given statistical error
Case study: digitization of a signal is to be carried out
Case study: only one sample of an already digitized signal is available
Choice of filter bandwidth
Rules
Bias error
Maximum statistical error
Optimum bandwidth
Probability that the measured PSD lies between r one standard deviation
Statistical error: other quantities
Peak hold spectrum
Generation of random signal of given PSD
Random phase sinusoid sum method
Inverse Fourier transform method
Using a window during the creation of a random signal from a PSDStatistical Properties of Random Vibration in the Time Domain
Distribution of instantaneous values
Properties of derivative process
Number of threshold crossings per unit time
Average frequency
Threshold level crossing curves
Moments
Average frequency of PSD defined by straight line segments
Linear-linear scales
Linear-logarithmic scales
Logarithmic-linear scales
Logarithmic-logarithmic scales
Fourth moment of PSD defined by straight line segments
Linear-linear scales
Linear-logarithmic scales
Logarithmic-linear scales
Logarithmic-logarithmic scales
Generalization: moment of order n
Linear-linear scales
Linear-logarithmic scales
Logarithmic-linear scales
Logarithmic-logarithmic scalesProbability Distribution of MAXIMA of Random Vibration
Probability density of MAXIMA
Expected number of MAXIMA per unit time
Average time interval between two successive MAXIMA
Average correlation between two successive MAXIMA
Properties of the irregularity factor
Variation interval
Calculation of irregularity factor for band-limited white noise
Calculation of irregularity factor for noise of form b GConst.f
Case study: variations of irregularity factor for two narrow band signals
Error related to the use of Rayleigh’s law instead of a complete probability density function
Peak distribution function
General case
Particularcase of a narrow band Gaussian process
Mean number of MAXIMA greater than the given threshold (by unit time)
Mean number of MAXIMA above given threshold between two times
Mean time interval between two successive MAXIMA
Mean number of MAXIMA above given level reached by signal excursion above this threshold
Time during which the signal is above a given value
Probability that a maximum is positive or negative
Probability density of the positive MAXIMA
Probability that the positive MAXIMA is lower than a given threshold
Average number of positive MAXIMA per unit of time
Average amplitude jump between two successive extremaStatistics of Extreme Values
Probability density of MAXIMA greater than a given value
Return period
Peak
p A expected among p N peaks
Logarithmic rise
Average maximum of p N peaks
Variance of maximum
Mode (most probable maximum value)
Maximum value exceeded with risk D
Application to the case of a centered narrow band normal process
Distribution function of largest peaks over duration T
Probability that one peak at least exceeds a given threshold
Probability density of the largest MAXIMA over duration T
Average of highest peaks
Mean value probability
Standard deviation of highest peaks
Variation coefficient
Most probable value
Median
Value of density at mode
Expected maximum
Average maximum
Maximum exceeded with given risk D
Wide band centered normal process
Average of largest peaks
Variance of the largest peaks
Variation coefficient
Asymptotic laws
Gumbel asymptote
Case study: Rayleigh peak distribution
Expressions for large values of p N
Choice of type of analysis
Study of the envelope of a narrow band process
Probability density of the MAXIMA of the envelope
Distribution of MAXIMA of envelope
Average frequency of envelope of narrow band noiseResponse of a One-Degree-of-Freedom Linear System to Random Vibration
Average value of the response of a linear system
Response of perfect bandpass filter to random vibration
The PSD of the response of a one-dof linear system
Rms value of response to white noise
Rms value of response of a linear one-dof system subjected to bands of random noise
Case where the excitation is a PSD defined by a straight line segment in logarithmic scales
Case where the vibration has a PSD defined by a straight line segment of arbitrary slope in linear scales
Case where the vibration has a constant PSD between two frequencies
Excitation defined by an absolute displacement
Case where the excitation is defined by PSD comprising n straight line segments
Rms value of the absolute acceleration of the response
Transitory response of a dynamic system under stationary random excitation
Transitory response of a dynamic system under amplitude modulated white noise excitationCharacteristics of the Response of a One-Degree-of-Freedom Linear System to Random Vibration
Moments of response of a one-degree-of-freedom linear system: irregularity factor of response
Moments
Irregularity factor of response to noise of a constant PSD
Characteristics of irregularity factor of response
Case of a band-limited noise
Autocorrelation function of response displacement
Average numbers of MAXIMA and minima per second
Equivalence between the transfer functions of a bandpass filter and a one-dof linear system
Equivalence suggested by D.M. Aspinwall
Equivalence suggested by K.W. Smith
Rms value of signal filtered by the equivalent bandpass filterFirst Passage at a Given Level of Response of a One-Degree-of-Freedom Linear System to a Random Vibration
Assumptions
Definitions
Statistically independent threshold crossings
Statistically independent response MAXIMA
Independent threshold crossings by the envelope of MAXIMA
Independent envelope peaks
S.H. Crandall method
D.M. Aspinwall method
Markov process assumption
W.D. Mark assumption
J.N. Yang and M. Shinozuka approximation
E.H. Vanmarcke model
Assumption of a two state Markov process
Approximation based on the mean clump size
AppendicesSummary of other Volumes in the Series