Chapter 1. Statistical Properties of a Random Process
Chapter 2. Random Vibration Properties in the Frequency Domain
Chapter 3. Rms Value of Random Vibration
Chapter 4. Practical Calculation of the Power Spectral Density
Chapter 5. Statistical Properties of Random Vibration in the Time Domain
Chapter 6. Probability Distribution of Maxima of Random Vibration
Chapter 7. Statistics of Extreme Values
Chapter 8. Response of a One-Degree-of-Freedom Linear System to Random Vibration
Chapter 9. Characteristics of the Response of a One-Degree-of-Freedom Linear System to Random Vibration
Chapter 10. First Passage at a Given Level of Response of a One-Degree-of-Freedom Linear System to a Random Vibration
List of Symbols
The list below gives the most frequent definition of the main symbols used in this book. Some of the symbols can have another meaning which will be defined in the text to avoid any confusion.
a Threshold value of
ℓ(t) or maximum of
ℓ(t) A Maximum of A (t) A(t) Envelope of a signal b Exponent c Viscous damping constant e(t) Narrow band white noise E() Expectation of… E1( ) First definition of error function E2( ) Second definition of error function Erf Error function E( ) Expected function of … f Frequency of excitation fsamp. Sampling frequency fmax Maximum frequency f0 Natural frequency g Acceleration due to gravity G Particular value of power spectral density G( ) Power spectral density for 0 ≤ f ≤ ∞ Measured value of G( ) G
ℓu( ) Cross-power spectral density h Interval (f/f0) or f2/f1 h(t) Impulse response H( ) Transfer function i k Stiffness K Number of subsamples
ℓ Value of
ℓ(t) Mean value of
ℓ(t) Average maximum of Np peaks
ℓrms Rms value of
ℓ(t) Rms value of
ℓ(t) Generalized excitation (displacement) First derivative of
ℓ(t) Second derivative of
ℓ(t) L Given value of
ℓ(t) Lrms Rms value of filtered signal L(Ω) Fourier transform of
ℓ(t) Fourier transform of m Mean M Number of points of PSD Ma Average number of maxima which exceeds threshold per unit time Mn Moment of order n n Order of moment or number of degrees of freedom na Average number of crossings of threshold a per unit time Average number of crossings of threshold
a with positive slope per unit time n0 Average number of zero-crossings per unit time Average number of zero-crossings with positive slope per second (average frequency) Average number of maxima per unit time N Number of curves or Number of points of signal or
Numbers of dB Np Number of peaks Average number of crossings of threshold
a with positive slope for given length of time Average number of zero-crossings with positive slope for given length of time Average number of positive maxima for given length of time p( ) Probability density pN ( ) Probability density of largest maximum over given duration P Probability PSD Power spectral density q qE Probability that a maximum is positive Probability that a maximum is negative q( ) Probability density of maxima of
ℓ(t) q(θ) Reduced response First derivative of q(θ) Second derivative of q(θ) Q Q factor (quality factor) Q( ) Distribution function of maxima of
ℓ(t) Q(u) Probability that a maximum is higher than a given threshold r Irregularity factor rms Root mean square (value) r(t) Temporal window R Slope in dB/octave or Ratio of the number of minima to the number of maxima RE( ) Auto-correlation function of envelope R
ℓu Cross-correlation function between
ℓ(t) and u(t) R(f) Fourier transform of r(t) R(t) Envelope of maxima of u(t) First derivative of R(t) R(
τ) Auto-correlation function s Standard deviation S0 Value of constant PSD S( ) Power spectral density for –∞ ≤ f ≤ +∞ t Time T Duration of sample of signal or duration of vibration Ta Average time between two successive maxima u Ratio of threshold a to rms value
ℓrms of
ℓ(t) u0 Initial value of u(t) Initial value of Average of highest peaks urms Rms value of u(t) Rms value of Rms value of u(t) Generalized response First derivative of u(t) Second derivative of u(t) v Ratio a/urms vrms Rms value of xrms Rms value of x(t) Absolute acceleration of base of one-degree-of-freedom system Rms value of Maximum value of yrms Rms value of y(t) Rms value of Rms value of zrms Rms value of z(t) Rms value of Rms value of α Time-constant of the probability density of the first passage of a threshold or
Risk of up-crossing or β 2 (1 – 2
ξ2) Variable of chi-square with n degrees of freedom δt Time step δ( ) Dirac delta function Δ
τ Effective time interval Δf Frequency interval between half-power points or frequency step of the PSD ΔF Bandwidth of analysis filter Δ
ℓ Interval of amplitude of
ℓ(t) Δt Time interval ε Statistical error or Euler’s constant (0.57721566490…) γ
ℓu Coherence function between
ℓ(t) and...
