List of Symbols

The list below gives the most frequent definition of the main symbols used in this work. Some of the symbols can have another meaning locally, this will be defined in the text to avoid any confusion.

a Threshold value of z (t) aerf Inverse error function b Parameter b of Basquin relationship N σb =C c Viscous damping constant C Basquin relationship constant (N σb = C) CoVm Coefficient of variation of mean distribution D Fatigue damage e Error E Exaggeration factor Mean of the environment EE Expected environment ES Selected environment erf Error function ERS Extreme response spectrum E( ) Expectation of … f Frequency of excitation fM Mean frequency f0 Natural frequency FDS Fatigue response spectrum Fm Maximum value of F (t) f(t) Frequency sweeping law F(t) External force applied to a system g Acceleration due to gravity G( ) Power spectral density for 0 ≤ f ≤ ∞ h Interval (f/f0) H Drop height H() Transfer function i I0( ) Zero order Bessel function J Damping constant k Stiffness or uncertainty factor kv Aging coefficient K Constant of proportionality between stress and deformation rms Rms value of (t) m Maximum value of (t) (t) Generalized excitation (displacement) ( ) First derivative of (t) (t) Second derivative of (t) m Mass MRS Maximum response spectrum Mn Moment of order n n Number of cycles undergone by test-bar or material or Number of measurements or Number of tests Mean number of up-crossings of threshold a with positive slope per second Mean number of zero-crossings with positive slope per second (mean frequency) Mean number of maxima per second N Number of cycles to failure or Mean number of envelope maxima per second or Number of peaks higher than a given threshold Mean number of positive maxima higher than a given threshold for a given duration pv Probability of correct operation related to aging psd Power spectrum density p( ) Probability density p(T) Probability density of first passage of a threshold during time T P( ) Distribution function q Q(u) Probability density of maxima Q Q factor (quality factor) Q(u) Probability that a maximum is higher than a given threshold r Irregularity factor rms Root mean square (value) R Extreme response spectrum Mean of strength Re Yield stress Rm Ultimate tensile strength RU Response spectrum with given up-crossing risk S Standard deviation SE Standard deviation of environment SR Standard deviation of resistance SRS Shock Response Spectrum t Time or Random variable of Student distribution law ts Sweeping duration T Duration of vibration TF Test factor T1 Time-constant of logarithmic frequency sweep TS Test severity u Ratio of threshold a to rms value zrms of z(t) or value of u(t) urms rms value of u(t) um Maximum value of u(t) u0 Threshold value of u(t) URS Up-crossing risk Response Spectrum u(t) Generalized response vi Impact velocity VE Variation coefficient of real environment VR Variation coefficient of strength of the material xm Maximum value of x(t) x(t) Absolute displacement of the base of a single degree-of-freedom system (t) Absolute velocity of base of a single degree-of-freedom system (t) Absolute acceleration of the base of a single degree-of-freedom system rms rms value of (t) m maximum value of (t) rms rms value of (t) rms rms value of (t) (t) Absolute response acceleration of the mass of a one-degree-of-freedom system Zrms rms value of z(t) zarms rms value of the response to a random vibration zm Maximum value of Z(t) zp Peak value of Z(t) zs Relative response displacement to a sinusoidal vibration zsrms rms value of the response to a sinusoidal vibration zsup The largest value of Z(t) rms rms value of (t) rms rms value of (t) z(t) Relative response displacement of the mass of a single degree-of-freedom system with respect to its base (t) Relative response velocity (t) Relative response acceleration α Risk of up-crossing δ Non-centrality parameter of the non-central t-distribution Δf Frequency interval between half-power points ΔN Number of cycles carried out between half-power points Δt Time spent between half-power points ΔV Velocity change ε Euler’s constant (0.577 215 662…) γ(t) Incomplete gamma function Γ( ) Gamma function η Dissipation (or loss) coefficient φ Phase π 3.141 592 65 … π0 Confidence level σ Stress σa Alternating stress σD Fatigue limit stress σm Mean stress σrms Rms stress value σmax Maximum...