THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

20.9 Random Vibrations 613expressed as follows in normalized form:p(x) = 1σ √ /σ 2 (20.47)2π e−0.5x2and the probability of the value of x falling between a and b iswhereP(a < x < b) =∫ bap(x) dx (20.48)p(x) = the probability density of the functionx = the amount the function differs from the meanσ = the standard deviationP = the probability of x falling within a particular rangeIt is apparent that the total area under the probability density curve must be unity:P(−∞ < x < ∞) =∫ +∞−∞p(x) dx = 1Root-Mean-Square Value and AutocorrelationThe time-average root-mean-square of a function is defined by√( ∫ 1 T)x rms = lim xT →∞ T2 (t) dt0(20.49)wherex rms = the root-mean-square of function xT = the time intervalThe temporal autocorrelation function describes, on the average, the way in whichthe instantaneous value of a function depends on previous values. It is given by( ∫ 1 T)R(τ) = lim x(t)x(t + τ) dt(20.50)T →∞ T 0whereR(τ) = autocorrelation functionτ = the time interval between measurementst = time