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

614 20. Vibration and Vibration ControlErgodic ProcessesA function may vary in such a manner that there is no narrow time interval that canbe truly representative of the function. However, if the time interval is sufficientlylong and the probability distribution functions are independent of the time intervalduring which they were measured, the function then represents a stationary process.This means that the root-mean-square value measured during one intervalshould be equal to the root-mean-square value measured during a later interval.The autocorrelation function will also be unaffected by a time shift. If root-meansquarevalues and the autocorrelation functions of a number of ensembles of dataare equal to the temporal values, then the process may be considered ergodic aswell as stationary.Spectral DensityThe power spectral density, also known as the root-mean-square spectral density,can be determined from the autocorrelation function as follows:S(ω) =∫ +∞−∞ R(τ)e−iωτ dτ2π(20.51)whereS(ω) = root-mean-square spectral density, in g 2 /(rad/s) or (m/s 2 ) 2 /(rad/s)ω = radial frequency, rad/sEquation (20.51) is used in analytical studies. The inverse relationship isR(τ) =∫ +∞−∞S(ω)e iωτ dωVibration measurements obtained from an FFT analyzer or another spectralinstrument may be expressed in dB re 1gordBre1m/s 2 , within each frequencyband. These values may be converted to root-mean-square spectral density W ( f )(usually expressed in g 2 /Hz or some other engineering units), where W ( f ) andS(ω) are related bywhereW ( f ) = 4π S(ω)W ( f ) = spectral density, units 2 /Hz), defined for positive frequencies onlyf = frequency, HzS(ω) = spectral density, units 2 /(rad/s), defined for both positive andnegative frequencies (a mathematical artifice)