By Sir James H. Jeans, Physics
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Extra info for An elementary treatise on theoretical mechanics
26] defines theoretically the PSD. In practice, this relation cannot be respected exactly since the calculation of G(f) would require an infinite integration time and an infinitely narrow bandwidth. NOTES. - The function G(f) is positive or zero whatever the value off. - The PSD was defined above for f ranging between 0 and infinity, which corresponds to the practical case. 27] - The pulsation Q = 2 n f is sometimes used as variable instead off. 30] 'j — /"• ^o^ r^? 32] f(f\ NOTE. 34] 1 / v where ck = — ^ctj - Pj ij 2 77ze power spectral density can also be defined from this development in a Fourier series.
The first possibility is to consider the temporal mean of the instantaneous values of the recording. We have: = lim — P4t)dt T->oo2T ~ T [1-38] if this limit exists. This limit may very well not exist for some or for all the samples and, if it exists, it may depend on the selected sample ^(t); but it does not depend on time(l}. 2. Quadratic mean - rtns value The vibration t(i) results in general in an oscillation of the mechanical system around its equilibrium position, so that the arithmetic mean of the instantaneous values can be zero if the positive and negative values are compensated.
14 Random vibration For a stationary process, the autocorrelation function is written: R(T)= lim - ( 0 ) N [L31] '*M NOTES. 35] lim — N-»oo 15 N Properties 1. 36] Indeed 2. 9. 1. 6. Sample of random signal 16 Random vibration Let us consider a sample l(t) of duration T of a recording. It can be interesting to study the statistical properties of the instantaneous values of the function g(t). The first possibility is to consider the temporal mean of the instantaneous values of the recording. We have: = lim — P4t)dt T->oo2T ~ T [1-38] if this limit exists.
An elementary treatise on theoretical mechanics by Sir James H. Jeans, Physics