HILBERT TRANSFORM GENERALIZATION OF A CLASSICAL RANDOM VIBRATION INTEGRAL

Integrals which represent the spectral moments of the stationary respo nse of a linear and time-invariant system under random excitation are considered. It is shown that these integrals can be determined through the solution of linear algebraic equations. These equations are deriv ed by considering differential equations for both the autocorrelation function of the system response and its Hilbert transform. The method can be applied to determine both even-order and odd-order spectral mom ents. Furthermore, if provides a potent generalization afa classical f ormula used in control engineering and applied mathematics. The applic ability of the derived formula is demonstrated by considering random e xcitations with, among others, the white noise, ''Gaussian,'' and Kana i-Tajimi seismic spectra. The results for the classical problem of a r andomly excited single-degree-of-freedom oscillator are given in a con cise and readily applicable format.