Hilbert Transform Generalization of a Classical Random Vibration Integral

Integrals which represent the spectral moments of the stationary response 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 derived 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 moments. Furthermore, it provides a potent generalization of a classical formula used in control engineering and applied mathematics. The applicability of the derived formula is demonstrated by considering random excitations with, among others, the white noise, “Gaussian,” and Kanai-Tajimi seismic spectra. The results for the classical problem of a randomly excited single-degree-of-freedom oscillator are given in a concise and readily applicable format.