Multiple timescale spectral analysis

• We propose a general method for the fast computation of statistical cumulants in a spectral analysis.
• Application of the proposed method provides several approximations for the same problem.
• The concept is illustrated with various examples demonstrating applicability to single and multi degree-of-freedom systems, linear and nonlinear systems, second and third order cumulants and cases where the natural frequencies of the homogenous problem are much smaller/much larger than those of the forcing.

Spectral analysis is a classical tool for the structural analysis of structures subjected to random excitations. The most common application of spectral analysis is the determination of the steady-state second order cumulant of a linear oscillator, under the action of a stationary loading prescribed by means of its power spectral density. There exists however a broad variety of such similar problems, extending the concept to multi degree-of-freedom systems, non Gaussian excitation, slightly nonlinear oscillators or even transient excitations. In this wide class of problems, the cumulants of the response are obtained as the result of the integral of corresponding spectra over the frequency space, which is possibly multidimensional. Application of standard numerical integration techniques may be prohibitive, a reason why the spectral approach is often left aside. Besides, many engineering problems involve a clear timescale separation, usually of those pertaining to the loading and to the mechanical behavior of the system. In these problems, a proper consideration of the timescale separation results in dropping the order of integration by one, at least. This offers the possibility to derive analytical solutions, whenever the order of integration drops to zero, or to make numerical integration competitive. The paper presents this general method, together with some applications in wind and marine engineering.