Stationary Response of Multidegree-of-Freedom Strongly Nonlinear Systems to Fractional Gaussian Noise

Abstract

The stationary response of multidegree-of-freedom (MDOF) strongly nonlinear system to fractional Gaussian noise (FGN) with Hurst index 1/2 < H < 1 is studied. First, the system is modeled as FGN-excited and -dissipated Hamiltonian system. Based on the integrability and resonance of the associated Hamiltonian system, the system is divided into five classes: partially integrable and resonant, partially integrable and nonresonant, completely integrable and resonant, completely integrable and nonresonant, and nonintegrable. Then, the averaged fractional stochastic differential equations (SDEs) for five classes of quasi-Hamiltonian systems with lower dimension and involving only slowly varying processes are derived. Finally, the approximate stationary probability densities and other statistics of two example systems are obtained by numerical simulation of the averaged fractional SDEs to illustrate the application and compared with those from original systems to show the advantages of the proposed procedure.

Response of Quasi-Integrable and Non-Resonant Hamiltonian Systems to Fractional Gaussian Noise

“…One is FSDEs in Eq. (24) that is obtained by using the stochastic averaging method for quasi-Hamiltonian systems excited by fGn [13], [14]. Another is Itô SDEs in Eq.…”

“…The stochastic averaging methods, including the stochastic averaging methods for quasi-Hamiltonian systems, are the powerful approximate analytical methods that have been widely used in nonlinear stochastic dynamics [4], [11], [12]. Recently, the stochastic averaging method for quasi-Hamiltonian systems excited by fGn [13], [14] has been developed based on averaging principal [15], [16]. The dimension of the averaged fractional stochastic differential equation (FSDE) is less than that of original system while the dynamical characteristics of averaged system keep the same as the original system.…”