Reliability of quasi integrable and non-resonant Hamiltonian systems under fractional Gaussian noise excitation
- PDF / 2,297,700 Bytes
- 8 Pages / 595.276 x 790.866 pts Page_size
- 75 Downloads / 198 Views
RESEARCH PAPER
Reliability of quasi integrable and non‑resonant Hamiltonian systems under fractional Gaussian noise excitation Q. F. Lü1 · W. Q. Zhu1 · M. L. Deng1 Received: 28 February 2020 / Revised: 30 March 2020 / Accepted: 7 May 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise (fGn) excitation is studied. Noting rather flat fGn power spectral density (PSD) in most part of frequency band, the fGn is innovatively regarded as a wide-band process. Then, the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged Itô stochastic differential equations (SDEs). Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation. The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation. Keywords Reliability · First passage time · Quasi integrable and non-resonant Hamiltonian systems · Fractional Gaussian noise
1 Introduction Different from well known Gaussian white noise, fractional Gaussian noise (fGn) has long range (or long-memory), strongly spatial and/or temporal memory, which makes fGn a proper mathematical model of some real noises. FGn has attracted considerable attention from many researchers because of its wide potential applications in natural science or engineering. In fact, fGn has been applied in many fields as a kind of stochastic noise model such as neural network model [1, 2], finance market model [3, 4], atmospheric turbulence model [5], oil slick model [6] and some others [7, 8]. However, the response of dynamic system to fGn is not Markov process and the classical Itô stochastic calculus and differential equation cannot be used. Although some stochastic calculus for fGn have been developed [9, 10], it is still very trick to the study dynamical system, especially nonlinear system, driven by fGn. Recently, Xu et al. [11–13] made a great effort on stochastic differential equations (SDEs) with respect to fractional * M. L. Deng [email protected] 1
Department of Mechanics, State‑Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, China
Brownian motion (fBm) and proposed a stochastic averaging principle using forward pathwise integral, which makes it easier to handle SDEs driven by fGn mathematically. Based on Xu’s averaging principle, Deng et al. [14] have developed the stochastic averaging method for multi degree of freedom (MDOF) quasi Hamiltonian systems excited by fGn but the response statistics are obtained still using numerical simulation. Reliability is of great significance for engineering systems. In stochastic dynamics, first-passage failure is the major failure model
Data Loading...
