Stochastic Dynamics of Nonlinear Systems Driven by Non-normal Delta-Correlated Processes

Abstract

In this paper, nonlinear systems subjected to external and parametric non-normal delta-correlated stochastic excitations are treated. A new interpretation of the stochastic differential calculus allows first a full explanation of the presence of the Wong-Zakai or Stratonovich correction terms in the Itoˆ’s differential rule. Then this rule is extended to take into account the non-normality of the input. The validity of this formulation is confirmed by experimental results obtained by Monte Carlo simulations.

Random vibrations of Rayleigh vibroimpact oscillator under Parametric Poisson white noise

“…[33], the corresponding Itô stochastic differential equation for systems with Poisson white noise excitation is obtained as follows:…”

“…One of the most common models for non-Gaussian random loadings is Poisson white noise. Responses of smooth systems subjected to Poisson white noise have been intensively investigated by using equivalent linearization [29,30], cell mapping and path integration [31,32], moment equations [33], stochastic averaging method [34][35][36], etc. For vibroimpact systems with elastic impacts under Poisson white noise, analytical study is relatively little [37], let alone that with inelastic impacts for inherent difficulties due to additional finite relations at the impact instants.…”

Response of Linear and Non-Linear Structural Systems under Gaussian or Non-Gaussian Filtered Input

“…In order to reduce the number of equations, the dynamic response is evaluated in a reduced space by means of the following coordinate transformation X=«l>Y (49) where cl» is the modal matrix of the undamped system, containing the first few eigenvectors of the matrix K-1 M normalized with respect toM; in this way the matrix cl» is of order n x m, m being the number of modes selected for the analysis (m << n). The matrix cl» possesses the following properties (50) where Im is the identity matrix of order m and 0 2 is the diagonal matrix listing the square of the natural radial frequencies co~ (i = 1, 2, ... , m ). Letting cl» T C cl» = B, B being an m x m symmetric matrix, Eq.…”

“…Using the extension of Ito differential rule for non-normal input process [50], we can write the moment equations of the response. As an example, the differential equations of the ftrst two moments are given as…”

Derivation of Fokker–Planck equations for stochastic systems under excitation of multiplicative non-Gaussian white noise

“…Marcus SDEs are often appropriate models in engineering and scientific practice, because they preserve certain physical quantities such as energy [13,14,11,19,20]. It is recently shown [19] that Marcus SDE is equivalent to the well known Di Paola-Falsone SDE [5,6] which is widely used in engineering and physics [9,22,20]. Comparison of Marcus integral and Stratonovich integral is recently discussed in [3] for systems with jump noise.…”