A novel mean square formulation of stochastic nonlinear dynamic systems based on Adomian decomposition. (January 2023)

• Record Type: Journal Article
• Authors: Panda, Satyam; Gogoi, Ankush; Hazra, Budhaditya; Pakrashi, Vikram
• Is Part Of: Probabilistic engineering mechanics. Volume 71(2023)
• Publication Date: 2023-01
• Subjects: Nonlinear system -- Bistable system -- Mean square response -- Moment closure -- Adomian decomposition method -- Stochastic differential equations; Engineering -- Statistical methods -- Periodicals; Mechanics, Applied -- Statistical methods -- Periodicals; Probabilities -- Periodicals; Ingénierie -- Méthodes statistiques -- Périodiques; Mécanique appliquée -- Méthodes statistiques -- Périodiques; Probabilités -- Périodiques; 620.100727
• DOI: 10.1016/j.probengmech.2022.103404
• Languages: English
• ISSNs: 0266-8920

Abstract

Abstract: An Adomian decomposition based mathematical framework to derive the mean square responses of nonlinear structural systems subjected to stochastic excitation is presented. The exact mean square response estimation of certain class of nonlinear stochastic systems is achieved using Fokker–Planck–Kolmogorov (FPK) equations resulting in analytical expressions or using Monte Carlo simulations. However, for most of the nonlinear systems, the response estimation using Monte Carlo simulations is computationally expensive, and, also, obtaining solution of FPK equation is mathematically exhaustive owing to the requirement to solve a stochastic partial differential equation. In this context, the present work proposes an Adomian decomposition based formalism to derive semi-analytical expressions for the second order response statistics. Further, a derivative matching based moment approximation technique is employed to reduce the higher order moments in nonlinear systems into functions of lower order moments without resorting to any sort of linearization. Three case studies consisting of Duffing oscillator with negative stiffness, Rayleigh Van-der Pol oscillator and a Pendulum tuned mass damper inerter system with linear auxiliary spring–damper arrangement subjected to white noise excitation are undertaken. The accuracy of the closed form expressions derived using the proposed framework is established by comparing the mean square responses of the systems with the exact solutions. The results demonstrate the robustness of the proposed framework for accurate statistical analysis of nonlinear systems under stochastic excitation. Highlights: An Adomian based formalism for mean square stochastic responses of nonlinear systems is proposed. Bi-stable nonlinear stochastic oscillators are considered. Moment equations are formed using Ito–Taylor based formulation. Comparison of the mean square responses with equivalent linearization and near-exact EM is presented. Application of the proposed framework for a passive control system is demonstrated.