Stability of a nonlinearly damped second-order system with randomly fluctuating restoring coefficient
type: Item
instance of: scholarly article
title: Stability of a nonlinearly damped second-order system with randomly fluctuating restoring coefficient
zbMATH Open document ID: 0586.34051
DOI: 10.1016/0020-7462(86)90008-9
author: J. J. H. Brouwers
published in: International Journal of Non-Linear Mechanics
publication date: 1986-01-01T00:00:00Z
review text: The author considers the second order ordinary differential equation \[ (*)\quad \ddot x+\epsilon g(\dot x)+\{1+\epsilon^{1/2}f(t)\}x=0. \] The damping force \(g(\dot x)\) is of the form \(g(\dot x)=\beta_ 0\dot x+\beta_{\alpha}\dot x| \dot x|^{\alpha}\) where \(\beta_{\alpha}>0\), \(\alpha\geq -1\). The parametric excitation f(t) is a mean zero stationary Gaussian process, and \(\epsilon\) is a small positive parameter. The initial conditions are \(x(0)=x_ 0\), \(\dot x(0)=\dot x_ 0\). For \(\epsilon \ll 1\), the solution of (*) may be approximated closely by a sinusoidal wave of slowly varying amplitude and phase. The amplitude a(t) satisfies a stochastic differential equation (as \(\epsilon >0)\). This equation is solved and various properties of a(t) are investigated.
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 3939785
zbMATH Keywords: second order ordinary differential equation
zbMATH Keywords: damping force
zbMATH Keywords: parametric excitation
zbMATH Keywords: mean zero stationary Gaussian process
zbMATH Keywords: small positive parameter
zbMATH Keywords: stochastic differential equation
reviewed by: Kenneth S. Miller
MaRDI profile type: Publication
full work available at URL: 0020-7462(86)90008-9
OpenAlex ID: W2159276575
description: scientific article; zbMATH DE number 3939785