Response and stability of strongly non-linear oscillators under wide-band random excitation

• Published in: International journal of non-linear mechanics Vol. 36; no. 8; pp. 1235 - 1250
• Main Authors: Zhu, W.Q., Huang, Z.L., Suzuki, Y.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.12.2001; Elsevier Science
• Subjects: Exact sciences and technology; Nonlinear dynamics and nonlinear dynamical systems; Nonlinear system; Physics; Stationary solution; Statistical physics, thermodynamics, and nonlinear dynamical systems; Stochastic averaging; Stochastic Hopf bifurcation; Stochastic stability; Wide-band random excitation; Stationary solution; Stochastic Hopf bifurcation; Stochastic stability; Wide-band random excitation; Stochastic averaging; Nonlinear system; Parametric excitation; Random excitation; Duffing oscillator; Stationary response; Harmonic function; Probability density; Asymptotic stability; Non linear oscillator; Stability criteria; Van der Pol oscillator; System with one degree of freedom; Hopf bifurcation; Nonlinear systems
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/S0020-7462(00)00093-7

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.