Random vibration analysis of time-delayed dynamical systems

Abstract

This paper presents random vibration analyses of time-delayed linear and nonlinear dynamical systems. The method of continuous time approximation (CTA) is applied to describe the system dynamics in a high dimensional state space without time delay. Spectral analysis of time-delayed linear systems is studied first. An exact solution of the power spectral density function of the linear system is obtained and used to check the validity of the CTA method. The methods of CTA and equivalent linearization are combined to obtain steady-state responses of the system. The analytical results of E[x2] and E[ẋ2] of a Duffing system with time delay are compared with those of extensive Monte Carlo simulations. It is found that when the system is weakly nonlinear under low-level random excitations, the solutions agree well. For the system with strong nonlinearity or under high-level random excitations, the method of equivalent linearization becomes less accurate and its prediction error grows with the nonlinearity and the excitation strength.