Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations

D N Hao,N D Anh

doi:10.1155/2017/4907520

Abstract

This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

Highlights

It is known that time delay in real active control systems is unavoidable due to the time spent in calculating and executing the control forces, performing online computation, and so on
The obtained results are compared to ones obtained from the Monte-Carlo simulation which is “often the sole tool available for assessing the accuracy of random vibration solutions generated by approximate methods of analysis,” as pointed out in Robert and Spanos (2003) [17]
The numerical simulation of the mean square response is obtained by 10,000-realization MonteCarlo simulation (MCS) with the time from zero to 300 seconds

Summary

Introduction

It is known that time delay in real active control systems is unavoidable due to the time spent in calculating and executing the control forces, performing online computation, and so on. In 2009, Li et al [2] studied effects of time delay in feedback control on the first-passage failure of controlled systems under stochastic excitation by using the stochastic averaging method for quasi-integrable Hamiltonian systems. In 2012, Liu and Zhu [3] proposed a procedure based on the stochastic averaging method for the time delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems subject to Gaussian white noise excitation to study the response and stability of systems. They converted the problem of time delay stochastic optimal control of quasi-integrable Hamiltonian systems into the problem of stochastic optimal control without time delay and the result problem is solved by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. We showed that this assumption is acceptable

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Conclusion