The subharmonic response of one third order of Duffing oscillator under harmonic and random excitations is in-vestigated for the first time by a technique combining the stochastic averaging method, the equivalent lineariza-tion method, and the technique of auxiliary function for Fokker-Planck equation. The averaged equations are line-ar zed so that the stationary density function of the approximate response can be found exactly by the technique of auxiliary function. The one third order subharmonic response obtained by the present technique is validated by numerical simulation. The significant contribution of this work is that it may lead to a new trend in investigating subharmonic oscillators in random nonlinear systems.
Duffing oscillator, subharmonic, averaging method, equivalent linearization, auxiliary function, harmonic excita-tion, random excitation
In this paper, we are concerned with the Duffing oscillator, which has been applied to model many mechanical systems and has attracted much attention as a typical nonlinear system. When the system is under only a harmonic excitation or random one, two popular tools used to study such a nonlinear system are the averaging method and equivalent linearization method, respectively. The former was originally given by Krylov and Bogolyubov [1] and then it was developed by Bogolyubov and Mitropolskiy [2-4] and was extended to systems under a random excitation with the works of Stratonovich [5], Khasminskii [6], and others, which were reviewed in survey paper by Mitropolskiy [3], Robert and Spanos [7] and Manohar [8]. The later, the stochastic equivalent linearization method, which has attracted many researchers due to its originality and capability for various applications in engineering, was first studies by Kazakov [9], who extended Krylov and Bogolyubov’s linearization technique [1] of deterministic problems to random problems. This method was also reviewed in some books by Roberts and Spanos [10], and Socha [11]
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