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Research Papers

On the Stochastic Dynamical Behaviors of a Nonlinear Oscillator Under Combined Real Noise and Harmonic Excitations

[+] Author and Article Information
Chen Kong

State Key Laboratory of Mechanics and Control
of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China
e-mail: kongchen_bill@126.com

Zhen Chen

State Key Laboratory of Mechanics and Control
of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China
e-mail: czkillua@icloud.com

Xian-Bin Liu

Professor
State Key Laboratory of Mechanics
and Control of Mechanical Structures,
Nanjing University of Aeronautics and
Astronautics,
Nanjing 210016, China
e-mail: xbliu@nuaa.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 27, 2016; final manuscript received September 3, 2016; published online December 5, 2016. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(3), 031015 (Dec 05, 2016) (9 pages) Paper No: CND-16-1305; doi: 10.1115/1.4034735 History: Received June 27, 2016; Revised September 03, 2016

The exit problem and global stability of a nonlinear oscillator excited by an ergodic real noise and harmonic excitations are examined. The real noise is assumed to be a scalar stochastic function of an n-dimensional Ornstein–Uhlenbeck vector process which is the output of a linear filter system. Due to the existence of t-dependent excitation, two two-dimensional Fokker–Planck–Kolmogorov (FPK) equations governing the van der Pol variables process and the amplitude-phase process, respectively, are obtained and discussed through a perturbation method and the spectrum representations of the FPK operator and its adjoint operator of the linear filter system, while the detailed balance condition and the strong mixing condition are removed. Based on these FPK equations, the global properties of one-dimensional nonlinear oscillators with external or (and) internal periodic excitations under external or (and) internal real noises can be examined. Finally, a Duffing oscillator excited by a parametric real noise and parametric harmonic excitations is presented as an example, and the mean first-passage time (MFPT) about the oscillator's exit behavior between limit cycles is obtained under both wide-band noise and narrow-band noise excitations. The analytical result is verified by digital simulation.

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Figures

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Fig. 1

The time history of x-component of Poincaré points on Poincaré section Σ={(x,x˙,t)|t=0, mod 2T}, where T stands for the period of the external excitation T=2π/ω

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Fig. 2

The stationary probability density function obtained with half-power bandwidth μ = 10 for (a) and μ = 100 for (b)

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Fig. 3

MFPT versus different half-power bandwidths μ

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Fig. 4

MFPT versus the inverse square of the noise intensity σ with different half-power bandwidths μ

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