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Research Papers

Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations

[+] Author and Article Information
Y. J. Wu

School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, P.R.C.yongjunwu@ecust.edu.cn

W. Q. Zhu1

Department of Mechanics, Zhejiang University, Hangzhou 310027, P.R.Cwqzhu@yahoo.com

1

Corresponding author.

J. Vib. Acoust 130(5), 051004 (Aug 12, 2008) (9 pages) doi:10.1115/1.2948382 History: Received January 11, 2007; Revised January 08, 2008; Published August 12, 2008

Physical and engineering systems are often subjected to combined harmonic and random excitations. The random excitation is often modeled as Gaussian white noise for mathematical tractability. However, in practice, the random excitation is nonwhite. This paper investigates the stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. By using generalized harmonic functions, a new stochastic averaging procedure for estimating stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations is developed. The damping can be linear and (or) nonlinear and the excitations can be external and (or) parametric. After stochastic averaging, the system state is represented by two-dimensional time-homogeneous diffusive Markov processes. The method of reduced Fokker–Planck–Kolmogorov equation is used to investigate the stationary response of the vibration system. A nonlinearly damped Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary external resonance, based on the stationary joint probability density of amplitude and phase difference, the stochastic jump of the Duffing oscillator and P-bifurcation as the system parameters change are examined for the first time. The agreement between the analytical results and those from Monte Carlo simulation of original system shows that the proposed procedure works quite well.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 7

Stationary probability density p(a) of system 39. (—) Analytical result; (● ▲) result from Monte Carlo simulation. The parameters are the same as those in Fig. 1 except E.

Grahic Jump Location
Figure 8

Stationary probability density p(a) of system 39. (—) Analytical result; (● ▲) results from Monte Carlo simulation. The parameters are the same as those in Fig. 1 except α.

Grahic Jump Location
Figure 4

Stationary joint probability density p(a,γ) of system 39. The parameters are the same as those in Fig. 1 except D2=0.0. (a) Analytical result; (b) result from Monte Carlo simulation.

Grahic Jump Location
Figure 5

Stationary joint probability density p(a,γ) of system 39. The parameters are the same as those in Fig. 1 except D2=8.0. (a) Analytical result; (b) result from Monte Carlo simulation.

Grahic Jump Location
Figure 6

Stationary probability density p(a) of system 39. (—) Analytical result; (● ▲) result from Monte Carlo simulation. The parameters are the same as those in Fig. 1 except Ω.

Grahic Jump Location
Figure 1

Stationary joint probability density p(a,γ) ((a) and (b)) and sample function ((c) and (d)) of system 39. ω0=1.0, α=2.0, β1=0.1, β2=0.0, E=0.2, Ω=1.5, ωi=5.0, ζi=0.5, and Di=2.0(i=1,2). (a) Analytical results; (b) results from Monte Carlo simulation; (c) displacement sample; (d) velocity sample.

Grahic Jump Location
Figure 2

Stationary joint probability density p(a,γ) of system 39. The parameters are the same as those in Fig. 1 except D1=2.5. (a) Analytical result; (b) result from Monte Carlo simulation.

Grahic Jump Location
Figure 3

Stationary joint probability density p(a,γ) of system 39. The parameters are the same as those in Fig. 1 except D1=0.5. (a) Analytical result; (b) result from Monte Carlo simulation.

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