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Research Papers

First-Crossing Problem of Weakly Coupled Strongly Nonlinear Oscillators Subject to a Weak Harmonic Excitation and Gaussian White Noises

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

Department of Engineering Mechanics,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: yj.wu@sjtu.edu.cn

H. Y. Wang

Shanghai Mitsubishi Elevator Co. Ltd,
649 Changhua Rd., Jingan District,
Shanghai 200041, China
e-mail: wanghaoyu@smec-cn.com

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 14, 2017; final manuscript received January 23, 2018; published online February 23, 2018. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 140(4), 041006 (Feb 23, 2018) (11 pages) Paper No: VIB-17-1258; doi: 10.1115/1.4039244 History: Received June 14, 2017; Revised January 23, 2018

We study first-crossing problem of two-degrees-of-freedom (2DOF) strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time (AFCT). A 2DOF Duffing-van der Pol oscillator is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.

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Copyright © 2018 by ASME
Topics: Resonance , Excitation
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Figures

Grahic Jump Location
Fig. 1

Failure probability of system (66). A10 = A20=0: —♦— both external and internal resonances. (ω01 = 2.0, ω02 = 2.01, Ω = 2.01; G10 = G20=0), — • — only external resonance. (ω01 = 2.0, ω02 = 1.0, Ω = 2.01; G10 = 0), —▲—only internal resonance. (ω01 = 2.0, ω02 = 2.01, Ω = 0.449; G20 = 0), and — ▪ —no resonance. (ω01 = 2.0, ω02 = 1.0, Ω = 0.449).

Grahic Jump Location
Fig. 2

Conditional PDF of first-crossing time of system (66)

Grahic Jump Location
Fig. 3

AFCT of system (66). The other parameters are the same as those in Fig. 1 except that A10 is a variable.

Grahic Jump Location
Fig. 4

AFCT of system (66). The other parameters are the same as those in Fig. 3 except that D21 = 0.0.

Grahic Jump Location
Fig. 5

AFCT of system (66) (both external and internal resonances), (G10 = G20 = 0): (a) analytical results and (b) results from digital simulation

Grahic Jump Location
Fig. 6

AFCT of system (66) (both external and internal resonances) (A20 = G20 = 0): (a) analytical results and (b) results from digital simulation

Grahic Jump Location
Fig. 7

AFCT of system (66) (both external and internal resonances) (A10 = A20 = 0.3167): (a) analytical results and (b) results from digital simulation

Grahic Jump Location
Fig. 8

AFCT of system (66) (only external resonance) (G10 = 1.885): (a) analytical results and (b) results from digital simulation

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Fig. 9

AFCT of system (66) (only internal resonance) (A20 = 0): (a) analytical results and (b) results from digital simulation

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Fig. 10

AFCT of system (66) (no resonance): (a) analytical results and (b) results from digital simulation

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Fig. 11

Response sample of X1 and X2 by digital simulation (D21 = 0.0): (a) sample of X1 and (b) sample of X2

Grahic Jump Location
Fig. 12

Response sample of X1 and X2 by digital simulation (D21=0.004): (a) sample of X1 and (b) sample of X2

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