Research Papers

Probabilistic Solution of Vibro-Impact Systems Under Additive Gaussian White Noise

[+] Author and Article Information
H. T. Zhu

Associate Professor
State Key Laboratory of Hydraulic Engineering Simulation and Safety,
Tianjin University,
Tianjin 300072, China
e-mail: htzhu@tju.edu.cn

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 30, 2012; final manuscript received March 16, 2014; published online April 18, 2014. Assoc. Editor: Steven W. Shaw.

J. Vib. Acoust 136(3), 031018 (Apr 18, 2014) (7 pages) Paper No: VIB-12-1363; doi: 10.1115/1.4027211 History: Received December 30, 2012; Revised March 16, 2014

This paper presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact systems under additive Gaussian white noise. The constraint is a unilateral zero-offset barrier. The vibro-impact system is first converted into a system without barriers using the Zhuravlev nonsmooth coordinate transformation. The stationary PDF of the converted system is governed by the Fokker–Planck equation which is solved by the exponential-polynomial closure (EPC) method. A vibro-impact Duffing oscillator with either elastic or lightly inelastic impacts is considered in a numerical analysis. Meanwhile, the level of nonlinearity in displacement is also examined in this study as well as the case of negative linear stiffness. Comparison with the simulated results shows that the EPC method can present a satisfactory PDF for displacement and velocity when the polynomial order is taken as 4 in the investigated cases. The tail of the PDF also works well with the simulated result.

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Grahic Jump Location
Fig. 2

Comparison of PDFs in Case 2 (r = 0.95, μ = 0.1, k0 = 1.0): (a) PDFs of displacement, (b) logarithmic PDFs of displacement, (c) PDFs of velocity, and (d) logarithmic PDFs of velocity

Grahic Jump Location
Fig. 3

Comparison of PDFs in Case 3 (r = 0.95, μ = 1.0, k0 = 1.0): (a) PDFs of displacement, (b) logarithmic PDFs of displacement, (c) PDFs of velocity, and (d) logarithmic PDFs of velocity

Grahic Jump Location
Fig. 1

Comparison of PDFs in Case 1 (r = 1, μ = 0.1, k0 = 1.0): (a) PDFs of displacement, (b) logarithmic PDFs of displacement, (c) PDFs of velocity, and (d) logarithmic PDFs of velocity

Grahic Jump Location
Fig. 4

Comparison of PDFs in Case 4 (r = 0.95, μ = 1.0, k0 = −1.0): (a) PDFs of displacement, (b) logarithmic PDFs of displacement, (c) PDFs of velocity, and (d) logarithmic PDFs of velocity




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