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.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Babitsky, V. I., 1998, Theory of Vibro-Impact Systems and Applications, Springer-Verlag, Berlin.
Ibrahim, R. A., 2009, Vibro-Impact Dynamics: Modeling, Mapping and Applications, Springer-Verlag, Berlin.
Holmes, P. J., 1982, “The Dynamics of Repeated Impacts With a Sinusoidally Vibrating Table,” J. Sound Vib., 84(2), pp. 173–189. [CrossRef]
Shaw, S. W., and Holmes, P., 1983, “Periodically Forced Linear Oscillator With Impacts: Chaos and Long-Period Motions,” Phys. Rev. Lett., 51(8), pp. 623–626. [CrossRef]
Shaw, S. W., and Holmes, P. J., 1983, “A Periodically Forced Impact Oscillator With Large Dissipation,” ASME J. Appl. Mech., 50(4a), pp. 849–857. [CrossRef]
Luo, G. W., and Xie, J. H., 1998, “Hopf Bifurcation of a Two-Degree-of-Freedom Vibro-Impact System,” J. Sound Vib., 213(3), pp. 391–408. [CrossRef]
Pilipchuk, V. N., 2001, “Impact Modes in Discrete Vibrating Systems With Rigid Barriers,” Int. J. Non-Linear Mech., 36(6), pp. 999–1012. [CrossRef]
Baratta, A., 1990, “Dynamics of a Single-Degree-of-Freedom System With a Unilateral Obstacle,” Struct. Saf., 8(1–4), pp. 181–194. [CrossRef]
Dimentberg, M. F., and Iourtchenko, D. V., 2004, “Random Vibrations With Impacts: A Review,” Nonlinear Dyn., 36(2–4), pp. 229–254. [CrossRef]
Dimentberg, M. F., and Menyailov, A. I., 1979, “Response of a Single-Mass Vibroimpact System to White-Noise Random Excitation,” ZAMM–J. Appl. Math. Mech., 59(12), pp. 709–716. [CrossRef]
Dimentberg, M. F., 1988, Statistical Dynamics of Nonlinear and Time-Varying Systems, Research Studies Press, Taunton, UK.
Huang, Z. L., Liu, Z. H., and Zhu, W. Q., 2004, “Stationary Response of Multi-Degree-of-Freedom Vibro-Impact Systems Under White Noise Excitations,” J. Sound Vib., 275(1–2), pp. 223–240. [CrossRef]
Sri Namachchivaya, N., and Park, J. H., 2005, “Stochastic Dynamics of Impact Oscillators,” ASME J. Appl. Mech., 72(6), pp. 862–870. [CrossRef]
Feng, J. Q., Xu, W., and Wang, R., 2008, “Stochastic Responses of Vibro-Impact Duffing Oscillator Excited by Additive Gaussian Noise,” J. Sound Vib., 309(3–5), pp. 730–738. [CrossRef]
Feng, J. Q., Xu, W., Rong, H. W., and Wang, R., 2009, “Stochastic Responses of Duffing-Van der Pol Vibro-Impact System Under Additive and Multiplicative Random Excitations,” Int. J. Non-Linear Mech., 44(1), pp. 51–57. [CrossRef]
Iourtchenko, D. V., and Song, L. L., 2006, “Numerical Investigation of a Response Probability Density Function of Stochastic Vibroimpact Systems With Inelastic Impacts,” Int. J. Non-Linear Mech., 41(3), pp. 447–455. [CrossRef]
Zhuravlev, V. F., 1976, “A Method for Analyzing Vibration-Impact Systems by Means of Special Functions,” Mech. Solids, 11(2), pp. 23–27.
Pilipchuk, V. N., 2011, “Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics,” http://arxiv.org/abs/1101.4597
Er, G. K., 1998, “An Improved Closure Method for Analysis of Nonlinear Stochastic Systems,” Nonlinear Dyn., 17(3), pp. 285–297. [CrossRef]
Er, G. K., 2000, “The Probabilistic Solutions to Nonlinear Random Vibrations of Multi-Degree-of-Freedom Systems,” ASME J. Appl. Mech., 67(2), pp. 355–359. [CrossRef]
Zhu, H. T., Er, G. K., Iu, V. P., and Kou, K. P., 2010, “Probability Density Function Solution of Nonlinear Oscillators Subjected to Multiplicative Poisson Pulse Excitation on Velocity,” ASME J. Appl. Mech., 77(3), p. 031001. [CrossRef]
Zhu, H. T., Er, G. K., Iu, V. P., and Kou, K. P., 2011, “Probabilistic Solution of Nonlinear Oscillators Excited by Combined Gaussian and Poisson White Noises,” J. Sound Vib., 330(12), pp. 2900–2909. [CrossRef]
Zhu, H. T., Er, G. K., Iu, V. P., and Kou, K. P., 2012, “Responses of Nonlinear Oscillators Excited by Nonzero-Mean Parametric Poisson Impulses on Displacement,” J. Eng. Mech., 138(5), pp. 450–457. [CrossRef]
Er, G. K., Guo, S. S., and Iu, V. P., 2012, “Probabilistic Solutions of the Stochastic Oscillators With Even Nonlinearity in Displacement,” ASME J. Vib. Acoust., 134(5), p. 054501. [CrossRef]
Guo, S. S., 2014, “Probabilistic Solutions of Stochastic Oscillators Excited by Correlated External and Parametric White Noises,” ASME J. Vib. Acoust., 136(3), p. 031003. [CrossRef]
Xia, Z., and Tang, J., 2013, “Characterization of Dynamic Response of Structures With Uncertainty by Using Gaussian Processes,” ASME J. Vib. Acoust., 135(5), p. 051006. [CrossRef]
Lutes, L. D., and Sarkani, S., 2004, Random Vibrations: Analysis of Structural and Mechanical Systems, Elsevier, New York.


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. 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. 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




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In