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Reliability and control of strongly nonlinear vibro-impact system under external and parametric Gaussian noises

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Abstract

This paper mainly focuses on reliability and the optimal bounded control for maximizing system reliability of a strongly nonlinear vibro-impact system. Firstly, the new stochastic averaging in which the impact condition is converted to the system energy is applied to obtain the averaged Itô stochastic differential equation, by which the associated Backward Kolmogorov (BK) equation and Generalized Pontryagin (GP) equation are derived. Then, the dynamical programming equations are obtained based on the dynamical programming principle, by which the optimal bounded control for maximizing system reliability is devised Finally, the effects of the bounded control and noise intensity on the reliability of the vibro-impact system are discussed in detail; meanwhile, the influence of impact conditions on the system’s reliability is also studied. The feasibility and effectiveness of the proposed analytical method are substantiated by numerical results obtained from Monte-Carlo simulation.

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References

  1. Ervin E K. Vibro-impact behavior of two orthogonal beams. J Eng Mech, 2009, 135: 529–537

    Article  Google Scholar 

  2. Jing H S, Young M. Random response of a single-degree-of-freedom vibro-impact system with clearance. Earthq Engng Struct Dyn, 1990, 19: 789–798

    Article  Google Scholar 

  3. Ibrahim R A. Vibro-impact Dynamics: Modeling, Mapping and Appcations. Berlin Heidelberg: Springer-Verlag, 2009

    Book  Google Scholar 

  4. Shaw S W, Holmes P J. A periodically forced impact oscillator with large dissipation. J Appl Mech, 1983, 50: 849–857

    Article  Google Scholar 

  5. Luo G W. Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops. Phys Lett A, 2004, 323: 210–217

    Article  MathSciNet  CAS  ADS  Google Scholar 

  6. Wagg D J. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator. Chaos Soliton Fract, 2004, 22: 541–548

    Article  ADS  Google Scholar 

  7. Dimentberg M F, Haenisch H G. Pseudo-linear vibro-impact system with a secondary structure: Response to a white-noise excitation. J Appl Mech, 1998, 65: 772–774

    Article  Google Scholar 

  8. Dimentberg M F, Iourtchenko D V, van Ewijk O. Subharmonic response of a quasi-isochronous vibro-impact system to a randomly disordered periodic excitation. Nonlinear Dyn, 1998, 17: 173–186

    Article  Google Scholar 

  9. Dimentberg M F, Iourtchenko D V. Towards incorporating impact losses into random vibration analyses: A model problem. Probab Eng Mech, 1999, 14: 323–328

    Article  Google Scholar 

  10. Zhuravlev V F. A method for analyzing vibration-impact systems by means of special function. Mech Solids, 1976, 11: 23–27

    MathSciNet  ADS  Google Scholar 

  11. Feng J, Xu W, Wang R. Stochastic responses of vibro-impact duffing oscillator excited by additive Gaussian noise. J Sound Vib, 2008, 309: 730–738

    Article  ADS  Google Scholar 

  12. Feng J, Xu W, Rong H, et al. Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations. Int J Non-Linear Mech, 2009, 44: 51–57

    Article  Google Scholar 

  13. Liu L, Xu W, Yue X, et al. Stochastic response of Duffing-Van der Pol vibro-impact system with viscoelastic term under wide-band excitation. Chaos Soliton Fract, 2017, 104: 748–757

    Article  MathSciNet  ADS  Google Scholar 

  14. Zhu H T. Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. J Sound Vib, 2014, 333: 954–961

    Article  ADS  Google Scholar 

  15. Zhu H T. Probabilistic solution of vibro-impact stochastic Duffing systems with a unilateral non-zero offset barrier. Physica A, 2014, 410: 335–344

    Article  MathSciNet  ADS  Google Scholar 

  16. Zhu H T. Response of a vibro-impact Duffing system with a randomly varying damping term. Int J Non-Linear Mech, 2014, 65: 53–62

    Article  Google Scholar 

  17. Gu X, Zhu W. A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations. J Sound Vib, 2014, 333: 2632–2642

    Article  ADS  Google Scholar 

  18. Yang G, Xu W, Feng J, et al. Response analysis of Rayleigh-Van der Pol vibroimpact system with inelastic impact under two parametric white-noise excitations. Nonlinear Dyn, 2015, 82: 1797–1810

    Article  MathSciNet  Google Scholar 

  19. Yang G, Xu W, Gu X, et al. Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises. Chaos Soliton Fract, 2016, 87: 125–135

    Article  MathSciNet  ADS  Google Scholar 

  20. Liu D, Li M, Li J. Probabilistic response and analysis for a vibro-impact system driven by real noise. Nonlinear Dyn, 2018, 91: 1261–1273

    Article  Google Scholar 

  21. Liu D, Li J, Meng Y. Probabilistic response analysis for a class of nonlinear vibro-impact oscillator with bilateral constraints under colored noise excitation. Chaos Soliton Fract, 2019, 122: 179–188

    Article  MathSciNet  ADS  Google Scholar 

  22. Chen L, Qian J, Zhu H, et al. The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators. J Sound Vib, 2019, 439: 260–270

    Article  ADS  Google Scholar 

  23. Ren Z, Xu W, Wang D. Dynamic and first passage analysis of ship roll motion with inelastic impacts via path integration method. Nonlinear Dyn, 2019, 97: 391–402

    Article  Google Scholar 

  24. Xu M. First-passage failure of linear oscillator with non-classical inelastic impact. Appl Math Model, 2018, 54: 284–297

    Article  MathSciNet  Google Scholar 

  25. Su M, Xu W, Yang G. Response analysis of van der Pol vibro-impact system with Coulomb friction under Gaussian white noise. Int J Bifurcat Chaos, 2018, 28: 1830043

    Article  MathSciNet  Google Scholar 

  26. Gan C B, Zhu W Q. First-passage failure of quasi-non-integrable-Hamiltonian systems. Int J Non-Linear Mech, 2001, 36: 209–220

    Article  MathSciNet  Google Scholar 

  27. Huang D, Zhou S, Litak G. Analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit. Nonlinear Dyn, 2019, 97: 663–677

    Article  Google Scholar 

  28. Huang D, Zhou S, Litak G. Theoretical analysis of multi-stable energy harvesters with high-order stiffness terms. Commun Nonlinear Sci Numer Simul, 2019, 69: 270–286

    Article  MathSciNet  ADS  Google Scholar 

  29. Zhu W Q. Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl Mech Rev, 2006, 59: 230–248

    Article  ADS  Google Scholar 

  30. Zhu W Q, Huang Z L, Deng M L. Feedback minimization of firstpassage failure of quasi non-integrable Hamiltonian systems. Int J Non-Linear Mech, 2002, 37: 1057–1071

    Article  Google Scholar 

  31. Li X P, Huan R H, Wei D M. Feedback minimization of the firstpassage failure of a hysteretic system under random excitations. Probab Eng Mech, 2010, 25: 245–248

    Article  Google Scholar 

  32. Wang S L, Jin X L, Wang Y, et al. Reliability evaluation and control for wideband noise-excited viscoelastic systems. Mech Res Commun, 2014, 62: 57–65

    Article  ADS  Google Scholar 

  33. Zakharova A, Vadivasova T, Anishchenko V, et al. Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator. Phys Rev E, 2010, 81: 011106

    Article  CAS  ADS  Google Scholar 

  34. Xu Y, Gu R, Zhang H, et al. Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Phys Rev E, 2011, 83: 056215

    Article  ADS  Google Scholar 

  35. Li C, Xu W, Feng J, et al. Response probability density functions of Duffing-Van der Pol vibro-impact system under correlated Gaussian white noise excitations. Physica A, 2013, 392: 1269–1279

    Article  MathSciNet  ADS  Google Scholar 

  36. Roberts J B. First-passage time for randomly excited non-linear oscillators. J Sound Vib, 1986, 109: 33–50

    Article  ADS  Google Scholar 

  37. Zhu W Q, Wu Y J. First-passage time of Duffing oscillator under combined harmonic and white-noise excitations. Nonlinear Dyn, 2003, 32: 291–305

    Article  Google Scholar 

  38. Zhu W Q. Recent developments and applications of the stochastic averaging method in random vibration. Appl Mech Rev, 1996, 49: S72–S80

    Article  ADS  Google Scholar 

  39. Zhu W Q, Deng M L. Optimal bounded control for minimizing the response of quasi non-integrable hamiltonian systems. Nonlinear Dyn, 2004, 35: 81–100

    Article  MathSciNet  CAS  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnic University, Xi’an, 710072, China

    Li Liu & Wei Xu

  2. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

    Li Liu

  3. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    GuiDong Yang & DongMei Huang

Authors
  1. Li Liu
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  2. Wei Xu
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  3. GuiDong Yang
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  4. DongMei Huang
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Corresponding author

Correspondence to Wei Xu.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11872305 & 11532011) and Natural Science Basic Research Plan in Shanxi Province of China (Grant No. 2018JQ1088).

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Liu, L., Xu, W., Yang, G. et al. Reliability and control of strongly nonlinear vibro-impact system under external and parametric Gaussian noises. Sci. China Technol. Sci. 63, 1837–1845 (2020). https://doi.org/10.1007/s11431-020-1626-5

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  • DOI: https://doi.org/10.1007/s11431-020-1626-5

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