Nonlinear Dynamics

, Volume 96, Issue 3, pp 2001–2011 | Cite as

Stochastic response and bifurcations of a dry friction oscillator with periodic excitation based on a modified short-time Gaussian approximation scheme

  • Qun HanEmail author
  • Xiaole Yue
  • Hongmei Chi
  • Shun Chen
Original Paper


In this paper, a modified short-time Gaussian approximation (STGA) scheme is incorporated into the generalized cell mapping (GCM) method to study the stochastic response and bifurcations of a nonlinear dry friction oscillator with both periodic and Gaussian white noise excitations. Firstly, the Fokker–Planck–Kolmogorov equation and the moment equations with Gaussian closure are derived. Because of the periodic force and the singularity of moment equations in the initial condition being a Dirac delta function, a set of novel STGA solutions of the conditional probability density functions (PDFs) over each small fraction of one period is then constructed in order to form the solution over the whole period. At last, the transient and steady-state response PDFs are computed by the GCM method. The accuracy of the results is demonstrated by the direct Monte Carlo simulations. The transient PDFs are presented when evolving from a Gaussian initial distribution to a non-Gaussian steady-state one. The effects of periodic excitation and dry friction damping on the steady-state stochastic response are particularly discussed. When the amplitude of periodic excitation is changed, both stochastic P-bifurcation and D-bifurcation are detected. It is also found that the increase in the dry friction damping coefficient makes the chaotic response disappear gradually, inducing stochastic D-bifurcation but no stochastic P-bifurcation.


Stochastic bifurcations Dry friction damping Periodic excitation Short-time Gaussian approximation Generalized cell mapping 



This work was funded by the National Natural Science Foundation of China (Grant Nos. 11672230, 11532011, 11702214) and the Fundamental Research Funds for the Central Universities (Grant Nos. 2662017QD024, 2662018JC007).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Hartog, J.P.D.: Forced vibrations with combined coulomb and viscous friction. Trans. ASME 53(9), 107–115 (1931)zbMATHGoogle Scholar
  2. 2.
    Pennestr, E., Rossi, V., Salvini, P., Valentini, P.P.: Review and comparison of dry friction force models. Nonlinear Dyn. 83(4), 1785–1801 (2016)zbMATHGoogle Scholar
  3. 3.
    Pierre, C., Ferri, A.A., Dowell, E.H.: Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. J. Appl. Mech. 52(4), 958–964 (1985)zbMATHGoogle Scholar
  4. 4.
    Stefaski, A., Wojewoda, J., Furmanik, K.: Experimental and numerical analysis of self-excited friction oscillator. Chaos Solitons Fractals 12(9), 1691–1704 (2001)zbMATHGoogle Scholar
  5. 5.
    Ding, Q., Chen, Y.: Analyzing resonant response of a system with dry friction damper using an analytical method. J. Vib. Control 14(8), 1111–1123 (2008)zbMATHGoogle Scholar
  6. 6.
    Feeny, B., Guran, A., Hinrichs, N., Popp, K.: A historical review on dry friction and stick-slip phenomena. Appl. Mech. Rev. 51(5), 321–341 (1998)Google Scholar
  7. 7.
    Bellido, F., Ramrez-Malo, J.B.: Periodic and chaotic dynamics of a sliding driven oscillator with dry friction. Int. J. Non-Linear Mech. 41(6–7), 860–871 (2006)zbMATHGoogle Scholar
  8. 8.
    Yang, F.H., Zhang, W., Wang, J.: Sliding bifurcations and chaos induced by dry friction in a braking system. Chaos Solitons Fractals 40(3), 1060–1075 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Pascal, M.: New limit cycles of dry friction oscillators under harmonic load. Nonlinear Dyn. 70, 1435–1443 (2016)MathSciNetGoogle Scholar
  10. 10.
    Feng, Q.: A discrete model of a stochastic friction system. Comput. Methods Appl. Mech. Eng. 192, 2339–2354 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Rong, H.W., Wang, X.D., Xu, W., Fang, T.: Resonance response of a single-degree-of-freedom nonlinear dry system to a randomly disordered periodic excitation. Acta Phys. Sin. 58(11), 7558–7564 (2009)Google Scholar
  12. 12.
    Chen, Y., Just, W.: First-passage time of Brownian motion with dry friction. Phys. Rev. E 89(2), 022103 (2014)Google Scholar
  13. 13.
    Tian, Y.P., Wang, Y., Jin, X.L., Huang, Z.L.: Optimal load resistance of a randomly excited nonlinear electromagnetic energy harvester with Coulomb friction. Smart Mater. Struct. 23(9), 095001 (2014)Google Scholar
  14. 14.
    Sun, J.J., Xu, W., Lin, Z.F., Zhou, Y.: Random responses analysis of friction systems with viscoelastic forces under Gaussian colored noise excitation. Appl. Math. Mech. 38(1), 109–117 (2017)Google Scholar
  15. 15.
    Sun, J.J., Xu, W., Lin, Z.F.: Research on the reliability of friction system under combined additive and multiplicative random excitations. Commun. Nonlinear Sci. Numer. Simul. 54, 1–12 (2018)MathSciNetGoogle Scholar
  16. 16.
    Kumar, P., Narayanan, S., Gupta, S.: Stochastic bifurcation analysis of a Duffing oscillator with Coulomb friction excited by Poisson white noise. Proc. Eng. 144, 998–1006 (2016)Google Scholar
  17. 17.
    Sun, J.-Q.: Random vibration analysis of a non-linear system with dry friction damping by the short-time Gaussian cell mapping method. J. Sound Vib. 180(5), 785–795 (1995)Google Scholar
  18. 18.
    Kapitaniak, T.: Stochastic response with bifurcations to non-linear Duffing’s oscillator. J. Sound Vib. 102(3), 440–441 (1985)MathSciNetGoogle Scholar
  19. 19.
    Kapitaniak, T.: Chaotic distribution of non-linear systems perturbed by random noise. Phys. Lett. A 116(6), 251–254 (1986)MathSciNetGoogle Scholar
  20. 20.
    Gan, C.B., Lei, H.: Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations. J. Sound Vib. 330(10), 2174–2184 (2011)Google Scholar
  21. 21.
    Xu, Y., Liu, Q., Guo, G.B., Xu, C., Liu, D.: Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance. Nonlinear Dyn. 89, 1579–1590 (2017)MathSciNetGoogle Scholar
  22. 22.
    Xu, Y., Ma, J.Z., Wang, H.Y., Li, Y.G., Kurths, J.: Effects of combined harmonic and random excitations on a Brusselator model. Eur. Phys. J. B 90(10), 194 (2017)MathSciNetGoogle Scholar
  23. 23.
    Chen, Z., Liu, X.B.: Noise induced transitions and topological study of a periodically driven system. Commun. Nonlinear Sci. Numer. Simul. 48, 454–461 (2017)MathSciNetGoogle Scholar
  24. 24.
    Sun, Y.H., Hong, L., Jiang, J.: Stochastic sensitivity analysis of nonautonomous nonlinear systems subjected to Poisson white noise. Chaos Solitons Fractals 104, 508–515 (2017)zbMATHGoogle Scholar
  25. 25.
    Kashani, H.: Analytical parametric study of bi-linear hysteretic model of dry friction under harmonic, impulse and random excitations. Nonlinear Dyn. 11, 1–13 (2017)Google Scholar
  26. 26.
    Liu, W.Y., Zhu, W.Q., Chen, L.C.: Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations. Probab. Eng. Mech. 53, 109–115 (2018)Google Scholar
  27. 27.
    Hsu, C.S.: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. Springer, New York (1987)zbMATHGoogle Scholar
  28. 28.
    Hong, L., Xu, J.X.: Chaotic saddles in Wada basin boundaries and their bifurcations by the generalized cell-mapping digraph (GCMD) method. Nonlinear Dyn. 32(4), 371–385 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Xu, W., He, Q., Fang, T., Rong, H.W.: Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise. Int. J. Non-Linear Mech. 39(9), 1473–1479 (2004)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Liu, X.M., Jiang, J., Hong, L., Tang, D.F.: Global bifurcation involving Wada boundary metamorphosis by a method of generalized cell mapping with sampling-adaptive interpolation. Int. J. Bifurc. Chaos 28(2), 1830003 (2018)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Yue, X.L., Xu, W., Wang, L., Zhou, B.C.: Transient and steady-state responses in a self-sustained oscillator with harmonic and bounded noise excitations. Probab. Eng. Mech. 30, 70–76 (2012)Google Scholar
  32. 32.
    Han, Q., Xu, W., Yue, X.L.: Exit location distribution in the stochastic exit problem by the generalized cell mapping method. Chaos Solitons Fractals 87, 302–306 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hong, L., Jiang, J., Sun, J.-Q.: Response analysis of fuzzy nonlinear dynamical systems. Nonlinear Dyn. 78(2), 1221–1232 (2014)MathSciNetGoogle Scholar
  34. 34.
    Sun, J.-Q., Hsu, C.S.: The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation. J. Appl. Mech. 57(4), 1018–1025 (1990)MathSciNetGoogle Scholar
  35. 35.
    Li, Z.G., Jiang, J., Hong, L.: Noise-induced transition in a piecewise smooth system by generalized cell mapping method with evolving probabilistic vector. Nonlinear Dyn. 88(2), 1473–1485 (2017)MathSciNetGoogle Scholar
  36. 36.
    Han, Q., Xu, W., Sun, J.-Q.: Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method. Phys. A Stat. Mech. Appl. 458, 115–125 (2016)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Wehner, M.F., Wolfer, W.G.: Numerical evaluation of path-integral solutions to Fokker–Planck equations. Phys. Rev. A 27(5), 2663–2670 (1983)Google Scholar
  38. 38.
    Risken, H.: The Fokker–Planck Equation, Methods of Solution and Application, 2nd edn. Springer, Berlin (1989)zbMATHGoogle Scholar
  39. 39.
    Sun, J.-Q.: Stochastic Dynamics and Control. Elsevier, Amsterdam (2006)Google Scholar
  40. 40.
    Wu, W.F., Lin, Y.K.: Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations. Int. J. Non-Linear Mech. 19(4), 349–362 (1984)zbMATHGoogle Scholar
  41. 41.
    Sun, J.-Q., Hsu, C.S.: Cumulant-neglect closure method for nonlinear systems under random excitations. J. Appl. Mech. 54(3), 649–655 (1987)zbMATHGoogle Scholar
  42. 42.
    Kunze, M.: On Lyapunov exponents for non-smooth dynamical systems with an application to a pendulum with dry friction. J. Dyn. Differ. Equ. 12(1), 31–116 (2000)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of ScienceHuazhong Agricultural UniversityWuhanChina
  2. 2.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anChina

Personalised recommendations