Dynamic Analysis of Stochastic Friction Systems Using the Generalized Cell Mapping Method

  • Shichao MaEmail author
  • Xin NingEmail author
  • Liang Wang
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 75)


Friction systems are a kind of typical non-smooth systems in the actual engineering and often generates complicated dynamics. It is difficult to handle this systems by conventional analysis methods directly. In this context, we investigate the stochastic responses of friction systems using the generalized cell mapping method under random excitation in this paper. To verify the accuracy and validate the applicability of the suggested approach, we present two classical nonlinear friction systems, i.e., Coulomb force model and Dahl force model as examples. Meanwhile, this method is in good agreement with Monte Carlo simulation method and the computation time is greatly reduced. In addition, further discussion finds that the adjustable parameters can induce the stochastic P-bifurcation in the two examples, respectively.


Friction systems Dynamic analysis Stochastic responses Cell mapping 



This work was supported by the National Science Foundation of China through the Grants (11872306, 11772256), the Central University Fundamental Research Fund (3102018zy043).


  1. 1.
    Awrejcewicz, J., Olejnik, P.: Analysis of dynamic systems with various friction laws. ASME Appl. Mech. Rev. 58, 389–411 (2005)CrossRefGoogle Scholar
  2. 2.
    Xu, W., Wang, L., Feng, J.Q., Qiao, Y., Han, P.: Some new advance on the research of stochastic non-smooth systems. Chin. Phys. B. 27, 110503-110501-110506 (2018)CrossRefGoogle Scholar
  3. 3.
    Astrom, K.J., de Wit, C.C.: Revisiting the LuGre friction model, stick-slip motion and friction dependence. IEEE Control Syst. Mag. 28, 101–114 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berger, E.J.: Friction modeling for dynamic system simulation. ASME Appl. Mech. Rev. 55, 535–577 (2002)CrossRefGoogle Scholar
  5. 5.
    Olsson, H., Astrom, K.J., de Wit, C.C., Gafvert, M., Lischinsky, P.: Friction models and friction compensation. Eur. J. Control. 4, 176–195 (1998)CrossRefGoogle Scholar
  6. 6.
    Baule, A., Touchette, H., Cohen, E.G.D.: Stick-slip motion of solids with dry friction subject to random vibrations and an external field. Nonlinearity 24, 351–372 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kumar, P., Narayanan, S., Gupta, S.: Stochastic bifurcation analysis of a Duffing oscillator with Coulomb friction excited by Poisson white noise. Procedia Eng. 144, 998–1006 (2016)CrossRefGoogle Scholar
  8. 8.
    Guerine, A., El Hami, A., Walha, L., Fakhfakh, T., Haddar, M.: Dynamic response of a Spur gear system with uncertain friction coefficient. Adv. Eng. Softw. 000, 1–10 (2016)zbMATHGoogle Scholar
  9. 9.
    Fang, Y.N., Liang, X.H., Zuo, M.J.: Effects of friction and stochastic load on transient characteristics of a spur gear pair. Nonlinear Dyn. 93, 599–609 (2018)CrossRefGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jin, X.L., Wang, Y., Huang, Z.L.: Approximately analytical technique for random response of LuGre friction system. Int. J. Non-Linear Mech. 104, 1–7 (2018)CrossRefGoogle Scholar
  12. 12.
    Hsu, C.S.: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear System. Springer, New York (1987)CrossRefGoogle Scholar
  13. 13.
    Hong, L., Xu, J.X.: Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys. Lett. A 262, 361–375 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Tongue, B.H., Gu, K.Q.: A theory basis for interpolated cell mapping. SIAM J. Appl. Math. 48(5), 1206–1214 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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, 785–795 (1995)CrossRefGoogle Scholar
  16. 16.
    Yue, X.L., Xu, W., Jia, W.T., Wang, L.: Stochastic response of a φ6 oscillator subjected to combined harmonic and Poisson white noise excitations. Phys. A 392, 2988–2998 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wang, L., Xue, L.L., Xu, W., Yue, X.L.: Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method. Int. J. Non-linear Mech. 96, 56–63 (2017)CrossRefGoogle Scholar
  18. 18.
    Wang, L., Ma, S.C., Sun, C.Y., Jia, W.T., Xu, W.: Stochastic response of a class of impact systems calculated by a new strategy based on generalized cell mapping method. ASME J. Appl. Mech. 85, 054502 (2018)CrossRefGoogle Scholar
  19. 19.
    Yue, X.L., Xu, Y., Xu, W., Sun, J.Q.: Probabilistic response of dynamicsl systems based on the global attractor with the compatible cell mapping method. Phys. A 516, 509–519 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhu, W.Q., Cai, G.Q.: Introduction to Stochastic dynamics. Science and Technology Press (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Northwestern Polytechnical University School of AstronauticsXi’anPeople’s Republic of China
  2. 2.National Key Laboratory of Aerospace Flight DynamicsXi’anPeople’s Republic of China
  3. 3.Northwestern Polytechnical University School of ScienceXi’anPeople’s Republic of China

Personalised recommendations