[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Vibration Testing and System Dynamics
ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

Analytical Periodic Motions and Bifurcations in a Frictional Oscillator on Displacement-Dependent Conveyor Systems

Journal of Vibration Testing and System Dynamics 10(4) (2026) 403--418 | DOI:10.5890/JVTSD.2026.12.007

Ziyi Li, Albert C. J. Luo

Department of Mechanical Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA

Download Full Text PDF

 

Abstract

In this paper, dynamics of a frictional oscillator on the displacement-related belt boundary is studied. To understand dynamics of such a frictional oscillator, analytical conditions of motion switching on the nonlinear boundary are developed. For each domain, the linear oscillator solutions are given. From such solutions and motion switching conditions, periodic motions of the frictional oscillator is obtained, and numerical results give illustrations of periodic motions with switchability conditions at the boundary. From such methodology, dynamics of nonlinear frictional oscillators is studied in sequel. The analytical framework developed in this paper can be applied for discontinuous dynamical systems with state-dependent boundaries, the analytical conditions for grazing, sliding, and passable motions can be developed.

References

  1. [1]  Den Hartog, J.P. (1931), Forced vibrations with Coulomb and viscous damping, Transactions of the American Society of Mechanical Engineers, 53, 107-115.
  2. [2]  Levitan, E.S. (1960), Forced oscillation of a spring-mass system having combined Coulomb and viscous damping, Journal of the Acoustical Society of America, 32, 1265-1269.
  3. [3]  Filippov, A.F. (1964), Differential equations with discontinuous right-hand side, American Mathematical Society Translations, Series 2, 42, 199-231.
  4. [4]  Filippov, A.F. (1988), Differential equations with discontinuous righthand sides, Kluwer Academic Publishers, Dordrecht.
  5. [5]  Aubin, J.P. and Cellina, A. (1984), Differential inclusions, Springer-Verlag, Berlin.
  6. [6]  Aubin, J.P. and Frankowska, H. (1989), Set-valued analysis, Birkhauser, Boston.
  7. [7]  Hundal, M.S. (1979), Response of a base excited system with Coulomb and viscous friction, Journal of Sound and Vibration, 64, 371-378.
  8. [8]  Shaw, S.W. (1986), On the dynamic response of a system with dry-friction, Journal of Sound and Vibration, 108, 305-325.
  9. [9]  Feeny, B.F. and Moon, F.C. (1994), Chaos in a forced dry-friction oscillator: experiments and numerical modeling, Journal of Sound and Vibration, 170, 303-323.
  10. [10]  Hinrichs, N., Oestreich, M., and Popp, K. (1997), Dynamics of oscillators with impact and friction, Chaos, Solitons and Fractals, 8(4), 535-558.
  11. [11]  Hinrichs, N., Oestreich, M., and Popp, K. (1998), On the modeling of friction oscillators, Journal of Sound and Vibration, 216(3), 435-459.
  12. [12]  Natsiavas, S. (1998), Stability of piecewise linear oscillators with viscous and dry friction damping, Journal of Sound and Vibration, 217, 507-522.
  13. [13]  Leine, R.I., Van Campen, D.H., De Kraker, A., and van Den Steen, L. (1998), Stick-slip vibrations induced by alternate friction models, Nonlinear Dynamics, 16, 41-54.
  14. [14]  Ko, P.L., Taponat, M.-C., and Pfaifer, R. (2001), Friction-induced vibration-with and without external disturbance, Tribology International, 34, 7-24.
  15. [15]  Andreaus, U. and Casini, P. (2002), Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle, International Journal of Non-Linear Mechanics, 37, 117-133.
  16. [16]  Thomsen, J.J. and Fidlin, A. (2003), Analytical approximations for stick-slip vibration amplitudes, International Journal of Non-Linear Mechanics, 38, 389-403.
  17. [17]  Pilipchuk, V.N. and Tan, C.A. (2004), Creep-slip capture as a possible source of squeal during decelerating sliding, Nonlinear Dynamics, 35, 258-285.
  18. [18]  Li, Y. and Feng, Z.C. (2004), Bifurcation and chaos in friction-induced vibration, Communications in Nonlinear Science and Numerical Simulation, 9, 633-647.
  19. [19]  Luo, A.C.J. (2005), A theory for non-smooth dynamical systems on connectable domains, Communications in Nonlinear Science and Numerical Simulation, 10(1), 1-55.
  20. [20]  Luo, A.C.J. (2006), Singularity and dynamics on discontinuous vector fields, Elsevier, Amsterdam.
  21. [21]  Luo, A.C.J. and Gegg, B.C. (2006), On the mechanism of stick and nonstick, periodic motions in a periodically forced, linear oscillator with dry friction, Journal of Vibration and Acoustics, 128(1), 97-105.
  22. [22]  Luo, A.C.J. and Gegg, B.C. (2006), Stick and non-stick periodic motions in a periodically forced, linear oscillator with dry friction, Journal of Sound and Vibration, 291, 132-168.
  23. [23]  Luo, A.C.J. and Gegg, B.C. (2007), An analytical prediction of sliding motions along discontinuous boundary in non-smooth dynamical systems, Nonlinear Dynamics, 49, 401-424.
  24. [24]  Gegg, B.C. and Luo, A.C.J. (2006), Dynamics of a harmonically excited oscillator with dry-friction on a sinusoidally time-varying, traveling surface, International Journal of Bifurcation and Chaos, 16(12), 3539-3566.
  25. [25]  Luo, A.C.J. and Gegg, B.C. (2006), Periodic motions in a periodically forced oscillator moving on an oscillating belt with dry friction, Journal of Computational and Nonlinear Dynamics, 1(3), 212-220.
  26. [26]  Gegg, B.C., Luo, A.C.J., and Suh, S.C. (2008), Grazing bifurcations of a harmonically excited oscillator moving on a time-varying translation belt, Nonlinear Analysis: Real World Applications, 9, 2156-2174.
  27. [27]  Gegg, B.C., Luo, A.C.J., and Suh, S.C. (2009), Sliding motions on a periodically time-varying, translation belt, Journal of Vibration and Control, 15(5), 671-703.
  28. [28]  Luo, A.C.J. (2009), Discontinuous dynamical systems on time-varying domains, Springer, London.
  29. [29]  Luo, A.C.J. (2012), Discontinuous dynamical systems, Higher Education Press/ Springer, Beijing/ New York.
  30. [30]  Luo, A.C.J. and Huang, J.Z. (2012), Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator, Nonlinear Analysis: Real World Applications, 13, 241-257.
  31. [31]  Zheng, S. and Fu, X.L. (2013), Chatter dynamics on impulse surfaces in impulsive differential systems, Journal of Applied Nonlinear Dynamics, 2(4), 373-396.
  32. [32]  Tang, X.W., Fu, X.L., and Sun, X.H. (2020), The dynamical behavior of a two degrees of freedom oblique impact system, Discontinuity, Nonlinearity, and Complexity, 9(1), 117-139.
  33. [33]  Guo, S. and Luo, A.C.J. (2020), An analytical prediction of periodic motions in a discontinuous dynamical system, Journal of Vibration Testing and System Dynamics, 4(4), 377-388.
  34. [34]  Luo, A.C.J. and Liu, C.P. (2021), Analytical dynamics of a discontinuous dynamical system with a hyperbolic boundary, Journal of Vibration Testing and System Dynamics, 5(3), 285-319.
  35. [35]  Niknam, A. and Farhang, K. (2021), Stick-slip instability in a compliant bistable double-slider mechanism, Journal of Applied Nonlinear Dynamics, 10(4), 775-789.
  36. [36]  Kammogne, A.S.T., Tchaptchet, M.N., Siewe, M.S., Louodop, P., and Kenne, G. (2024), Effective control scheme of PWM universal motor with Coulomb friction based on particle swarm optimization, Journal of Applied Nonlinear Dynamics, 13(1), 155-175.
  37. [37]  Kuo, C.-W. and Suh, C.S. (2016), Mitigating grazing bifurcation and vibro-impact instability in time-frequency domain, Journal of Applied Nonlinear Dynamics, 5(2), 169-184.
  38. [38]  Akhmet, M.U. and Kivilcim, A. (2017), An impact oscillator with a grazing cycle, Discontinuity, Nonlinearity, and Complexity, 6(2), 105-111.

Copyright © 2011-2026 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates