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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 Dynamics of a Discontinuous Dynamical System with a Hyperbolic Boundary

Journal of Vibration Testing and System Dynamics 5(3) (2021) 285--319 | DOI:10.5890/JVTSD.2021.09.009

Albert C. J. Luo , Chuanping Liu

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

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Abstract

In this paper, analytical dynamics of a discontinuous system with a hyperbolic control boundary is studied. The analytical conditions for flow switchability of the discontinuous system are developed from a flow passability theory at boundaries, which are for a better understanding of dynamics of such a discontinuous dynamical system. With the analytical switchability conditions, the sliding and non-sliding motions are described through generic mappings. Periodic motions in the discontinuous dynamical system with hyperbolic boundary are studied through mapping structures. The bifurcation trees of periodic motions are presented, and the corresponding stability and bifurcation of periodic motions are analyzed. The grazing bifurcation for a flow to the boundary is discussed as periodic motion switching. Numerical simulations are completed for illustration of complex periodic motions and flow switchability at the hyperbolic boundary. This paper is dedicated to Nail Ibragimov as a good friend and colleague for 20 years friendship with Albert Luo.

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, The Journal of the Acoustical Society of America, 32, 1265-1269.
  3. [3]  Filippov, A.F. (1964), Differential equations with discontinuous right-hand side. Am. Math. Soc. Trans. Ser., 42(2), 199--231.
  4. [4]  Filippov, A.F. (1988), Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht,
  5. [5]  Aizerman, M.A. and Pyatnitsky, E.S. (1974), Fundamentals of the theory of discontinuous systems, Series 1, Automatic and Remote Control, 35, 1066--1079.
  6. [6]  Aizerman, M.A. and Pyatnitsky, E.S. (1974), Fundamentals of the theory of discontinuous systems, Series 2, Automatic and Remote Control, 35: 1241--1262.
  7. [7]  Utkin, V. (1977), Variable structure systems with sliding modes, IEEE, Transactions on Automatic Control, 22(2), pp.212-222.
  8. [8]  Shaw, S.W. and Holmes, P.J. (1983), A periodically forced piecewise linear oscillator, Journal of Sound and Vibration, 90(1), 129-155.
  9. [9]  Han, R.P.S., Luo, A.C.J., and Deng, W. (1995), Chaotic motion of a horizontal impact pair, Journal of Sound and Vibration, 181, 231-250.
  10. [10]  Luo, A.C.J. and Han, R.P.S. (1996) The dynamics of a bouncing ball with a sinusoidally vibrating table revisited, Nonlinear Dynamics, 10(1), 1-18.
  11. [11]  Luo, A.C.J. (2005), The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation, Journal of Sound and Vibration, 283, 723-748.
  12. [12]  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(1), 41-51.
  13. [13]  Bazhenov, V.A., Lizunov, P.P., Pogorelova, O.S., Postnikova, T.G., and Otrashevskaia, V.V. (2015), Stability and bifurcations analysis for 2-DOF vibro-impact system by parameter continuation method. Part I: loading curve, Journal of Applied Nonlinear Dynamics, 4(4), 357--370.
  14. [14]  Bazhenov, V.V., Pogorelova, O.S., and Postnikova, T.G. (2019), Breakup of closed curve-quasiperiodic route to chaos in vibro-impact system, Discontinuity, Nonlinearity, and Complexity, 8(3), 299-311.
  15. [15]  Akhmet, M.U. and K\i v\i lc\i m, A. (2018), van der Pol oscillators generated from grazing dynamics, Discontinuity, Nonlinearity, and Complexity, 7(3), 259-274.
  16. [16]  Luo, A.C.J. (2005), A theory for non-smooth dynamic systems on the connectable domains, Communications in Nonlinear Science and Numerical Simulation, 10(1), 1-55.
  17. [17]  Luo, A.C.J. (2005), Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamic systems, Journal of Sound and Vibration, 28, 443-456.
  18. [18]  Luo, A.C.J. and Gegg, B.C. (2006), Stick and non-stick periodic motions in periodically forced oscillators with dry friction, Journal of Sound and Vibration, 291, 132-168.
  19. [19]  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
  20. [20]  Luo, A.C.J. and Gegg, B.C. (2006), Dynamics of a harmonically excited oscillator with dry-friction on a sinusoidally time-varying, traveling surface, International Journal of Bifurcation and Chaos, 16, 3539-3566.
  21. [21]  Luo, A.C.J. (2008), A theory for flow switchability in discontinuous dynamical systems, Nonlinear Analysis: Hybrid Systems, 2(4), 1030-1061.
  22. [22]  Luo, A.C.J. and Rapp, B.M. (2009), Flow switchability and periodic motions in a periodically forced, discontinuous dynamical system, Nonlinear Analysis: Real World Applications, 10(5), 3028-3044.
  23. [23]  Luo, A.C.J. and Rapp, B.M. (2010), On motions and switchability in a periodically forced, discontinuous system with a parabolic boundary. Nonlinear Analysis: Real World Applications, 11(14), 2624-2633.
  24. [24]  Luo, A.C.J. (2012), Discontinuous Dynamical Systems, HEP/Springer: Beijing/ Berlin.
  25. [25]  Luo, A.C.J. and Huang, J.Z. (2017), Complex dynamics of bouncing motions on boundaries and corners in a discontinuous dynamical system, Journal of Computational and Nonlinear Dynamics, 12(6), 061014 (11).
  26. [26]  Li, L. and Luo, A.C.J. (2016), Periodic orbits in a second-order discontinuous system with an elliptic boundary, International Journal of Bifurcation and Chaos, 26(13), 1650224.
  27. [27]  Tang, X., Fu, X., and Sun, X. (2019), Periodic motion for an oblique impact system with single degree of freedom, Journal of Vibration Testing and Systems Dynamics, 3(1), 71-89.
  28. [28]  Tang, X., Fu, X., and Sun, X. (2020), The dynamical behavior of a two degrees of freedom oblique impact system, Discontinuity, Nonlinearity, and Complexity, 9(1), 117-139.
  29. [29]  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.
  30. [30]  Guo, S. and Luo, A.C.J. (2021), Constructed limit cycles in a discontinuous dynamical system with multiple vector fields, Journal of Vibration Testing and System Dynamics, 5(1), 33-51.
  31. [31]  Guo, S. and Luo, A.C.J. (2021) A parameter study on periodic motions in a discontinuous dynamical system with two circular boundaries, Discontinuity, Nonlinearity, and Complexity, 10(2), 289-309.
  32. [32]  Luo, A.C.J. and Li, L.P. (2016), Periodic orbits and bifurcations in discontinuous system with a hyperbolic boundary, International Journal of Dynamics and Control, 5, 513-529.
  33. [33]  Luo, A.C.J. and Liu, C.P. (2020), Analytical periodic motions in a discontinuous system with a switching hyperbola, International Journal of Dynamics and Control, DOI:10.1007/s40435-020-00648-5.