---

Journal of Vibration Testing and System Dynamics 5(1) (2021) 33--51 | DOI:10.5890/JVTSD.2021.03.003

Siyu Guo, Albert C.J. Luo

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

Download Full Text PDF

 

Abstract

In this paper, limit cycles in a discontinuous dynamical system with different vector fields in different domains are constructed. Such limit cycles are obtained through algebraic equations based on specific mapping structures. The stability and bifurcations of limit cycles are studied through eigenvalue analysis. The grazing and sliding bifurcations on the bifurcation trees are presented. Once a grazing or sliding bifurcation occurs, a limit cycle switches to a new limit cycle or vanishes. Such bifurcations (i.e., saddle-node bifurcation, Neimark bifurcation, period-doubling bifurcation, grazing bifurcation, and sliding bifurcation) are for onset and vanishing of limit cycles at specific parameters. This study presents how to construct limit cycles in discontinuous dynamical systems and how to develop the corresponding mathematical conditions of motion switchability at boundaries.

References

1. [1]	Hartog, J.P.D. (1931), Forced vibrations with combined coulomb and viscous friction, Transaction of the American Society of Mechanical Engineering, 53, 107-115.

2. [2]	Filippov, A.F. (1964), Differential equations with discontinuous right-hand side, American Mathematical Society Translations, Series 2 42,199-231.

3. [3]	Filippov, A.F. (1967), Classical solutions of differential equations with multi-valued right-hand side, SIAM Journal on Control, 5(4), 609-621.

4. [4]	Filippov, A.F. (1988), Differential Equations with Discontinuous with Right-hand Sides, Kluwer Academic, Netherlands.

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), 212-222.

8. [8]	Masri, S.F. (1970), General motion of impact dampers, The Journal of the Acoustical Society of America, 47(1B), 229-237.

9. [9]	Shaw, S.W. and Holmes, P.J. (1983), A periodically forced piecewise linear oscillator, Journal of sound and vibration, 90(1), 129-155.

10. [10]	Leine, R.I. and Van Campen, D.H. (2002), Discontinuous bifurcations of periodic solutions, Mathematical and computer modelling, 36(3), 259-273.

11. [11]	Kowalczyk, P. and Piiroinen, P.T. (2008), Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator, Physica D: Nonlinear Phenomena, 237(8), 1053-1073.

12. [12]	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.

13. [13]	Bazhenov, V.V., Pogorelova, O.S., and Postnikova, T.G. (2019), Breakup of closed curve-quasiperiodic route to chaos in vibro-impact dystem, Discontinuity, Nonlinearity, and Complexity, 8(3), 299-311.

14. [14]	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.

15. [15]	Luo, A.C.J. (2006), Singularity and Dynamics on Discontinuous Vector Fields, Elsevier, Amsterdam.

16. [16]	Luo, A.C.J. and Chen. L.D. (2005), Periodic motion and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons {$\&$ Fractals}, 24, 567--78.

17. [17]	Luo, A.C.J. and Gegg, B.C. (2006), Stick and non-stick periodic motions in a periodically forced oscillator with dry-friction, Journal of Sound and Vibration, 291, 132--68.

18. [18]	Luo, A.C.J. and Gegg, B.C. (2006), On the mechanism of stick and non-stick, periodic motions in a forced linear oscillator including dry friction, ASME Journal of Vibration and Acoustics, 128, 97-105.

19. [19]	Luo, A.C.J. (2008), A theory for flow switchability in discontinuous dynamical systems, Nonlinear Analysis: Hybrid systems, 4, 1030-1061.

20. [20]	Luo, A.C.J. (2011), On flow barriers and switchability in discontinuous dynamical systems, International Journal of Bifurcation and Chaos, 21(1), 1-76.

21. [21]	Luo, A.C.J. (2012), Discontinuous Dynamical Systems, HEP/ Springer: Beijing/Berlin.

22. [22]	Huang, J.Z. and Luo, A.C.J. (2017), Complex dynamics of bouncing motions at boundaries and corners in a discontinuous dynamical system, ASME Journal of Computational and Nonlinear Dynamics, 12, 061014 (11 pages).

23. [23]	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(3), 71-89.

24. [24]	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.

25. [25]	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.

26. [26]	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), in press.