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Journal of Vibration Testing and System Dynamics 4(4) (2020) 377--388 | DOI:10.5890/JVTSD.2020.12.006

Siyu Guo, Albert C. J. Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

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Abstract

Periodic motions in a discontinuous dynamical system are studied. The discontinuous dynamical system consists of three distinct linear dynamical systems with two different circular boundaries. Analytical conditions for switching and sliding motions at the two circular boundaries are developed. From such analytical conditions, periodic motions in discontinuous dynamical systems can be determined through specific mapping structures. Illustrations give periodic motions in discontinuous dynamical systems, which are different from the continuous dynamical systems. In different domains, the motions are different, and the discontinuity of the periodic motions at the boundaries is observed, and the sliding motion on the boundaries is also a portion of periodic motion. Such periodic motions in discontinuous dynamical systems do not have any Fourier series. The methodology presented in this paper can be applied to other discontinuous dynamical systems. The circular boundaries can be treated different energy levels as control conditions in engineering systems.

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

3. [3]	Aizerman, M.A. and Pyatnitsky, E.S. (1974), Fundamentals of the theory of discontinuous systems, Series 1, Automatic and Remote Control, 35, 1066-1079.

4. [4]	Aizerman, M.A. and Pyatnitsky, E.S. (1974), Fundamentals of the theory of discontinuous systems, Series 2, Automatic and Remote Control, 35, 1241-1262.

5. [5]	Utkin, V. (1977), Variable structure systems with sliding modes, IEEE Transactions on Automatic Control, 22(2), 212-222.

6. [6]	Leine, R.I. and Van Campen, D.H. (2002), Discontinuous bifurcations of periodic solutions, Mathematical and computer modelling, 36(3), 259-273.

7. [7]	Galvanetto, U. (2004), Sliding bifurcations in the dynamics of mechanical systems with dry friction---remarks for engineers and applied scientists, Journal of Sound and Vibration, 276(1-2), 121-139.

8. [8]	Kowalczyka, 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.

9. [9]	Nordmark, A.B. (1991), Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145(2), 279-297.

10. [10]	Kleczka, M., Kreuzer, E., and Schiehlen, W. (1992), Local and global stability of a piecewise linear oscillator, Philosophical Transactions of the Royal Society of London, Series A: Physical and Engineering Sciences, 338(1651), 533-546.

11. [11]	Leine, R.I., Van Campen, D.H., and Van de Vrande, B.L. (2000), Bifurcations in nonlinear discontinuous systems. Nonlinear Dynamics, 23(2), 105-164.

12. [12]	Luo, A.C.J. (2005), A theory for non-smooth dynamical systems on connectable domains, Communication in Nonlinear Science and Numerical Simulation, 10, 1-55

13. [13]	Luo, A.C.J. (2012), Discontinuous Dynamical Systems. Beijing: Higher Education Press.

14. [14]	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), Article No.: 1650224.

15. [15]	Huang, J.Z. and Luo, A.C.J. (2017), Complex dynamics of bouncing motions on boundaries and corners in a discontinuous dynamical system. Journal of Computational {$\&$ Nonlinear Dynamics}, 12(6), Article No.: 061014 (11 pages).