ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

The Dynamical Behavior of a Two Degrees of Freedom Oblique Impact System

Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 117--139 | DOI:10.5890/DNC.2020.03.009

Xiaowei Tang$^{1}$,$^{2}$, Xilin Fu$^{1}$, Xiaohui Sun$^{1}$

$^{1}$ School of Mathematics and Statistics, Shandong Normal University, Ji’nan, 250014, P.R. China

$^{2}$ Mathematical School, Qilu Normal University, Ji’nan, 250014, P.R. China

Abstract

The oblique impact phenomena is quite common in practical engineering. In this paper, by using the theory of discontinuous dynamical systems, we studied the complex dynamics behaviors of a two degrees of freedom oblique impact system. We can see that the dynamics of oblique impact is more complex than that of the direct impact. The occurrence or disappearance conditions of sticking motion and grazing motion on the separation boundaries are given in Section 3. The conditions here are necessary and sufficient, which generate better results than those obtained with only sufficient conditions. The results appropriately interpret the physical phenomenon of this oblique impact system, hence validate our conclusions. As a supplement, we also give the analytic conditions of the existence of periodic motions. Numerical simulations for sticking motion and grazing motion are presented at last.

References

1.  [1] Foale, S. and Bishop, S.R. (1994), Bifurcations in impact oscillations, Nonlinear Dynamics, 6(3), 285-299.
2.  [2] Shaw, S.W. and Holmes, P.J. (1983), A periodically forced piecewise linear oscillator, Journal of Sound and Vibration, 90(1), 129-155.
3.  [3] Shaw, S.W. and Holmes, P.J. (1983), A periodically forced impact oscillator with large dissipation, Journal of Applied Mechanics, 50(4), 849-857.
4.  [4] Knudsena, J. andMassiha, A.R. (2003), Dynamic stability of weakly damped oscillators with elastic impacts and wear, Journal of Sound and Vibration, 263(1), 175-204.
5.  [5] Nordmark, A.B. (1991), Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145(2), 279-297.
6.  [6] Luo, G.W., Yu, J.N., Yao, H.M., and Xie, J.H. (2006), Periodic-impact motions and bifurcations of the vibratory system with a clearance, Chinese Journal of Mechanical Eengineering, 42(2), 87-95.
7.  [7] 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(1-2), 132-168.
8.  [8] Wagg, D.J. (2004), Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator, Chaos, Solitons and Fractals, 22(3), 541-548.
9.  [9] Wagg, D.J. (2005), Periodic sticking motion in a two-degree-of-freedom impact oscillator, International Journal of Non-Linear Mechanics, 40(8), 1076-1087.
10.  [10] Han,W., Hu, H.Y., and Jin, D.P. (2007), Experimental study on dynamics of an oblique-impact vibrating system of two degrees of freedom, Nonlinear Dynamics, 50(3), 551-573.
11.  [11] Han,W. and Hou, Z.Q. (2005),Mapping and bifurcation analysis on the periodic motions of the oblique-impact vibrating systems, Journal of Naval Aeronautical Engineering Institute, 20(3), 301-305.
12.  [12] Zhai, H. and Ding, Q. (2013), Stability and nonlinear dynamics of a vibration system with oblique collisions, Journal of Sound and Vibration, 332, 3015-3031.
13.  [13] 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.
14.  [14] 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, 285(1-2), 443-456.
15.  [15] Luo, A.C.J. and O’Connor, Dennis (2007), Nonlinear dynamics of a gear transmission system part I: Mechanism of impacting chatter with stick, Proceedings of the ASME 2007 International Design Engineering Technical Conference and Computers and Information in Engineering Conference, 2007-34881, 251-259.
16.  [16] Luo, A.C.J., and O’Connor, D. (2007), Nonlinear dynamics of a gear transmission system part II: Periodic impacting chatter and stick, Proceedings of the ASME 2007 International Design Engineering Technical Conference and Computers and Information in Engineering Conference, 2007-43192, 1987-2001.
17.  [17] Luo, A.C.J. and O’Connor, D. (2009), Impact Chatter in a gear transmission system with two oscillators, Journal of Multi-body Dynamics, 223(3), 159-188.
18.  [18] Luo, A.C.J. and O’Connor, D. (2009), Periodic motions and chaos with impacting chatter with stick in a gear transmission system, International Journal of Bifurcation and Chaos, 19(6), 2093-2105.
19.  [19] Luo, A.C.J. (2008), A theory for flow switchability in discontinuous dynamical systems, Nonlinear Analysis: Hybrid systems, 2, 1030-1061.
20.  [20] Luo, A.C.J. and Bing, X. (2009), An analytical prediction of periodic flows in the Chua’s circuit system, International Journal of Bifurcation and Chaos, 19(7), 2165-2180.
21.  [21] Luo, A.C.J. and Guo, Y. (2009),Motion switching and chaos of a particle in a generalized Fermi-acceleration oscillator, Mathematical Problems in Engineering, Paper No. 298906.
22.  [22] Luo, A.C.J. and Guo, Y.(2010), Switching Mechanism and Complex Motions in an Extended Fermi-Acceleration Oscillator, Journal of Computational and Nonlinear Dynamics, 5(4), 1-14.
23.  [23] Guo, Y. and Luo, A.C.J. (2011), Discontinuity and bifurcation analysis of motions in a Fermi oscillator under dual excitations, Journal of Vibro engineering, 13(1), 66-101.
24.  [24] Luo, A.C.J. and Guo, Y.(2013), Vibro-Impact Dynamics,Wiley, New York.
25.  [25] Bazhenov,V.A., Lizunov, P.P., Pogorelova, O.S., et al (2015), Stability and bifurcations analysis for 2-DOF vibroimpact system by parameter continuation method. Part I: Loading curve, Journal of Applied Nonlinear Dynamics, 4(4), 357- 370.
26.  [26] Bazhenov, V.A., Lizunov, P.P., Pogorelova, O.S., and Postnikova, T.G. (2016), Numerical bifurcation analysis of discontinuous 2-DOF vibroimpact system. Part 2: Frequency-amplitude response, Journal of Applied Nonlinear Dynamics, 5(3), 269-281.
27.  [27] Zhang, Y.Y. and Fu, X.L.(2015),On periodicmotions of an inclined impact pair, Communications in Nonlinear Science and Numerical Simulation, 20(3), 1033-1042.
28.  [28] Zheng, S.S. and Fu, X.L. (2015), Chatter dynamic analysis for a planing model with the effect of pulse, Journal of Applied Analysis and Computation, 5(4), 767-780.