Stick motions and grazing flows in an inclined impact oscillator

Abstract In this paper, the dynamics of an inclined impact oscillator under periodic excitation are investigated using the flow switchability theory of the discontinuous dynamical systems. Different domains and boundaries for such system are defined according to the impact discontinuity. Based on above domains and boundaries, the analytical conditions of the stick motions and grazing motions for the inclined impact oscillator are obtained mathematically, from which it can be seen that such oscillator has more complicated and rich dynamical behaviors. The numerical simulations are given to illustrate the analytical results of complex motions, and several period-1 motions period-2 motion and chaotic motion of the ball in the inclined impact oscillator are also presented. There are more theories about such impact pair to be discussed in future.

[1]  Xilin Fu,et al.  Synchronization of Two Different Dynamical Systems under Sinusoidal Constraint , 2014, J. Appl. Math..

[2]  Albert C. J. Luo,et al.  Discontinuous Dynamical Systems on Time-varying Domains , 2009 .

[3]  F. Peterka,et al.  Bifurcations and transition phenomena in an impact oscillator , 1996 .

[4]  C. Bapat.  Impact-pair under periodic excitation , 1988 .

[5]  A. Luo,et al.  Switching Mechanism and Complex Motions in an Extended Fermi-Acceleration Oscillator , 2010 .

[6]  Xilin Fu,et al.  Chatter dynamic analysis for Van der Pol Equation with impulsive effect via the theory of flow switchability , 2014, Commun. Nonlinear Sci. Numer. Simul..

[7]  Albert C. J. Luo,et al.  A theory for flow switchability in discontinuous dynamical systems , 2008 .

[8]  P. J. Holmes.  The dynamics of repeated impacts with a sinusoidally vibrating table , 1982 .

[9]  Karl Popp,et al.  Dynamics of oscillators with impact and friction , 1997 .

[10]  G. W. Luo,et al.  Controlling bifurcation and chaos of a plastic impact oscillator , 2009 .

[11]  Albert C. J. Luo,et al.  The dynamics of a bouncing ball with a sinusoidally vibrating table revisited , 1996 .

[12]  A. Luo,et al.  Analytical Dynamics of a Ball Bouncing on a Vibrating Table , 2012 .

[13]  Albert C. J. Luo,et al.  Period-doubling induced chaotic motion in the LR model of a horizontal impact oscillator , 2004 .

[14]  Anil K. Bajaj,et al.  Periodic motions and bifurcations in dynamics of an inclined impact pair , 1988 .

[15]  G. Luo,et al.  Analyses of impact motions of harmonically excited systems having rigid amplitude constraints , 2007 .

[16]  Anil K. Bajaj,et al.  On the dynamics and stability of an inclined impact pair , 1986 .

[17]  P. Holmes,et al.  A periodically forced piecewise linear oscillator , 1983 .

[18]  The dynamics of a symmetric impact oscillator between two rigid stops , 2011 .

[19]  H. I. Weber,et al.  Mathematical modeling and experimental investigation of an embedded vibro-impact system , 2011 .

[20]  Xilin Fu,et al.  On periodic motions of an inclined impact pair , 2015, Commun. Nonlinear Sci. Numer. Simul..

[21]  Albert C. J. Luo,et al.  Vibro-impact Dynamics , 2013 .

[22]  Bogusław Radziszewski,et al.  Dynamics of impacts with a table moving with piecewise constant velocity , 2009 .

[23]  Albert C. J. Luo,et al.  A theory for non-smooth dynamic systems on the connectable domains , 2005 .

[24]  A. Luo,et al.  Grazing phenomena and fragmented strange attractors in a harmonically forced, piecewise, linear system with impacts , 2006 .

[25]  Sami F. Masri,et al.  On the stability of the impact damper. , 1966 .

[26]  V. Babitsky,et al.  Excitation of Vibro-Impact Systems by Periodic Impulses , 1999 .

[27]  Bogusław Radziszewski,et al.  Bouncing ball dynamics: Simple model of motion of the table and sinusoidal motion , 2013, 1302.0369.

[28]  Yu Guo,et al.  Parametric Analysis of bifurcation and Chaos in a periodically Driven Horizontal Impact Pair , 2012, Int. J. Bifurc. Chaos.

[29]  N. Popplewell,et al.  Stable periodic motions of an impact-pair , 1983 .

[30]  C. N. Bapat.  The general motion of an inclined impact damper with friction , 1995 .

[31]  Albert C. J. Luo,et al.  Periodic motions in a simplified brake system with a periodic excitation , 2009 .

[32]  Albert C. J. Luo,et al.  Analytical Dynamics of Complex Motions in a Train Suspension System , 2011 .