On discontinuous dynamical behaviors of a 2-DOF impact oscillator with friction and a periodically forced excitation

Abstract In this paper, a 2-DOF (two-degree-of-freedom) impact oscillator with friction and a periodically forced excitation is investigated via using the flow switchability theory in discontinuous dynamical systems. Based on the discontinuities caused by friction and impact between the two masses, the phase space is partitioned into different boundaries and domains. Using the G-functions and vector fields, the analytical conditions of grazing motions and passable motions are discussed, and the appearing and vanishing conditions of sliding motions and side-stick motions are also developed. The periodic motions with stick or non-stick are described through the generic mappings. For better understanding of the analytical conditions of periodic motions, grazing motions, stick motions and passable motions, the velocity and displacement time-histories, G-function responses and trajectories are presented. The investigation on such a 2-DOF impact oscillator with friction may be helpful for achieving optimal design of the single row cylindrical roller bearing systems. Besides, it has an important significance to the noise suppression in mechanical systems with clearance.

[1]  Balakumar Balachandran,et al.  Nonlinear dynamics of milling processes , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  Tomasz Kapitaniak,et al.  Dynamics of impact oscillator with dry friction , 1996 .

[3]  Zhaoxia Yang,et al.  Analysis of dynamical behaviors of a 2-DOF vibro-impact system with dry friction , 2018, Chaos, Solitons & Fractals.

[4]  Analysis of dynamical behaviors of a double belt friction-oscillator model , 2016 .

[5]  Albert C. J. Luo,et al.  Mechanism of Impacting Chatter with Stick in a Gear Transmission System , 2009, Int. J. Bifurc. Chaos.

[6]  Balakumar Balachandran,et al.  Dynamics of an Elastic Structure Excited by Harmonic and Aharmonic Impactor Motions , 2003 .

[7]  R. I. Zadoks,et al.  A NUMERICAL STUDY OF AN IMPACT OSCILLATOR WITH THE ADDITION OF DRY FRICTION , 1995 .

[8]  Jean W. Zu,et al.  Dynamics of a dry friction oscillator under two-frequency excitations , 2004 .

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

[10]  Steven W. Shaw,et al.  The Dynamics of a Harmonically Excited System Having Rigid Amplitude Constraints, Part 2: Chaotic Motions and Global Bifurcations , 1985 .

[11]  C. Glocker,et al.  Application of the nonsmooth dynamics approach to model and analysis of the contact-impact events in cam-follower systems , 2012 .

[12]  Martin Bohner,et al.  Impulsive differential equations: Periodic solutions and applications , 2015, Autom..

[13]  Xiaodi Li,et al.  Stabilization of Delay Systems: Delay-Dependent Impulsive Control , 2017, IEEE Transactions on Automatic Control.

[14]  Jinjun Fan,et al.  On discontinuous dynamics of a periodically forced double-belt friction oscillator , 2018 .

[15]  Jean W. Zu,et al.  A numerical study of a dry friction oscillator with parametric and external excitations , 2005 .

[16]  A. H. Nayfeh,et al.  Nonlinear motions of beam-mass structure , 1990 .

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

[18]  M. Oestreich,et al.  Bifurcation and stability analysis for a non-smooth friction oscillator , 1996 .

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

[20]  P. L. Ko,et al.  Friction-induced vibration — with and without external disturbance , 2001 .

[21]  P. Flores,et al.  Development of a biomechanical spine model for dynamic analysis , 2012, 2012 IEEE 2nd Portuguese Meeting in Bioengineering (ENBENG).

[22]  Li Shuangshuang.  Passable motions and stick motions of friction-induced oscillator with 2-DOF on a speed-varying belt , 2016 .

[23]  L. Manevitch,et al.  Oscillatory models of vibro-impact type for essentially non-linear systems , 2008 .

[24]  Yunqing Zhang,et al.  Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints , 2009 .

[25]  A. Luo,et al.  On possible infinite bifurcation trees of period-3 motions to chaos in a time-delayed, twin-well Duffing oscillator , 2018 .

[26]  Earl H. Dowell,et al.  Multi-Harmonic Analysis of Dry Friction Damped Systems Using an Incremental Harmonic Balance Method , 1985 .

[27]  P. Perlikowski,et al.  Bifurcation analysis of non-linear oscillators interacting via soft impacts , 2017 .

[28]  D. Bainov,et al.  Impulsive Differential Equations: Periodic Solutions and Applications , 1993 .

[29]  Tomasz Kapitaniak,et al.  Hard versus soft impacts in oscillatory systems modeling , 2010 .

[30]  Paulo Flores,et al.  Contact Force Models for Multibody Dynamics , 2016 .

[31]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[32]  Albert C. J. Luo,et al.  On Discontinuous Dynamics of a Freight Train Suspension System , 2014, Int. J. Bifurc. Chaos.

[33]  Brandon C. Gegg,et al.  Grazing phenomena in a periodically forced, friction-induced, linear oscillator , 2006 .

[34]  C. J. Begley,et al.  A Detailed Study of the Low-Frequency Periodic Behavior of a Dry Friction Oscillator , 1997 .

[35]  Xin Wu,et al.  Use of degeneration to stabilize near grazing periodic motion in impact oscillators , 2019, Commun. Nonlinear Sci. Numer. Simul..

[36]  Xiaodi Li,et al.  Input-to-state stability of non-linear systems with distributed-delayed impulses , 2017 .

[37]  A. Luo,et al.  Periodic orbits and bifurcations in discontinuous systems with a hyperbolic boundary , 2017 .

[38]  Ge Chen,et al.  On Dynamical Behavior of a Friction-Induced Oscillator with 2-DOF on a Speed-Varying Traveling Belt , 2017 .

[39]  Brandon C. Gegg,et al.  On the Mechanism of Stick and Nonstick, Periodic Motions in a Periodically Forced, Linear Oscillator With Dry Friction , 2006 .

[40]  Steven W. Shaw,et al.  A Periodically Forced Impact Oscillator With Large Dissipation , 1983 .

[41]  Jinde Cao,et al.  An Impulsive Delay Inequality Involving Unbounded Time-Varying Delay and Applications , 2017, IEEE Transactions on Automatic Control.

[42]  A. Luo,et al.  Bifurcation trees of period-3 motions to chaos in a time-delayed Duffing oscillator , 2017 .

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

[44]  Jinjun Fan,et al.  Discontinuous dynamical behaviors in a vibro-impact system with multiple constraints , 2018 .

[45]  Xilin Fu,et al.  Stick motions and grazing flows in an inclined impact oscillator , 2015 .

[46]  David E. Stewart,et al.  Rigid-Body Dynamics with Friction and Impact , 2000, SIAM Rev..

[47]  E. Levitan.  Forced Oscillation of a Spring‐Mass System having Combined Coulomb and Viscous Damping , 1959 .

[48]  Safya Belghith,et al.  Diversity in the nonlinear dynamic behavior of a one-degree-of-freedom impact mechanical oscillator under OGY-based state-feedback control law: Order, chaos and exhibition of the border-collision bifurcation , 2018, Mechanism and Machine Theory.

[49]  Ugo Andreaus,et al.  Dynamics of three‐block assemblies with unilateral deformable contacts. Part 1: contact modelling , 1999 .

[50]  Xilin Fu,et al.  Flow switchability of motions in a horizontal impact pair with dry friction , 2017, Commun. Nonlinear Sci. Numer. Simul..

[51]  Ping Liu,et al.  Analysis of discontinuous dynamical behavior of a class of friction oscillators with impact , 2018, International Journal of Non-Linear Mechanics.

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

[53]  Olav Egeland,et al.  Dynamic modelling and force analysis of a knuckle boom crane using screw theory , 2019 .

[54]  Ugo Andreaus,et al.  Dynamics of three-block assemblies with unilateral deformable contacts. Part 2: Actual application , 1999 .

[55]  Ugo Galvanetto,et al.  Non-linear dynamics of a mechanical system with a frictional unilateral constraint , 2009 .

[56]  D. O’Regan,et al.  Sufficient conditions for pulse phenomena of nonlinear systems with state-dependent impulses , 2016 .

[57]  Liping Li,et al.  Periodic Orbits in a Second-Order Discontinuous System with an Elliptic Boundary , 2016, Int. J. Bifurc. Chaos.

[58]  Yujin Wang,et al.  Dynamics of a rolling robot of closed five-arc-shaped-bar linkage , 2018 .

[59]  Jinjun Fan,et al.  Analysis of dynamical behaviors of a friction-induced oscillator with switching control law , 2017 .

[60]  P. Casini,et al.  Dynamics of friction oscillators excited by a moving base and/or driving force , 2001 .

[61]  Balakumar Balachandran,et al.  Dynamics of Elastic Structures Subjected to Impact Excitations , 1999 .

[62]  Paulo Flores,et al.  Study of the friction-induced vibration and contact mechanics of artificial hip joints , 2014 .

[63]  Xilin Fu,et al.  Periodic Motion of the van der Pol Equation with Impulsive Effect , 2015, Int. J. Bifurc. Chaos.

[64]  Dingguo Zhang,et al.  Multiple frictional impact dynamics of threshing process between flexible tooth and grain kernel , 2017, Comput. Electron. Agric..

[65]  Xiaodi Li,et al.  Stability of nonlinear differential systems with state-dependent delayed impulses , 2016, Autom..

[66]  Xiaodi Li,et al.  Impulsive Control for Existence, Uniqueness, and Global Stability of Periodic Solutions of Recurrent Neural Networks With Discrete and Continuously Distributed Delays , 2013, IEEE Transactions on Neural Networks and Learning Systems.

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

[68]  K. P. Byrne,et al.  Analysis of a random repeated impact process , 1981 .

[69]  Oleg Gendelman.  Modeling of inelastic impacts with the help of smooth-functions , 2006 .

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

[71]  Albert C. J. Luo,et al.  Discontinuous Dynamical Systems , 2012 .

[72]  S. Rahmanian,et al.  Bifurcation in planar slider–crank mechanism with revolute clearance joint , 2015 .

[73]  Ugo Andreaus,et al.  Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle , 2002 .

[74]  Ping Liu,et al.  Analysis of Discontinuous Dynamical Behaviors of a Friction-Induced Oscillator with an Elliptic Control Law , 2018 .

[75]  Xiaoli Zhang,et al.  Effect of delayed impulses on input-to-state stability of nonlinear systems , 2017, Autom..