Discontinuous Dynamic Analysis of a Class of 2-DOF Oscillators With Strong Nonlinearity Under a Periodic Excitation

Starting from the car suspension system, the nonlinear characteristics of a class of two-degree-of-freedom oscillators with strong nonlinearity under a periodic excitation are discussed by the switching theory of flow in discontinuous dynamical systems. Based on the discontinuous forces and different motions of the two masses, the phase plane of each mass is composed of stick domain, nonstick domain (or free domain) and separation boundary in absolute and relative coordinates, respectively. The swithching criteria between the stick and nonstick motions and the conditions of grazing motion in two different regions are developed via the G-functions and switching control laws. The mapping dynamics theory is used to give the four-dimensional transformation set and four-dimensional mapping, and the conditions for periodic motions are explored. In addition, the stick motions, two kinds of grazing motions, periodic motions for this system and a comparison of the velocities, accelerations (or forces responses) of the two masses under the two conditions of control force are simulated numerically. The results show that the stability and comfort of the vehicle can be improved by adjusting the control force, which is generated by the control unit of system or exerted by the external excitation. For further investigating the influence of system parameters on dynamical behaviors, the stick and grazing bifurcation scenarios varying with driving frequency or amplitude are also developed, which can provide useful information for parameter selection of vibration systems with clearance and the optimal design of vehicle suspension systems. This paper also has important reference value for practical applications in other industries or machinery with elastic impacts.

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

[2]  Jinjun Fan,et al.  On discontinuous dynamics of a 2-DOF friction-influenced oscillator with multiple elastic constraints , 2019, International Journal of Non-Linear Mechanics.

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

[4]  Farhan A. Salem,et al.  Vehicle Suspension Systems Control: A Review , 2013 .

[5]  Tingwen Huang,et al.  Persistence of delayed cooperative models: Impulsive control method , 2019, Appl. Math. Comput..

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

[7]  Albert C. J. Luo,et al.  Period-1 Evolutions to Chaos in a Periodically Forced Brusselator , 2018, Int. J. Bifurc. Chaos.

[8]  Ekaterina Pavlovskaia,et al.  The nature of the normal form map for soft impacting systems , 2008 .

[9]  Ayman A. Aly,et al.  Fuzzy Control of a Quarter-Car Suspension System , 2009 .

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

[11]  J. M. T. Thompson,et al.  Chaos after period-doubling bifurcations in the resonance of an impact oscillator , 1982 .

[12]  Sami F. Masri.  Electric-analog studies of impact dampers , 1967 .

[13]  Anders Karlström,et al.  An analytical model for train-induced ground vibrations from railways , 2006 .

[14]  E. Gottzein,et al.  Magnetic suspension control systems for the MBB high speed train , 1975 .

[15]  Bernard Brogliato,et al.  Impacts in Mechanical Systems: Analysis and Modelling , 2000 .

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

[17]  Chun-Liang Lin,et al.  Optimal design for passive suspension of a light rail vehicle using constrained multiobjective evolutionary search , 2005 .

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

[19]  Fuad E. Alsaadi,et al.  Robust set stabilization of Boolean control networks with impulsive effects , 2018, Nonlinear Analysis: Modelling and Control.

[20]  Bernard Brogliato,et al.  Some perspectives on the analysis and control of complementarity systems , 2003, IEEE Trans. Autom. Control..

[21]  Albert C. J. Luo,et al.  The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation , 2005 .

[22]  A. Zbiciak,et al.  Dynamic analysis of a soft-contact problem using viscoelastic and fractional-elastic rheological models , 2015 .

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

[24]  Jianlong Qiu,et al.  Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback , 2019, Nonlinear Analysis: Hybrid Systems.

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

[26]  Toshio Yoshimura,et al.  ACTIVE SUSPENSION OF PASSENGER CARS USING SLIDING MODE CONTROLLERS (BASED ON REDUCED MODELS) , 2014 .

[27]  H. Nijmeijer,et al.  Dynamics and Bifurcations ofNon - Smooth Mechanical Systems , 2006 .

[28]  Soumitro Banerjee,et al.  Border collision bifurcations in a soft impact system , 2006 .

[29]  J. M. T. Thompson,et al.  Complex dynamics of compliant off-shore structures , 1983, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

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

[31]  Liqing Li,et al.  Existence of Positive Solutions for p-Laplacian Dynamic Equations with Derivative on Time Scales , 2013, J. Appl. Math..

[32]  Paul H. Calamai,et al.  Application of genetic algorithms to the design optimization of an active vehicle suspension system , 1998 .

[33]  G. Yeh.  Forced Vibrations of a Two‐Degree‐of‐Freedom System with Combined Coulomb and Viscous Damping , 1964 .

[34]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

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

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

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

[38]  T D Gillespie,et al.  Fundamentals of Vehicle Dynamics , 1992 .

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

[40]  J. Ambrósio,et al.  Dynamic Analysis for Planar Multibody Mechanical Systems with Lubricated Joints , 2004 .

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

[42]  Jianhua Shen,et al.  State‐dependent switching control of delayed switched systems with stable and unstable modes , 2018, Mathematical Methods in the Applied Sciences.

[43]  B. Brogliato.  Nonsmooth Impact Mechanics: Models, Dynamics and Control , 1996 .

[44]  Balakumar Balachandran,et al.  A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008 , 2008 .

[45]  Zhaoxia Yang,et al.  On discontinuous dynamical behaviors of a 2-DOF impact oscillator with friction and a periodically forced excitation , 2019, Mechanism and Machine Theory.

[46]  Ekaterina Pavlovskaia,et al.  Experimental study of impact oscillator with one-sided elastic constraint , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[47]  A. Luo,et al.  On periodic solutions of a second-order, time-delayed, discontinuous dynamical system , 2018, Chaos, Solitons & Fractals.

[48]  Hassan Sayyaadi,et al.  A new model in rail–vehicles dynamics considering nonlinear suspension components behavior , 2009 .

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

[50]  Qiang Wang,et al.  Bifurcations of periodic motion in a three-degree-of-freedom vibro-impact system with clearance , 2017, Commun. Nonlinear Sci. Numer. Simul..

[51]  Yeong-Bin Yang,et al.  Steady-state response and riding comfort of trains moving over a series of simply supported bridges , 2003 .

[52]  Xiaodi Li,et al.  Review of stability and stabilization for impulsive delayed systems. , 2018, Mathematical biosciences and engineering : MBE.

[53]  Mike Blundell,et al.  The influence of rubber bush compliance on vehicle suspension movement , 1998 .

[54]  Shen-Haw Ju,et al.  Experimentally investigating finite element accuracy for ground vibrations induced by high-speed trains , 2008 .

[55]  Jinjun Fan,et al.  Analysis of dynamical behaviors of a 2-DOF friction oscillator with elastic impacts and negative feedbacks , 2020, Nonlinear Dynamics.

[56]  Albert C. J. Luo,et al.  Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamic systems , 2005 .

[57]  J. H. Xie,et al.  Symmetry and bifurcations of a two-degree-of-freedom vibro-impact system , 2008 .

[58]  Xiaojing Xu,et al.  Finite-time stability analysis of stochastic switched boolean networks with impulsive effect , 2019, Appl. Math. Comput..

[59]  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.

[60]  Xiaodi Li,et al.  Persistent impulsive effects on stability of functional differential equations with finite or infinite delay , 2018, Appl. Math. Comput..

[61]  Albert C. J. Luo,et al.  Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts , 2005 .

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

[63]  Xilin Fu,et al.  Discontinuous dynamics of a class of oscillators with strongly nonlinear asymmetric damping under a periodic excitation , 2018, Commun. Nonlinear Sci. Numer. Simul..

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

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

[66]  Fuad E. Alsaadi,et al.  Output tracking control of Boolean control networks with impulsive effects , 2018 .

[67]  J. P. Den Hartog,et al.  LXXIII. Forced vibrations with combined viscous and coulomb damping , 1930 .

[68]  Paul I. Ro,et al.  A sliding mode controller for vehicle active suspension systems with non-linearities , 1998 .

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

[70]  Vincent Acary,et al.  Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics , 2008 .

[71]  Albert C. J. Luo,et al.  On Nonlinear Dynamics of Simplified Brake Dynamical Systems , 2007 .