Lyapunov exponents of impact oscillators with Hertz׳s and Newton׳s contact models

Abstract In this paper, investigations of a harmonically excited one-degree-of-freedom mechanical system having an amplitude constraint are presented. The contact between the oscillated mass and the barrier is modeled by Hertz׳s law with a non-linear damping as well as by Newton׳s law. The influence of the frequency of excitation force on the system׳s behavior is studied in a wide range of the control parameter by determining and analyzing the corresponding spectra of Lyapunov exponents. The dynamical behaviors of two systems with impacts: a system with Hertz׳s undamped impacts and a system with perfectly elastic hard impacts, which are equivalent in the sense of the same rate of impact energy dissipation, are compared and strong qualitative and quantitative similarities are observed. As an application example, a simple cantilever beam system with impacts is considered and the combined effects of the nonlinearities due to beam deflection and impacts of Hertz׳s as well as Newton׳s types are investigated.

[1]  Andrzej Stefański,et al.  Estimation of the largest Lyapunov exponent in systems with impacts , 2000 .

[2]  Y. Yue,et al.  Lyapunov exponents and coexistence of attractors in vibro-impact systems with symmetric two-sided rigid constraints , 2009 .

[3]  C. Wu,et al.  A robust method on estimation of Lyapunov exponents from a noisy time series , 2011 .

[4]  Sawada,et al.  Measurement of the Lyapunov spectrum from a chaotic time series. , 1985, Physical review letters.

[5]  M. Belhaq,et al.  Control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator , 2011 .

[6]  V. Bazhenov,et al.  Numerical investigations of the dynamic processes in vibroimpact systems in modeling impacts by a force of contact interaction , 2008 .

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

[8]  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.

[9]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[10]  L. Pust,et al.  Impact Oscillator with Hertz's Model of Contact , 2003 .

[11]  Ni Qiao,et al.  Bifurcations and chaos in a forced cantilever system with impacts , 2006 .

[12]  A. Dabrowski,et al.  Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product , 2012 .

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

[14]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

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

[16]  Ugo Galvanetto.  Numerical computation of Lyapunov exponents in discontinuous maps implicitly defined , 2000 .

[17]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[18]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[19]  I. Shimada,et al.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems , 1979 .

[20]  Marian Wiercigroch,et al.  Cumulative effect of structural nonlinearities: chaotic dynamics of cantilever beam system with impacts , 2005 .

[21]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[22]  M. A. Aizerman,et al.  On the stability of periodic motions , 1958 .

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

[24]  Qishao Lu,et al.  A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems , 2006 .

[25]  P. Müller.  Calculation of Lyapunov exponents for dynamic systems with discontinuities , 1995 .

[26]  The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system , 2011 .

[27]  Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops , 2011 .

[28]  Ugo Andreaus,et al.  Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator , 2010 .

[29]  G. Luo,et al.  Dynamic analysis and suppressing chaotic impacts of a two-degree-of-freedom oscillator with a clearance , 2009 .

[30]  Tomasz Kapitaniak,et al.  Dynamics of a two-degree-of-freedom cantilever beam with impacts , 2009 .

[31]  Iberê L. Caldas,et al.  Calculation of Lyapunov exponents in systems with impacts , 2004 .

[32]  José Manoel Balthazar,et al.  Suppressing grazing chaos in impacting system by structural nonlinearity , 2008 .

[33]  Marian Wiercigroch,et al.  Experimental Study of a Symmetrical Piecewise Base-Excited Oscillator , 1998 .

[34]  Ulrich Parlitz,et al.  Identification of True and Spurious Lyapunov Exponents from Time Series , 1992 .

[35]  Raouf A. Ibrahim,et al.  Vibro-Impact Dynamics: Modeling, Mapping and Applications , 2009 .

[36]  Ekaterina Pavlovskaia,et al.  Bifurcation analysis of an impact oscillator with a one-sided elastic constraint near grazing , 2010 .

[37]  The effect of discretization on the numerical simulation of the vibrations of the impacting cantilever beam , 2010 .

[38]  Arne Nordmark,et al.  Non-periodic motion caused by grazing incidence in an impact oscillator , 1991 .

[39]  Steven R. Bishop,et al.  Bifurcations in impact oscillations , 1994 .