Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method

Abstract In this paper, an explicit formulation of the incremental harmonic balance (IHB) scheme for computation of periodic solutions of a harmonically excited oscillator which is asymmetric with both stiffness and viscous damping piecewise linearities is derived. Analysis of dynamical behavior as bifurcation and chaos of the non-linear vibration system considered is effectively carried out by the IHB procedure, showing that the system exhibits chaos via the route of period-doubling bifurcation, with coexistence of multiple periodic attractors observed and analyzed by the interpolated cell mapping method. In addition, numerical simulation by the IHB method is compared with that by the fourth order Runge–Kutta numerical integration routine, which shows that this method is in many respects distinctively advantageous over classical approaches, and especially excels in performing parametric studies.

[1]  S. Narayanan,et al.  Bifurcation and chaos of an articulated loading platform with piecewise non-linear stiffness using the incremental harmonic balance method , 2000 .

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

[3]  S. Narayanan,et al.  Bifurcation and chaos in escape equation model by incremental harmonic balancing , 2000 .

[4]  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.

[5]  L. Chua,et al.  Hyper chaos: Laboratory experiment and numerical confirmation , 1986 .

[6]  Y. K. Cheung,et al.  Amplitude Incremental Variational Principle for Nonlinear Vibration of Elastic Systems , 1981 .

[7]  W. S. Zhang,et al.  Periodic forced vibration of unsymmetrical piecewise-linear systems by incremental harmonic balance method , 1991 .

[8]  Hiroyuki Kumano,et al.  Forced Vibrations in an Unsymmetric Piecewise-Linear System Excited by General Periodic Force Functions , 1979 .

[9]  S. Lau,et al.  Nonlinear Vibrations of Piecewise-Linear Systems by Incremental Harmonic Balance Method , 1992 .

[10]  E. H. Twizell,et al.  Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise-linearity , 2001 .

[11]  Chaos in a four-variable piecewise-linear system of differential equations , 1988 .

[12]  Benson H. Tongue,et al.  Interpolated Cell Mapping of Dynamical Systems , 1988 .

[13]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[14]  J. M. T. Thompson,et al.  Subharmonic Resonances and Chaotic Motions of a Bilinear Oscillator , 1983 .

[15]  S. Lau,et al.  Solution Diagram Of Non-Linear Dynamic Systems By The IHB Method , 1993 .

[16]  J. J. Stoker.  Nonlinear Vibrations in Mechanical and Electrical Systems , 1950 .

[17]  L. Chua,et al.  The simplest dissipative nonautonomous chaotic circuit , 1994 .

[18]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[19]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[20]  S. T. Noah,et al.  Forced periodic vibration of unsymmetric piecewise-linear systems , 1988 .