Periodic and chaotic motions of an unsymmetrical oscillator in nonlinear structural dynamics

Abstract The chaotic dynamics of a harmonically-excited single-degree-of-freedom unsymmetric system in continuum nonlinear structural dynamics is studied in some detail through computer simulations. The quadratic and cubic nonlinearities occurring in the motion equation model the geometrical and mechanical characteristics of a suspended elastic cable vibrating in its plane and actually exhibiting a single equilibrium position. Regions of different periodic or chaotic responses in the forcing control parameter space are obtained. Three main frequency ranges are distinguished, two located in the neighbourhood of the order one-half and one-third subharmonic resonances of the system, the third one covering the zones of order three and two superharmonic resonances. Several types of bifurcations and strange attractors are observed and identified through various dynamic measures, both qualitative and quantitative. The algorithmic experience about their use in situations with different “chaoticity” is augmented.

[1]  A. Maewal.  Chaos in a Harmonically Excited Elastic Beam , 1986 .

[2]  F. Moon.  Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractors , 1980 .

[3]  Earl H. Dowell,et al.  Numerical simulations of periodic and chaotic responses in a stable duffing system , 1987 .

[4]  Francis C. Moon,et al.  Criteria for chaos of a three-well potential oscillator with homoclinic and heteroclinic orbits , 1990 .

[5]  Francesco Benedettini,et al.  Planar non-linear oscillations of elastic cables under superharmonic resonance conditions , 1989 .

[6]  P. F. Meier,et al.  Evaluation of Lyapunov exponents and scaling functions from time series , 1988 .

[7]  The approximate approach to chaos phenomena in oscillators having single equilibrium position , 1990 .

[8]  Raymond H. Plaut,et al.  Oscillations and instability of a shallow-arch under two-frequency excitation , 1985 .

[9]  Anil K. Bajaj,et al.  Amplitude modulated and chaotic dynamics in resonant motion of strings , 1989 .

[10]  Edward R. Vrscay,et al.  Chaotic motion under parametric excitation , 1989 .

[11]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

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

[13]  Raymond H. Plaut,et al.  Period Doubling and Chaos in Unsymmetric Structures Under Parametric Excitation , 1989 .

[14]  Periodic Solutions Leading to Chaos in an Oscillator with Quadratic and Cubic Nonlinearities , 1991 .

[15]  John Dugundji,et al.  Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation , 1971 .

[16]  Benson H. Tongue,et al.  Existence of chaos in a one-degree-of-freedom system , 1986 .

[17]  J. Rudowski,et al.  On an approximate criterion for chaotic motion in a model of a buckled beam , 1987 .

[18]  P. Holmes,et al.  A nonlinear oscillator with a strange attractor , 1979, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[19]  B. H. Tongue.  Characteristics of Numerical Simulations of Chaotic Systems , 1987 .

[20]  J. Yorke,et al.  The liapunov dimension of strange attractors , 1983 .

[21]  G. Rega,et al.  Non-linear dynamics of an elastic cable under planar excitation , 1987 .

[22]  Mohammed S El Naschie,et al.  Stress, Stability and Chaos in Structural Engineering: An Energy Approach , 1990 .

[23]  Earl H. Dowell,et al.  On the Understanding of Chaos in Duffings Equation Including a Comparison With Experiment , 1986 .

[24]  Fabrizio Vestroni,et al.  Parametric analysis of large amplitude free vibrations of a suspended cable , 1984 .

[25]  Wanda Szemplińska-Stupnicka.  Secondary resonances and approximate models of routes to chaotic motion in non-linear oscillators , 1987 .

[26]  Francis C. Moon,et al.  The fractal dimension of the two-well potential strange attractor , 1985 .