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Chaos / Volume 22 / Issue 1 / REGULAR ARTICLES

Chaos 22, 013118 (2012); http://dx.doi.org/10.1063/1.3683480 (10 pages)

Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling

G. Sigalov1, O. V. Gendelman2, M. A. AL-Shudeifat1, L. I. Manevitch3, A. F. Vakakis1, and L. A. Bergman1

1College of Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
2Faculty of Mechanical Engineering, Technion–Israel Institute of Technology, Haifa, Israel
3Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia

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(Received 3 August 2011; accepted 17 January 2012; published online 14 February 2012)

We show that nonlinear inertial coupling between a linear oscillator and an eccentric rotator can lead to very interesting interchanges between regular and chaotic dynamical behavior. Indeed, we show that this model demonstrates rather unusual behavior from the viewpoint of nonlinear dynamics. Specifically, at a discrete set of values of the total energy, the Hamiltonian system exhibits non-conventional nonlinear normal modes, whose shape is determined by phase locking of rotatory and oscillatory motions of the rotator at integer ratios of characteristic frequencies. Considering the weakly damped system, resonance capture of the dynamics into the vicinity of these modes brings about regular motion of the system. For energy levels far from these discrete values, the motion of the system is chaotic. Thus, the succession of resonance captures and escapes by a discrete set of the normal modes causes a sequence of transitions between regular and chaotic behavior, provided that the damping is sufficiently small. We begin from the Hamiltonian system and present a series of Poincaré sections manifesting the complex structure of the phase space of the considered system with inertial nonlinear coupling. Then an approximate analytical description is presented for the non-conventional nonlinear normal modes. We confirm the analytical results by numerical simulation and demonstrate the alternate transitions between regular and chaotic dynamics mentioned above. The origin of the chaotic behavior is also discussed.

© 2012 American Institute of Physics

Lead Paragraph

The paper studies transient dynamics of lightly damped system consisting of linear oscillator and eccentric rotator. Such systems are considered as viable candidates for use in applications related to vibration suppression and mitigation in structural engineering; therefore, it is rather important to achieve clear understanding of how the energy of the oscillations is damped out. We reveal rather unusual pattern of this transient behavior—as the energy decreases, the system “switches” between regular and chaotic motions. We demonstrate that the regular motion occurs at certain energy levels, at which the system is captured into special resonant oscillatory states (nonlinear normal modes). By simple analytic estimation, we are able to predict these special energy levels of the regular motion. This result helps one to design the vibration absorbers with required dynamic behavior.

Article Outline

  1. INTRODUCTION
  2. THE HAMILTONIAN SYSTEM
  3. RELAXATION PROCESS: ALTERNATE CHAOTIC AND REGULAR DYNAMICS IN THE WEAKLY DISSIPATIVE SYSTEM
  4. CONCLUDING REMARKS

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KEYWORDS and PACS

Keywords

chaos, nonlinear dynamical systems

PACS

  • 05.45.Pq

    Numerical simulations of chaotic systems

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3683480

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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