Research Papers

Energy Exchange and Localization in Essentially Nonlinear Oscillatory Systems: Canonical Formalism

[+] Author and Article Information
O. V. Gendelman

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 3200003, Israel
e-mail: ovgend@tx.technion.ac.il

T. P. Sapsis

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 17, 2016; final manuscript received October 5, 2016; published online October 26, 2016. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 84(1), 011009 (Oct 26, 2016) (9 pages) Paper No: JAM-16-1247; doi: 10.1115/1.4034930 History: Received May 17, 2016; Revised October 05, 2016

Over recent years, a lot of progress has been achieved in understanding of the relationship between localization and transport of energy in essentially nonlinear oscillatory systems. In this paper, we are going to demonstrate that the structure of the resonance manifold can be conveniently described in terms of canonical action–angle (AA) variables. Such formalism has important theoretical advantages: all resonance manifolds may be described at the same level of complexity, appearance of additional conservation laws on these manifolds is easily proven both in autonomous and nonautonomous settings. The harmonic balance-based complexification approach, used in many previous studies on the subject, is shown to be a particular case of the canonical formalism. Moreover, application of the canonic averaging allows treatment of much broader variety of dynamical models. As an example, energy exchanges in systems of coupled trigonometrical and vibro-impact oscillators are considered.

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Grahic Jump Location
Fig. 1

Pair of impact oscillators coupled by linear spring

Grahic Jump Location
Fig. 2

Time series for displacements in coupled vibro-impact system (Fig. 1) with nonzero initial velocity of one impactor: (a) ε=0.058 and (b) ε=0.059. Red (thin solid) line— u1(t) and blue (thick points) line— u2(t). Plot (c) presents kinetic energy of both oscillators for ε=0.059 : green (thin solid) line—oscillator 1 and brown (thick points)—oscillator 2.

Grahic Jump Location
Fig. 3

Phase portraits of the averaged system with Hamiltonian (32) for (a) κ=0.22, (b) κ=24/π4=0.2464, and (c) κ=0.26. The thick red line denotes the limiting phase trajectory (LPT).

Grahic Jump Location
Fig. 4

Phase portrait of the effective Hamiltonian (43) on the sphere, shown in terms of color and contours in spherical coordinates, for constant ε=0.1 and growing values of N. The red trajectory describes the transition of the LPT through saddle.

Grahic Jump Location
Fig. 5

Transition from energy exchange to localization in coupled trigonometric oscillators, described by Hamiltonian (41), ε=0.1. The figures correspond to different sets of initial conditions q1(0)=A,q2(0)=p1(0)=p2(0)=0 : (a) A = 0.1, (b) A = 0.5, (c) A = 0.55, and (d) A = 0.58. q1(t) —red (thin solid) line and q2(t) —black (thick point) line. Equation (45) predicts transition from the beatings to localization for Acr=0.578.

Grahic Jump Location
Fig. 6

Transition from energy exchange to localization in coupled trigonometric oscillators, described by Hamiltonian (41), ε=2.29. The figures correspond to different sets of initial conditions q1(0)=A,q2(0)=p1(0)=p2(0)=0 : (a) A = 1.2, (b) A = 1.225, and (c) A = 1.235. q1(t) —red (thin solid) line and q2(t) —black (thick point) line. Equation (45) predicts transition from the beatings to localization for Acr=1.258.



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