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Research Papers

Boundary for Complete Set of Attractors for Forced–Damped Essentially Nonlinear Systems

[+] Author and Article Information
Itay Grinberg

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mails: gitay@tx.technion.ac.il;
grinbergitay@gmail.com

Oleg V. Gendelman

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: ovgend@tx.technion.ac.il

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 6, 2014; final manuscript received March 12, 2015; published online March 31, 2015. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 82(5), 051004 (May 01, 2015) (6 pages) Paper No: JAM-14-1470; doi: 10.1115/1.4030045 History: Received October 06, 2014; Revised March 12, 2015; Online March 31, 2015

Forced–damped essentially nonlinear oscillators can have a multitude of dynamic attractors. Generically, no analytic procedure is available to reveal all such attractors. For many practical and engineering applications, however, it might be not necessary to know all the attractors in detail. Knowledge of the zone in the state space (or the space of initial conditions), in which all the attractors are situated might be sufficient. We demonstrate that this goal can be achieved by relatively simple means—even for systems with multiple and unknown attractors. More specifically, this paper suggests an analytic procedure to determine the zone in the space of initial conditions, which contains all attractors of the essentially nonlinear forced–damped system for a given set of parameters. The suggested procedure is an extension of well-known Lyapunov functions approach; here we use it for analysis of stability of nonautonomous systems with external forcing. Consequently, instead of the complete state space of the problem, we consider a space of initial conditions and define a bounded trapping region in this space, so that for every initial condition outside this region, the dynamic flow will eventually enter it and will never leave it. This approach is used to find a special closed curve on the plane of initial conditions for a forced–damped strongly nonlinear oscillator with single-degree-of-freedom (single-DOF). Solving the equations of motion is not required. The approach is illustrated by the important benchmark example of x2n potential, including the celebrated Ueda oscillator for n = 2. Another example is the well-known model of forced–damped oscillator with double-well potential. We also demonstrate that the boundary curve, obtained by analytic tools, can be efficiently “tightened” numerically, yielding even stricter estimation for the zone of the existing attractors.

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Figures

Grahic Jump Location
Fig. 1

Family of boundary curves for Ueda oscillator (see Sec. 3 for details). Multiple arrows describe various directions of the phase flow for different time instances.

Grahic Jump Location
Fig. 2

Boundary curve (dashed) for Ueda oscillator with k = 0.08, B = 0.2, α = 0.092,β = -0.5, and c = 0.007. Closed curves inside the boundary curve correspond to known attractors of the Ueda oscillator for this choice of parameters.

Grahic Jump Location
Fig. 3

Boundary curve (dashed) for potential x2n with n = 7. Forcing function g is a square wave, k = 0.08, B = 0.2, α = 0.092,β = -0.5, and c = 0.007. Solid black curve corresponds to attractor of the system.

Grahic Jump Location
Fig. 4

Sketch of the step of numeric tightening process

Grahic Jump Location
Fig. 5

Illustration of efficiency of numeric tightening of the boundary curve for potential x2n, n = 2, k = 1, B = 0.4, and ε = 0.0001

Grahic Jump Location
Fig. 6

Numeric tightening of the boundary curve for double-well potential x4-x2, k = 2, b = 0.91, and ε = 0.000001

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