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Research Papers

Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Analytical Study

[+] Author and Article Information
L. I. Manevitch

Institute of Chemical Physics,
RAS,
Kosygin str. 4, Moscow, Russia

G. Sigalov

College of Engineering,
University of Illinois at Urbana–Champaign,
Champaign, IL 61820
e-mail: sigalov@illinois.edu

F. Romeo

Department of Structural and
Geotechnical Engineering,
Sapienza University of Rome,
Rome 00185, Italy

A. Vakakis

College of Engineering,
University of Illinois at Urbana–Champaign, Champaign, IL 61820

1Corresponding author.

Manuscript received June 10, 2013; final manuscript received July 8, 2013; accepted manuscript posted July 31, 2013; published online September 23, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(4), 041011 (Sep 23, 2013) (9 pages) Paper No: JAM-13-1235; doi: 10.1115/1.4025150 History: Received June 10, 2013; Revised July 08, 2013; Accepted July 31, 2013

We present an analytical study of the conservative and dissipative dynamics of a two-degree-of-freedom (DOF) system consisting of a linear oscillator coupled to a bistable light attachment. The main objective of the paper is to study the beneficial effect of the bistability on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to an appropriately designed attachment. As a numerical study of the problem has shown in a companion paper (Romeo, F., Sigalov, G., Bergman, L. A., and Vakakis, A. F., 2013, “Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study,” J. Comput. Nonlinear Dyn. (submitted)) there is an essential difference in the system's behavior when compared to the conventional case of a monostable attachment. On the other hand, some similarity to the behavior of an oscillator with rotator attachment has been revealed. It relates, in particular, to the generation of nonconventional nonlinear normal modes and to the existence of two qualitatively different types of dynamics. We find that all numerical results can be explained in the framework of fundamental (1:1) and superharmonic (1:3) resonances (for large energies), as well as a subharmonic resonance (for small energies). This allows us to use the concept of limiting phase trajectories (LPTs) introduced earlier by one of the authors, and to derive accurate analytical approximations to the dynamics of the problem in terms of nonsmooth generating functions.

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References

Figures

Grahic Jump Location
Fig. 5

Slow dynamics on the LPT of the system with |k-μ/2-σ| = 1.146 > 1 (corresponding to the responses shown in Fig. 4)

Grahic Jump Location
Fig. 4

System with weak energy exchanges between the LO and NES, with |k-μ/2-σ| = 1.146 > 1. (a) Poincaré map showing the LPT (b) transient responses corresponding to the LPT (c) trajectories in the phase plane (Δ,Θ) conserving the adiabatic invariant K defined by Eq. (15). The LPT (shown by the arrow) has a small range in Θ meaning that (nearly) complete energy exchange between the LO and the NES is not possible in this case during the nonlinear beat.

Grahic Jump Location
Fig. 3

Slow dynamics on the LPT of the system with |k-μ/2-σ| = 0.885 < 1 (corresponding to the responses shown in Fig. 2)

Grahic Jump Location
Fig. 2

System with LPT optimally designed for intense energy exchange between the LO and NES, with |k-μ/2 - σ| = 0.885 < 1. (a) Poincaré map showing the LPT (b) transient responses corresponding to the LPT (c) trajectories in the phase plane (Δ,θ) conserving the adiabatic invariant K defined by Eq. (15); the trajectory with the largest range in θ is the LPT indicating that complete energy exchange between the LO and the NES occurs in this case during a cycle of the nonlinear beat.

Grahic Jump Location
Fig. 6

Analytical approximation based on Eq. (24) for the LPT depicted in Fig. 3

Grahic Jump Location
Fig. 7

Numerical simulation of the displacements of the linear oscillator, u1(τ), and the relative response, u2(τ), for the dissipative system (30) originally at rest after an impulse is applied to the linear oscillator, and system parameters, ɛ = 0.316, c = 1, σ = 0.158, v0 = 1.9, ɛ3η = ɛ3≈0.0316; all three stages considered in the analytical study are realized in this case

Grahic Jump Location
Fig. 1

Leading-order slow flow (12) reduced on a two-torus for g = -0.787 and (a) 2k = 1.5 (b) 2k = 3.5

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