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Research Papers

High-Frequency Dynamic Overshoot in Linear and Nonlinear Periodic Media

[+] Author and Article Information
Yijing Zhang

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign, Urbana, IL 61801
e-mail: yzhng123@ill

Alexander F. Vakakis

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 3, 2016; final manuscript received July 13, 2016; published online September 1, 2016. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 12(1), 011012 (Sep 01, 2016) (11 pages) Paper No: CND-16-1114; doi: 10.1115/1.4034272 History: Received March 03, 2016; Revised July 13, 2016

We study the transient responses of linear and nonlinear semi-infinite periodic media on linear elastic foundations under suddenly applied, high-frequency harmonic excitations. We show that “dynamic overshoot” phenomena are realized whereby, due to the high-rate of application of the high-frequency excitations, coherent traveling responses are propagating to the far fields of these media; and this, despite the fact that the high frequencies of the suddenly applied excitations lie well within the stop bands of these systems. For the case of a linear one-dimensional (1D) spring-mass lattice, a leading-order asymptotic approximation in the high frequency limit of the suddenly applied harmonic excitation shows that the transient dynamic overshoot is expressed in terms of the Green's function at its free end. Then, a two-dimensional (2D) strongly nonlinear granular network is considered, composed of two semi-infinite, ordered homogeneous granular lattices mounted on linear elastic foundations and coupled by weak linear coupling terms. A high-frequency harmonic excitation is applied to one of the granular lattices—designated as the “excited lattice”, with the other lattice designated as the “absorbing” one. The resulting dynamic overshoot phenomenon consists of a “pure” traveling breather, i.e., of a single propagating oscillatory wavepacket with a localized envelope, resulting from the balance of discreteness, dispersion, and strong nonlinearity. The pure breather is asymptotically studied by a complexification/averaging technique, showing nearly complete but reversible energy exchanges between the excited and absorbing lattices as the breather propagates to the far field. Verification of the analytical approximations with direct numerical simulations is performed.

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Figures

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Fig. 1

The linear spring-particle lattice under suddenly applied harmonic excitation

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Fig. 2

Forced transient response of system (1) for excitation frequency, (a) in the lower AZ, (b) the PZ, and (c) the upper AZ of the lattice; in each case the velocity time series of the seven leading particles (offset in the plot for clarity), and the spatio-temporal evolution of the instantaneous kinetic energy of the 40 leading particles are shown

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Fig. 3

Comparison of the responses of the exact system (2) and the impulsively excited model (7): (a) velocity time series of the leading seven particles of the exact system (---------) and simplified system ( - - - - - - -), and (b) spatio-temporal evolution of the instantaneous kinetic energies of the 40 leading particles in the exact system (left) and the model (7) (right)

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Fig. 4

The two-dimensional weakly coupled granular network under suddenly applied, high-frequency periodic excitation

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Fig. 5

Numerical simulation of the dynamic overshoot in the granular network (8): (a) velocity time series of the first seven granules of the excited and absorbing granules, and (b) spatio-temporal evolution of the instantaneous kinetic energies of granules 2-40 of the excited and absorbing chains; the velocity times series in (a) are shifted horizontally for clarity

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Fig. 6

Fourier spectra of the velocity time series of the first, fourth, and seventh granules of the excited and absorbing chains depicted in Fig. 5 (the pure breather develops after the first granule of the excited lattice); note that the ultrahigh excitation frequency appears only in the responses of the first granules and is filtered out in the responses of the other granules

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Fig. 7

Analytical approximations (13) and (14) for the dynamic overshoot (pure breather): (a) velocity time series of the first seven granules of the excited and absorbing granules, and (b) spatio-temporal evolution of the instantaneous kinetic energies of granules 2-40 of the excited and absorbing chains; the velocity times series in (a) are shifted horizontally for clarity

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Fig. 8

Fourier spectra of the analytical approximations of the velocity time series of the first, fourth, and seventh granules of the excited and absorbing chains depicted in Fig. 7 (the pure breather develops after the first granule of the excited lattice); these resultsshould be compared to the numerical results of Fig. 6

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