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Research Papers

Analysis and Experimental Estimation of Nonlinear Dispersion in a Periodic String

[+] Author and Article Information
Kevin L. Manktelow, Michael J. Leamy

G.W. Woodruff School of Mechanical
Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0405

Massimo Ruzzene

D. Guggenheim School of Aerospace
Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0405

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 22, 2013; final manuscript received March 4, 2014; published online April 15, 2014. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 136(3), 031016 (Apr 15, 2014) (8 pages) Paper No: VIB-13-1292; doi: 10.1115/1.4027137 History: Received August 22, 2013; Revised March 04, 2014

Wave dispersion in a string carrying periodically distributed masses is investigated analytically and experimentally. The effect of the string's geometric nonlinearity on its wave propagation characteristics is analyzed through a lumped parameter model yielding coupled Duffing oscillators. Dispersion frequency shifts are predicted that correspond to the hardening behavior of the nonlinear chain and that relate well to the backbone of individual Duffing oscillators. Experiments conducted on a string of finite length illustrate the relation between measured resonances and the dispersion properties of the medium. Specifically, the locus of resonance peaks in the frequency/wavenumber domain outlines the dispersion curve and highlights the existence of a frequency bandgap. Moreover, amplitude-dependent resonance shifts induced by the string nonlinearity confirm the hardening characteristics of the dispersion curve. Analytical and experimental results provide a critical link between nonlinear dispersion frequency shifts and the backbone curves intrinsic to nonlinear frequency response functions. Moreover, the study confirms that amplitude-dependent wave properties for nonlinear periodic systems may be exploited for tunability of wave transport characteristics such as frequency bandgaps and wave speeds.

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Figures

Grahic Jump Location
Fig. 1

Nonlinear spring mass chain with cubic nonlinear interaction (δ denotes the relative displacement of neighboring masses)

Grahic Jump Location
Fig. 2

Amplitude-dependent dispersion in the monoatomic chain is strongly related to the Duffing backbone curve; when the wavenumber is π/3 the two are equal.

Grahic Jump Location
Fig. 3

Detail of bead and wire (a) and periodic string fixed to two upright aluminum I-beams (b)

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Fig. 4

Test article and measurement hardware configuration

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Fig. 5

FRF corresponding to the velocity response of bead 14

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Fig. 6

Snapshots of the forced dynamic deflection shapes (black circles) and theoretical deflection shapes (gray squares) near the resonant frequencies (labeled 1–14) and near the defect frequency (labeled accordingly). The 16 black markers indicate LDV measurement points that include the 14 beads and the two end points. The gray markers indicate theoretical deflection shapes expected for the theoretical model provided in Eq. (5). Note that no theoretical defect mode is plotted.

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Fig. 7

The analytical model accurately captures the expected Bloch-wave dispersion, revealing a band gap in the neighborhood of 250 Hz. Markers correspond to the experimentally measured natural frequencies of the system (pictured on the right subfigure). Dashed and solid lines indicate dispersion relationships for periodic string and wire models, respectively. The inclusion of bending stiffness (wire model) improves the fit at higher frequencies.

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Fig. 8

Sine sweep response recorded at the seventh bead is representative of the transition through resonance and jump phenomena typical of a Duffing-like system. The right subfigure reveals an essentially monochromatic response near the peak response.

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Fig. 9

STFT spectrogram showing higher harmonic generation and the jump response occurring during sweep through resonance

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Fig. 10

Several frequency up-sweeps performed from low (400 mVpp) to high (5 Vpp) excitation levels

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Fig. 11

Jump phenomena and associated backbone curve

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Fig. 12

Experimental dispersion shift and analytical dispersion diagram. The left subfigure depicts the typical Brillouin diagram with an additional third axis denoting amplitude (shift not to scale on left subfigure). The right subfigure depicts a zoom of the experimental backbone curve AB (black) and theoretical backbone curve for a simplified model (gray).

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