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Research Papers

Limits of the Kelvin Voigt Model for the Analysis of Wave Propagation in Monoatomic Mass-Spring Chains

[+] Author and Article Information
Antonio Palermo

Department of Civil, Chemical, Environmental
and Materials Engineering–DICAM,
University of Bologna,
Viale del Risorgimento 2,
Bologna 40136, Italy
e-mail: antonio.palermo6@unibo.it

Alessandro Marzani

Department of Civil, Chemical, Environmental
and Materials Engineering–DICAM,
University of Bologna,
Viale del Risorgimento 2,
Bologna 40136, Italy
e-mail: alessandro.marzani@unibo.it

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 7, 2015; final manuscript received October 26, 2015; published online December 8, 2015. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 138(1), 011022 (Dec 08, 2015) (9 pages) Paper No: VIB-15-1117; doi: 10.1115/1.4031999 History: Received April 07, 2015; Revised October 26, 2015

In this study, the effect of energy dissipation on harmonic waves propagating in one-dimensional monoatomic linear viscoelastic mass-spring chains is investigated. In particular, first dispersion laws in terms of wavenumber, attenuation, and wave propagation velocities (phase, group, and energy) for a generic viscoelastic mass-spring chain are derived from the homologous linear elastic (LE) expressions in force of the correspondence principle. A new formula for the energy velocity is introduced to account for energy dissipation. Next, such relations are specified for the Kelvin Voigt (KV) and the standard linear solid (SLS) rheological models. The analysis of the KV mass-spring chain in the high-frequency regime proves that the so-called wavenumber-gap is not related to energy dissipation, as assumed in previous studies, but is due to the nonphysical rigid behavior of the model at high frequencies. The SLS mass-spring chain, in fact, does not show any wavenumber-gap and at high frequencies recovers the wavenumber dispersion curve of the LE system. The behavior of the energy velocity for the different mass-spring chains confirms this conclusion.

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References

Figures

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Fig. 1

Viscoelastic mass-spring chain: (a) uniform rod and discrete mass-spring chain and (b) mass-spring unit cell

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Fig. 2

Normalized time-averaged kinetic energies (modal mass). Time-averaged kinetic energies 〈T〉 and 〈T1〉 are evaluated for a mass-spring chain with mass m = 4, length a = 1, and considering a displacement amplitude |un|=1.

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Fig. 3

Viscoelastic mechanical models: (a) KV model, (b) SLS model, and (c) GM model

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Fig. 4

Propagation velocities of an LE, a KV, and an SLS 1D continuous system whose mechanical parameters are defined in Table 1

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Fig. 5

Dispersion curves of an LE, a KV, and an SLS mass-spring chain. (a) ω versus ξr and (b) ω versus ξi. The considered mechanical parameters are defined in Table 2.

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Fig. 6

Dispersion curves of an LE, a KV, and an SLS mass-spring chain in the high-frequency regime. (a) ω versus ξr and (b) ω versus ξi, the squares refer to the values estimated with the approximated expression of Eq. (29).

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Fig. 7

Mass-dashpot chain

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Fig. 8

Velocities of an LE, a KV, and an SLS mass-spring chain: (a) vp and ve and (b) vg and ve

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Fig. 9

ve versus ω for the LE, the KV, and the SLS mass-spring chains

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Fig. 10

Representation of the complex roots Xi of the characteristic equation

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Fig. 11

(a) Real and (b) imaginary parts of the complex moduli for an SLS (E1 = 1, E2 = 1, and η = 0.08, from Table 2) and a GM models (E3 = 0.5, E4 = 0.5, and η2 = 0.02): Data for the GM havebeen derived from those of the SLS according to Eqs. (B4) and (B5)

Tables

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