Damping-induced interplay between vibrations and waves in a forced non-dispersive elastic continuum with asymmetrically placed local attachments

Blanchard, Antoine; McFarland, D. Michael; Bergman, Lawrence A.; Vakakis, Alexander F.

doi:10.1098/rspa.2014.0402

Cited by 14 publications

(3 citation statements)

References 30 publications

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“…Recently, a nonresonant damped side branch has been used for spatial separation (localization) of acoustic traveling and standing waves in a duct with constant cross-sectional area (CSA), where the acoustic impedance of the side branch required to produce this localization is found analytically as a function of wavenumber and side-branch location. [1][2][3] In structural dynamics, spatial separation of traveling and standing waves has also been studied for a non-dispersive taut string with an attached spring-dashpot support and a vibration absorber, 4,5 a dispersive taut string on a partial viscoelastic foundation, 6 and an Euler-Bernoulli beam with one or two spring-dashpot supports. 7 The ducts, strings and beams stud-ied for localization of waves were all assumed to have constant geometries, as in the case of the duct 1,2 shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions for Side-branch Impedance Producing Spatial Localization of Acoustic Waves in Ducts with Varying Cross Sections

Xiao¹,

Lu²,

McFarland³

et al. 2021

The International Journal of Acoustics and Vibration

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Analytical mathematical models and solutions for spatial localization of acoustic waves through an impedance discontinuity produced by an intermediate damped side branch are studied in stationary media in ducts with varying cross sections. Three specific geometries, namely, with polynomial, sinusoidal, and exponential longitudinal variations, are investigated. The sound fields inside the ducts are modeled by Webster's horn equation. Traveling-wave solutions are obtained by appropriate transformations. The side-branch impedances required for spatial localization (confinement) of traveling and standing waves are found analytically and verified numerically using three-dimensional finite element analysis. The impact of the longitudinal variation of the duct's cross-sectional area (CSA) on the side-branch impedance is examined. It was found that the required side-branch resistance changes more than the reactance with the variation of the duct CSA. A measure of a traveling wave is defined to quantitatively examine the spatial localization of acoustic waves. It was found that the CSA corrections on the side-branch impedances are important. The results of this study reveal the quantitative relationships between the side-branch impedance and the CSA variations for zero reflection from the impedance discontinuity. The mathematical approach presented is potentially helpful for a design of a full anechoic termination and energy localization in duct systems.

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Section: Introductionmentioning

confidence: 99%

Analytical Solutions for Side-branch Impedance Producing Spatial Localization of Acoustic Waves in Ducts with Varying Cross Sections

Xiao¹,

Lu²,

McFarland³

et al. 2021

The International Journal of Acoustics and Vibration

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“…In particular, it can give rise to mode complexity, i. e., a non-trivial phase lag between the oscillations of different material points of a system [5,2]. Depending on the system parameters and the degree of asyncronicity of the oscillation, the dynamics may change completely from standing waves to a traveling waves [2,3]. It should be noted that non-unison vibrations can also occur in undamped nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Global complexity effects due to local damping in a nonlinear system in 1:3 internal resonance

Krack,

Bergman,

Vakakis

2021

Preprint

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It is well-known that nonlinearity may lead to localization effects and coupling of internally resonant modes. However, research focused primarily on conservative systems commonly assumes that the near-resonant forced response closely follows the autonomous dynamics. Our results for even a simple system of two coupled oscillators with a cubic spring clearly contradict this common belief. We demonstrate analytically and numerically global effects of a weak local damping source in a harmonically forced nonlinear system under condition of 1:3 internal resonance: The global motion becomes asynchronous, i. e., mode complexity is introduced with a non-trivial phase difference between the modal oscillations. In particular, we show that a maximum mode complexity with a phase difference of 90 • is attained in a multi-harmonic sense. This corresponds to a transition from generalized standing to traveling waves in the system's modal space. We further demonstrate that the localization is crucially affected by the system's damping. Finally, we propose an extension of the definition of mode complexity and mode localization to nonlinear quasi-periodic motions, and illustrate their application to a quasi-periodic regime in the forced response.

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“…Non-classical damping, on the other hand, can generally lead to mode complexity; and if it is strong enough, it may even lead to a transition from standing to traveling waves, for example, as was recently shown for a locally damped linear string. 5,6 However, the extent of motion complexity is typically expected to remain small if damping is relatively weak. It should be noted that non-classical damping is quite common in many engineering applications: For instance, mechanical joints or attachments subject to dry friction may introduce a rather local dissipation source which leads to a highly non-classical damping distribution in the system.…”

Section: Introductionmentioning

confidence: 99%

Motion complexity in a non-classically damped system with closely spaced modes: From standing to traveling waves

Krack

Bergman

Vakakis

2015

Proceedings of the Institution of Mechanical Engineers, Part K:

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This article addresses the phenomenon of motion complexity in a periodically oscillating system, i.e. the occurrence of non-trivial phase lags among the system's coordinates. Specifically, the steady-state forced response of a linear, weakly damped, self-adjoint system is studied, for which the extent of motion complexity is typically expected to be small. Yet, it is shown that under the condition of closely spaced modes, weak non-classical damping may lead to a transition from standing waves to traveling waves. A system of two oscillators weakly coupled via a linear spring-damper element is considered. The emergence of these motions is related to the distribution of the applied forces and the effect of adding nonlinearity in the form of a cubic spring to the system is investigated. Moreover, it is demonstrated that under certain conditions, the traveling wave response is critical, i.e. it is associated with a resonance or anti-resonance.

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Damping-induced interplay between vibrations and waves in a forced non-dispersive elastic continuum with asymmetrically placed local attachments

Cited by 14 publications

References 30 publications

Analytical Solutions for Side-branch Impedance Producing Spatial Localization of Acoustic Waves in Ducts with Varying Cross Sections

Analytical Solutions for Side-branch Impedance Producing Spatial Localization of Acoustic Waves in Ducts with Varying Cross Sections

Global complexity effects due to local damping in a nonlinear system in 1:3 internal resonance

Motion complexity in a non-classically damped system with closely spaced modes: From standing to traveling waves

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