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# Strongly nonlinear acoustics of one-dimensional granular sonic vacua

#### Kalkunte Raghunath, Jayaprakash

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https://hdl.handle.net/2142/44278

## Description

- Title
- Strongly nonlinear acoustics of one-dimensional granular sonic vacua
- Author(s)
- Kalkunte Raghunath, Jayaprakash
- Issue Date
- 2013-05-24T22:06:21Z
- Director of Research (if dissertation) or Advisor (if thesis)
- Vakakis, Alexander F.
- Doctoral Committee Chair(s)
- Vakakis, Alexander F.
- Committee Member(s)
- Dankowicz, Harry
- Ostoja-Starzewski, Martin
- Bergman, Lawrence A.

- Department of Study
- Mechanical Sci & Engineering
- Discipline
- Theoretical & Applied Mechans
- Degree Granting Institution
- University of Illinois at Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(s)
- Granular Media
- Solitary Waves
- Nonlinear Normal Modes
- Periodic Granular Media
- Granular Dimers
- Nonlinear Resonance
- Nonlinear Anti-resonance

- Abstract
- This research is concerned with the nonlinear dynamics and acoustics of one dimensional ordered granular media. In particular the considered granular systems are composed of discrete elastic spherical beads that mutually interact through strongly nonlinear Hertzian force interaction law. In this work we primarily consider the granular chains in the limit of zero pre-compression, thus leading to complete absence of linear acoustics and zero speed of sound (as defined in the classical sense), hence their characterization as ‘sonic vacua’. Furthermore, we sparingly incorporate a dissipative mechanism between interacting neighboring beads. Due to the absence of cohesive interaction forces, the interaction between the beads is strongly nonlinear and non-smooth owing to possible bead separations and ensuing collisions. However, with the application of pre-compression the interaction between beads becomes weakly nonlinear and smooth and the granular media can no longer be considered as ‘sonic vacua’. Hence, the dynamics and acoustics of these material systems are highly tunable with varying pre-compression. The first part of the study is primarily focused on the oscillatory dynamics of finite dimensional homogeneous granular chains. We initiate the study by investigating the existence of nonlinear normal modes (NNMs) in these systems with fixed boundary conditions. The realized modes which have energy dependent frequency, when represented on frequency-energy plots (FEP) divide these plots into two mutually exclusive regions separated by the out of phase NNM which corresponds to the highest possible oscillation frequency. All the NNMs realized are situated in the region below the out-of-phase NNM, which is denoted as propagation band; the complementary region is then the attenuation band. When the chain is harmonically base excited, frequencies in the propagation band are spatially extended whereas those in the attenuation band are spatially localized wherein the beads experience constant compression. The dynamics in the attenuation zone is weakly nonlinear and has been studied analytically. Furthermore, the existence of frequency bands has been experimentally verified. This happens to be the first exposition on the frequency band zones in homogeneous granular chains from both theoretical and experimental perspectives. The second part of the study is concerned with periodic ordered diatomic (dimer) granular chains consisting of spherical beads of two types. Initially we consider the most simple dimer chain wherein each bead of type 1 is preceded and followed by a bead of type 2; such chains are denoted as 1:1 dimers and the dynamics of such chains is governed by a single parameter (ε) scaling the mass of the two types of beads. Due to the periodic variation of the masses, a propagating pulse loses energy in the form of radiating waves in its trail and thus the pulse attenuates in dimer chains with an arbitrary value of mass ratio. Interestingly, at certain discrete values of mass ratios, the energy leakage from the propagating pulse ceases and the pulse propagates without attenuation, thus such pulses are called solitary waves. At the particular mass ratios where solitary waves are realized, these waves form an important energy transfer mechanism and any arbitrary pulse eventually disintegrates into a train of solitary waves. This is the first exposition of the realization of solitary waves in 1:1 dimer granular chains. Moreover, we show that these chains support a countable infinity of solitary waves parameterized by energy. The contrasting (but more intuitive) effect of substantial energy radiation from the propagating pulse is also observed at a discrete set of mass ratios. This phenomenon is designated as resonance. A complete analytical formulation for these two phenomena is provided. Furthermore, these phenomena have been experimentally verified. The experimental results show good correspondence with the theoretical results thus validating the theoretically predicted existence of nonlinear resonances in granular media. This happens to be the first experimental investigation of the resonances in uncompressed dimer chains. The spatial periodicity of traveling waves radiated in the trail of the propagating pulse (at arbitrary mass ratio) is found to depend only on the specific mass ratio of the dimer. The effect of the mass ratio on the realization of traveling waves, and in turn their significance to the resonance and pulse attenuation is studied by considering reduced order dimer chains composed of finite number of beads with periodic boundary conditions. Interesting bifurcations of the traveling waves have been discovered and correlation between the bifurcations and resonances is noted. We further study the dynamics of a general class of 1:N (N≥2) dimer chains. The dynamics of these chains is governed by two non-dimensional parameters, the mass ratio (ε) and the stiffness ratio (α) scaling the respective properties of the two types of beads. We report on a countable infinity of traveling solitary waves and resonances and prove numerically and asymptotically their existence in the 1:2 dimer chain. These solitary waves studied in homogeneous and 1:1 dimer chains possess symmetric velocity waveforms, In contrast, the traveling solitary waves velocity waveforms of the 1:2 dimers of the heavy beads are symmetric, whereas those of the light beads are non-symmetric. Interestingly, we show that no such solitary waves or resonances can be realized in general 1:N granular dimers with N>2. The final part of this study is concerned with the nonlinear dynamics of granular containers. These are granular setups composed of different types of homogeneous chains positioned in alternating configurations. Depending on the properties of the different types of homogeneous granular chains of the container, the wave energy can be entrapped in an appropriately designed intermediate layer. The primary aim of this study is to apply the binary collision approximation (BCA) for the analytic estimation of the amplitude of the scattered pulses (solitary waves) in the granular container. In addition, we provide a numerical study showing the qualitatively different dynamics of these containers depending on the frequency and amplitude of an applied harmonic excitation.
- Graduation Semester
- 2013-05
- Permalink
- http://hdl.handle.net/2142/44278
- Copyright and License Information
- Copyright 2013 Jayaprakash Kalkunte Raghunath

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