Alqawasmeh, Qusi Ibrahim Ahmad
(2016)

*Dispersion and energy relations in periodic chains and grids.*
[Laurea magistrale], Università di Bologna, Corso di Studio in

Civil engineering [LM-DM270], Documento full-text non disponibile

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## Abstract

In this study wave propagation, dispersion relations, and energy relations for linear elastic periodic systems are analyzed. In particular, the dispersion relations for monoatomic chain of infinite dimension are obtained analytically by writing the Block-type wave equation for a unit cell in order to capture the dynamic behavior for chains under prescribed vibration. By comparing the discretized model (mass-spring chain) with the solid bar system, the nonlinearity of the dispersion relation for chain indicates that the periodic lattice is dispersive in contrast to the continuous rod, which is non dispersive.
Further investigations have been performed considering one-dimensional diatomic linear elastic mass-spring chain. The dispersion relations, energy velocity, and group velocity have been derived. At certain range of frequencies harmonic plane waves do not propagate in contrast with monoatomic chain. Also, since the diatomic chain considered is a linear elastic chain, both of the energy velocity and the group velocity are identical. As long as the linear elastic condition is considered the results show zero flux condition without residual energy. In addition, this paper shows that the diatomic chain dispersion relations are independent on the unit cell scheme.
Finally, an extension for the study covers the dispersion and energy relations for 2D- grid system. The 2x2 grid system show a periodicity of the dispersion surface in the wavenumber domain. In addition, the symmetry of the surface can be exploited to identify an Irreducible Brillouin Zone (IBZ). Compact representations of the dispersion properties of multidimensional periodic systems are obtained by plotting frequency as the wave vector’s components vary along the boundary of the IBZ, which leads to a widely accepted and effective visualization of bandgaps and overall dispersion properties.

Abstract

In this study wave propagation, dispersion relations, and energy relations for linear elastic periodic systems are analyzed. In particular, the dispersion relations for monoatomic chain of infinite dimension are obtained analytically by writing the Block-type wave equation for a unit cell in order to capture the dynamic behavior for chains under prescribed vibration. By comparing the discretized model (mass-spring chain) with the solid bar system, the nonlinearity of the dispersion relation for chain indicates that the periodic lattice is dispersive in contrast to the continuous rod, which is non dispersive.
Further investigations have been performed considering one-dimensional diatomic linear elastic mass-spring chain. The dispersion relations, energy velocity, and group velocity have been derived. At certain range of frequencies harmonic plane waves do not propagate in contrast with monoatomic chain. Also, since the diatomic chain considered is a linear elastic chain, both of the energy velocity and the group velocity are identical. As long as the linear elastic condition is considered the results show zero flux condition without residual energy. In addition, this paper shows that the diatomic chain dispersion relations are independent on the unit cell scheme.
Finally, an extension for the study covers the dispersion and energy relations for 2D- grid system. The 2x2 grid system show a periodicity of the dispersion surface in the wavenumber domain. In addition, the symmetry of the surface can be exploited to identify an Irreducible Brillouin Zone (IBZ). Compact representations of the dispersion properties of multidimensional periodic systems are obtained by plotting frequency as the wave vector’s components vary along the boundary of the IBZ, which leads to a widely accepted and effective visualization of bandgaps and overall dispersion properties.

Tipologia del documento

Tesi di laurea
(Laurea magistrale)

Autore della tesi

Alqawasmeh, Qusi Ibrahim Ahmad

Relatore della tesi

Correlatore della tesi

Scuola

Corso di studio

Ordinamento Cds

DM270

Parole chiave

chains,grids,diatomic chains,lattice,energy velocity,group velocity,dispersion relations,unit cell

Data di discussione della Tesi

14 Luglio 2016

URI

## Altri metadati

Tipologia del documento

Tesi di laurea
(NON SPECIFICATO)

Autore della tesi

Alqawasmeh, Qusi Ibrahim Ahmad

Relatore della tesi

Correlatore della tesi

Scuola

Corso di studio

Ordinamento Cds

DM270

Parole chiave

chains,grids,diatomic chains,lattice,energy velocity,group velocity,dispersion relations,unit cell

Data di discussione della Tesi

14 Luglio 2016

URI

Gestione del documento: