Traveling waves in trimer granular lattice I: Bifurcation structure of traveling waves in the unit-cell model

• Published in: Communications in nonlinear science & numerical simulation Vol. 38; pp. 8 - 22
• Main Authors: K.R., Jayaprakash, Shiffer, A., Starosvetsky, Y.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2016
• Subjects: Bifurcations; Dynamical systems; Granular trimers; Mathematical models; Nonlinear dynamics; Nonlinear normal modes; Nonlinearity; Traveling waves; Trimers; Unit cell; Granular trimers; Nonlinear normal modes; Traveling waves; Bifurcations
• DOI: 10.1016/j.cnsns.2016.02.002
• ISSN: 1007-5704

•We consider periodic motions of one unit cell of granular trimer under zero pre-compression with periodic boundary conditions.•We study the evolution of traveling waves and standing waves in granular trimer.•This study unveils the peculiar bifurcation structure of traveling waves emerging in granular trimer.•The explored bifurcation structure is governed by two system parameters and is energy independent. Present paper is the first one in the series devoted to the dynamics of traveling waves emerging in the uncompressed, tri-atomic granular crystals. This work is primarily concerned with the dynamics of one-dimensional periodic granular trimer (tri-atomic) chains in the state of acoustic vacuum. Each unit cell consists of three spherical particles of different masses subject to periodic boundary conditions. Hertzian interaction law governs the mutual interaction of these particles. Under the assumption of zero pre-compression, this interaction is modeled as purely nonlinear, which means the absence of linear force component. The dynamics of such chains is governed by the two system parameters that scale the mass ratios between the particles of the unit cell. Such a system supports two different classes of periodic solutions namely the traveling and standing waves. The primary objective of the present study is the numerical analysis of the bifurcation structure of these solutions with emphasis on the dynamics of traveling waves. In fact, understanding of the bifurcation structure of the traveling wave solutions emerging in the unit-cell granular trimer is rather important and can shed light on the more complex nonlinear wave phenomena emerging in semi-infinite trimer chains.