Travelling and standing envelope solitons in discrete non-linear cyclic structures. (15th December 2016)

• Record Type: Journal Article
• Authors: Grolet, Aurelien; Hoffmann, Norbert; Thouverez, Fabrice; Schwingshackl, Christoph
• Is Part Of: Mechanical systems and signal processing. Volume 81(2016)
• Page Start: 75
• Page End: 87
• Publication Date: 2016-12-15
• Subjects: Non-linear dynamics -- Cyclic system -- Travelling waves -- Envelope soliton -- Multiple scales; Structural dynamics -- Periodicals; Vibration -- Periodicals; Constructions -- Dynamique -- Périodiques; Vibration -- Périodiques; Structural dynamics; Vibration; Periodicals; 621
• DOI: 10.1016/j.ymssp.2016.02.062
• Languages: English
• ISSNs: 0888-3270

Abstract

Abstract: Envelope solitons are demonstrated to exist in non-linear discrete structures with cyclic symmetry. The analysis is based on the Non-Linear Schrodinger Equation for the weakly non-linear limit, and on numerical simulation of the fully non-linear equations for larger amplitudes. Envelope solitons exist for parameters in which the wave equation is focussing and they have the form of shape-conserving wave packages propagating roughly with group velocity. For the limit of maximum wave number, where the group velocity vanishes, standing wave packages result and can be linked via a bifurcation to the non-localised non-linear normal modes. Numerical applications are carried out on a simple discrete system with cyclic symmetry which can be seen as a reduced model of a bladed disk as found in turbo-machinery. Abstract : Highlights: Use of continuum approximation to derive a non-linear Schrodinger equation for the evolution of travelling waves. Analytic expression for envelope soliton in a cyclic structure as a particular solution of the NLSE. Numerical application on a simple non-linear cyclic system which can be seen as a simple approximation of a bladed disk. Both travelling and standing soliton patterns arise which have the property that the higher the amplitude is, the more localised the solution gets. Use of continuation techniques to show that standing solitons are linked to (standing) non-linear normal modes through bifurcation. Contribution to the field of solitons in the context of structural mechanics.