Chapter 18 Solitons in Cyclic and Symmetric Structures Filipe Fontanela, Aurelien Grolet, Loic Salles, and Norbert Hoffmann Abstract This research focuses on localised states arising from modulationally unstable plane waves in non-conservative cyclic and symmetric structures. The main application is on vibrations of bladed-disks of aircraft engines experiencing nonlinear effects, such as large displacements, friction dissipation, and/or complex fluid-structure interactions. The investigation is based on a minimal model composed of a chain of linearly damped Duffing oscillators under external travelling wave excitation. The computed results are based on two strategies: (1) a Non-Linear Schrödinger Equation (NLSE) approximation; and (2) the periodic and quasi-periodic Harmonic Balance Methods (HBM). In both cases, the results show that unstable plane waves may self-modulate, leading to stable and unstable single and multiple solitons configurations. Keywords Localised vibrations · Solitons · Cyclic structures · Non-linear Schrödinger equation · Harmonic balance methods 18.1 Introduction Localisation of vibrations is a very important topic in rotating machines, such as bladed-disks of aircraft engines, due to high cycle fatigue. In the linear regime, localised vibrations arise in ideally periodic structures due to inherent inhomogeneities resulting e.g. from the manufacturing processes or wear. However, in real applications, when structures e.g. experience large deformations induced by strong excitations, their behaviours deviate from the linear regime due to non-linearities. It is well-known that, in the non-linear regime, energy localisation may arise even in perfect period structures due to bifurcations. This research focuses on the non-linear dynamics of cyclic and symmetric structures excited by travelling waves. This excitation is very common in turbomachinery applications since it can be generated due to unbalances or aerodynamic excitations. The findings are based on two different strategies: (1) a non-conservative NLSE approximation; and (2) a fully numeric periodic and quasi-period HBM approach. The results show that stable solitons may emerge from unstable homogeneous solutions, leading to localised vibrations which move along the structure preserving their shapes. 18.2 Physical System The physical system under investigation consists of Ns unitary masses, cyclically connected to each other by linear springs !2 c, and attached to the ground by linear springs !2 0, viscous dampers 2, and cubic springs . The displacement for the nth degree of freedomun is written as Run C 2Pun C! 2 0un ! 2 c.un 1 CunC1 2un/ C u 3 n Dfn; (18.1) F. Fontanela ( ) · L. Salles Imperial College London, London, UK e-mail: ff515@ic.ac.uk; ffontanela@gmail.com A. Grolet Arts et Métiers ParisTech, Lille, France N. Hoffmann Imperial College London, London, UK Hamburg University of Technology, Hamburg, Germany © The Society for Experimental Mechanics, Inc. 2019 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-74280-9_18 175
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