Abstract
In this paper we develop a fully numerical approach to compute quasi-periodic vibrations bifurcating from nonlinear periodic states in cyclic and symmetric structures. The focus is on localised oscillations arising from modulationally unstable travelling waves induced by strong external excitations. The computational strategy is based on the periodic and quasi-periodic harmonic balance methods together with an arc-length continuation scheme. Due to the presence of multiple localised states, a new method to switch from periodic to quasi-periodic states is proposed. The algorithm is applied to two different minimal models for bladed disks vibrating in large amplitudes regimes. In the first case, each sector of the bladed disk is modelled by a single degree of freedom, while in the second application a second degree of freedom is included to account for the disk inertia. In both cases the algorithm has identified and tracked multiple quasi-periodic localised states travelling around the structure in the form of dissipative solitons.
Highlights
The emergence of localised vibrations in cyclic and symmetric structures is an important phenomenon due to potential problems induced by high cycle fatigue
Mechanical components with cyclic symmetry are very common in the aerospace industry, where e.g. bladed disks of aircraft engines [1], space antennas [2] and reflectors [3] are usually composed of ideally identical substructures assembled in a cyclic and symmetric configuration
This paper introduces a fully numerical approach to compute quasi-periodic states bifurcating from periodic vibrations due to nonlinearities
Summary
The emergence of localised vibrations in cyclic and symmetric structures is an important phenomenon due to potential problems induced by high cycle fatigue. In the linear framework localised vibrations arise due to the lack of symmetry induced e.g. by manufacturing variability or wear This topic is well-known in the aerospace industry as a mistuning problem, and studies have mainly focused on effective modelling techniques [4,5], experimental characterisation together with model identification [6,7], and even the use of intentional mistuning to reduce the level of vibrations in real applications [8,9]. This paper introduces a fully numerical approach to compute quasi-periodic states bifurcating from periodic vibrations due to nonlinearities. The focus is on localised states resulting from the emergence of modulation instability for travelling wave excitations in cyclic and symmetric structures This investigation has been motivated by recent findings in Refs.
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