Solitons in cyclic and symmetric structures
Creators: name: Fontanela, Filipe
Publisher: name: Imperial College London
Year: 2019

description: The main aim of this thesis is to investigate the emergence of localised vibrations in cyclic and perfectly symmetric structures due to the effect of nonlinearity. The application is on vibrations of bladed disks of aircraft engines, i.e. mechanical structures composed of ideally symmetric sectors assembled in a cyclic configuration. Firstly, a minimal model of a bladed disk vibrating under the effect of geometric nonlinearities is introduced. The system is composed of a chain of Duffing oscillators, and the cubic-type stiffening effect represents, physically, a first-order correction for an underlying nonlinear stress-strain dependence. The analysis starts from a weakly nonlinear and slowly varying approximation, where the system can be studied using insights from a continuum wave mechanics approach. The investigation shows that homogeneous solutions might become unstable when nonlinear effects are large, leading to strong and stable localised vibrations. In order to solve more complex physical models, where the weakly nonlinear and slowly varying approximation might not be valid, a fully numerical approach is proposed. The strategy is based on the periodic and quasi-periodic harmonic balance methods, and thus the displacement of each degree of freedom of the model is assumed to be written as a truncated Fourier series. Firstly, the same chain of Duffing oscillators investigated before is readdressed, and the results obtained from the fully numerical implementation are discussed. In the following, a more complex physical model for a bladed disk, where each sector is modelled by two degrees of freedom, is addressed numerically. The results show similar features compared to the simple chain of Duffing oscillators, i.e. large engine order excitations might lead to localised vibrations due to geometric nonlinearities. In order to study a real geometry, a dummy bladed disk is introduced and solutions of this physical system are computed from fully nonlinear finite elements. The results show that localised solutions might bifurcate from homogeneous states when the displacement levels are about 10\% of the blade thickness. Finally, the effect of impacts is considered. The main motivation is to investigate the possibility of energy localisation when nonlinearities go beyond the cubic-type stiffening effects. The results show that homogeneous states might self-modulate and localise, spontaneously, through envelope dynamics even if nonlinearity arises due to non-smooth interactions. Lastly, the emergence of localised vibrations due to nonlinear effects is investigated experimentally. A test rig composed of two weakly coupled beams impacting against rigid stoppers is designed. The results show that strong localised regimes might be observed, and the final configuration depends on the underlying initial conditions only. A minimal model with piecewise linear springs is then developed in order to study, phenomenologically, the observed results. The theoretical analysis shows that the measured localised regimes might be seen as results of bifurcated normal modes if the system is analysed from a modal analysis perspective.