Master thesis and internship[BR]- Master's thesis : Dynamics of Classical Nonlinear Systems and Blade-tip/Casing Interface and Computation of Nonlinear Modes Using the Harmonic Balance Method [BR]- Integration Internship

Abstract

One of the main goal that is pursued by aeronautical engineers is to reduce the environmental footprint of the current aircraft while reducing their fuel consumption. Therefore, aircraft engine manufacturers try to design more efficient engines. One alternative to increase this efficiency is to reduce the operating clearance between the rotating and the static parts within the turbomachine, which promotes contact phenomena. In particular, blade-tip/casing interactions and their dangerous consequences push the engineers to find some methods to simulate these contacts. This thesis aimed to better understand the dynamics of nonlinear systems through the application of one of these methods: a frequency based approach relying on the harmonic balance method (HBM). This HBM involved important surrounding methods required to solve the nonlinear equations of motion. In particular, the alternating frequency time process enabled to handle the nonlinearity by sampling a period of oscillation into several time intervals N and by evaluating the nonlinear effort at each time step. This parameter turned out to be crucial in the rest of the methodology as it controlled aliasing errors which could be detrimental to the good representation of the frequency response curve (CRF ) of a system. Four classical systems composed by few degrees of freedom were studied: the Duffing oscillator, a system built by the linear coupling between two Duffing oscillators, the Laxalde wiper and the Jeffcott rotor. By using an appropriate Newton-Raphson iterative solver and some suitable continuation technique, the equations of motion were solved, which enabled to analyze the CRF of those systems. It was observed that the response curves of nonlinear systems have nothing to do with that of linear systems. In fact, it has been noticed that the resonance peaks were modified compared to the linear case and that multiple resonances seemed to exist within the pulsation range of interest. This latter observation enabled to introduce the concept of secondary resonances. One of the main goal of this work was also to try to apply the methodology to those blade-tip/casing contact interactions. For that purpose, the HBM method was conducted on an industrial application (the NASA rotor 37). As a consequence, the handling of a contact nonlinearity (numerical treatment, modeling,...) was introduced and the analysis of the CRF showed that the nonlinear resonances of the different degrees of freedom (radial displacements of the different contact nodes) were not occurring at the exact same frequency, underlying the complexity and the effects that a contact nonlinearity could imply. In addition, the comparison with time integration results showed a good match but a large underestimation of the nonlinear resonance by time integration methods, which further reinforced the benefit of using a frequency based approach as well. Some parametric studies were also led on the classical systems and showed that for cubic nonlinearities, an hardening of those systems is to be noticed with the increasing nonlinear stiffness. This work ended by the computation of nonlinear modes which were represented as backbone curves crossing the nonlinear resonances of the forced responses computed with different values of the external forcing. The computation of those modes has been divided into a conservative and a non conservative case. In the first case, the free responses of the conservative systems were computed by removing the damping matrix from the equations. It was observed that the nonlinear modes were accurately computed even if some discrepancies were noticed and could be attributed to the absence of this damping matrix. This accuracy turned out to increase with the increasing number of harmonics. In the non conservative case, the introduction of a damping factor was added to account for the dissipative character of the nonlinearity. For systems such as the Laxalde system, the nonlinear mode was nicely predicted by the modified algorithm. Nevertheless, the biggest complexity lay in the convergence of the solution. In order to enhance the chances of convergence, the initialization of the solver should be carefully done, the idea being to approximate the nonlinear mode by the associated linear mode at energy levels for which the nonlinearity is less active. As explained in this document, for contact nonlinearities, the complexity was even greater due to the necessity to initialize the solver at very low energy levels.