Suivi numérique des bifurcations pour l'analyse paramétrique de la dynamique non-linéaire des rotors

Abstract : Generally speaking, the rotating systems utilized in the energy production have a small rotor-stator gap, are able to run during long periods, and are mounted on hydrodynamic bearings. Rotor-stator interactions in case of blade loss, crack propagation due to fatigue, and a variable stiffness due to the nonlinear restoring forces of the bearings can make the rotordynamics nonlinear and the responses complicated: significant amplitude and frequency shifts are introduced, sub- and super-harmonics appear, and hysteresis occurs. It is of great importance to understand, predict and control this complicated dynamics. Due to the large number of DOFs and the broad range of study frequency, the computation time for solving the equations of motion by a temporal integration method can be quite prohibitive. It becomes particularly disadvantageous at the design stage where a parametrical study need to be quickly performed. An alternative numerical method, which is general and effective at the same time, is proposed in order to analyse the nonlinear response of the rotors at steady state. Firstly, the periodic responses of nonlinear rotors are calculated in the frequency domain by combining harmonic balance method (HBM) and alternating frequency-time (AFT). With the help of continuation method, all dynamic equilibrium solutions of nonlinear systems are determined for the range of study frequency. Then, Floquet exponents which are the eigenvalues of Jacobian are sought for stability analysis of periodic solutions. Then the local stability of the periodic solution is analysed through the bifurcation indicators which are based on the evolution of Floquet exponents. The bifurcations of periodic solution branch, such as limit point, branch point, and Neimark-Sacker bifurcation, are thus detected. By declaring a system parameter (friction coefficient, rotor / stator gap, excitation amplitude, ...) as a new variable, applying once again the continuation method to the augmented system determines directly the bifurcation's evolution as a function of this parameter. Thus, parametric analysis of the nonlinear dynamic behaviour is achieved, the stability boundary or the regime change boundary is directly determined. Numerous developments are implemented in the calculation code Cast3M.
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Lihan Xie. Suivi numérique des bifurcations pour l'analyse paramétrique de la dynamique non-linéaire des rotors. Mécanique [physics.med-ph]. Université de Lyon, 2016. Français. ⟨NNT : 2016LYSEI018⟩. ⟨tel-02191438⟩

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