Asymptotic numerical method and polynomial chaos expansion for the study of stochastic non-linear normal modes

Nonlinear normal mode (NNM) analysis is one emerging technique to analyse the nonlinear vibration of bladed-disk. It links the resonance frequency to the energy present in the system, and allows a simple identification of internal resonances in the structure. Non-linear vibration analysis is traditionally carried out under the assumption that the mechanical properties and forcing function are deterministic. Since every mechanical system is by nature uncertain a truly accurate nonlinear dynamic analysis requires the inclusion of random variables in the response predictions. The propagation of random input uncertainties in a NNM analysis is the main aim of the presented work. The Asymptotic Numerical Method (ANM) will be used to calculate the NNMs for a contact problem in a computationally efficient way. The stochastic NNM permits to quantify the effect of uncertainties on the resonance frequency and the change in mode shape due to non-linearities, leading to the calculation of uncertain internal resonances. The proposed method is initially applied to a simple spring-mass system to demonstrate the effects of uncertainty on the NNM predictions. In a second step a blade-casing interaction with localized contact non-linearity is investigated with a real geometry. The resulting NNMs show the presence of internal resonance for both cases.