Energy transfer between nodal diameters of cyclic symmetric structures exhibiting polynomial nonlinearities: Cyclic condition and analysis

Abstract

The recent trend of new design for large slender blades in turboengines facilitates structural large deformation and geometrical nonlinear vibratory effects. The cyclic symmetry properties of such structures, combined with these nonlinearities give rise to specific complex phenomena such as internal resonances or energy localization. Robust and efficient methods have been developed to recover the solutions of such problems but they currently lack proper tools to analyze the results. In this paper, a formula is provided to compute polynomial nonlinear forces directly in the cyclic (spectral) domain of a cyclic symmetric structure. This new approach enables to express the equation of motion directly in the spectral domain, and offers two main advantages. It first provides a reduction of the system by determining a priori which nodal diameters will be coupled and by giving a closed-form expression for a direct evaluation of the cyclic nonlinear force. The proposed approach also facilitates results interpretations. The method is applied to a simplified bladed disk with a cubic nonlinearity that models symmetric large deformation, although the analytical development is valid for any polynomial nonlinearity. A detailed analysis of the different phenomena occurring in the cyclic structure excited along a particular nodal diameter is provided. More precisely, a multiple scales analysis is performed and yields interesting insights of internal resonances between different nodal diameters. This analytical method is completed with several numerical simulations involving the harmonic balance method and bifurcation algorithms. Both of these analyses recover coherent results on the prediction of different internal resonances for traveling or standing wave excitations.