Multi-parameter perturbation methods for the eigensolution sensitivity analysis of nearly-resonant non-defective multi-degree-of-freedom systems

Abstract

The dynamic behavior of structural systems may be strongly characterized by the occurrence of multiple internal resonances for particular combinations of the mechanical parameters. The linear models governing these resonant or nearly-resonant systems tend to exhibit high sensitivity of the eigenvalues and eigenvectors to small parameter modifications. This pathological condition is recognized as a source of relevant phenomena, such as frequency veering and mode localization or hybridization. The paper presents the generalization of uniformly valid perturbation methods to perform eigensolution sensitivity analyses in multi-degree-of-freedom Hamiltonian systems with a generic number of close eigenvalues. The leading idea is to treat systematically nearly-resonant systems as multi-parameter perturbations of a perfectly-resonant, non-defective – though a priori unknown – reference system. Given a single nearly-resonant system, a multi-parameter perturbation method is presented to achieve a two-fold objective: first, identify a close resonant system suited to serve as a starting point for sensitivity analyses (inverse problem); second, to approximate asymptotically the eigensolution of all the nearly-resonant systems which may arise from its generic perturbation (direct problem). The conditions of existence and uniqueness of the inverse problem solution are discussed. The direct problem solution is analyzed with a focus on the eigensolution sensitivity to parameter perturbations with different physical meanings, such as a slight geometric disorder or weak elastic coupling in periodic structures. Finally, the procedure is verified on a prototypical structural system describing the section dynamics of a suspended bridge.