A multi-parameter perturbation solution for the inverse eigenproblem of nearly-resonant N-dimensional Hamiltonian systems

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 pathologic 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 N-dimensional Hamiltonian systems with a generic number of close eigenvalues. The leading idea is to systematically treat 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 twofold objective: first, identify a close resonant system suited to serve as a starting point for sensitivity analyses (inverse problem); second, asymptotically approximate the eigensolution of all the nearly-resonant systems which may arise from its generic perturbation (direct problem). 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. Besides the particular class of periodic systems, the work findings apply to a number of internally-resonant engineering structures in which components with different stiffness properties are assembled together, as may happen when a rigid main structure is joined with a set of flexible identical sub-structures. Typical examples in the civil and mechanical engineering fields are cable-stayed bridges, made of a rigid deck supported by several flexible cable stays, and bladed disks, in which several flexible radial blades are attached to a rigid rotor-disk