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 me- chanical 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 rel- evant phenomena, such as frequency veering and mode localization or hybridization. The paper presents the generalization of uniformly valid perturbation methods to per- form eigensolution sensitivity analyses in N-dimensional Hamiltonian systems with a generic number of close eigenvalues. The leading idea is to systematically treat nearly-resonantsystemsasmulti-parameterperturbationsofaperfectly-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 start- ing point for sensitivity analyses (inverse problem); second, asymptotically approx- imate 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 physi- cal 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 flex- ible radial blades are attached to a rigid rotor-disk.