Keywords: resonant systems, inverse eigenproblem, perturbation methods.

The analytical models of several structural systems show that eigenvalues and eigenvectors possess high sensitivity to the physical parameters. Following the eigensolution loci versus one or more significant parameters may represent a matter of theoretical and practical interest in stability analysis, design optimization, model updating and vibration control. The intersection or closeness between two or more eigenvalue loci define the resonance or near-resonance regions of the parameter space. In these regions significant phenomena, including frequency veering and mode localization or hybridization, may affect the system free dynamics, while important energy transfers among the resonant modes may characterize the forced response.

In the civil and mechanical engineering fields, an exact frequency tuning can be easily realized in nominally periodic or symmetric systems, considered perfect structures. If slight imperfections disturb the symmetry and periodicity, small frequency shifts occur, originating nearly-resonant structures [1]. Small geometric defects and weak linear interactions, denoted respectively as disorder and coupling terms, are common imperfections affecting perfect (uncoupled and ordered) systems. In this context, perturbation methods may represent an efficient tool to perform eigensolution sensitivity analyses. According to perturbation strategies, known a reference system, the eigensolution of a slightly perturbed system can be approximated constructing asymptotical expansions. With respect to traditional approaches [1], which consider only the coupling as perturbation (single-parameter perturbation), a refined approach [2] retains both the disorder and the coupling as independent perturbations of the uncoupled and ordered system (multi-parameter perturbation).

This paper presents a systematization of the multi-parameter approach for eigensolution sensitivity analyses of N-dimensional Hamiltonian systems in the nearly-resonance regions of the parameter space. The proposed algorithm requires minimal sufficient information, consisting of the eigensolution (not necessarily complete) of a single nearly-resonant system, as could be derived from experimental measures. The nearly-resonant system is postulated to arise from the unknown perturbation of a perfectly-resonant, thought itself unknown, system. First, the eigensolution sensitivity of the perfect system, with respect to a generic multi-parameter perturbation, is analyzed (task I: Direct problem). The direct problem solution gives the approximated eigenpairs of each nearly-resonant system which could arise from the perturbation. Second, the unknown perfect system to be perturbed is identified, starting from the partial knowledge of the experimental system (task II: Inverse problem). Questions regarding the existence and uniqueness of the inverse problem solution are discussed. Finally, the entire procedure is successfully applied to perform the eigensolution sensitivity analysis of a prototypal structural system, characterized by frequency veering and mode hybridization phenomena.

1

C. Pierre, "Mode localization and eigenvalue loci veering phenomena in disordered structures", J. Sound & Vibration, 126(3), 485-502, 1988. doi:10.1016/0022-460X(88)90226-X

2

A. Luongo, "Eigensolutions of perturbed nearly defective matrices", J. Sound & Vibration, 185(3), 377-395, 1995. doi:10.1006/jsvi.1995.0387

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