criteria for predicting veerings and crossings in both continuous and discretized models. They showed that the key point to differentiate crossing and veering phenomena is the coupling factor, and they suggested a simple example: coupled oscillator. To summarize, Perkins and Mote [3] used perturbation theory to derive “coupling factors” to quantify the eigenfunctions coupling. Pierre [4] explained how localization and veering are related to two kind of “coupling”: the physical coupling between component structures, and the modal coupling seen between mode shapes through parameter perturbations. He asserted localization and veering occur when modal coupling is of the same order or greater than physical coupling. His studies showed that, in structures with close eigenvalues, small structural irregularities result in both strong localization of mode shapes and abrupt veering away of the loci of the eigenvalues when these are plotted against a parameter representing the system disorder. Regarding coupling factors in curve veering and curve crossing, there are some fields in which these phenomena are studied. For example, modal analysis of bridges with aeroelastic effects [5] and vibration analysis of rotating cantilever beams [6] are strongly influenced by coupling problems. Balmès [7] analysed the eigenvector transformations of a three degree of freedom lumped-mass system. The simplest cyclic symmetric spring-mass system predicts a double mode and it shows the effect of a parameter variation on the variables of the parameterization. Balmès pointed out that only very particular conservative structures present eigenvalues veer and the exchange of mode shape properties happens as a rotation in a fixed subspace, similar to that of the example [7]. Only three types of conservative structure have been identified as allowing truly multiple modes (allowing the eigenvalues to be equal and therefore modal crossing with instantaneous rotation on mode shapes): 1. symmetric or cyclic structures, where it is allowed through algebraic properties of the group of symmetric properties, 2. multi-dimensional substructures for which motions in different dimensions uncouple, such as plates having a bending and a torsional mode at the same frequency, 3. structures with fully uncoupled substructures. Mode localization and curve veering phenomena have been studied in the frequency domain by Mugan [8]. The singular-value decomposition (SVD) was employed to study the effects of localization phenomena on input-output relationships, and power and energy transmission ratios of structures. The problem of measuring the phenomena of eigenvalue curve veering and mode localization has been studied also by Liu [9]. He suggested to define a critical value for the derivative of the eigenvectors or for the second derivative of the eigenvalues, above which the modes would be deemed to be veering. Adhikari [10] cited examples where veering is influenced, and sometimes even suppressed, by the effect of damping. The use of expression for derivative of undamped modes can give rise to erroneous results even when the modal damping is quite low. Young [11] dealt with the problem using two simple examples, concerning with the theory of continuous bodies. The inadequacy of approximate methods has been shown to be one source of couplings, while recently it has been found that there exist two different kinds of coupling responsible for the occurrence: implicit and explicit couplings. The former is generated by the incompleteness of the admissible function used in the approximate approach, while the latter is induced by the interaction between the main component and sub-component of structure. Curve veering can be observed in systems with explicit coupling where exact solutions are available. More recently Du Bois, Adhikari and Lieven [12] presented a detailed experimental and numerical investigation on veering and crossing phenomena. Despite the widespread acceptance of veering theory, supported by poor experimental data, they developed an experimental structure made up of redundant truss. The transverse stiffness of the beams is influenced by the applied axial load. This structural stiffness modulation is used to provide the parametric variation for the experiment. Also a FE model have been developed to compare experimental and numerical data. In particular the counterintuitive variations of the mode shapes in these regions have been confirmed. The investigation has highlighted the impact of veering on model updating and modal correlation algorithms, as well as any discipline concerned with the analysis of closely spaced modes. The analysis of mode shape transformations in terms of eigenvector rotations is found to be a valuable tool in quantifying the dynamic behaviour, and this is expected to find application in a wide range of parametric modal analyses. In literature there are many articles that talk about repeated eigenvalues, but in term of algorithms for computing the derivatives of eigenvalues and eigenvectors. For example in [13] it is shown an algorithms for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues. 326
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