Finally, in [14] D’Ambrogio and Fregolent proposed an extension of Modal Assurance Criterion (MAC) for coincident or close eigenvalues problems. They consider the correlation between a modal vector and subspace spanned by several modal vectors, instead of the usual correlation between two modal vectors. At the end of this introduction it is possible to say that literature are very full of articles that deal with close eigenvalues, double eigenvalues, curve veering and crossing phenomena and mode localization, and a lot of fields use this concept to study physical phenomena. But, on the other end, there are not very experimental studies, except [12], that focus on this issues. This is the reason that have switched on idea to identify tunable and simple test bench with two coincident and/or close eigenvalues without symmetric properties or completely uncoupled dynamic systems. After a review about theory of curve veering and crossing phenomena in lumped parameters systems and continuous systems, this paper presents a simple test rig for experimentally testing coincident or close eigenvalues with crossing or veering phenomena. The main aims of the test rig design are: to obtain a simple and tunable test rig for experimental validation of its dynamic behaviour; to understand curve crossing and curve veering taking into account uncertainty and variability of the structure, concerning natural frequencies and mode shapes with an experimental viewpoint; to comprehend possible energy paths with close or coincident modes, also to analyse possible dissipation strategies locally far from sources (application as damping systems). OVERVIEW OF CURVE CROSSING AND CURVE VEERING PHENOMENA Eigenvalues are often plotted versus a system parameter creating a family of root loci. Two converging loci either do or do not intersect. It is necessary to distinguish between curve crossing (coincident eigenvalues) and curve veering (close eigenvalues) phenomena. The former occurs when one eigenvalue curve intersects another curve and the dynamics behaviour is characterized by coincident eigenfrequencies: modes order changes, whereas the eigenfunctions (or eigenvectors in lumped systems) remain associate to the corresponding eigenvalues. In the latter two loci approach each other and abruptly diverge without meeting. Moreover an important characteristic of curve veering is that the eigenfunctions associated with the eigenvalues on each locus are interchanged during veering in a rapid but continuous way. In order to identify curve crossing and curve veering the MAC index has been used [15], applying it before and after the occurrence of phenomena. With MAC is possible to evaluate the modes correlation in the transition area. To better understand curve crossing and curve veering phenomena and to explain the utility of MAC to distinguish one of other, two different cases of crossing and veering phenomena are taken into account: a monodimensional lumped system (see Balmès’s system in [7]); a two-dimensional system concerning an axial-symmetric system, such as a bell. Figure 1 shows the cyclic and symmetric system of Balmès [7]. Without loosing generality, unitary masses and springs are assumed. Figure 1 – Lumped cyclic and symmetric monodimensional system. 327
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