Structural Dynamics with Coincident Eigenvalues: Modelling and Testing

Structural Dynamics with Coincident Eigenvalues: Modelling and Testing Elvio Bonisoli, Cristiana Delprete and Marco Esposito Politecnico di Torino, Corso Duca degli Abruzzi, 24 - 10129, Torino, Italy John E. Mottershead Department of Engineering, University of Liverpool, 1.19 Harrison-Hughes Building, The Quadrangle, Liverpool L69 3GH, United Kingdom ABSTRACT Theory of curve crossing and curve veering phenomena is well known in structural dynamics, but only few papers have used test bench to demonstrate and validate this eigenvalues behaviour. The aim of this paper is to present a theoretical and experimental analysis on a nonsymmetric experimental structure with eigenvalues curve veering and crossing phenomena. Starting from literature examples, detailed numerical models on lumped parameters systems and continuous systems with coincident and/or close eigenvalues are examined in order to developed a numerical FE model suitable to describe a tunable and simple test rig with coincident eigenvalues and curve veering phenomena without symmetric properties or completely uncoupled dynamic systems. The test bench is made of simple beams and masses properly linked together. The angle of an intermediate beam is used as tunable physical parameter to vary the eigenvalues of the system and to couple two bending modes or bending and torsional modes. Numerical and experimental results are compared, and sensitivity of mode shapes to variation of system parameters is discussed. INTRODUCTION There have been extensive research works on veering and crossing phenomena in dynamic systems. The behaviour is generally well understood. In the former, as the eigenvalues change under a parameter variation, converging roots loci get closer and then suddenly veer away. During this process all the properties of the involved modes are swapped, leading to a curious behaviour in the so called “transition zone”. In the latter eigenvalues loci do not veer away but intersect without any swap of modal properties. Therefore, when two eigenvalues loci approach each other, they can cross or abruptly diverge. Theoretical studies of this behaviour have been reported for half a century but despite this heritage, explicit references to experimental results are scarce. Moreover, literature presents mainly symmetric or uncoupled lumped systems with coincident eigenvalues properties and eigenvalues curve veering and crossing are not deeply analysed with experimental viewpoint. One of the earlier observer of these phenomena, in structural dynamics, was Leissa [1], that cited further examples to draw attention to the possibility of fallacious artefacts in numerical models, and demonstrated that veering could be artificially induced through inadequate approximations and discretizations. Furthermore, to explain that in the veering away region mode shapes and nodal patterns must undergo sudden changes, Leissa used this sentence: “figuratively speaking, a dragonfly one instant, a butterfly the next, and something indescribable in between”. The rapid change in the eigenfunctions during the veering has raised doubt on the validity of many approximate solutions. Later, Kuttler and Sigillito [2] used an example of fixed membrane problem on rectangle to confirm the existence of curve veering in accurate mathematical models. Perkins and Mote [3] presented an exact mathematical solution of elementary eigenvalue problem to confirm the existence of curve veering physical phenomena. Thus, the purpose of their study was to validate the existence of curve veering in continuous models by presenting an exact eigensolution, which veers, and to derive simple 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. 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. To obtain curve crossing and curve veering k23 e k31 have been used as parameters. This allows the system to lose symmetry property, and desired phenomena to set up. The different behaviour is described through the following two cases:   k23 = 1 N/m  curve crossing (coincident eigenvalues), k23 = 1.02 N/m  curve veering (close eigenvalues). It is straightforward that crossing and veering phenomena are tightly linked with certain physical parameters of the system. This last concept is used to design the test rig. Solving the eigenvalue problem for each value of k31 and plotting eigenvalues curves of mode 2 and mode 3 versus this parameter, leads to the results displayed in Figure 2. In the first case of perfect cyclic and symmetric system (k23 = 1 N/m), applying MAC between eigenvectors close to the crossing point (k31 = 0.99 N/m and k31 = 1.01 N/m) it possible to easily depict the mode reversal (Figure 3 on the left). During curve crossing phenomena, involved modes maintain their peculiarities, but they do swap: mode 2 become mode 3 and viceversa. Orthogonality of eigenvectors is guaranteed, due to mass matrix proportional to identity. This change is only a marginal effect because order is only linked with frequency value and not with substantial mode characteristics. This is confirmed by observing eigenvectors before and after occurrence of crossing phenomena in Figure 4 on the left: every mode maintains its property and changes only the identification number. In the exact configuration of crossing, system shows two coincident eigenvalues and the associated eigenspace has size two, while in all other cases, where eigenvalues are very close but not coincident, each eigenspace has size one. In the second case of non perfect symmetric system (k23 = 1.02 N/m), solving the eigenvalue problem for each value of k31 and plotting eigenvalues curves of mode 2 and mode 3 versus this parameter, leads to the results displayed in Figure 2 on the right. Starting from k31 = 0.9 N/m and by increasing this stiffness value, eigenvalues curves get closer but they abruptly veer away. In this case there are not coincident eigenvalues, not even for k31 = 1. Applying MAC for two different values of k31 (k31 = 0.99 N/m) and after (k31 = 1.01 N/m), where the distance between the two eigenvalues is minimum, it is possible to obtain the result reported on the right of Figure 3. It is straightforward to understand that the modes undergo a modal properties change, maintaining the initial rank. The modal assurance criterion confirms the alteration of mode shape by varying one of the parameter of the system. This result agrees to the Leissa’s sentence [1], in fact the modes shapes across the curve veering mix their dynamic properties in a continuous manner. The strange and sudden variation of the 2nd and 3rd mode shapes can be seen in Figure 4 on the right. Presence of symmetry could be justify because results come from physical system with symmetry and cyclic properties, despite of the abrupt change is very clear. Leissa said that was very difficult to identify a continuity between modes shapes [1], as confirmed in Figure 4. In fact, despite the step of k31 has been refined, modes shapes suddenly change. Curve Veering: k 12 = 1 N/m; k 23 = 1.02 N/m Curve Crossing: k 12 = 1 N/m; k 23 = 1 N/m 3.3 2 3 3.15 2 3 3.25 3.2 3.1 Eigenvalues [rad2/s 2] Eigenvalues [rad2/s 2] 3.15 3.05 3 2.95 3.1 3.05 3 2.95 2.9 2.9 2.85 2.8 0.9 2.85 0.92 0.94 0.96 0.98 1 1.02 k 31 [N/m] 1.04 1.06 1.08 1.1 2.8 0.9 0.92 0.94 0.96 0.98 1 1.02 k 31 [N/m] 1.04 Figure 2 – Natural frequencies of the lumped system versus k31 parameter: curve crossing with k23 = 1 N/m (left), curve veering with k23 = 1.02 N/m (right). 1.06 1.08 1.1 100 100 3.5 80 3 2.5 60 2 40 1.5 1 20 Modes kFEA 31 = 1.01 N/m Modes kFEA 31 = 1.01 N/m 3.5 80 3 2.5 60 2 40 1.5 1 20 0.5 0.5 0 1 2 3 4 0 0 1 2 3 0 4 Modes kEMA 31 = 0.99 N/m Modes kEMA 31 = 0.99 N/m Figure 3 – MAC index between mode-shapes of different k31 parameter: curve crossing with k23 = 1 N/m (left), curve veering with k23 = 1.02 N/m (right) involving the 2nd and 3rd modes. EigenVectors 1.5 Mode 1 - 1 = 0 rad2/s 2 EigenVectors k 31 = 0.85 : 1.15 N/m. k 12 = 1 N/m; k 23 = 1.02 N/m 1.5 Mode 2 - 2 = 3 rad2/s 2 Mode 3 - 3 = 3.02 rad2/s 2 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 1 2 D.o.F. 3 Mode 1 Mode 2 Mode 3 -1 0.5 1 1.5 2 D.o.F. 2.5 3 3.5 Figure 4 – Mode-shapes of the lumped system: curve crossing with perfect symmetric system (left), curve veering with non-perfect symmetric system (right). In order to study a symmetric mono-axial system and to simulate its dynamics behaviour, a simple FE model of a bell has been developed. Figure 5 shows the CAD model and the corresponding FE model, made of bar elements. This system has been implemented in Matlab to easily obtain frequencies and modes shapes. In Figure 6 are displayed the first four mode shapes: couples of two coincident eigenvalues with rotated eigenvectors come out, due to axial-symmetric properties. To obtain crossing and veering phenomena it has been defined to alter stiffness parameters of one or more bar elements. In particular, Figure 5 shows the corresponding nodes of constant and variable stiffness parameters. Lumped mass matrix is taken into account. Like in the example of Balmès, it is necessary to set up a variable parameter and use another parameter as a discriminant between crossing and veering phenomena. kBar represents constant stiffness value that should have a perfect axial-symmetric model. Figure 7 shows the first 16 eigenfrequencies curves varying the circumferential stiffness k2-3 parameter. The different behaviour is described through the following two cases:   k2-3 = kBar k2-3 = 1.3 kBar  curve crossing (coincident eigenvalues),  curve veering (close eigenvalues). Crossing phenomena is present for k2-3 = 100% kBar (Figure 7 on the left), in fact this value allows the model to possess axial-symmetric property and then to have coincident eigenvalues. In case of different stiffness, i.e. k2-3 = 100% kBar, curve veering appears in some couples of modes (Figure 7 on the right). When it is imposed a stiffness parameter variation, different by a circumferential variable parameter like k2-3, it will be obtained a curve veering phenomena. When symmetric characteristics are not present, most of crossing phenomena involving mode-shapes of Figure 6 became veering, but some curve crossing between bending and torsional modes are still evident. In Figure 7 letter “C” and ellipse specify curve crossing point, whilst letter “V” and rectangle specify curve veering point. Similarly, to order eigenfrequencies a swap matrix obtained from MAC across every point of veering and/or crossing phenomena is adopted. Figure 5 – Bell and corresponding FE model. MODE 5 - Freq = 685.23 Hz 3 2 2 1 1 y axis [m] y axis [m] MODE 4 - Freq = 685.23 Hz 3 0 -1 -1 -2 -2 -3 -3 -2 -1 0 x axis [m] 1 2 -3 -3 3 3 2 2 1 1 y axis [m] 3 0 -1 -2 -2 -2 -1 0 x axis [m] 1 -1 0 x axis [m] 1 2 3 2 3 0 -1 -3 -3 -2 MODE 7 - Freq = 702.53 Hz MODE 6 - Freq = 702.53 Hz y axis [m] 0 2 3 -3 -3 -2 -1 0 x axis [m] 1 Figure 6 – First mode-shapes of the bell system (red lines) superimposed on the underformed shapes(blue lines). Bell Eigenvalues. 16 D.o.f. Bell Eigenvalues. 16 D.o.f. 1600 1600 1400 1400 V C 1200 C 1200 C 800 C 600 C 400 C 400 200 C 200 0 V 1000 Freq. [Hz] Freq. [Hz] 1000 20 40 60 80 100 800 600 120 140 0 V C V 20 k 2-3 [%] 40 60 80 100 120 140 k 2-3 [%] Figure 7 – Natural frequencies of the structure versus angle configuration . At the end of this paragraph it is possible to assert that curve crossing and curve veering are phenomena tightly linked with physical parameters and their occurrence depends on symmetric or nonsymmetric properties. In particular, a slight loss of symmetry allows to obtain curve veering in a system with only curve crossing phenomena. Also in these last systems, crossing phenomena may be again present, if different eigenspaces with independent eigenvalues are interacting. The properties of symmetry is not, therefore, a fundamental requirement to have a system with coincident or very closed modes. Finally, MAC is an efficient tool to discriminate crossing and veering phenomena. TEST RIG DESCRIPTION Main ideas of the test bench are to obtain a structure without symmetric and cyclic properties but with dynamics behaviour characterised by coincident eigenvalues (curve crossing) and/or close eigenvalues (curve veering). Observing the dynamic instability of flutter phenomena, the structure shown in Figure 8 has a very simplified wing shape, where a sensitive geometrical parameter is chosen to change its dynamic properties. Figure 8 – Sketch of the test rig. The sketch of the system is composed by three beam elements and three lumped masses. The test bench uses the angle of the intermediate beam as a tunable physical parameter to modify the system eigenvalues, in order to couple two bending modes or a bending with a torsional mode. In particular, a coupling between torsional and bending modes is expected at a particular configuration. According to the boundary configurations with the intermediate beam horizontal or vertical, the eigenproblem for different  values predicts sensitive changes in the bending modes. Drawing eigenvalues curves versus this physical parameter demonstrates the dynamic properties of the test bench. Neglecting structural damping in the FE model, the system equations for a n-dofs structure result: Mx  Kx  0 (1) where the mass matrix M and the stiffness matrix K  are real, symmetric and positive definite. Therefore, the eigenproblem results: K   Mφ  0 2 (2) where the n eigenvalues  2 and eigenvectors φ are evaluated through determining no trivial solutions, therefore:   det K    2 M  0 (3) In order to evaluate curve crossing or veering phenomena, the evaluation of M and K  matrices are obtained through a FE model developed in Matlab. Beam finite elements with six degree of freedom for each node are adopted as shown in Figure 9. This choice is suitable to evaluate the dynamic properties of three-dimensional structures that may be approximated with truss schemes. These beam elements are used to discretize the physical structure as shown in Figure 9. Beam finite elements in 2D and 3D representation are sketched with cyan boxes, while lumped masses are identified by red balls. A very flexible FE model toolbox has been developed to represent different configurations and to simulate the sensitivity of different structural parameters, such as lumped masses, beam sections and lengths. Axis y [m] 0.2 0 0-0.2 0.1 Axis x [m] 0.2 0.3  0.4 0.5 0.6 0.2 Axis z [m] 0 -0.2 Figure 9 – FE model of the structure. 1 2 3 4 5 6 7 8 9 10 11 M1 12 13 14 15 16 17 18 19 20 21 M2 22 23 24 25 26 27 28 29 30 31 M3 The design aim is to define suitable configurations where it is possible to see a high number of curve veering and curve crossing phenomena with respect to the angle parameter. The eigenvalues graph shown in Figure 10 represents the expected dynamic behaviour in the frequency range up to 220 Hz. The mode-shapes related to the configuration   0 are used for labelling the curves and for following the eigenfrequencies modifications; different colours are adopted for distinguishing crossing (“C” symbols) and veering phenomena (“V” symbols). MAC index is adopted to follow the mode-shapes and to distinguish crossing from veering. Another result that can be obtained by the FE model is the analysis of modes shapes, especially across the curve veering or curve crossing points. In Figure 11 MAC index between eigenvectors with  = 44° and 45° are plotted on the left. They involve the 6 th mode (2nd torsional) and the 7th mode (bending); a typical crossing phenomena is evinced. Therefore this case produces a order change of modes, the modes shapes remain the same and the corresponding curves intersect each other. The same approach is used between eigenvectors with  = 68° and 69° on the right of Figure 11, where two bending modes are involved in a curve veering phenomena. The 5th and 6th bending modes are very similar first and after occurrence of veering phenomena, in fact the MAC index is not suitable to distinguish two different modes. Increasing the angular resolution between the eigenvector comparison produces always diagonal MAC index, corresponding to not crossing phenomena. Non zeros values of off-diagonal MAC terms are due to non perfect orthogonal properties of eigenvectors. The mass matrix is not proportional to identity because, although beam elements are equal, translational and rotational inertial terms are different and lumped masses are taken into account. Finally, even for the test bench curve crossing and curve veering present the same peculiarity described in the last paragraph. Therefore the design of the test bench allows to validate the theoretical results about crossing and veering and to study, in the future, energy transfer paths. Mode shapes for  = 0° 220 Mode 1 - 1 Byx Mode 2 - 1 Bzx Mode 3 - 1 T Mode 4 - 2 Bzx Mode 5 - 2 Byx Mode 6 - 2 T Mode 7 - 3 Bzx Mode 8 - 3 T Mode 9 - 4 Bzx Mode 10 - 5 Bzx V 200 180 160 Freq [Hz] 140 C 120 C 100 C 80 V 60 C 40 V 20 0 0 20  [deg] 40 60 80 Figure 10 – Natural frequencies of the structure versus angle configuration . 100 18 100 18 16 16 14 12 60 10 8 40 6 4 20 2 80 ModesFEA  = 68° ModesFEA  = 44° 80 14 12 60 10 8 40 6 4 20 2 0 5 Modes  = 45° 10 15 20 0 EMA 0 Modes  = 69° 5 10 15 20 0 EMA Figure 11 – MAC index between mode-shapes of different angle configuration: curve crossing involving the 6th and 7th modes (left), curve veering involving the 5th and 6th modes (right). TEST RIG DESIGN AND EXPERIMENTAL RESULTS Some practical approximations and physical dimensions are neglected or not simulated in the FE model, therefore to design and build the test rig some engineering choices are adopted to reproduce the numerical behaviour in physical reality. The most important characteristic is to have a tunable parameter in the structure to vary and tune the angle  of the middle beam. A manual rotary table is chosen in the first lumped mass. It guarantees the rotation of the middle beam and controls the orthogonality of the third beam by means of a clamp with a device integrated in the second lumped mass. One main peculiarity of the test bench is the flexibility, in fact the structure allows different lumped masses and beam element dimensions in a simple way. Figure 12 shows 3D drawings and photographs of the lumped masses in which is possible to see the rotary table and other parts of the group. An exploded view of the group shows rotational pins, jaws and twice parts necessary to obtain the tunable angular parameter. Beam elements used to join the lumped masses and to obtain correct stiffness properties are chosen in order to reproduce the dynamic properties predicted in Figure 10. These elements are interchanging with others that have rectangular section but with different dimensions and length. This is a further flexibility characteristic of the test bench. To obtain the reciprocal position (parallel or orthogonal) between the first and the third beam of the test bench during variation of the angular parameter , an alignment profile is developed (transparent light blue profile of Figure 12). This profile must be used only during the tuning of test bench, to guarantee alignment, but it must be removed during the experimental test. To constrain the test bench, a bracket that joins the main structure with a seismic mass is designed. This bracket has its first bending mode over 250 Hz, in order to not interact in the frequency range of experimental testes. After the assembly of the test bench, a preliminary experimental campaign has been conduct to validate theoretical results. Three three-axial accelerometers are set on every lumped mass (Figure 13), and subsequently rowing hammer technique is implemented for data acquisition. A mean of 10 impact tests are used for the measurement of FRF for each testing configuration. In particular 19 different configurations are taken into account, from  = 0° to  = 90° with a step of 5°. An updating procedure is now necessary to validate the numerical result of Figure 10. By means of experimental data two main results are obtained for the FE model: the calibration of equivalent length of beam elements, considering lumped masses and the updated stiffness parameter for the analytical FE data with respect to the experimental modes. To calibrate beam elements in the model, torsional modes are used because they are not influenced by length of actual masses, but only by section and length of beams. To be more confident with respect to the numerical model, the following step is to use a commercial FE model integrated in a CAD software, like SolidWorks-Cosmos, to predict more accurately the dynamic behaviour with respect to the angle parameter . Mass 1 Mass 2 Mass 3 Figure 12 – 3D drawings and photographs of the test rig. Mass 1 Mass 3 Mass 2 Figure 13 – Experimental setup of the test rig. CONCLUSIONS The paper presents a theoretical analysis and an experimental test rig on a nonsymmetric structure with eigenvalues curve veering and crossing phenomena. Detailed numerical models on lumped parameters systems and continuous systems with coincident and/or close eigenvalues are examined to developed a numerical FE model suitable to describe the tunable and simple test rig with coincident eigenvalues and curve veering phenomena without symmetric properties or completely uncoupled dynamic systems. A consistent numerical FE model has been developed to design the structure. The initial comparison between the experimental data obtained through the test rig and the numerical results seems to be suitable for validate the methodology. The test bench is useful to investigate curve veering phenomena with an experimental overview. It allows to completely control the dynamic behaviour through a physical system parameters, and it could also be a consistent tool, due to its form similar to a wing, to understand flutter dynamic instability through coupling bending wing mode with torsional one. BIBLIOGRAPHY [1] Leissa W., “On a curve veering aberration”, Journal of Applied Mathematics and Physics (ZAMP), 25, 1974, pp. 99-111. [2] Kutter J.R., Sigillito V.G., “On curve veering”, Journal of Sound and Vibration, 75, 1981, pp. 585-588. [3] Perkins N.C., Mote C.D.Jr., “Comments on curve veering in eigenvalue problems”, Journal of Sound and Vibration, 106(3), 1986, pp. 451-463. [4] Pierre C., “Mode localization and eigenvalue loci veering phenomena in disordered structures”, Journal of Sound and Vibration, 126(3), 1988, pp. 485-502. [5] Chen X., Kareem A., “Curve Veering of Eigenvalue Loci of Bridges with Aeroelastic Effects”, Journal of Engineering Mechanics, 129(2), 2003, pp. 146-159. [6] Yoo H.H., Shin S.H., “Vibration analysis of rotating cantilever beams”, Journal of Sound and Vibration, 212(5), 1998, pp. 807-828. [7] Balmès E., “High modal density, curve veering, localization: a different perspective on the structural response”, Journal of Sound and Vibration, 161(2), 1993, pp. 358-363. [8] Mugan A., “Effect of mode localization on INPUT-OUTPUT directional properties of structures”, Journal of Sound and Vibration, 258(1), 2002, pp. 45-63. [9] Liu X.L., “Behaviour of derivatives of eigenvalues and eigenvectors in curve veering and mode localization and their relation to close eigenvalues”, Journal of Sound and Vibration, 256(3), 2002, pp. 551-564. [10] Adhikari S., “Rates of change of eigenvalues and eigenvectors in damped dynamic system”, AIAA Journal, 39(11), 1999, pp. 1452-1457. [11] Young L.J., Hwang M. C., "Curve Veering Phenomena One-Dimensional Eigenvalue Problems", Proceedings of the Eighteenth National Conference of the Chinese Society of Mechanical Engineers, 12, 2001, Taipei, Taiwan, R.O.C., pp. 239-246. [12] Du Bois J.L., Adhikari S., Lieven N.A.J., “Eigenvalue curve veering in stressed structures: An experimental study”, Journal of Sound and Vibration, 322, 2009, pp. 1117-1124. [13] Lane D.R., “Eigenvector Derivatives with Repeated Eigenvalues”, AIAA Journal, 27(4), 1989, pp. 486491. [14] D’Ambrogio W., Fregolent A., “Higer-order MAC for the correlation of close and multiple mode”, Mechanical Systems and Signal Processing, 17(3), 2003, pp. 559-610. [15] Allemang R.J., Brown D.L., “A correlation coefficient for modal vector analysis”, Proceedings of 1st IMAC, Orlando, Florida, USA, 1982, pp. 110-116.