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: 0 2 φ K M (2) where the n eigenvalues 2 and eigenvectors φ are evaluated through determining no trivial solutions, therefore: 0 det 2 K M (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. 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.2 0 0.2 -0.2 0 0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 M3 M1 M2 Axis y [m] Axis z [m] Axis x [m] Figure 9 – FE model of the structure. 332
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