0 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 EMA FEA 0 20 40 60 80 100 0 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 EMA FEA 0 20 40 60 80 100 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 2 nd and 3rd modes. 1 2 3 -1 -0.5 0 0.5 1 1.5 EigenVectors D.o.F. Mode 1 - 1 = 0 rad2/s2 Mode 2 - 2 = 3 rad2/s2 Mode 3 - 3 = 3.02 rad2/s2 0.5 1 1.5 2 2.5 3 3.5 -1 -0.5 0 0.5 1 1.5 D.o.F. EigenVectors k 31 = 0.85 : 1.15 N/m. k 12 = 1 N/m; k 23 = 1.02 N/m Mode 1 Mode 2 Mode 3 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 curve crossing (coincident eigenvalues), k2-3 = 1.3 kBar curve veering (close eigenvalues). Modes k31 = 0.99 N/m Modes k31 = 0.99 N/m Modes k31 = 1.01 N/m Modes k31 = 1.01 N/m 329
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