76 G. CanbaloMglu and H.N. Özgüven Table 7.4 Material properties used in the original FE model Modulus of elasticity (E) (GPa) Poisson’s ratio ( ) Density (¡) (kg/m3) Beam1 200 0.3 7,950 Beam2 200 0.3 7,950 Table 7.5 Comparison of the natural frequencies obtained from theoretical FE model with the actual values Mode number Actual natural frequency (from FE model of simulated test system) (Hz) Natural frequency calculated from theoretical FE model (Hz) 1 43.1 41.8 2 185.8 180.3 3 258.2 250.4 Fig. 7.6 Comparison of the linear responses obtained from theoretical FE model and linear model of the actual system On the other hand, the theoretical linear FE model of the system is built by taking the modulus of elasticity and density of the beams slightly different from the ones used in simulated test (Table 7.4). The effect of these variations on the first three natural frequencies of the linear part is given in Table 7.5. Then, the driving point FRFs at the tip of Beam1 in Z (transverse) direction are calculated by using the theoretical FE model, and just to see the effect of using slightly different modulus of elasticity and density in the theoretical model, these FRFs are compared with the exact values obtained from the FE model of the simulated test system (Fig. 7.6). Note that in practical applications the linear FRFs of the actual nonlinear system cannot be directly measured, and it is the intension of this study to calculate these FRFs from measured FRFs of the nonlinear system. As can be seen in Table 7.5 and Fig. 7.6, there are discrepancies between the natural frequencies and linear responses obtained from theoretical FE model and actual values. In order to improve these results, FE model needs to be updated. However, the major problem in practical applications is that when the structure is nonlinear the FRF curve for the linear part cannot be accurately obtained especially when there is friction type of nonlinearity along with other types of nonlinearities. In the theoretical linear FE model of the system, height of the Beam1(h1), height of the Beam2(h2), length of the Beam2 (L2), modulus of elasticity of the Beam1 and Beam2 are decided to be the candidates for updating parameters. Selection of the updating parameters is carried out based on the sensitivity of natural frequencies to these parameters. By changing one parameter while keeping all the others constant, several different parameter sets are built. Running several FE analyses for these parameter sets, percentage changes in the natural frequencies with respect to percentage changes in parameters and the average sensitivity values are estimated for the first three flexural natural frequencies. The results are given in Figs. 7.7, 7.8, and 7.9. As can be seen in Figs. 7.7 and 7.9, first and third flexural natural frequencies have the highest sensitivity values to the height and length of Beam2. On the other hand, second flexural natural frequency has the highest sensitivity to the height
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