A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies Systemparameters Mass [kg] Stiffness [N/m] Mass [kg] Stiffness [N/m] mA1 =10 kA11 =2· 10 3 mB1 =2 kB1 =1· 10 3 mA2 =3 kA12 =2· 10 3 mB2 =4 kB2 =1· 10 3 mA3 =3 kA2 =1· 10 3 mA3 =8 kB3 =2· 10 3 mA4 =6 kA3 =1· 10 3 mB4 =5 mA5 =2 kA41 =0.5· 10 3 mA6 =2 kA42 =1· 10 3 mA7 =4 kA5 =0.2· 10 3 Table 1: Parameters of mass-spring systems both 100 and 500 frequency points, corresponding to a frequency resolution of 0.5Hz and 0.1Hz respectively, on the interval from0 to 50 Hz. The results are listed in table 2 below and are pretty satisfactory, with errors on the identified frequencies limited to 3% with respect to the true eigenfrequencies. As expected, the higher frequency resolution analysis gave slightly more accurate results. Mode no. True freq. Identified freq. Difference Identified freq. Difference [Hz] Δf =0.5Hz [Hz] [%] Δf =0.1Hz [Hz] [%] ω1 0 0 0 0 0 ω2 7.02 7.25 3.3 7.05 0.5 ω3 15.26 15.25 -0.1 15.25 -0.1 ω4 16.21 16.25 0.2 16.25 0.2 ω5 19.53 19.75 1.1 19.55 0.1 ω6 24.70 24.75 0.2 24.65 -0.2 ω7 28.67 28.75 0.3 28.65 -0.1 ω8 30.69 30.75 0.2 30.65 -0.1 ω9 39.58 39.75 0.4 39.55 -0.1 Table 2: Identified eigenfrequencies of assembled system With the eigenfrequencies identified, the next step is to obtain the associated mode shapes in the way described in the previous section. As for the frequencies, the accuracy of the identified mode shapes is dependent on the frequency resolution of the models. Using the same two frequency resolutions as for the identification of the eigenfrequencies, the corresponding mode shapes are identified and compared to the true mode shapes using the MAC criterion [1]. The results are shown in figure 12. For the case of the coarse frequency spacing of 0.5 Hz, it can be seen that the results are not so good for the closely spaced modes. Increasing the frequency resolution to 0.1Hz solves this problemand all the modes are accurately identified; all MAC values are larger than 0.99 with respect to the true mode shapes obtained analytically. 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Identified modes MAC, df = 0.5 Hz True modes 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Identified modes MAC, df = 0.1 Hz True modes 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 12: MAC values for identifiedmodes with frequency resolution of 0.5 Hz (left) and 0.1 Hz (right) 343
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