35 Project-Oriented Validation on a Cantilever Beam Under Vibration Active Control 359 0 10 20 30 40 50 60 -1.5 -1 -0.5 0 0.5 Iteration Times Convergence Error, % freq1 freq2 Fig. 35.7 Evolution of frequencies in the updating procedure Table 35.2 Modal parameters of the cantilever beam with one fixed end and rubber springs ModeNo. EMAfreq (Hz) EMA freq after updating (Hz) FEA freq before updating (Hz) FEA freq and MAC after updating (Hz) Error (Hz) 1 48.93 49.08 48.68 48.76 0.95 0.31 2 149.23 147.09 152.48 147.37 0.95 0.28 35.2.3 Verification the Identified Supporting Stiffness of the Cantilever Beam with One Fixed End and Rubber Springs After the EMA of the cantilever beam with one fixed end and rubber springs, the values of the FE model were identified by the RS based method illustrated in Fig. 35.1. Figure 35.7 gives the evolution of frequencies in the updating procedure. All parameters converged well. The identified rubber spring stiffness is 408 N/mm, damper is 0.05 N s/mm. See Table 35.2. The spring stiffness 408 N/mm is checked by using a static stiffness experiment. In the example, D-optimal design with 18 runs was employed in regression of the five order polynomials for the two parameters model. Then the stiffness and lumped damping of the supporting parameters were obtained using the presented approach. The errors between EMA frequencies after updating with the error coefficient and the updated FEA frequencies are near 0.3 Hz, the resolution frequency of the EMA is 0.25 Hz, so the results is consistent. The MAC data is more than 0.95. The FE model of cantilever beam has good prediction. 35.2.4 Identified Mean Supporting Stiffness of the Cantilever Beam with One Fixed End and with an Electromagnetic Actuator After the EMA of the cantilever beam with an electromagnetic actuator, the mean values of the FE model were identified by the RS based method illustrated in Fig. 35.1, Fig. 35.8 give the FRFs excited by sweep sine force. Two bending modal frequencies and damping ratios were used as response feature in the problem. The identified support stiffness in lower frequency near 50 Hz is 525 N/mm, damper is 0.1 N s/mm. The identified support stiffness in higher frequency near 165 Hz is 3,750 N/mm, damper is 0.1 N s/mm. The FE model with the identified parameters gives good prediction of modal frequencies. See Table 35.3. The mean value of relative errors of modal frequencies is less than0.1%. The errors between EMA frequencies after updating with the error coefficient and the updated FEA frequencies are less 0.25 Hz, (the resolution frequency of the EMA is 0.25 Hz), so the results is consistent. The FE model of this cantilever beam with electromagnetic actuator has good prediction.
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