392 A.J. Mazzei and R.A. Scott Xd Amplitude 1 0.8 0.6 0.4 0.2 0 Amplitude 1 0.5 −0.5 −1 0 1 0.8 0.6 0.4 0.2 0 Xd wd,2 = 6.4939 wd,1 = 3.8660 1 0.8 0.6 0.4 0.2 0 Fig. 36.5 Frequencies and mode shapes via FDM approach (fixed-fixed, ten elements) Amplitude 1 0.8 0.6 0.4 0.2 0 Xd Xd 0 0.2 0.4 0.6 0.8 1 Amplitude 0.6 0.4 0.2 −0.2 −0.4 −0.6 −0.8 −1 0 wd,2 = 5.4351 wd,1 = 1.8652 0 0.2 0.4 0.6 0.8 1 Fig. 36.6 Frequencies and mode shapes via FDM approach (free-fixed, ten elements) Using this approach the first and second frequencies, for the fixed-fixed case, are approximately !d,1 D4.14 and !d,2 D7.44, respectively. The changes in the sign of the deflections can be seen in Fig. 36.7 (first frequency) and Fig. 36.8 (second frequency). They show the deflections before and after going through a resonance. The first and second frequencies for the free-fixed case are approximately!d,1 D2.29and!d,2 D5.65. The changes in the sign of the deflections can be seen in Fig. 36.9 (first frequency) and Fig. 36.10 (second frequency). Note that the mode shapes cannot be readily obtained using this method (here the previously discussed FDM is used for that).
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