620 Y.S. Lee et al. Fig. 59.2 Construction of lnjƒm.x; t/j from the experimental measurements for the VI beam: (a) mD10 and (b) mD2 [20] x/L t 0 0.2 0.4 0.6 0.8 1 0.16 0.17 0.18 0.19 0.2 −50 0 50 x/L t 0 0.2 0.4 0.6 0.8 1 0.16 0.17 0.18 0.19 0.2 −100 −50 0 50 100 a b Fig. 59.3 Spatiotemporal IMOs: (a) Linear beam (eighth mode); and (b) VI beam (eighth mode) [20] Finally, we can derive the complex forcing amplitude in Eq. (59.2) as ƒm.x; t/ j.O m m/! 2 m NAm m.x/e j me m!mt (59.8) Taking the logarithm of the absolute value of both sides of Eq. (59.8) decouples the time-/spatially-dependent variables of such that lnjƒm.x; t/j constant m!mt Clnj m.x/j (59.9) where the constant is lnj.O m m/!2 m NAme j mj. That is, the spatiotemporal variations of lnjƒm.x; t/j for the linear beam will be linearly decaying in time while keeping the form of the mode shape function at every instant. Referring to the earlier work [19, 20] regarding the details of EMD analysis of the measured acceleration signals, we now construct this slowly varying forcing amplitudefor each nonlinear interaction model from the experimental measurements (cf. Fig. 59.2a, b depict ƒm.x; t/ for the tenth and second modes of the VI beam). The IMOs corresponding to the high-frequency (HF) modes (typically, above the fourth mode) tend to maintain their linear dynamics in between impacts (although the overall dynamics is strongly nonlinear), those for the low-frequency (LF) modes exhibit strongly nonlinear modal interactions independent of vibro-impact patterns in a global fashion [19, 20]. This is because the HF modes are more sensitive to the vibro-impact nonlinearity, whereas the LF modes are insensitive to local changes of the mode shapes due to defects in the structure. Finally, we construct the spatiotemporal IMO (STIMO) solutions (cf. Fig. 59.3), which may clarify the global and local effects of nonlinear modal interactions in HF and LF IMOs. For example, the eighth STIMO for the linear beam model
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