59 Empirical Slow-Flow Identification for Structural Health Monitoring and Damage Detection 619 Table 59.1 Positions of the accelerometers and rigid stops of the VI beam x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 xSTP Positions (mm) 131 263 395 527 657 787 917 1,052 1,215 1,311 1,185 Table 59.2 The ten leading natural frequencies (in Hz) of the linear cantilever beam in Fig. 59.1 First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth !n (Hz) 3.7 23.2 64.9 126.9 209.4 314.7 433.9 580.7 751.3 926.7 Applying impulsive excitations at position x3 and measuring the acceleration responses at the ten positions, we perform an experimental modal analysis (EMA [8]) to extract basic modal properties from the linear beam. For example, Table 59.2 shows the ten leading natural frequencies of the linear cantilever beam in Fig. 59.1. However, for the VI beam setup, traditional EMA based on Fourier transform fails to provide physically meaningful information due to strongly nonlinear (and nonsmooth) vibro-impacts. To get rid of nonsmooth dynamics due to vibro-impact nonlinearity, we perform EMD analysis on the VI beam responses and then adopt the local approach to NSI [1–3] applied to the resulting smooth components. Since we postprocess time series measured at numerous positions along a structure, it is reasonable to expand the idea of 1-D temporal slow flows and hence 1-D temporal intrinsic modal oscillators (IMOs [2]) to multi-dimensional spatiotemporal slow flows and multi-dimensional IMOs. Let W.x; t/ be a time series measured at x over the whole structure, which we suppose can be expressed as a sum of the intrinsic modes Wm.x; t/; i.e., we write W.x; t/ N X mD1 Wm.x; t/ (59.1) where Wm.x; t/ denotes the spatiotemporal intrinsic mode oscillating at or near the fast frequency!m, andx 2 R3 ingeneral defined by structural boundary conditions. For a linear system, Wm.x; t/ simply implies the m-th normal mode vibration. Then, without loss of generality, we can extend the expression for an 1-D IMO such that RWm.x; t/ C2 m!m PWm.x; t/ C! 2 mWm.x; t/ Re ƒm.x; t/e j!mt (59.2) where the dot denotes partial differentiation with respect to time t, and the spatiotemporal variations of complex forcing functionƒm.x; t/ can be computed as ƒm.x; t/ 2hP'm.x; t/ C O m!m'm.x; t/i (59.3) where the spatiotemporal slowly-varying slow flow can be expressed as 'm.x; t/ Dj!m OAm.x; t/ej O m.x;t/. In practice, Wm.x; t/ can be constructed by augmenting a set of intrinsic mode functions (IMFs, or equivalently, a set of IMO solutions) with the same dominant fast frequency !m column by column, instead of directly solving the partial differential equation (59.2). That is, we write Wm.x; t/ cm.x1; t/ : : :cm.x2; t/ : : : : : :cm.xr; t/ : : : (59.4) where cm.xr; t/ denotes the m-th IMF (or IMO solution) decomposed from the measurement at the r-th sensor position xr. Then, the slowly varying variables are computed straightforwardly through a complexification as OAm.x; t/ DpWm.x; t/ 2 CHŒWm.x; t/ 2; O m.x; t/ Dtan 1fHŒWm.x; t/ =Wm.x; t/g !mt (59.5) where HŒ denotes Hilbert transformation. Considering the acceleration responses can be approximated as a sum of all IMO solutions, the spatiotemporal slowly varying envelope and phase for the linear beam can be computed as OAm.x; t/ D NAm m.x/e m!mt ; O m.x; t/ D m Dconstant (59.6) where m.x/ is the m-th mode shape function, and m is the modal damping factor. Therefore, the spatiotemporal slow flow can be obtained as 'm.x; t/ Dj!m NAm m.x/e j me m!mt ; P'm.x; t/ D j m! 2 m NAm m.x/e j me m!mt (59.7)
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