A very useful feature of the Hamiltonian FEP is its relation to the transient dynamics of the corresponding weakly damped system. As discussed in Vakakis et al. [31] the dynamics of the weakly damped system can be closely related to the underlying Hamiltonian dynamics: indeed, for weak damping the transient dynamics tracks specific branches of periodic orbits in the FEP. As energy decreases due to damping dissipation sudden transitions may occur as the damped response jumps from the neighborhood of one branch of periodic solutions to another. The sequence of branches of periodic orbits tracked by the damped dynamics in the FEP is ultimately dictated by the initial conditions and the level of damping in the system. Using as an example system (5), the close correspondence between the weakly damped and Hamiltonian dynamics can best be demonstrated by superimposing on the Hamiltonian FEP the wavelet spectra of the time series corresponding to the difference ( ) ( ) y t v t . This is done in the left column of Fig. 4 for the 1:3 transient resonance capture depicted of Fig. 1. In the same figure we depict the transient responses ( ) v t and ( ) y t together with their wavelet spectra. We see that during 1:3 transient resonance capture the dynamics tracks closely the 1:3 subharmonic ‘tongue’ 13 S [31], so a relatively simple topological picture of the transition emerges. Using, however, a different set of initial conditions we get drastically different dynamics as evidenced by the high-frequency transition depicted in the right column of Fig. 4. In this case the dynamics is initiated on the higher energy superharmonic tongue 21 U so the damped dynamics tracks a completely different set of branches in the FEP. This results in a complicated multi-frequency nonlinear transition which, however, can be analyzed through theoretical and numerical slow-fast partitioning of the dynamics. It is evident though that performing NSI based only on either one of the measured time series we would miss a component of the dynamics. Moreover, even if both transitions are analyzed, NSI would still be incomplete as there would still exist dynamics not captured by the transitions of Fig. 4. The previous example highlights the important challenges that the analyst is faced with when performing NSI. The first challenge is to address the (generic) feature of nonlinear systems to exhibit qualitatively different responses with varying energy and/or initial conditions. To address this challenge one needs to adopt a global approach for identifying the basic (essential) dynamical features of a system over broad frequency and energy ranges. The second challenge is to be able to identify complex multi-frequency transitions (such as the ones depicted in Fig. 4) for fixed sets of initial conditions (or energy). This dictates a local approach to NSI whereby a specific nonlinear transition is considered and the task is to identify the nonlinear modal interactions that govern this transition. The NSI methodology presented in this paper addresses both of the above challenges by proposing a combined global/local approach to NSI: global features of the dynamics are identified in the frequency-energy domain by constructing FEPs, whereas local transitions (such as the ones depicted in Fig. 4) are identified by constructing appropriate slow-flow models. In this way we ensure that both the global and local requirements of NSI are addressed. The added benefit of this approach is that it is based on direct analysis of measured time series which contain complete information of the nonlinear dynamics to be identified. 3.2 Methodology Based on the previous discussion we summarize the methodology for NSI of dynamical systems. The methodology has global and local components and relies on direct processing of measured time series. The central assumption of the method is that the measured dynamics can be decomposed in terms of slowly modulated fast oscillations, which is a reasonable assumption for non-chaotic measured data. The basic elements of the method are outlined below: (i) Measure time series from a number of sensors throughout the system under transient excitation, and perform empirical mode decomposition (EMD) of the measured time series. Extract the intrinsic mode functions (IMFs) at each sensing location. Hilbert-transform the computed IMFs to extract their instantaneous frequencies and compare them to wavelet transform (WT) spectra of the corresponding time series; thus, determine the dominant IMFs and the corresponding fast frequencies in the dynamics at each sensing location. This will identify the basic time scales and the dimensionality of the dynamics. (ii) Based on the correspondence between the measured dominant IMFs and the underlying slow-flow dynamics of the system (cf. Section 2), relate the slow components of the dominant IMFs to the slow flow dynamics. Using the dominant IMFs reconstruct the time series and depict it in a frequency-energy plot; under the assumption of weak dissipation this will reconstruct a portion of the FEP of the dynamics of the system under investigation; no a priori model is assumed for this reconstruction (nonparametric global component of NSI). (iii) For a given nonlinear transition in the FEP define a parametric reduced-order slow-flow model of the system with the dimensionality of the dynamics, and identify its parameters; thus construct a local slow-flow model of the dynamics (parametric local aspect of NSI). (iv) By varying the excitation and/or the initial conditions, consider different nonlinear transitions of the system over different frequency and energy ranges, and construct the corresponding portions of the FEP of the system together with the associated local slow-flow models. 519
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