(v) The final outcomes of the NSI are: (a) a frequency-energy plot of the global dynamics depicting the possible coexisting families of solutions and their bifurcations over the frequency and energy ranges of interest (global result); and (b) the corresponding local slow flow models of the dynamics describing nonlinear transitions on the FEP (local results). The approach summarized above addresses in a systematic way a fundamental limitation of current nonlinear system identification methods: their inability to allow that the responses of nonlinear systems may depend crucially on initial conditions and/or the applied excitations. Instead, the NSI methodology takes into account that nonlinear systems may change their dynamics with energy and possibly possess numerous co-existing solutions (attractors). The added flexibility of ‘probing’ the dynamics over different frequency and energy ranges in order to extract different local models is important when identifying systems capable of strongly nonlinear dynamical behavior. The global aspect of our method, namely FEP construction, can be applied to both discrete and continuous nonlinear dynamical systems irrespective of dimensionality. By constructing FEPs we can identify global features of the dynamics, e.g., ranges of frequencies and energies where the system possesses linearized responses (corresponding to nearly horizontal branches of solutions in the FEP), coexisting branches of strongly nonlinear solutions, bifurcation points signifying the limits of response branches, etc. In addition, it is well established that forced resonances of nonlinear systems occur in neighborhoods of free periodic solutions (or nonlinear normal modes); hence, by identifying the FEPs we gain understanding of the structure of nonlinear (fundamental or subharmonic) resonances in the forced dynamics. Finally, we point out the added benefit of considering transient instead of steady-state responses in our NSI method. Indeed, analyzing transient responses is an efficient way of probing the dynamics (as the damped transitions in the FEP plots of Fig. 4 demonstrate; i.e., steady-state motions would appear merely as isolated dots in these plots as they would correspond to fixed frequencies and energies). Performing transient tests allows us to effectively probe the dynamics of a system and to depict these results in compact form in an FEP. 4. APPLICATIONS 4.1 Global Aspect of NSI: FEP Construction The instantaneous frequency of an identified dominant IMF, ( ) ( ) k mc t , can be computed directly from expression (10) as, ( ) ( ) ˆ ˆ ( ) ( ) k k m m t t . The corresponding instantaneous energy of the IMF can be expressed as a sum of kinetic and potential energies as ( ) ( ) 2 2 ( ) 2 ( ) ( ) ( ) / 2 k k k m m m m E t c t c t . If the mass distribution for the system is known then the instantaneous mechanical energy of the system can be estimated as a summation of the energies of the IMFs multiplied by appropriate mass factors, ( ) 1 1 ( ) ( ) n N k tot k m k m E t m E t (12) where , 1, , km k n corresponds to the mass distribution of the system among components (and can be deduced from the physical configuration of the system), and is a factor used to match the exact initial conditions of the damped transition with the approximate initial conditions satisfied by the IMFs (this can be directly deduced from the measured time series). If the system is linear then 1 , so 1 accounts for the energy of the nonlinear terms. Using these expressions a partial construction of the frequency-energy plot (FEP) can be made (corresponding to the studied transition) and a global picture of the dynamics deduced. By considering different nonlinear transitions we may construct different regions of the FEP and perform global identification of the dynamics over broad frequency and energy ranges. As a demonstration of global nonlinear identification, in Fig. 5 we provide a partial reconstruction of the FEP for the strongly nonlinear system of coupled oscillators (5), and compare it to the exact result. The reconstructions were performed for the two damped transitions depicted in Fig. 4. In this case, the mass distribution of the system is 1 1 m (LO) and 2 0.05 m (NES), and the correction factor is computed as 1.5 . In the left of Fig. 5 the FEP reconstructions are performed using the instantaneous frequency and energy estimates derived above, whereas the right is based on direct wavelet transform spectra of the identified dominant IMFs. In both cases we demonstrate the efficacy of performing global nonlinear system identification based on the identified dominant IMFs of the responses. 520
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