( ) ( ) ( ) ( ) ˆ ˆ ( ) 2 ( ) ( ) m m m m k k k m k t t t (14) where the forcing terms on the right-hand-sides are the slow components of the nonlinear modal interactions in (13). The slow complex amplitudes ( ) ˆ ( ) m k t in (14) can be extracted directly from the identified dominant IMFs ( ) ( ) k mc t , so that the slow components of the nonlinear modal interactions, ( ) ( ) m k t , can be computed once the damping coefficients are estimated. This estimation is performed using an optimization process by imposing the requirement that each of the equations in (13) reproduces the corresponding slow component of the IMF. Further details are provided in Lee et al. [24]. It follows that we can directly compute the nonlinear modal interactions (i.e., the forcing terms) in (14) and computationally reconstruct the slow-flow dynamics for this transition. Additional applications of the outlined NSI methodology were performed in identifying and reduced-order modeling of the strongly nonlinear interactions that trigger aeroelastic instability (flutter) of an in-flow wing [25]; in identifying the nonlinear interactions governing passive flutter suppression of a wing with an attached nonlinear energy sink [24]; and in the study of nonlinear interactions of a flexible component with an attached essentially nonlinear substructure [26]. Current work focuses in extending the methodology to dynamical systems with closely spaced modes and in systems with non-smooth nonlinearities. This second aspect is discussed below. 5. CONCLUDING REMARKS We presented the basic elements of a time-domain nonlinear system identification (NSI) method based on the close correspondence between analytical and empirical slow-flow dynamics, and modeling the local and global nonlinear dynamics of both smooth and non-smooth dynamical systems. Since the NSI method requires no a priori system information but instead relies only on direct analysis of measured time series (i.e., it is a purely output-based approach), it holds promise of broad applicability to a general class of nonlinear systems. The derived nonlinear interaction models (NIMs) are sets of intrinsic modal oscillators (IMOs) that result from direct empirical slow-flow analysis of the time series. Finally, the instantaneous frequencies and total energies in the modal space are calculated with the help of the established NIMs. The resulting frequency-energy plots (FEPs) provide a global model of the nonlinear dynamics. We provided two examples that demonstrate that the branches obtained from the NSI method can approximate those calculated from the corresponding mathematical model with reasonable accuracy. ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant Nos. CMMI-0927995 and CMMI-0928062. REFERENCES [1] Ewins DJ. Modal Testing: Theory and Practice. Research Studies Press, UK, 1990. [2] Brandon JA. Some insights into the dynamics of defective structures. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 1998; 212(6): 441-454. [3] Nayfeh AH, Mook D. Nonlinear Oscillations. Wiley Interscience: New York, 1985. [4] Kerschen G, Golinval J-C, Vakakis AF, Bergman LA. The method of proper orthogonal decomposition for order reduction of mechanical systems: An overview. Nonlinear Dynamics 2005; 41(1-3): 147-170. [5] Kerschen G, Worden K, Vakakis AF, Golinval J-C. Past, present and future of nonlinear system identification in structural dynamics. Mechanical Systems and Signal Processing 2005; 20(3): 505-592. [6] Moon FC. Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. Wiley Interscience: New York, 2004. [7] Nichols JM, Virgin LN. System identification through chaotic interrogation. Journal of Sound and Vibration 2003; 17(4): 871-881. [8] Feeny BF, Yuan C-M, Cusumano JP. Parametric identification of an experimental magneto-elastic oscillator. Journal of Sound and Vibration 2001; 247(5): 785-806. [9] Silva W. Identification of nonlinear aeroelastic systems based on the Volterra theory: Progress and opportunities. Nonlinear Dynamics 2005; 39: 25-62. [10] Masri S, Caughey T. A nonparametric identification techanique for nonlinear dynamic systems. Transactions of the ASME, Journal of Applied Mechanics 1979; 46: 433-441. [11] Leontaritis IJ, Billings SA. Input-output parametric models for nonlinear systems. Part I. Deterministic nonlinear systems. International Journal of Control 1985; 41: 303-328. [12] Leontaritis IJ, Billings SA. Input-output parametric models for nonlinear systems. Part II. Stochastic nonlinear systems, International Journal of Control 1985; 41: 329-344. 522
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