and coalesce into the backbone branch of another NNM, thereby realizing an internal resonance between the two modes. This is briefly illustrated in Figure 12 regarding the 3:1 tongue emanating from the backbone of the first wing torsional mode (see Figure 10). Modal shapes are given at three different locations on the tongue (see (a), (b) and (c) in Figure 12). When the energy gradually increases along the tongue, a smooth transition from the first wing torsional mode to a higher tail torsional mode clearly occurs. Interestingly, Figure 12(b) shows an inherently nonlinear mode with no counterpart in the underlying linear system. It corresponds to a 3:1 internal resonance as evidenced by the evolution of the time series and the frequency content, also represented in Figure 12, of the periodic motions along the tongue. A third harmonic progressively appears, and the structure vibrates according to a subharmonic motion characterized by two dominant frequency components. The relative importance of the third harmonic grows along the tongue, until the mode transition is realized. Similarly, two other tongues corresponding to a 5:1 and a 9:1 internal resonance between this first wing torsional mode and higher modes are observed in the FEP of Figure 10. Moreover, the FEP of Figure 11 reveals the presence of a 9:1 internal resonance between the second wing torsional mode and another higher mode of the aircraft. We note that the practical realization of these internal resonances is questionable in view of the low frequency changes. 5 CONCLUSION In this paper, a numerical method for the computation of NNMs of mechanical structures was introduced. The approach targets the computation of the undamped modes of structures discretized by finite elements and relies on the continuation of periodic solutions. This computational approach turns out to be capable of dealing with complex real-world structures, such as the full-scale aircraft studied herein. Through a reduced-order model accurate in the [0-100Hz] range, the NNMs were indeed computed accurately even in strongly nonlinear regimes and with a reasonable computational burden. Internal resonances were also computed by the algorithm and were briefly discussed. REFERENCES [1] A.F. Vakakis, L.I. Manevitch, Y.V. Mikhlin, V.N. Pilipchuk, A.A. Zevin, Normal Modes and Localization in Nonlinear Systems, John Wiley & Sons, New York (1996). [2] A.F. Vakakis, Non-linear normal modes (NNMs) and their applications in vibration theory: An overview, Mechanical Systems and Signal Processing, Vol. 11, No. 1 (1997), pp. 3-22. [3] G. Kerschen, M. Peeters, J.C. Golinval, A.F. Vakakis, Nonlinear normal modes, Part I: A useful framework for the structural dynamicist, Mechanical Systems and Signal Processing, Vol. 23, No. 1 (2009), pp. 170-194. [4] J.C. Slater, A numerical method for determining nonlinear normal modes, Nonlinear Dynamics, Vol. 10, No. 1 (1996), pp. 19-30. [5] E. Pesheck, Reduced-order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds, PhD Thesis, University of Michigan, Ann Arbor (2000). [6] Y.S. Lee, G. Kerschen, A.F. Vakakis, P.N. Panagopoulos, L.A. Bergman, D.M. McFarland, Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Physica D-Nonlinear Phenomena, Vol. 204, No. 1-2 (2005), pp. 41-69. [7] R. Arquier, S. Bellizzi, R. Bouc, B. Cochelin, Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes, Computers & Structures, Vol. 84, No. 24-25 (2006), pp. 1565-1576. [8] R. Seydel, Practical Bifurcation and Stability Analysis, from Equilibirum to Chaos, Springer-Verlag, 2nd Edition (1994). [9] A.H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods,WileyInterscience, New York (1995). [10] E. Doedel, AUTO, Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, (2007). 241
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