Nonlinear Dynamics, Volume 1

326 T. L. Hill et al. 34.5 Conclusions This paper has introduced the concept of energy transfer rate for nonlinear normal modes. This describes the rate at which energy must be transferred between the underlying linear modes of a nonlinear system, given a time-delay perturbation to a perfect forcing (i.e., a forcing that allows the system to respond precisely on an NNM). It is demonstrated that the energy transfer rate relates to the significance of an NNM, and a high rate means that the response is unlikely to be observed in a typical forced response. This is verified by considering the time taken for an NNM response to converge to a steady-state response, given a time-delay perturbation in a perfect forcing. It is seen that responses with a high energy transfer rate are seen to correspond to high convergence times. The key feature of the energy transfer rate is the efficiency of its computation. This provides significant advantage when compared to existing approaches which rely on expensive simulations, or analytical derivations. References 1. Rand, R.H.: A direct method for non-linear normal modes. Int. J. Non Linear Mech. 9(5), 363–368 (1974) 2. Montaldi, J., Roberts, M., Stewart, I.: Existence of nonlinear normal modes of symmetric hamiltonian systems. Nonlinearity3(3), 695 (1990) 3. Lewandowski, R.: On beams membranes and plates vibration backbone curves in cases of internal resonance. Meccanica 31(3), 323–346 (1996) 4. Nayfeh, A.H., Chin, C., Nayfeh, S.A.: Nonlinear normal modes of a cantilever beam. J. Vib. Acoust. 117(4), 477–481 (1995) 5. Gendelman, O.V.: Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn. 37(2), 115–128 (2004) 6. Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. J. Sound Vib. 298(4), 958–981 (2006) 7. Cammarano, A., Hill, T.L., Neild, S.A., Wagg, D.J.: Bifurcations of backbone curves for systems of coupled nonlinear two mass oscillator. Nonlinear Dyn. 77(1–2), 311–320 (2014) 8. Rosenberg, R.M.: Normal modes of nonlinear dual-mode systems. J. Appl. Mech. 27, 263–268 (1960) 9. Rosenberg, R.M.: The normal modes of nonlinear n-degree-of-freedom systems. J. Appl. Mech. 29(1), 7–14 (1962) 10. Shaw, S., Pierre, C.: Non-linear normal modes and invariant manifolds. J. Sound Vib. 150(1), 170–173 (1991) 11. Shaw, S., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164(1), 85–124 (1993) 12. Haller, G., Ponsioen, S.: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dyn. 86(3), 1493–1534 (2016) 13. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes, part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23(1), 170–194 (2009). Special Issue: Non-linear structural dynamics 14. Hill, T.L., Cammarano, A., Neild, S.A., Wagg, D.J.: Interpreting the forced responses of a two-degree-of-freedom nonlinear oscillator using backbone curves. J. Sound Vib. 349, 276–288 (2015) 15. Hill, T.L., Cammarano, A., Neild, S.A., Wagg, D.J.: Out-of-unison resonance in weakly nonlinear coupled oscillators. Proc Royal Soc Lond A Math Phys Eng Sci 471, 2173 (2014) 16. Hill, T.L., Neild, S.A., Cammarano, A.: An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures. J Sound Vib. 379, 150–165 (2016) 17. Shaw, A.D., Hill, T.L., Neild, S.A., Friswell, M.I.: Periodic responses of a structure with 3:1 internal resonance. Mech. Syst. Signal Process. 81, 19–34 (2016) 18. Detroux, T., Noël, J., Kerschen, G., Virgin, L.N.: Experimental study of isolated response curves in a two-degree-of-freedom nonlinear system. In: Proceedings of the International Modal Analysis Conference (2016) 19. Hill, T.L., Neild, S.A., Cammarano, A., Wagg, D.J.: The influence of phase-locking on internal resonance from a nonlinear normal mode perspective. J. Sound Vib. 379, 135–149 (2016) 20. Hill, T.L., Cammarano, A., Neild, S.A., Barton, D.A.W.: Identifying the significance of nonlinear normal modes. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 473, 2199 (2017) 21. Hill, T., Neild, S., Cammarano, A., Barton, D.: The significance of nonlinear normal modes for forced responses. In: Nonlinear Dynamics, vol. 1, pp. 135–142. Springer (2017) 22. Hill, T., Cammarano, A., Neild, S., Wagg, D.: An analytical method for the optimisation of weakly nonlinear systems. Proc. EURODYN2014, 1981–1988 (2014) 23. Wagg, D.J., Neild, S.A.: Beams. In: Nonlinear Vibration with Control. Solid Mechanics and Its Applications, vol. 218, pp. 261–312. Springer, Dordrecht (2015) 24. Hill, T.L.: Modal interactions in nonlinear systems. PhD thesis, University of Bristol (2016) 25. Doedel, E.J., with major contributions from Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Dercole, F., Oldeman, B.E., Paffenroth, R.C., Sandstede, B., Wang, X.J., Zhang, C.: AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University, Montreal (2008). Available at: http://cmvl.cs.concordia.ca/ 26. Neild, S.A., Champneys, A.R., Wagg, D.J., Hill, T.L., Cammarano, A.: The use of normal forms for analysing nonlinear mechanical vibrations. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 373, 2051 (2015)

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