Non-Linear Normal Modes in Dynamics - Discrete Systems

Abstract:

The aim of the paper is to inform about main features of Non-linear Normal Modes (NNM) as a powerful tool for investigation of multi-degree of freedom (MDOF) dynamic systems. In particular, it is shown how this concept can be used to investigate forced resonances of non-linear symmetric systems including non-linear localization of vibrational energy. NNMs can provide a valuable tool for understanding essentially non-linear dynamic phenomena having no counterparts in linear theory and which do not enable analysis using linearized procedures. Discrete MDOF systems are considered in this study. A couple of possible approaches are outlined together with some demonstrations of numerical results.

• Periodical: Applied Mechanics and Materials (Volume 821)
• Pages: 254-265
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.821.254
• Online since: January 2016
• Authors: Jiří Náprstek*, Cyril Fischer
• Keywords: Discretization, Multi-Scale Method, Non-Linear Dynamical System, Non-Linear Normal Modes

References

[1] A.M. Lyapunov, The General Problem of the Stability of Motion (Doctoral dissertation), University of Kharkov, 1892 (in Russian).

[2] R.M. Rosenberg, Normal modes of non-linear dual-mode systems. Jour. Applied Mechanics, 27 (1960) 263-268.

[3] R.M. Rosenberg, On non-linear vibrations of systems with many degrees of freedom. Advances in Applied Mechanics, 9 (1966) 155-242.

DOI: 10.1016/s0065-2156(08)70008-5

[4] R. Rand, Non-linear normal modes in two-degree-of-freedom system. Jour. of Applied Mechanics, 6 (1971) 545-547.

DOI: 10.1016/0020-7462(71)90049-7

[5] R. Devaney, Homoclinic orbits in Hamiltonian systems. Jour. Differential Equations, 21 (1976) 431-438.

DOI: 10.1016/0022-0396(76)90130-3

[6] J. Carr, Applications of Centre Manifold Theory. Springer, Berlin, Heidelberg, (1981).

[7] P. Boxler, A stochastic version of center manifold theory. Probability Theory and Related Fields, 83(4) (1989) 509-545.

DOI: 10.1007/bf01845701

[8] A.F. Vakakis, T. Nayfeh, M. King, A multiple-scales analysis of non-linear, localized modes in a cyclic periodic system. Transactions of the ASME, 60 (1993) 388-397.

DOI: 10.1115/1.2900806

[9] A.F. Vakakis, Non-linear normal modes (NNMs) and their applications in vibrati8on theory: an overwiev. Mechanical Systems and Signal Processing, 11(1) (1997) 3-22.

DOI: 10.1006/mssp.1996.9999

[10] A.H. Nayfeh, S.A. Nayfeh, On nonlinear modes of continuous systems. Jour. Vibration and Acoustics, 116 (1994) 129-136.

DOI: 10.1115/1.2930388

[11] A.H. Nayfeh, C. Chin, S.A. Nayfeh, Nonlinear normal modes of a cantilever beam. Jour. Vibration and Acoustics, 117 (1995) 477-481.

DOI: 10.1115/1.2874486

[12] S.W. Shaw, Normal modes of vibration for non-linear continuous systems. Jour. Sound and Vibration, 169(3) (1994) 319-347.

DOI: 10.1006/jsvi.1994.1021

[13] M.E. King, A.F. Vakakis, A very complicated structure of resonances in a non-linear system with cyclic symmetry: non-linear forced localization. Nonlinear Dynamics, 7 (1995) 85-104.

DOI: 10.1007/bf00045127

[14] M.S. Jolly, R. Rosa, Computation of non-smooth local centre manifolds. IMA Journal of Numerical Analysis, 25(4) (2005) 698-725.

DOI: 10.1093/imanum/dri013

[15] S. Lenci, G. Rega, Dimension reduction of homoclinic orbits of buckled beams via the non-linear modes technique. Int. Jour. Non-Linear Mechanics, 42 (2007) 515-528.

DOI: 10.1016/j.ijnonlinmec.2007.02.004

[16] G. Kerschen, M. Peeters, J.C. Golinval, A.F. Vakakis, Non-linear normal modes, Part I: A useful framework for the structural dynamicist. Mechanical Systems and Signal Processing, 23 (2009) 170-194.

DOI: 10.1016/j.ymssp.2008.04.002

[17] D. Jiang, C. Pierre, S.W. Shaw, The construction of non-linear normal modes for systems with internal resonance. Int. Jour. Non-Linear Mechanics, 40 (2009) 729-746.

DOI: 10.1016/j.ijnonlinmec.2004.08.010

[18] A.F. Vakakis(edt. ), Normal Modes and Localization in Non-Linear Systems. Kluwer, Dordrecht, (2001).

[19] J. Náprstek, Combined analytical and numerical approaches in Dynamic Stability analyses of engineering systems. Jour. Sound and Vibration, 338 (2015) 2-41.

DOI: 10.1016/j.jsv.2014.06.029

[20] G. Iooss, M. Adelmeyer, Topics in Bifurcation Theory and Applications. World Scientific Publishing, London, (1992).

[21] Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory (3rd edition). Springer, Berlin, Heidelberg, New York, (2004).

[22] G.W. Whitehead, Elements of homotopy theory. Springer, Berlin, Heidelberg, (1978).

[23] M. Byrtus, V. Zeman, Alternative methods for vibration analysis of friction blade coupling. In: Proc. Dynamics of Machines 2012 (L. Pešek edt. ), IT ASCR, Prague, 2012, pgs. 23-28.

[24] A.F. Vakakis, L.I. Manevitch, Y.V. Mikhlin, V.N. Pilipchuk, A.A. Zevin, Normal modes and localization in non-linear systems. Wiley, New York, (1996).

DOI: 10.1002/9783527617869

[25] A. Weinstein, Normal modes for nonlinear Hamiltonian systems. Investigations in Mathematics, 20 (1973) 47-57.

[26] J.K. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Communications in Pure and Applied Mathematics, 29 (1976) 727-747.

DOI: 10.1002/cpa.3160290613

[27] N. Bogolyubov, Y. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations. Gordon & Breach Scientific Publication, New York, (1961).