Document Type : Original Article

Authors

1 Department of Aerospace Engineering, Amirkabir University of Tchnology, Tehran, IRAN

2 Department of Aerospace Engineering , Amirkabir University of Technology,Tehran, IRAN

3 Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, IRAN

Abstract

In the last decade, nonlinear normal modes have attracted the attention of many researchers, and many methods and algorithms have been proposed to calculate them. Among the proposed methods, the combination of the shooting method and the continuation of the periodic solution is the strongest methods. However, the computational cost of the method has still limited its application. In this paper, an updated formula is used to reduce the computational costs of the method. Using this updated formula significantly reduced the computation time so that the computational speed of nonlinear normal modes increased tenfold. Also, as the power of nonlinear terms increases in the system, the efficiency of the updated formula increases. In order to evaluate the accuracy of the proposed method, a system with two degrees of freedom was studied, and it was observed that the results obtained are consistent with the results in other works.

Keywords

[1] A.F. Vakakis, Analysis and identification of linear and nonlinear normal modes in vibrating systems, in:  Mechanical engineering, California Institute of Technology, California, 1991.
[2]A.F. Vakakis, Non-similar normal oscillations in a strongly non-linear discrete system, Publisher, City, 1992.
[3]A.F. Vakakis, L.I. Manevitch, Y.V. Mikhlin, V.N.P. Chuk, A.A. Zevin, Normal Modes and Localization in Nonlinear Systems John Wiley & Sons, 1996.
[4] S. Shaw, C. Pierre, Non-linear normal modes and invariant manifolds, Publisher, City, 1991.
[5]S. Shaw, C. Pierre, On nonlinear normal modes., Publisher, City, 1992.
[6]S.W. Shaw, C. Pierre, Normal modes for non-linear vibratory systems, Publisher, City, 1993.
[7]Y.V. Mikhlin, K.V. Avramov, Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments, Publisher, City, 2011.
[8]K.V. Avramov, Y.V. Mikhlin, Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems, Publisher, City, 2013.
[9]L. Renson, G. Kerschen, B. Cochelin, Numerical computation of nonlinear normal modes in mechanical engineering, Publisher, City, 2016.
[10]J.P. Noël, G. Kerschen, Nonlinear system identification in structural dynamics: 10 more years of progress, Publisher, City, 2017.
[11]A.M. Lyapunov, The General Problem of the Stability of Motion, Publisher, City, 1947.
[12]A. Weinstein, Normal modes for nonlinear hamiltonian systems, Publisher, City, 1973.
[13]J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Publisher, City, 1976.
[14]H. Kauderer, Nichtlineare Mechanik Springer- Verlag, 1958.
[15]R.M.Rosenberg, On Nonlinear Vibrations of Systems with Many Degrees of Freedom, Publisher, City, 1966.
[16]G. Haller, S. Ponsioen, Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction, Publisher, City, 2016.
[17]E. Pesheck, Reduced order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds, in, The University of Michigan., 2001.
[18]S.W. Shaw, C. Pierre, Normal modes of vibration for non-linear continuous systems, Publisher, City, 1994.
[19]L. Renson, G. Deliége, G. Kerschen, An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems, Publisher, City, 2014.
[20]G.I. Cirillo, A. Mauroy, L. Renson, G. Kerschen, R. Sepulchre, A spectral characterization of nonlinear normal modes, Publisher, City, 2016.
[21]S. Ponsioen, T. Pedergnana, G. Haller, Automated computation of autonomous spectral submanifolds for nonlinear modal analysis, Publisher, City, 2018.
[22]R. Seydel, Practical Bifurcation and Stability Analysis, 3rd ed., Springer-Verlag, New York, 2010.
[23] W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000.
[24]J.C. Slater, A numerical method for determining nonlinear normal modes, Publisher, City, 1996.
[25]Y.S. Lee, G. Kerschen, A.F. Vakakis, P. Panagopoulos, L. Bergman, D.M. McFarland, Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Publisher, City, 2005.
[26]G. Kerschen, M. Peeters, J.C. Golinval, A.F. Vakakis, Nonlinear normal modes, Part I: A useful framework for the structural dynamicist, Publisher, City, 2009.
[27]M. Peeters, R. Viguié, G. Sérandour, G. Kerschen, J.C. Golinval, Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques, Publisher, City, 2009.
[28]R.J. Kuether, M.S. Allen, A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models, Publisher, City, 2014.
[29]R.J. Kuether, B.J. Deaner, J.J. Hollkamp, M.S. Allen, Evaluation of Geometrically Nonlinear Reduced-Order Models with Nonlinear Normal Modes, Publisher, City, 2015.
[30]L. Renson, J.P. Noël, G. Kerschen, Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes, Publisher, City, 2015.
[31]A.F. Vakakis, Designing a Linear Structure with a Local Nonlinear Attachment for Enhanced Energy Pumping, Publisher, City, 2003.
[32]K.V. Avramov, O.V. Gendelman, On interaction of vibrating beam with essentially nonlinear absorber, Publisher, City, 2010.
[33]M.A. Al-Shudeifat, N.E. Wierschem, L.A. Bergman, A.F. Vakakis, Numerical and experimental investigations of a rotating nonlinear energy sink, Publisher, City, 2017.
[34]M. Kurt, M. Eriten, D.M. McFarland, L.A. Bergman, A.F. Vakakis, Methodology for model updating of mechanical components with local nonlinearities, Publisher, City, 2015.
[35]S. Peter, A. Grundler, P. Reuss, L. Gaul, R.I. Leine, Towards Finite Element Model Updating Based on Nonlinear Normal Modes, in: G. Kerschen (Ed.) Nonlinear Dynamics, Volume 1, Springer International Publishing, Cham, 2016, pp. 209-217.
[36]D.A. Ehrhardt, M.S. Allen, T.J. Beberniss, S.A. Neild, Finite element model calibration of a nonlinear perforated plate, Publisher, City, 2017.
[37]C.I. VanDamme, M. Allen, J.J. Hollkamp, Nonlinear Structural Model Updating Based Upon Nonlinear Normal Modes, in:  2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, 2018.
[38]M. Song, L. Renson, J.-P. Noël, B. Moaveni, G. Kerschen, Bayesian model updating of nonlinear systems using nonlinear normal modes, Publisher, City, 2018.
[39] W. Lacarbonara, B. Carboni, G. Quaranta, Nonlinear normal modes for damage detection, Publisher, City, 2016.
[40]F.B. Deuflhard P, Kunkel P, Efficient Numerical Pathfollowing Beyond Critical Points, Publisher, City, 1987.