Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods

The aim of the present paper is to compare two different methods available for reducing the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD), and an asymptotic approximation of the nonlinear normal modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the partial differential equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed. The response is investigated also for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation. Bifurcation diagrams of Poincare maps obtained from direct time integration and calculation of the maximum Lyapunov exponent have been used to characterize the system. © 2007 Elsevier Ltd. All rights reserved.

[1]  Christophe Pierre,et al.  Nonlinear normal modes for vibratory systems under harmonic excitation , 2005 .

[2]  H. Poincaré,et al.  Les méthodes nouvelles de la mécanique céleste , 1899 .

[3]  C. Pierre,et al.  A NEW GALERKIN-BASED APPROACH FOR ACCURATE NON-LINEAR NORMAL MODES THROUGH INVARIANT MANIFOLDS , 2002 .

[4]  Christophe Pierre,et al.  Non-linear normal modes and invariant manifolds , 1991 .

[5]  Nadine Aubry,et al.  The dynamics of coherent structures in the wall region of a turbulent boundary layer , 1988, Journal of Fluid Mechanics.

[6]  M. P. Païdoussis,et al.  Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method , 2003 .

[7]  Marco Amabili,et al.  Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections , 2003 .

[8]  O. Thomas,et al.  Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry , 2006 .

[9]  A. H. Nayfeh,et al.  Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems , 2003 .

[10]  Marco Amabili,et al.  Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures , 2006 .

[11]  D. Decker,et al.  Topics in bifurcation theory , 1978 .

[12]  S. Zahorian.  Principal‐Components Analysis for Low Redundancy Encoding of Speech Spectra , 1979 .

[13]  Christophe Pierre,et al.  Normal Modes for Non-Linear Vibratory Systems , 1993 .

[14]  Alexander F. Vakakis,et al.  Interaction Between Slow and Fast Oscillations in an Infinite Degree-of-Freedom Linear System Coupled to a Nonlinear Subsystem: Theory and Experiment , 1999 .

[15]  M. Païdoussis,et al.  Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction , 2003 .

[16]  G. Kerschen,et al.  The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview , 2005 .

[17]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[18]  Cyril Touzé,et al.  Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes , 2004 .

[19]  P. Coullet,et al.  A simple global characterization for normal forms of singular vector fields , 1987 .

[20]  Alexander F. Vakakis,et al.  An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems , 1994 .

[21]  Gaëtan Kerschen,et al.  On the exploitation of chaos to build reduced-order models , 2003 .

[22]  Christophe Pierre,et al.  The construction of non-linear normal modes for systems with internal resonance , 2005 .

[23]  Lawrence Sirovich,et al.  The use of the Karhunen-Loegve procedure for the calculation of linear Eigenfunctions , 1991 .

[24]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[25]  Alexander F. Vakakis,et al.  Normal modes and localization in nonlinear systems , 1996 .

[26]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[27]  R. M. Rosenberg,et al.  On Nonlinear Vibrations of Systems with Many Degrees of Freedom , 1966 .

[28]  Robert Bouc,et al.  A new formulation for the existence and calculation of nonlinear normal modes , 2005 .

[29]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[30]  Marco Amabili,et al.  Effect of the geometry on the non-linear vibration of circular cylindrical shells , 2002 .

[31]  D. A. Evensen,et al.  Nonlinear flexural vibrations of thin-walled circular cylinders , 1967 .

[32]  A. Vakakis,et al.  PROPER ORTHOGONAL DECOMPOSITION (POD) OF A CLASS OF VIBROIMPACT OSCILLATIONS , 2001 .

[33]  Joseph C. Slater,et al.  A numerical method for determining nonlinear normal modes , 1996 .

[34]  Michael P. Païdoussis,et al.  A cantilever conveying fluid: coherent modes versus beam modes , 2004 .

[35]  Y. Mikhlin.  Matching of local expansions in the theory of non-linear vibrations , 1995 .

[36]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[37]  Ioannis T. Georgiou,et al.  Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods , 2005 .

[38]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[39]  M. P. Païdoussis,et al.  A compact limit-cycle oscillation model of a cantilever conveying fluid , 2003 .

[40]  Marco Amabili,et al.  Chaotic vibrations of circular cylindrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method , 2006 .

[41]  Claude-Henri Lamarque,et al.  Analysis of non-linear dynamical systems by the normal form theory , 1991 .