Chaotic vibrations of circular cylindrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method

Abstract The geometric nonlinear response of a water-filled, simply supported circular cylindrical shell to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency is investigated. The response is investigated 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 Lyapunov exponents and Lyapunov dimension have been used to study the system. By increasing the excitation amplitude, the response undergoes (i) a period-doubling bifurcation, (ii) subharmonic response, (iii) quasi-periodic response and (iv) chaotic behaviour with up to 16 positive Lyapunov exponents (hyperchaos). The model is based on Donnell's nonlinear shallow-shell theory, and the reference solution is obtained by the Galerkin method. The proper orthogonal decomposition (POD) method is used to extract proper orthogonal modes that describe the system behaviour from time-series response data. These time-series have been obtained via the conventional Galerkin approach (using normal modes as a projection basis) with an accurate model involving 16 degrees of freedom (dofs), validated in previous studies. The POD method, in conjunction with the Galerkin approach, permits to build a lower-dimensional model as compared to those obtainable via the conventional Galerkin approach. Periodic and quasi-periodic response around the fundamental resonance for fixed excitation amplitude, can be very successfully simulated with a 3-dof reduced-order model. However, in the case of large variation of the excitation, even a 5-dof reduced-order model is not fully accurate. Results show that the POD methodology is not as “robust” as the Galerkin method.

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

[2]  M. P. Païdoussis,et al.  NON-LINEAR DYNAMICS AND STABILITY OF CIRCULAR CYLINDRICAL SHELLS CONTAINING FLOWING FLUID, PART II: LARGE-AMPLITUDE VIBRATIONS WITHOUT FLOW , 1999 .

[3]  Marco Amabili,et al.  A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach , 2003 .

[4]  M. P. Païdoussis,et al.  NON-LINEAR DYNAMICS AND STABILITY OF CIRCULAR CYLINDRICAL SHELLS CONTAINING FLOWING FLUID. PART I: STABILITY , 1999 .

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

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

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

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

[9]  Marco Amabili,et al.  Non-linear vibrations of doubly curved shallow shells , 2005 .

[10]  J. Cusumano,et al.  Period-infinity periodic motions, chaos, and spatial coherence in a 10 degree of freedom impact oscillator , 1993 .

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

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

[13]  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 .

[14]  Michael P. Païdoussis,et al.  A horizontal fluid-conveying cantilever: spatial coherent structures, beam modes and jumps in stability diagram , 2005 .

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

[16]  J. Cusumano,et al.  Bifurcation and Modal Interaction in a Simplified Model of Bending-Torsion Vibrations of the Thin Elastica , 1995 .

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

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

[19]  F. Busse.  An exploration of chaos: J. Argyris, G. Faust and M. Haase, Elsevier, Amsterdam, 1994, 722 pp., ISBN 0-444-82002-7 (hardbound), 0-444-82003-5 (paperback) , 1994 .

[20]  Earl H. Dowell,et al.  Modal equations for the nonlinear flexural vibrations of a cylindrical shell , 1968 .

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

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

[23]  M. P. Païdoussis,et al.  NON-LINEAR DYNAMICS AND STABILITY OF CIRCULAR CYLINDRICAL SHELLS CONTAINING FLOWING FLUID. PART III: TRUNCATION EFFECT WITHOUT FLOW AND EXPERIMENTS , 2000 .