Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method

The nonlinear (large-amplitude) response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of their lowest natural frequencies is investigated. The shell is assumed to be completely filled with an incompressible and inviscid fluid at rest. Donnell's nonlinear shallow-shell theory is used, and the 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, validated in previous studies. The POD method, in conjunction with the Galerkin approach, permits a lower-dimensional model as compared to those obtainable via the conventional Galerkin approach. Different proper orthogonal modes computed from time series at different excitation frequencies are used and solutions are compared. Some of these sets of modes are capable of describing the system behaviour over the whole frequency range around the fundamental resonance with good accuracy and with only 3 degrees of freedom. They allow a drastic reduction in the computational effort, as compared to using the 16 degree-of-freedom model necessary when the conventional Galerkin approach is used.

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

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

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

[6]  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 .

[7]  C. D. Babcock,et al.  Nonlinear Vibration of Cylindrical Shells , 1975 .

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

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

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

[11]  Eelco Jansen.  Non-stationary flexural vibration behaviour of a cylindrical shell , 2002 .

[12]  Paulo B. Gonçalves,et al.  Non-linear vibration analysis of fluid-filled cylindrical shells , 1988 .

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

[14]  J. H. Ginsberg,et al.  Large Amplitude Forced Vibrations of Simply Supported Thin Cylindrical Shells , 1973 .

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

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

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

[18]  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 .

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

[20]  Aouni A. Lakis,et al.  Non-linear free vibration analysis of laminated orthotropic cylindrical shells , 1998 .

[21]  V. D. Kubenko,et al.  Nonlinear problems of the vibration of thin shells (review) , 1998 .

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

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