Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells

2005 ◽  
Vol 41 (1-3) ◽  
pp. 129-145 ◽  
Author(s):  
Paulo B. Gonçalves ◽  
ZenÓn J. G. N. DEL PRADO
Keyword(s):  
Nonlinear Vibration ◽  
Cylindrical Shells ◽  
Instability Analysis ◽  
Galerkin Models ◽  
Low Dimensional

Nonlinear Vibrations of Partially Fluid-Filled Cylindrical Shells

2010 ◽  
Author(s):  
Paulo B. Gonc¸alves ◽  
Frederico M. A. da Silva ◽  
Zeno´n J. G. N. del Prado
Keyword(s):  
Nonlinear Vibration ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Hydrodynamic Pressure ◽  
Vibration Modes ◽  
Internal Fluid ◽  
Irrotational Motion ◽  
Low Dimensional ◽  
The Galerkin Method ◽  
Modal Solution

The present work investigates the nonlinear dynamic behavior and instabilities of partially fluid-filled cylindrical shell subjected to lateral pressure. Donnell shallow shell theory is employed to model the shell. The fluid is modeled as non-viscous and incompressible and its irrotational motion is described by a velocity potential which satisfies the Laplace equation. A discrete low-dimensional model for the nonlinear vibration analysis of thin cylindrical shells is derived to study the shell vibrations. First, a general expression for the nonlinear vibration modes that satisfy all the relevant boundary, continuity and symmetry conditions is derived using a perturbation procedure validated in previous studies and then the Galerkin method is used to discretize the equations of motion. The same modal solution is used to derive the hydrodynamic pressure on the shell wall. The influence played by the height of the internal fluid on the natural frequencies, nonlinear shell response and bifurcations is examined.

On Low-Dimensional Galerkin Modeling of Confined Twin-Jet Flow and Heat Transfer

2001 ◽  
Author(s):  
H. Gunes ◽  
K. Gocmen ◽  
L. Kavurmacioglu
Keyword(s):  
Heat Transfer ◽  
Jet Flow ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Temperature Fields ◽  
Transitional Flow ◽  
Flow And Heat Transfer ◽  
Galerkin Models ◽  
Velocity And Temperature Fields ◽  
Low Dimensional

Abstract The two-dimensional incompressible non-isothermal confined twin-jet flow has been numerically studied in the transitional flow regime by a finite volume technique. Results have been obtained for the velocity and temperature distributions close to the onset of temporal oscillations. Next, the proper orthogonal decomposition (POD) is applied to the instantaneous flow and temperature data to obtain POD-based basis functions for both velocity and temperature fields. These basis functions are capable to identify the coherent structures in the velocity and temperature fields. The low-dimensional Galerkin models of the full Navier-Stokes and energy equations are constructed by the Galerkin projection onto basis functions. Since the low-dimensional Galerkin models are much easier to analyze than the full governing equations, basic insights into important mechanisms of dynamically complex flow and heat transfer (e.g. flow instabilities) can be easily studied by these models. The numerical implications, the validity of the models and their performance characteristics are discussed.

Parametrically Excited Vibrations and Auto-Parametric Resonance in Cylindrical Shells

2000 ◽  
Author(s):  
A. A. Popov ◽  
J. M. T. Thompson ◽  
F. A. McRobie
Keyword(s):  
Internal Resonance ◽  
Cylindrical Shells ◽  
Mode Interaction ◽  
Geometric Description ◽  
Modal Interactions ◽  
Structural Behaviour ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Shell Vibration ◽  
Low Dimensional

Abstract Vibrations of cylindrical shells parametrically excited by external axial forcing or by internal auto-parametric resonances are considered. A Rayleigh-Ritz discretization of the von Kármán-Donnell equations through symbolic computations leads to low dimensional models of shell vibration. After applying methods and ideas of modern dynamical systems theory, complete bifurcation diagrams are constructed and analyzed with an emphasis on modal interactions and their relevance to structural behaviour. In the case of free shell vibrations, the Hamiltonian and a transformation into action-angle coordinates followed by averaging provides readily a geometric description of the interaction between concertina and chequerboard modes. It was established that the interaction should be most pronounced when there are slightly less than 2 N square chequerboard panels circumferentially, where N is the ratio of shell radius to thickness. The two mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards, and at internal resonance, significant energy transfer occurs between the modes. The regular and chaotic features of this interaction are studied analytically and numerically.

The Role of Modal Coupling on the Non-Linear Response of Cylindrical Shells Subjected to Dynamic Axial Loads

2000 ◽  
Author(s):  
Paulo B. Gonçalves ◽  
Zenón J. G. N. Del Prado
Keyword(s):  
Dynamic Instability ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Parametric Instability ◽  
Compressive Load ◽  
Basins Of Attraction ◽  
Dynamic Component ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Low Dimensional

Abstract This paper discusses the dynamic instability of circular cylindrical shells subjected to time-dependent axial edge loads of the form P(t) = P0+P1(t), where the dynamic component p1(t) is periodic in time and P0 is a uniform compressive load. In the present paper a low dimensional model, which retains the essential non-linear terms, is used to study the non-linear oscillations and instabilities of the shell. For this, Donnell’s shallow shell equations are used together with the Galerkin method to derive a set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. To study the non-linear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, stable and unstable fixed points, bifurcation diagrams and basins of attraction. Particular attention is paid to two dynamic instability phenomena that may arise under these loading conditions: parametric instability and escape from the pre-buckling potential well. The numerical results obtained from this investigation clarify the conditions, which control whether or not instability may occur. This may help in establishing proper design criteria for these shells under dynamic loads, a topic practically unexplored in literature.

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