Frederico M. A. Silva

Transient Stability of Empty and Fluid-Filled Cylindrical Shells

In the present work a qualitatively accurate low dimensional model is used to study the non-linear dynamic behavior of shallow cylindrical shells under axial loading. The dynamic version of the Donnell non-linear shallow shell equations are discretized by the Galerkin method. The shell is considered to be initially at rest, in a position corresponding to a pre-buckling configuration. Then, a harmonic excitation is applied and conditions to escape from this configuration are sought. By defining steady state and transient stability boundaries, frequency regimes of instability may be identified such that they may be avoided in design. Initially a steady state analysis is performed; resonance response curves in the forcing plane are presented and the main instabilities are identified. Finally, the global transient response of the system is investigated in order to quantify the degree of safety of the shell in the presence of small perturbations. Since the initial conditions, or even the shell parameters, may vary widely, and indeed are often unknown, attention is given to all possible transient motions. As parameters are varied, transient basins of attraction can undergo quantitative and qualitative changes; hence a stability analysis which only considers the steady-state and neglects this global transient behavior, may be seriously non-conservative. Keywords : Cylindrical shells, fluid-structure interaction, parametric instability, nonlinear vibrations

Influence of physical and geometrical system parameters uncertainties on the nonlinear oscillations of cylindrical shells

This work investigates the influence of physical and geometrical system parameters uncertainties and excitation noise on the nonlinear vibrations and stability of simply-supported cylindrical shells. These parameters are composed of both deterministic and random terms. Donnells non-linear shallow shell theory is used to study the non-linear vibrations of the shell. To discretize the partial differential equations of motion, first, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all modes that couple with the linear modes through the quadratic and cubic nonlinearities. Then, a particular solution is selected which ensures the convergence of the response up to very large deflections. Finally, the in-plane displacements are obtained as a function of the transversal displacement by solving the in-plane equations analytically and imposing the necessary boundary, continuity and symmetry conditions. Substituting the obtained modal expansions into the equation of motion and applying the Galerkins method, a discrete system in time domain is obtained. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the physical and geometrical system parameters. Special attention is given to the influence of the uncertainties on the parametric instability and escape boundaries.

Nonlinear Dynamics, Safety, and Control of Structures Liable to Interactive Unstable Buckling

The nonlinear dynamics of two archetypal structural systems exhibiting interactive modal post-buckling behavior is addressed, the discrete Augusti’s model and a reduced-order model of the axially loaded cylindrical shell. The uncoupled models exhibit a stable post-buckling response. However, the modal interaction leads to unstable post-buckling paths that entail a complex dynamic behavior and imperfection sensitivity, with a marked influence on the dynamic integrity and safety. Perfect and imperfect Augusti’s models are investigated in terms of static buckling, linear vibrations, nonlinear normal modes, local and global nonlinear response to harmonic excitation, dynamic integrity, control of global bifurcations aimed at increasing the load carrying capacity. Then, as an example of a continuous system exhibiting strong modal coupling and interaction, a two-degree-of-freedom model of the thin-walled cylindrical shell is investigated in terms of global behavior and dynamic integrity. The influence of uncertainties on the nonlinear response and dynamic integrity is also shortly addressed. The chapter shows how a judicious use of the tools of nonlinear dynamics sheds light on the actual safety of structural systems liable to unstable buckling under static and dynamic loads.

Nonlinear vibrations of fluid-filled functionally graded cylindrical shell considering a time-dependent lateral load and static preload

In the recent years, functionally gradient materials (FGMs) have gained considerable attention with possible applications in several engineering fields, especially in a high-temperature or hazardous environment. In this work, the nonlinear vibrations of a simply supported fluid-filled functionally graded cylindrical shell subjected to a lateral time-dependent load and axial static preload are analyzed. To model the shell, the Donnell nonlinear shell theory is used. The fluid is assumed to be incompressible, nonviscous, and irrotational. A new function to describe the variation in the volume fraction of the constituent material through the shell thickness is proposed, extending the concept of sandwich structures to a functionally graded material. Material properties are graded along the shell thickness according to the proposed volume fraction power law distribution. A consistent reduced order model derived from a perturbation technique is used to describe the displacements of the shell and, the Galerkin method is applied to derive a set of coupled nonlinear ordinary differential equations of motion. Results show the influence of the variation of the two constituent materials along the shell thickness, internal fluid, static preload, and shell geometry on the natural frequencies, nonlinear frequency–amplitude relation, resonance curves, and bifurcation scenario of the FG cylindrical shell.

Influence of Modal Coupling on the Nonlinear Vibration of Simply Supported Cylindrical Panels

The aim of this paper is to analyse the influence of the nonlinear modal coupling on the nonlinear vibrations of a simply supported cylindrical panel excited by a time dependent transversal load. The cylindrical panel is modeled by the Donnell nonlinear shallow shell theory and the lateral displacement field is based on a perturbation procedure. The axial and circumferential displacement are described in terms of the obtained lateral displacement, generating a precise low-dimensional model that satisfies all transversal boundary conditions. The discretized equations of motion in time domain are determined by applying the standard Galerkin method. Various numerical techniques are employed to obtain the cylindrical panel resonance curves, bifurcation scenario and basins of attraction. The results show the influence of geometry and the nonlinear modal coupling on the nonlinear response of the cylindrical panel.

Influence of Physical and Geometrical Uncertainties in the Parametric Instability Load of an Axially Excited Cylindrical Shell

This work investigates the influence of Young’s modulus, shells thickness, and geometrical imperfection uncertainties on the parametric instability loads of simply supported axially excited cylindrical shells. The Donnell nonlinear shallow shell theory is used for the displacement field of the cylindrical shell and the parameters under investigation are considered as uncertain parameters with a known probability density function in the equilibrium equation. The uncertainties are discretized as Hermite-Chaos polynomials together with the Galerkin stochastic procedure that discretizes the stochastic equation in a set of deterministic equations of motion. Then, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all nonlinear modes that couple with the linear modes. So, a particular solution is selected which ensures the convergence of the response up to very large deflections. Applying the standard Galerkin method, a discrete system in time domain that considers the uncertainties is obtained and solved by fourth-order Runge-Kutta method. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the parameters. Special attention is given to the influence of the uncertainties on the parametric instability and time response, showing that the Hermite-Chaos polynomial is a good numerical tool.

Nonlinear Dynamics of Functionally Graded Cylindrical Shells With Internal Fluid

This work analyzes the nonlinear vibrations of a simply supported functionally graded cylindrical shell considering the effects of an internal fluid and static preloading. The cylindrical shell is subjected to a time dependent axial loading. The fluid is considered to be incompressible, non-viscous and irrotational and its effect on the shell wall is obtained using the potential flow theory. The shell is modeled by Donnell nonlinear shallow shell theory. The axial and circumferential displacement fields are described in terms of lateral displacement, thus generating a low-dimensional model, while the lateral displacement field is determined by a perturbation procedure which provides a general expression for the nonlinear vibration modes. These modal expansions satisfy the boundary and symmetry conditions of the problem. The discretized equations of motion are obtained by applying the Galerkin method. Various numerical techniques are employed to obtain the resonance curves and time responses of the cylindrical shell, showing the influence of the geometry, the internal fluid, static preloading and functionally graded material law on the shell dynamics and stability.Copyright

ON THE GALERKIN - ITERATIVE METHOD APPLIED TO THE NON-LINEAR VIBRATIONS OF RECTANGULAR PLATES

In this work, by using the Galerkin-iterative method, the non-linear free and forced vibrations of rectangular plates are investigated. This method gives the eigenfunctions for the lateral displacements of plates with different sets of boundary conditions. Special attention is given to the frequency-amplitude relations and to the parametric instability boundaries of rectangular plates in the force control space.

Nonlinear Vibrations of Partially Fluid-Filled Cylindrical Shells

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.Copyright