Paulo B. Gonçalves

Non-linear vibration analysis of fluid-filled cylindrical shells

Abstract A theoretical analysis is presented for determining the elastic non-linear vibrations of a prestressed thin-walled cylindrical shell filled with an ideal fluid. For the vibrations of the shell itself, the dynamic version of the Sanders non-linear equations for the case of moderately small rotations is employed. Modal expansions are used for the displacements of the shell middle surface that are required to satisfy the “classical simply supported” boundary conditions and the circumferential periodicity condition. The fluid is taken as non-viscous and incompressible, and the coupling between the deformable shell and this medium is taken into account. The velocity potential is expanded in terms of harmonic functions which satisfy the Laplace equation term by term. The Galerkin method is used to reduce the problem to a system of coupled algebraic non-linear equations for the modal amplitudes. Solutions are presented to show the effects of fluid and shell parameters on the non-linear vibrations of the shell.

Numerical methods for analysis of plates on tensionless elastic foundations

Abstract A numerical methodology for analysis of plates resting on tensionless elastic foundations, described either by the Winkler model or as an elastic half-space, is presented in this paper. The contact surface is assumed unbonded and frictionless. The finite element method is used to discretize the plate and foundation. To overcome the difficulties in solving the plate–foundation equilibrium equations together with the inequality constraints due to the frictionless unilateral contact condition, a variational formulation equivalent to these equations is presented from which three alternative linear complementary problems (LCP) are derived and solved by Lemke’s complementary pivoting algorithm. In the first formulation, the LCP variables are the plate displacements and the elastic foundation reaction, in the second, the LCP is derived in terms of the elastic foundation reaction and, in the third formulation, the variables are the elastic foundation displacements and the gap between the bodies. Once the LCP is solved the no-contact regions where the plate lifts up away from the foundation and the sub-grade reaction, as well as the plate displacements and stresses, can be easily obtained. The methodology is illustrated by three examples and the results are compared with existing analytical and numerical results found in the literature.

Nonlinear Oscillations and Stability of Parametrically Excited Cylindrical Shells

Based on Donnell shallow shell equations, the nonlinear vibrations and dynamic instability of axially loaded circular cylindrical shells under both static and harmonic forces is theoretically analyzed. First the problem is reduced to a finite degree-of-freedom one by using the Galerkin method; then the resulting set of coupled nonlinear ordinary differential equations of motion are solved by the Runge–Kutta method. To study the nonlinear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, Lyapunov exponents, 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 excitation of flexural modes and escape from the pre-buckling potential well. Calculations are carried out for the principal and secondary instability regions associated with the lowest natural frequency of the shell. Special attention is given to the determination of the instability boundaries in control space and the identification of the bifurcational events connected with these boundaries. The results clarify the importance of modal coupling in the post-buckling solution and the strong role of nonlinearities on the dynamics of cylindrical shells.

Finite axisymmetric deformations of an initially stressed fluid-filled cylindrical membrane

This paper investigates the large deformations of an extended fluid-filled cylindrical membrane. The static case and the behaviour of the membrane rotating at a constant angular velocity are both considered. A detailed experimental analysis was carried out involving different geometries, and initial axial forces and the influence of the axial force and the fluid volume were investigated. An apparatus was developed to support vertically the extended cylindrical membrane while it is filled with liquid. The membrane used in these experiments is composed of an isotropic, homogeneous and elastic rubber, which is modelled as a neo-Hookean incompressible material, described by a single elastic constant. This constant was obtained by comparing the experimental and numerical solutions for the membrane under traction. The differential equilibrium equations for this specific problem and material were derived and solved by the shooting method. When the extended membrane was filled with liquid, it was observed that the height of liquid increased initially as the volume of liquid inside the membrane increased until a certain critical height was reached after which it remained constant or decreased slightly with increasing volume, up to the moment when the membrane lost its stability into a non-symmetric mode. These experimental results are, as shown in the paper, in satisfactory agreement with the theory.

Oscillations of a beam on a non-linear elastic foundation under periodic loads

The complexity of the response of a beam resting on a nonlinear elastic foundation makes the design of this structural element rather challenging. Particularly because, apparently, there is no algebraic relation for its load bearing capacity as a function of the problem parameters. Such an algebraic relation would be desirable for design purposes. Our aim is to obtain this relation explicitly. Initially, a mathematical model of a flexible beam resting on a non-linear elastic foundation is presented, and its non-linear vibrations and instabilities are investigated using several numerical methods. At a second stage, a parametric study is carried out, using analytical and semi-analytical perturbation methods. So, the influence of the various physical and geometrical parameters of the mathematical model on the non-linear response of the beam is evaluated, in particular, the relation between the natural frequency and the vibration amplitude and the first period doubling and saddle-node bifurcations. These two instability phenomena are the two basic mechanisms associated with the loss of stability of the beam. Finally Melnikovs method is used to determine an algebraic expression for the boundary that separates a safe from an unsafe region in the force parameters space. It is shown that this can be used as a basis for a reliable engineering design criterion.

Dynamic non-linear behavior and stability of a ventricular assist device

This paper investigates the non-linear dynamic behavior and stability of the internal membrane of a ventricular assist device (VAD). This membrane separates the blood chamber from the pneumatic chamber, transmitting the driving cyclic pneumatic loading to blood flowing from the left ventricle into the aorta. The membrane is a thin, nearly spherical axi-symmetric shallow cap made of polyurethane and reinforced with a cotton mesh. Experimental evidence shows that the reinforced membrane behaves as an isotropic elastic material and exhibits both membrane and flexural stiffness. So, the membrane is modeled as an isotropic pressure loaded shallow spherical shell and its dynamic behavior and snap-through buckling considering different types of dynamic excitation relevant to the understanding of the VAD behavior is investigated. Based on Marguerre kinematical assumptions, the governing partial differential equations of motion are presented in the form of a compatibility equation and a transverse motion equation. The results show that the shell, when subjected to compressive pressure loading, may loose its stability at a limit point, jumping to an inverted position. If the compressive load is removed, the shell jumps back to its original configuration. This non-linear behavior is the key feature in the VAD behavior.

Influence of modal coupling on the nonlinear dynamics of Augusti's model

The non-linear behavior and stability under static and dynamic loads of an inverted spatial pendulum with rotational springs in two perpendicular planes, called Augusti’s model, is analyzed in this paper. This 2DOF lumped-parameter system is an archetypal model of modal interaction in stability theory representing a large class of structural problems. When the system displays coincident buckling loads, several post-buckling paths emerge from the bifurcation point (critical load) along the fundamental path. This leads to a complex potential energy surface. Herein, we aim to investigate the influence of nonlinear modal interactions on the dynamic behavior of Augusti’s model. Coupled/uncoupled dynamic responses, bifurcations, escape from the pre-buckling potential well, stability and space–time-varying displacements, attractor-manifold-basin phase portraits are numerically evaluated with the aim of enlightening the system complex response. The investigation of basins evolution due to variation of system parameters leads to the determination of erosion profiles and integrity measures which enlighten the loss of safety of the structure due to penetration of eroding fractal tongues into the safe basin.Copyright

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

Non-linear lower bounds for shell buckling design

Abstract Based on rigorous parametric non-linear elastic buckling analyses, the present work underlines the developments towards theoretical lower bounds of existing experimental buckling loads for some of the most practical shell geometries and loading cases, namely circular cylindrical shells under external pressure and/or axial loads and spherical caps under external pressure. Simple equations and formulae are presented and their predictions for buckling loads are compared with available test results and values prescribed by some of the existing design codes. These explicit lower bounds are close and non-conservative estimates of buckling loads of imperfect shells and as such are proposed as a consistent and rational basis for design of these shell structures.

Influence of Uncertainties on the Dynamic Buckling Loads of Structures Liable to Asymmetric Postbuckling Behavior

Structural systems liable to asymmetric bifurcation usually become unstable at static load levels lower than the linear buckling load of the perfect structure. This is mainly due to the imperfections present in real structures. The imperfection sensitivity of structures under static loading is well studied in literature, but little is know on the sensitivity of these structures under dynamic loads. The aim of the present work is to study the behavior of an archetypal model of a harmonically forced structure, which exhibits, under increasing static load, asymmetric bifurcation. First, the integrity of the system under static load is investigated in terms of the evolution of the safe basin of attraction. Then, the stability boundaries of the harmonically excited structure are obtained, considering different loading processes. The bifurcations connected with these boundaries are identified and their influence on the evolution of safe basins is investigated. Then, a parametric analysis is conducted to investigate the influence of uncertainties in system parameters and random perturbations of the forcing on the dynamic buckling load. Finally, a safe lower bound for the buckling load, obtained by the application of the Melnikov criterion, is proposed which compare well with the scatter of buckling loads obtained numerically.