The post-processed Galerkin method applied to non-linear shell vibrations

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary differential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the post-processing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-difference technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.

NONDEGENERATE UMBILICS, THE PATH FORMULATION and GRADIENT BIFURCATION PROBLEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Problemas de bifurcação de corank2 com dois parâmetros e a formulação por caminhos

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Static analysis of infinite laminated cylindrical shell panel

This paper presents an approximate three-dimensional elasticity solution for an infinitely long, cross-ply laminated circular cylindrical shell panel with simply supported boundary conditions, subjected to an arbitrary discontinuous transverse loading. The solution is based on the principal assumption that the ratio of the thickness of the lamina to its middle surface radius is negligible compared to unity. The validity of this assumption and the range of application of this approximate solution have been established through a comparison with an exact solution. Results of classical and first-order shear deformation shell theories have been compared with the results of the present solution to bring out the accuracy of these theories. It is also shown that for very shallow shell panels the definition of a thin shell should be based on the ratio of thickness to chord width rather than the ratio of thickness to mean radius.

Analysis of a long thick orthotropic circular cylindrical shell panel

An exact three-dimensional elasticity solution has been obtained for an infinitely long, thick transversely isotropic circular cylindrical shell panel, simply supported along the longitudinal edges and subjected to a radial patch load. Using a set of three displacement functions, the boundary value problem is reduced to Bessel's differential equation. Numerical results are presented for different thickness to mean radius ratios and semicentral angles of the shell panel. Classical and first-order shear deformation orthotropic shell theories have been examined in comparison with the present elasticity solution.

Static analysis of functionally graded cylindrical and conical shells or panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded conical and cylindrical shells subjected to mechanical loadings. Several types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the conical or cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally conical and cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.

The mechanical behaviour of cylindrical GRP panels

Glass reinforced plastic (GRP) is now an established material for the fabrication of sonar windows. Its good mechanical strength, light weight, resistance to corrosion and acoustic transparency, are all properties which fit it for this application. This thesis describes a study, undertaken at the Royal Naval Engineering College, Plymouth, into the mechanical behaviour of a circular cylindrical sonar panel. This particular type of panel would be used to cover a flank array sonar in a ship or submarine. The case considered is that of a panel with all of its edges mechanically clamped and subject to pressure loading on its convex surface. A comprehensive program of testing, to determine the orthotropic elastic properties of the laminated composite panel material is described, together with a series of pressure tests on 1:5 scale sonar panels. These pressure tests were carried out in a purpose designed test rig, using air pressure to provide simulated hydrostatic and hydrodynamic loading. Details of all instrumentation used in the experimental work are given in the thesis. The experimental results from the panel testing are compared with predictions of panel behaviour obtained from both the Galerkin solution of Flugge's cylindrical shell equations (orthotropic case), and finite element modelling of the panels using PAFEC. A variety of appropriate panel boundary conditions are considered in each case. A parametric study, intended to be of use as a preliminary design tool, and based on the above Galerkin solution, is also presented. This parametric study considers cases of boundary conditions, material properties, and panel geometry, outside of those investigated in the experimental work Final conclusions are drawn and recommendations made regarding possible improvements to the procedures for design, manufacture and fixing of sonar panels in the Royal Navy.

A quasi stationary approach to drying kinetics of cylindrical food particulates in a heat pump asisted fluid bed dryer

Experiments were undertaken to study drying kinetics of moist cylindrical shaped food particulates during fluidised bed drying. Cylindrical particles were prepared from Green beans with three different length:diameter ratios, 3:1, 2:1 and 1:1. A batch fluidised bed dryer connected to a heat pump system was used for the experimentation. A Heat pump and fluid bed combination was used to increase overall energy efficiency and achieve higher drying rates. Drying kinetics, were evaluated with non-dimensional moisture at three different drying temperatures of 30, 40 and 50o C. Numerous mathematical models can be used to calculate drying kinetics ranging from analytical models with simplified assumptions to empirical models built by regression using experimental data. Empirical models are commonly used for various food materials due to their simpler approach. However problems in accuracy, limits the applications of empirical models. Some limitations of empirical models could be reduced by using semi-empirical models based on heat and mass transfer of the drying operation. One such method is the quasi-stationary approach. In this study, a modified quasi-stationary approach was used to model drying kinetics of the cylindrical food particles at three drying temperatures.

Defining the flammability of cylindrical metal rods through characterization of the thermal effects of the ignition promoter

All relevant international standards for determining if a metallic rod is flammable in oxygen utilize some form of “promoted ignition” test. In this test, for a given pressure, an overwhelming ignition source is coupled to the end of the test sample and the designation flammable or nonflammable is based upon the amount burned, that is, a burn criteria. It is documented that (1) the initial temperature of the test sample affects the burning of the test sample both (a) in regards to the pressure at which the sample will support burning (threshold pressure) and (b) the rate at which the sample is melted (regression rate of the melting interface); and, (2) the igniter used affects the test sample by heating it adjacent to the igniter as ignition occurs. Together, these facts make it necessary to ensure, if a metallic material is to be considered flammable at the conditions tested, that the burn criteria will exclude any region of the test sample that may have undergone preheating during the ignition process. A two-dimensional theoretical model was developed to describe the transient heat transfer occurring and resultant temperatures produced within this system. Several metals (copper, aluminum, iron, and stainless steel) and ignition promoters (magnesium, aluminum, and Pyrofuze®) were evaluated for a range of oxygen pressures between 0.69 MPa (100 psia) and 34.5 MPa (5,000 psia). A MATLAB® program was utilized to solve the developed model that was validated against (1) a published solution for a similar system and (2) against experimental data obtained during actual tests at the National Aeronautics and Space Administration White Sands Test Facility. The validated model successfully predicts temperatures within the test samples with agreement between model and experiment increasing as test pressure increases and/or distance from the promoter increases. Oxygen pressure and test sample thermal diffusivity were shown to have the largest effect on the results. In all cases evaluated, there is no significant preheating (above about 38°C/100°F) occurring at distances greater than 30 mm (1.18 in.) during the time the ignition source is attached to the test sample. This validates a distance of 30 mm (1.18 in.) above the ignition promoter as a burn length upon which a definition of flammable can be based for inclusion in relevant international standards (that is, burning past this length will always be independent of the ignition event for the ignition promoters considered here. KEYWORDS: promoted ignition, metal combustion, heat conduction, thin fin, promoted combustion, burn length, burn criteria, flammability, igniter effects, heat affected zone.

What shapes player performance in soccer? Empirical findings from a panel analysis

In this article, we investigate the pay-performance relationship of soccer players using individual data from eight seasons of the German soccer league Bundesliga. We find a nonlinear pay-performance relationship, indicating that salary does indeed affect individual performance. The results further show that player performance is affected not only by absolute income level but also by relative income position. An additional analysis of the performance impact of team effects provides evidence of a direct impact of team-mate attributes on individual player performance.

The rate-limiting mechanism for the heterogeneous burning of cylindrical iron rods

This paper presents the findings of an investigation into the rate-limiting mechanism for the heterogeneous burning in oxygen under normal gravity and microgravity of cylindrical iron rods. The original objective of the work was to determine why the observed melting rate for burning 3.2-mm diameter iron rods is significantly higher in microgravity than in normal gravity. This work, however, also provided fundamental insight into the rate-limiting mechanism for heterogeneous burning. The paper includes a summary of normal-gravity and microgravity experimental results, heat transfer analysis and post-test microanalysis of quenched samples. These results are then used to show that heat transfer across the solid/liquid interface is the rate-limiting mechanism for melting and burning, limited by the interfacial surface area between the molten drop and solid rod. In normal gravity, the work improves the understanding of trends reported during standard flammability testing for metallic materials, such as variations in melting rates between test specimens with the same cross-sectional area but different crosssectional shape. The work also provides insight into the effects of configuration and orientation, leading to an improved application of standard test results in the design of oxygen system components. For microgravity applications, the work enables the development of improved methods for lower cost metallic material flammability testing programs. In these ways, the work provides fundamental insight into the heterogeneous burning process and contributes to improved fire safety for oxygen systems in applications involving both normal-gravity and microgravity environments.