Natural frequencies of rectangular orthotropic plates with a pair of parallel edges simply supported

Solutions of the exact characteristic equations for the title problem derived earlier by an extension of Bolotin's asymptotic method are considered. These solutions, which correspond to flexural modes with frequency factor, R, greater than unity, are expressed in convenient forms for all combinations of clamped, simply supported and free conditions at the remaining pair of parallel edges. As in the case of uniform beams, the eigenvalues in the CC case are found to be equal to those of elastic modes in the FF case provided that the Kirchoff's shear condition at a free edge is replaced by the condition. The flexural modes with frequency factor less than unity are also investigated in detail by introducing a suitable modification in the procedure. When Poisson's ratios are not zero, it is shown that the frequency factor corresponding to the first symmetric mode in the free-free case is less than unity for all values of side ratio and rigidity ratios. In the case of one edge clamped and the other free it is found that modes with frequency factor less than unity exist for certain dimensions of the plate—a fact hitherto unrecognized in the literature.

Natural frequencies of circumferentially truncated sector plates with simply supported straight edges

The classical Rayleigh-Ritz method in conjunction with suitable co-ordinate transformations is found to be effective for accurate estimation of natural frequencies of circumferentially truncated circular sector plates with simply supported straight edges. Numerical results are obtained for all the nine combinations of clamped, simply supported and free boundary conditions at the circular edges and presented in the form of graphs. The analysis confirms an earlier observation that the plate behaves like a long rectangular strip as the width of the plate in the radial direction becomes small.

Natural frequencies of circumferentially truncated sector plates with simply supported straight edges

The classical Rayleigh-Ritz method in conjunction with suitable co-ordinate transformations is found to be effective for accurate estimation of natural frequencies of circumferentially truncated circular sector plates with simply supported straight edges. Numerical results are obtained for all the nine combinations of clamped, simply supported and free boundary conditions at the circular edges and presented in the form of graphs. The analysis confirms an earlier observation that the plate behaves like a long rectangular strip as the width of the plate in the radial direction becomes small.

Effect of membrane action on the plastic collapse load of circular orthotropic slabs with fixed edges

In the case of reinforced concrete slabs fixed at the boundaries, considerable enhancement in the load carrying capacity takes place due to compressive membrane action. In this paper a method is presented to analyse the effects of membrane action in fixed orthotropic circular slabs, carrying uniformly distributed loads. Depending on the radial moment capacity being greater or less than the circumferential moment capacity, two cases of orthotropy have been considered. Numerical results are worked out for certain assumed physical parameters and for different coefficients of orthotropy. Variations of load and bending moments with the central deflection are presented.

Free vibration of rectangular plates of arbitrary thickness with one or more edges clamped

Frequencies of free vibration of rectangular plates of arbitrary thickness, with different support conditions, are calculated by using the Method of Initial Functions (MIF), proposed by Vlasov. Sixth and fourth order MIF theories are used for the solution. Numerical results are presented for three square plates for three thickness ratios. The support conditions considered are (i) three sides simply supported and one side clamped, (ii) two opposite sides simply supported and the other two sides clamped and (iii) all sides clamped. It is found that the results produced by the MIF method are in fair agreement with those obtained by using other methods. The classical theory gives overestimates of the frequencies and the departures from the MIF results increase for higher modes and larger thickness ratios.

The effects of habitat edges and trampling intensity on vegetation in urban forests

Although changes in urban forest vegetation have been documented in previous Finnish studies, the reasons for these changes have not been studied explicitly. Especially, the consequences of forest fragmentation, i.e. the fact that forest edges receive more solar radiation, wind and air-borne nutrients than interiors have been ignored. In order to limit the change in urban forest vegetation we need to know why it occurs. Therefore, the effects of edges and recreational use of urban forests on vegetation were investigated together in this thesis to reveal the relative strengths of these effects and to provide recommendations for forest management. Data were collected in the greater Helsinki area (in the cities of Helsinki, Vantaa and Espoo, and in the municipalities of Sipoo and Tuusula) and in the Lahti region (in the city of Lahti and in the municipality of Hollola) by means of systematic and randomized vegetation and soil sampling and tree measurements. Sample plots were placed from the forest edges to the interiors to investigate the effects of forest edges, and on paths of different levels of wear and off these paths to investigate the effects of trampling. The natural vegetation of mesic and sub-xeric forest site types studied was sensitive both to the effects of the edge and to trampling. The abundances of dwarf shrubs and bryophytes decreased, while light- and nitrogen-demanding herbs and grasses - and especially Sorbus aucuparia – were favoured at the edges and next to the paths. Results indicated that typical forest site types at the edges are changing toward more nitrophilic vegetation communities. Covers of the most abundant forest species decreased considerably – even tens of percentages – from interiors to the edges indicating strong edge effects. These effects penetrated at least up to 50 m from the forest edges into the interiors, especially at south to west facing open edges. The effects of trampling were pronounced on paths and even low levels of trampling decreased the abundances of certain species considerably. The effects of trampling extended up to 8 m from path edges. Results showed that the fragmentation of urban forest remnants into small and narrow patches should be avoided in order to maintain natural forest understorey vegetation in the urban setting. Thus, urban forest fragments left within urban development should be at least 3 ha in size, and as circular as possible. Where the preservation of representative original forest interior vegetation is a management aim, closed edges with conifers can act as an effective barrier against solar radiation, wind and urban load, thereby restricting the effects of the edge. Tree volume at the edge should be at least 225-250 m3 ha-1 and the proportion of conifers (especially spruce) 80% or more of the tree species composition. Closed, spruce-dominated edges may also prevent the excessive growth of S. aucuparia saplings at urban forest edges. In addition, closed edges may guide people’s movements to the maintained paths, thus preventing the spontaneous creation of dense path networks. In urban areas the effects of edges and trampling on biodiversity may be considerable, and are important to consider when the aim of management is to prevent the development of homogeneous herb-grass dominated vegetation communities, as was observed at the investigated edges.

On the Structure of Contractible Edges in k-connected Partial k-trees

Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2-connected graph. We also show that there are at least |V(G)| + k - 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees.

Non-contractible non-edges in 2-connected graphs

Any pair of non-adjacent vertices forms a non-edge in a graph. Contraction of a non-edge merges two non-adjacent vertices into a single vertex such that the edges incident on the non-adjacent vertices are now incident on the merged vertex. In this paper, we consider simple connected graphs, hence parallel edges are removed after contraction. The minimum number of nodes whose removal disconnects the graph is the connectivity of the graph. We say a graph is k-connected, if its connectivity is k. A non-edge in a k-connected graph is contractible if its contraction does not result in a graph of lower connectivity. Otherwise the non-edge is non-contractible. We focus our study on non-contractible non-edges in 2-connected graphs. We show that cycles are the only 2-connected graphs in which every non-edge is non-contractible. (C) 2010 Elsevier B.V. All rights reserved.

Breakup of drops around the edges of Rushton turbine

A detailed experimental and simulation study has been carried out in the present work to understand drop breakup in regions around the edge of the Rushton turbine in agitated vessels. The effect of impeller speed, impeller size, interfacial tension, and the viscosities of the two phases is studied on drop breakup through their effect on dmax, the size of the largest drop in the system, and the whole size distribution. The measurements were carried out using Galai particle size analyser and optical microscope. Experimental analysis shows that the dmax, maximum stable drop diameter varies with impeller tip velocity to the power -1. The variation of dmax with interfacial tension is studied using different surfactants. The effect of viscosity ratio, achieved by changing the dispersed phase viscosity, on dmax is captured. For the same dmax values obtained from two different dispersed phases show that the wider drop size distribution is observed for higher dispersed phase viscosity.

Detection of edges from projections

In a number of applications of computerized tomography, the ultimate goal is to detect and characterize objects within a cross section. Detection of edges of different contrast regions yields the required information. The problem of detecting edges from projection data is addressed. It is shown that the class of linear edge detection operators used on images can be used for detection of edges directly from projection data. This not only reduces the computational burden but also avoids the difficulties of postprocessing a reconstructed image. This is accomplished by a convolution backprojection operation. For example, with the Marr-Hildreth edge detection operator, the filtering function that is to be used on the projection data is the Radon transform of the Laplacian of the 2-D Gaussian function which is combined with the reconstruction filter. Simulation results showing the efficacy of the proposed method and a comparison with edges detected from the reconstructed image are presented

Linear dynamics and active control of an elastically supported travelling string

In this paper, the linear dynamics and active control of a string travelling with uniform velocity is presented. Discrete elastic supports are introduced along the length of the string. Finite element formulation is adopted to obtain the governing equations of motion. The velocity of translation introduces gyroscopic terms in the system equations. The effect of translation and the discrete elastic supports on the free vibration solution is studied. The solution is utilized in actively controlling the string vibrations due to an initial disturbance. The control, affected in modal space, is optimal with respect to a quadratic performance index. Numerical results are presented to demonstrate the effectiveness of the control strategy in regulating the travelling string vibrations.