single layer plate theories applied to the flexural vibration of completely free thick laminates two different refined single layer plate theories are used to analyze the flexural vibration of thick symmetrically laminated rectangular plates with all edges free both theories account for parabolic distribution of the transverse shear strains through the thickness of the plate the first theory includes the transverse normal effect and employs a three dimensional constitutive law whereas the second neglects this effect and employs plane stress reduced stiffnesses numerical results are obtained by using the ritz method characteristic orthogonal polynomials are used in the series representing the displacement functions convergence studies are shown to verify the accuracy of the approximate method the accuracy of the results is further assessed by comparison with results from three dimensional finite element analyses numerical results are presented for various plates including cross ply and angle ply laminated plates the effect of transverse shear and normal deformation on the plate natural frequencies is illustrated by comparing the frequencies of the two higher order theories and the classical plate theory for decreasing length to thickness ratio

a variable kinematic ritz formulation for vibration study of quadrilateral plates with arbitrary thickness a new variable kinematic ritz method applied to free vibration analysis of arbitrary quadrilateral thin and thick isotropic plates is presented carrera s unified formulation and the versatile pb 2 ritz method are properly combined to build a powerful yet simple modeling and solution framework the proposed technique allows to generate arbitrarily accurate ritz solutions from a large variety of refined two dimensional plate theories by expanding so called ritz fundamental nuclei of the plate mass and stiffness matrices theoretical development of the present methodology is described in detail convergence and accuracy of the method are examined through several examples on thin moderately thick and very thick plates of rectangular skew trapezoidal and general quadrilateral shapes with an arbitrary combination of clamped free and simply supported edges present results are compared with existing three dimensional solutions from open literature maximum and average differences of various higher order plate theories and three dimensional results are also presented with the aim of providing useful guidelines on the choice of appropriate plate theory to get a desired accuracy on frequency parameters

vibration of thick skew plates based on mindlin shear deformation plate theory the free flexural vibration of thick skew plates is considered the total energy functional is derived based on the first order shear deformation plate theory by mindlin the newly developed pb 2 rayleigh ritz method is applied to obtain the solution the method features the versatile pb 2 ritz functions for approximating the deflection and rotation functions which comprise the product of 1 a two dimensional polynomial function p 2 and 2 a basic function b the basic function is the key to ensure the automatic satisfaction of the geometric boundary conditions at the outset since it is formed from the product of the boundary equations raised to appropriate powers natural frequency parameters for skew mindlin plates with different combinations of boundary conditions are presented the solutions where possible are verified by comparisons with those available in the open literature the effect of the poisson ratio on vibration frequencies of plates with free edges is also investigated

relaxation methods for problems with strictly convex separable costs and linear constraints we consider the minimization problem with strictly convex possibly nondifferentiable separable cost and linear constraints the dual of this problem is an unconstrained minimization problem with differentiable cost which is well suited for solution by parallel methods based on gauss seidel relaxation we show that these methods yield the optimal primal solution and under additional assumptions an optimal dual solution to do this it is necessary to extend the classical gauss seidel convergence results because the dual cost may not be strictly convex and may have unbounded level sets

generalized scalings satisfying linear equations abstract we unify and generalize a broad class of problems referred in the literature as scaling problems by extending the applicability of a formulation suggested recently by bapat and raghavan specifically let a r n b r m and c r m n be given where a is strictly positive a c scaling of the vector a is defined to be a vector a r n with a i a i m k 1 u c kj k for some strictly positive vector u r m the problem of finding a c scaling of the vector a which satisfies the linear system cx b will be called a generalized scaling problem in this paper it is shown that previously studied matrix scaling problems e g finding scalings with prespecified row sums and column sums or finding scalings with row sums equaling the corresponding column sums or finding scalings of multidimensional matrices with prespecified margins are special instances of generalized scaling problems generalized scaling problems are reduced to convex optimization problems and the reduction is used to characterize solutions to develop necessary and sufficient conditions for their existence to establish uniqueness results and to characterize approximate solutions

an axiomatic approach to proportionality between matrices given a matrix p 0 what does it mean to say that a matrix f of the same dimension whose row and column sums must fall between specific limits is proportional to p this paper gives an axiomatic solution to this question in two distinct contexts first for any real allocation matrix f second for any integer constrained apportionment matrix f in the case of f real the solution turns out to coincide with what has been variously called biproportional scaling and diagonal equivalence and has been much used in econometrics and statistics in the case of f integer the problem arises in the simultaneous apportionment of seats to regions and to parties and also in the rounding of tables of census data

universal unification of life death evolution post evolution and extinction abstract a general law of universal maximally rapid entropy decrease unifies metabolism mortality and evolution yields their quantitatively accurate universal laws and singularities proves that accidental premature mortality is relatively small and quantifies it predicts that evolution may be continuous or followed by weak and strong spurts leads to a definition of post evolution and considers its probability suggests an absolute instability of technological terrestrial and extra terrestrial civilizations and allows for computer simulations of mortality and evolution

the historical bases of the rayleigh and ritz methods abstract rayleigh s classical book theory of sound was first published in 1877 in it are many examples of calculating fundamental natural frequencies of free vibration of continuum systems strings bars beams membranes plates by assuming the mode shape and setting the maximum values of potential and kinetic energy in a cycle of motion equal to each other this procedure is well known as rayleigh s method in 1908 ritz laid out his famous method for determining frequencies and mode shapes choosing multiple admissible displacement functions and minimizing a functional involving both potential and kinetic energies he then demonstrated it in detail in 1909 for the completely free square plate in 1911 rayleigh wrote a paper congratulating ritz on his work but stating that he himself had used ritz s method in many places in his book and in another publication subsequently hundreds of research articles and many books have appeared which use the method some calling it the ritz method and others the rayleigh ritz method the present article examines the method in detail as ritz presented it and as rayleigh claimed to have used it it concludes that although rayleigh did solve a few problems which involved minimization of a frequency these solutions were not by the straightforward direct method presented by ritz and used subsequently by others therefore rayleigh s name should not be attached to the method

flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables a set of characteristic orthogonal polynomials in two variables is used as deflection functions to obtain the natural frequencies and mode shapes of polygonal plates by the rayleigh ritz method the set of orthogonal polynomials is constructed by employing the gram schmidt orthogonalization procedure the first six natural frequencies are numerically evaluated for triangular plates of different configurations by using these orthogonal functions and the corresponding mode shapes are plotted results are compared with those obtained previously by other methods

three dimensional vibration analysis of thick rectangular plates using chebyshev polynomial and ritz method this paper describes a method for free vibration analysis of rectangular plates with any thicknesses which range from thin moderately thick to very thick plates it utilises admissible functions comprising the chebyshev polynomials multiplied by a boundary function the analysis is based on a linear small strain three dimensional elasticity theory the proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions a very simple and general programme has been compiled for the purpose for a plate with geometric symmetry the vibration modes can be classified into symmetric and antisymmetric ones in that direction in such a case the computational cost can be greatly reduced while maintaining the same level of accuracy convergence studies and comparison have been carried out taking square plates with four simply supported edges as examples it is shown that the present method enables rapid convergence stable numerical operation and very high computational accuracy parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail with respect to different thickness side ratios aspect ratios and poissons ratios these results may serve as benchmark solutions for validating approximate two dimensional theories and new computational techniques in future 2002 elsevier science ltd all rights reserved

a comparative study of algorithms for matrix balancing the problem of adjusting the entries of a large matrix to satisfy prior consistency requirements occurs in economics urban planning statistics demography and stochastic modeling these problems are called matrix balancing problems we describe five applications of matrix balancing and compare the algorithmic and computational performance of balancing procedures that represent the two primary approaches for matrix balancing matrix scaling and nonlinear optimization the algorithms we study are the ras algorithm a diagonal similarity scaling algorithm and a truncated newton algorithm for network optimization we present results from computational experiments with large scale problems based on producing consistent estimates of social accounting matrices for developing countries

vibrations of skewed cantilevered triangular trapezoidal and parallelogram mindlin plates with considering corner stress singularities based on the mindlin shear deformation plate theory a method is presented for determining natural frequencies of skewed cantilevered triangular trapezoidal and parallelogram plates using the ritz method considering the effects of stress singularities at the clamped re entrant corner the admissible displacement functions include polynomials and corner functions the admissible polynomials form a mathematically complete set and guarantee the solution convergent to the exact frequencies when sufficient terms are used the corner functions properly account for the singularities of moments and shear forces at the re entrant corner and accelerate the convergence of the solution detailed convergence studies are carried out for plates of various shapes to elucidate the positive effects of corner functions on the accuracy of the solution the results obtained herein are compared with those obtained by other investigators to demonstrate the validity and accuracy of the solution copyright 2005 john wiley sons ltd

analytical 3 d p element for quadrilateral plates part 1 thick isotropic plate structures an analytical three dimensional 3 d p version element for the vibration analysis of arbitrary quadrilateral thick plates is presented with the additional hierarchical shape functions and analytically integrated element matrices the computed accuracy is considerably improved the computed natural frequencies of cantilever and simply supported square plates show that the convergence rate of the present element is very fast with respect to the number of hierarchical terms and it can predict very accurate modes the element is applicable to the free vibration analysis of quadrilateral polygonal plates as well as 3 d space structures the continuous wavelet transform cwt is applied for the identification of damping ratios based on the rayleigh damping model the damped vibration response is obtained a simple experiment is performed to verify the predicted vibration responses the results show that the proposed element is also efficient for the vibration response analysis of plates

free vibration and stability of thick elastic plates subjected to in plane forces abstract natural frequencies and buckling loads of a simply supported thick plate subjected to in plane initial tensile and or compressive forces are analysed by using the method of power series expansion of displacement components a set of fundamental dynamic equations of a two dimensional higher order plate theory is derived through hamilton s principle several sets of truncated approximate theories are applied to solve the eigenvalue problems of a thick plate in order to assure the accuracy of the present theory convergence properties of the minimum natural frequency and the buckling load for the out of plane problem of a simply supported square plate are examined in detail it is noticed that the present approximate theories can predict the frequencies and buckling loads of an extremely thick plate more accurately compared to other refined theories and classical plate theory

algorithms for proportional matrices in reals and integers letr be the set of nonnegative matrices whose row and column sums fall between specific limits and whose entries sum to some fixedh 0 closely related axiomatic approaches have been developed to ascribe meanings to the statements the real matrixf r and the integer matrixa r are proportional to a given matrixp 0

on the three dimensional vibrations of the cantilevered rectangular parallelepiped a solution is presented for the three dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces this problem apparently is previously unsolved in the published literature the ritz method is used with displacements assumed in the form of algebraic polynomials convergence is studied numerical results are given for the first five frequencies of each of the four symmetry classes of vibration for five thick parallelepiped configurations including the cube contour plots are exhibited for the modal displacements of the cube the effects of varying poisson s ratio are also observed

a hierarchical functions set for predicting very high order plate bending modes with any boundary conditions in this paper a new hierarchical functions set is proposed to predict flexural motion of plate like structures in the medium frequency range this functions set is built from trigonometric functions instead of polynomials as classically encountered in the literature it is shown that such a trigonometric set presents all the advantages of a classical hierarchical polynomials set and additional ones which are of interest if very high order functions are intended to be used it is stated that this new trigonometric set can be used at very high orders up to 2048 without taking care of computer round off errors while the polynomials set fail at order 46 because of the limited numerical dynamics of computers this trigonometric set can be easily implemented on a computer it does not require quadruple precision pre computed arrays only a very low number which does not depend on the function order of basic operations is needed when calling such functions moreover it is shown that this trigonometric set presents a better convergence rate than polynomials when predicting high order natural flexural modes of rectangular plates with any boundary conditions

scalings of matrices which have prespecified row sums and column sums via optimization abstract the problem of scaling a matrix so that it has given row and column sums is transformed into a convex minimization problem in particular we use this transformation to characterize the existence of such scaling or corresponding approximations we obtain new results as well as new streamlined proofs of known results

analysis of thick isotropic and cross ply laminated plates by radial basis functions and a unified formulation in this paper we combine carrera s unified formulation and a radial basis function collocation technique for predicting the static deformations and free vibration behavior of thin and thick isotropic and cross ply laminated plates through numerical experiments the capability and efficiency of this collocation technique for static and vibration problems are demonstrated and the numerical accuracy and convergence are thoughtfully examined

matrix scaling entropy minimization and conjugate duality i existence conditions abstract we have four primary objectives in this paper first we introduce a problem called truncated matrix scaling that generalizes two well studied matrix scaling problems diagonal similarity scaling and fixed row column equivalence scaling second we derive necessary and sufficient conditions for the attainment of the infimum in the fenchel dual of a class of convex optimization problems third we show that existence of a solution for truncated matrix scaling is equivalent to the attainment of the infimum in a corresponding dual optimization problem we thereby derive necessary and sufficient conditions for the existence of a solution for truncated matrix scaling fourth we derive known existence conditions for similarity and equivalence scaling from the conditions for truncated matrix scaling

guidelines and recommendations to construct theories for metallic and composite plates this work has evaluated the refinement of some classical theories such as the kirchhoff and reissner mindlin theories adding generalized displacement variables up to fourth order to the taylor type expansion in the thickness plate direction isotropic orthotropic and laminated plates have been analyzed varying the thickness ratio orthotropic ratio and stacking sequence of the layout higher order theories have been implemented according to the compact scheme known as the carrera unified formulation the results have been restricted to simply supported orthotropic plates subjected to harmonic distributions of transverse pressure for which closed form solutions are available for a given plate problem isotropic orthotropic or laminated the effectiveness of each employed generalized displacement variable has been established comparing the error obtained accounting for and removing the variable in the plate governing equations a number of theories have therefore been constructed imposing a given error with respect to the available best results guidelines and recommendations that are focused on the proper selection of the displacement variables that have to be retained in refined plate theories are then furnished it has been found that the terms that have to be used according to a given error vary from problem to problem but they also vary when the variable that has to be evaluated displacement stress components is changed diagrams errors in terms of geometrical and orthotopic ratios and graphical schemes have been built to establish the appropriate theories with respect to the data of the problem under consideration

an extension of a theorem of darroch and ratcliff in loglinear models and its application to scaling multidimensional matrices abstract let c c ij be an m n matrix with real entries let b be any nonzero m vector let k c b 0 be bounded let x x 1 x 2 x n y y 1 y 2 y n be two nonnegative vectors with y k and x j 0 y j 0 for any coordinate j then it is shown that there exists a k and positive numbers z 1 z 2 z m such that j x j m i 1 z c ij i for all j th theorem slightly generalizes a theorem of darroch and ratcliff in loglinear models with a completely different proof technique the proof relies on an extension of a topological theorem of kronecker to set valued maps and the duality theorem of linear programming many theorems in scaling of matrices and multidimensional matrices are direct consequences of this theorem the main idea is to associate a suitable zero one matrix of transportation with any multidimensional matrix some motivations for scaling applications are also discussed

relaxation methods for network flow problems with convex arc costs we consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs for the case where all arc costs are strictly convex we study the convergence of a dual gauss seide type relaxation method that is well suited for parallel computation we then extend this method to the case where some of the arc costs are linear as a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in bertsekas 1 and bertsekas and tseng 2

a study of transverse normal stress effect on vibration of multilayered plates and shells this paper evaluates transverse normal stress zzeffect on vibration of multilayered structures to this purpose a mixed plate model initially introduced by toledano and murakami has been extended to dynamics analysis of double curved shells these models allow both continuous interlaminar transverse shear and normal stresses as well as the zigzag form of the displacement distribution in the shell thickness directions to be modelled governing equations have been derived by employing a reissner s mixed theorem classical models on the basis of standard displacement formulations have been considered for comparison purposes the evaluations of transverse stress effects have been conducted by comparing constant linear and higher order distributions of transverse displacement components in the plate thickness directions free vibrational response of layered simply supported plates cylindrical and spherical shells made of isotropic as well as orthotropic layers has been analyzed the numerical investigation carried out and comparison with earlier results has concluded that 1 the possibility of describing interlaminar continuous transverse normal stress makes the mixed theories more attractive with respect to other available modelling 2 any refinements of classical models are meaningless unless the effects of interlaminar continuous transverse shear and normal stresses are both taken into account in a multilayered shell theory

on the scaling of multidimensional matrices abstract elementary proofs are given for theorems of bapat and raghavan on the scaling of nonnegative multidimensional matrices theorems of sinkhorn and of brualdi parter and schneider are derived as corollaries for positive two dimensional matrices hilbert s projective metric and a theorem of g birkhoff are used to prove that sinkhorn s original iterative procedure converges geometrically the ratio of convergence is estimated from the given data

max balancing weighted directed graphs and matrix scaling a weighted directed graph g is a triple v a g where v a is a directed graph and g is an arbitrary real valued function defined on the arc set a let g be a strongly connected simple weighted directed graph we say that g is max balanced if for every nontrivial subset of the vertices w the maximum weight over arcs leaving w equals the maximum weight over arcs entering w we show that there exists a up to an additive constant unique potential pv for v v such that v a gp is max balanced where gap pu ga pv for a u v a we describe an o v 2 a algorithm for computing p using an algorithm for computing the maximum cycle mean of g finally we apply our principal result to the similarity scaling of nonnegative matrices

matrix scaling entropy minimization and conjugate duality ii the dual problem we present a matrix scaling problem calledtruncated scaling and describe applications arising in economics urban planning and statistics we associate a dual pair of convex optimization problems to the scaling problem and prove that the existence of a solution for the truncated scaling problem is characterized by the attainment of the infimum in the dual optimization problem we show that optimization problems used by bacharach 1970 bachem and korte 1979 eaves et al 1985 marshall and olkin 1968 and rothblum and schneider 1989 to study scaling problems can be derived as special cases of the dual problem for truncated scaling we present computational results for solving truncated scaling problems using dual coordinate descent thereby showing that truncated scaling provides a framework for modeling and solving large scale matrix scaling problems

three dimensional free vibration analysis of isotropic rectangular plates using the b spline ritz method abstract this paper presents three dimensional free vibration analysis of isotropic rectangular plates with any thicknesses and arbitrary boundary conditions using the b spline ritz method based on the theory of elasticity the proposed method is formulated by the ritz procedure with a triplicate series of b spline functions as amplitude displacement components the geometric boundary conditions are numerically satisfied by the method of artificial spring to demonstrate the convergence and accuracy of the present method several examples with various boundary conditions are solved and the results are compared with other published solutions by exact and other numerical methods based on the theory of elasticity and various plate theories rapid stable convergences as well as high accuracy are obtained by the present method the effects of geometric parameters on the vibrational behavior of cantilevered rectangular plates are also investigated the results reported here may serve as benchmark data for finite element solutions and future developments in numerical methods

dual ascent methods for problems with strictly convex costs and linear constraints a unified approach consider problems of the form text p qquad min f x ex geqq b where f is a strictly convex possibly nondifferentiable function and e and b are respectively a matrix and a vector a popular method for solving special cases of p e g network flow entropy maximization quadratic program is to dualize the constraints ex geqq b to obtain a differentiable maximization problem and then apply an iterative ascent method to solve it this method is simple and can exploit sparsity thus making it ideal for large scale optimization and in certain cases for parallel computation despite its simplicity however convergence of this method has been shown only under certain very restrictive conditions and only for certain special cases of p in this paper a block coordinate ascent method is presented for solving p that contains as special cases both dual coordinate ascent methods and dual gradient methods it is shown under certain mild assumptions on f and p that this method

interval constrained matrix balancing abstract we define two interval constrained matrix balancing problems the models encompass several known formulations of problems that appear in regional input output analysis estimation of traffic over transportation and telecommunications networks and estimation of social accounting matrices we develop primal dual row action algorithms for the solution of these models and establish their convergence both algorithms generalize existing scaling algorithms for equality constrained matrix balancing problems one of the algorithms is a generalization of the well known procedure for matrix balancing called ras that can handle the range constraints and is thus called range ras the structure of the problem makes the algorithm suitable for implementation on massively parallel computers details of our implementation on a connection machine cm 2 are given numerical results for the solution of problems of size up to 500 500 are given and the performance of the algorithm is compared with that of ras