three dimensional vibration analysis of rectangular plates based on differential quadrature method abstract this paper presents the formulation and numerical analysis of the three dimensional elasticity plate model using the differential quadrature dq method the governing equations in terms of displacement stress displacement relations and boundary conditions for the three dimensional plate model are first presented these equations are then normalized and discretized using the dq procedure example problems on the free vibration of rectangular plates with generic boundary conditions are selected two types of mesh pattern the uniform mesh pattern and the cosine mesh pattern were used and their convergence characteristics were studied the cosine mesh was then chosen as the better mesh pattern pertaining to the problems solved the solutions calculated using the cosine mesh pattern were then compared where possible with the exact or the numerical or the analytical solutions it is found that the differential quadrature method yields accurate results for the plate problems under the current investigation

an extended operator trigonometry the operator trigonometry of symmetric positive definite spd matrices is extended to arbitrary invertible matrices a and to arbitrary invertible bounded operators a on a hilbert space some background and motivation for these results is provided

positivity properties of viscosity solutions of a degenerate parabolic equation we consider the problem i u t u u u 2 in r n r u x 0 u 0 x in r n where is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on r n

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

on a class of forward backward parabolic equations existence of solutions abstract we study the initial boundary value problem u t u x x u t x x in 0 t u u t 0 in 0 t u u 0 in 0 where is an interval and u 0 is a nonnegative radon measure on the map is increasing in 0 and decreasing in for some 0 and satisfies 0 0 the regularizing map is increasing and bounded we prove existence of suitably defined nonnegative radon measure valued solutions the solution class is natural since smooth initial data may generate solutions which become measure valued after finite time

some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor abstract the study of the inefficiency of the ordinary least squares estimator olse with one regressor by watson 1951 required a lower bound for the efficiency defined as the ratio of the variance of the best linear unbiased estimator blue to the variance of the olse such a lower bound was provided by the cassels inequality 1951 which we note is closely related to five other inequalities including the well known inequality usually attributed to kantorovich 1948 but which was established already by frucht 1943 the main purpose in this paper is to show how these six inequalities are related with a historical perspective we present some proofs and conclude that all six inequalities are essentially equivalent in the sense that any one inequality implies the other five we identify conditions for equality in each inequality and present the six continuous integral analogues we end the paper with english translations of the seminal papers by frucht 1943 and schweitzer 1914 respectively from the spanish and hungarian and a fairly extensive bibliography

self similar intermediate asymptotics for a degenerate parabolic filtration absorption equation the equation tu u xx2u c 1 xu 2 is known in literature as a qualitative mathematical model of some biological phenomena here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock therefore we refer to this equation as a filtration absorption equation a family of self similar solutions to this equation is constructed numerical investigation of the evolution of non self similar solutions to the cauchy problems having compactly supported initial conditions is performed numerical experiments indicate that the self similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state identical zero an open problem caused by the nonuniqueness of the solution of the cauchy problem is discussed

analytical three dimensional elasticity solutions to some plate problems and some observations on mindlin s plate theory abstract solutions are obtained using the exact three dimensional theory of elasticity to i the eigenvalue problem or buckling under biaxial compression or the free lateral vibration of a simply supported rectangular plate with orthotropic stress strain properties and ii the static response of the same plate to a lateral load that varies sinusoidally in two directions the eigenvalue in problem i and the lateral deflections of both the surface and middle plane of the plate as well as the bending strains in problem ii are obtained in the form of series expansions in even powers of the plate thickness exact algebraic expressions are presented for the first two coefficients in the case of orthotropic plates additional coefficients can be obtained if required and are given for the much simpler case of isotropic plates in all cases the first term agrees with classical plate theory the solutions are compared with those obtained from mindlin s plate theory in neither problem is it found to be possible in general to choose values for mindlin s effective shear moduli to make the mindlin solution agree with the first two terms of the exact solution there are however two exceptions to this namely a restricted class of orthotropic materials embracing all isotropic ones in which the elastic constants satisfy a certain condition and the case of cylindrical bending when the mindlin plate reduces to a timoshenko beam of wide rectangular cross section in both these exceptional cases appropriate values for the effective shear moduli are obtained

three dimensional vibration analysis of a cantilevered parallelepiped exact and approximate solutions the paper investigates the hypothetical assumption of neglecting transverse normal stress in vibration analysis for cantilevered thick plates and rectangular parallelepiped the analysis solves the three dimensional elasticity energy functional including as well as excluding transverse normal stress and obtains free vibration solutions for a cantilevered parallelepiped although it is widely accepted the omission of transverse normal stress is well justified in kirchhoff love thin plate theory and higher order thick plate models the transverse normal stress effects and thickness extent to which the thick plate models apply as the thickness increases are practically unknown the inconsistency of assuming constant transverse normal displacement through thickness for thick plate models is also addressed the paper concludes that for a rectangular parallelepiped with thickness exceeding a certain limit there is considerable discrepancy if transverse normal stress is neglected

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

numerical aspects for free vibration of thick plates part i formulation and verification abstract this paper investigates the numerical aspects for free vibration analysis of a class of thick relative thickness ratio t b 0 2 shear deformable rectangular plates with all possible combinations of edge conditions these thick plate vibration frequencies most probably cannot be accurately determined using the classical thin plate theory or the mindlin plate theory the effect of transverse shear strain has been considered by using a higher order plate theory without the need for a shear correction factor unlike some domain discretization methods this analysis assumes a single global element for the entire plate the strain and kinetic energy integrals of the global plate element are formulated and the energy functional is minimized to yield the natural frequencies a set of boundary compliant admissible shape functions is developed to obtain solutions which are unattainable using the classical trigonometric admissible functions the numerical efficiency of symmetry classification is investigated and on certain computational platforms more than 80 of the execution time can be saved while maintaining the same level of accuracy

free vibration of thick layered rectangular plates with point supports by finite layer method in this paper the free vibration of thick isotropic and laminated composite rectangular plates with point supports is analyzed by the finite layer method a new set of two dimensional basic functions which satisfies the kinematic boundary conditions at the edges of the plate and the zero displacement conditions at point supports is developed to describe the variation of three dimensional displacements in the plane of a thin finite layer one dimensional linear or quadratic shape functions are adopted to describe the variation of the displacements through the thickness layer the governing eigenvalue equation of the plate is derived via the conventional displacement method numerical results for the three dimensional vibration of rectangular plates with point supports are presented herein for the first time the eigenfrequencies of simply supported rectangular plates with a central point support are studied in detail by considering the variations of aspect ratio side to thickness ratio properties of materials number of laminates and stacking sequences comparison with known thin plate and mindlin plate solutions is carried out to verify the applicability and accuracy of the proposed method

accretive operators and cassels inequality abstract let a b 0 and let z m n r such that z lies into the operator ball of diameter ai bi then for all positive definite a m n r tr az a b 2 ab tr az several related reverse inequalities are considered extending some known results

natural vibration of rectangular plates considered as tridimensional solids a variational method developed by the authors named wem for one and two dimensional boundary value problems is extended to three dimensions the availability of a classical solution obtained through the inverse method which is briefly included in an appendix allows the confirmation of the exactness of the alternative solution herein presented a prismatic solid supported by shear diaphragms at four consecutive faces is analysed in particular the numerical values of natural transversal frequencies enable the evaluation of the degree of approximation involved in using mindlin s theory for thick plates also comparison is made with results from a three dimensional ritz formulation

some further matrix extensions of the cauchy schwarz and kantorovich inequalities with some statistical applications abstract the well known cauchy schwarz and kantorovich inequalities may be expressed in terms of vectors and a positive definite matrix we consider what happens to these inequalities when the vectors are replaced by matrices the positive definite matrix is allowed to be positive semidefinite singular and the usual inequalities are replaced by lowner partial orderings some examples in the context of linear statistical models are presented

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

three dimensional vibration of rectangular plates variance of simple support conditions and influence of in plane inertia plate models developed on the basis of the three dimensional elasticity theory are crucial to the correct solutions in dynamics problems in a three dimensional setting there exist numerous possible definitions of boundary conditions this paper attempts to identify the variance of three dimensional simple support conditions existing in practice the hard simple support condition assumes vanishing normal stresses at the edges and zero transverse and tangential displacement at the peripheries the soft simple support on the other hand imposes zero transverse displacement with vanishing normal and shear stresses at the boundaries to quantify the relative effects of each boundary condition on the vibratory responses of thick plates a global three dimensional ritz formulation is employed for analysis this technique uses sets of finite polynomial series in cartesian co ordinates to approximate the global normal mode variations of the plate these separable one dimensional polynomials are orthogonally generated within the plate domain such that all the essential edge conditions are identically satisfied the accuracy of the proposed three dimensional ritz method is assured from the convergence and comparison studies numerical experiments have been conducted to study firstly the effects of geometric parameters on the overall normal mode characteristics of simply supported plates and secondly the effects of in plane inertia on the vibration frequencies of plates with different thicknesses the three dimensional deformed mode shapes for selected cases have also been computed these have served to describe more vividly the normal mode characteristics of different types of simply supported rectangular plates

three dimensional vibration of rectangular plates effects of thickness and edge constraints abstract a continuum three dimensional ritz formulation is presented for the free vibration analysis of homogeneous isotropic thick rectangular plates the method is applied specifically to investigate the effects of boundary constraints and thickness ratios on the vibration responses of these plates a general switching of transverse vibratory motions to surface parallel modes at the higher thickness ratio is manifested vividly through three dimensional deformed mode shape plots the present formulation is based on the linear three dimensional small strain elasticity energy principle in the solution process sets of self generating polynomial shape functions are used to derive the governing eigenvalue equation in essence these functions comprise a class of orthogonal polynomial functions and a basic function which is chosen to satisfy the essential geometric boundary conditions at the outset they are found very suitable for the computationally intensive task of 3 d modelling due to the ease of generation and numerical implementation several numerical examples have been computed to demonstrate the accuracy and efficiency of the method the influences of plate thickness and boundary constraints on the overall vibratory characteristics of thick plates are also discussed in detail

three dimensional elasticity solutions for free vibrations of rectangular plates by the differential quadrature method this paper presents accurate three dimensional elasticity solutions for free vibrations of six types of plates having free lateral surfaces two opposite sides simply supported and two other sides having combinations of simply supported clamped and free boundary conditions the solution methodology adopted in the present work first reduces the spatial variables in the governing elasticity equations to two by using displacement functions in the forms which satisfy the boundary conditions of the assumed simply supported sides the eigenvalue equations are then formulated from the reduced governing equations and the boundary conditions via the differential quadrature method the numerical results include the first nine natural frequencies of the six plate configurations for combinations of three aspect ratios and two thickness ratios these results are supported by appropriate convergence studies and comparisons with the results of other authors

convexity conditions of kantorovich function and related semi infinite linear matrix inequalities the kantorovich function x tax x ta 1x where a is a positive definite matrix is not convex in general from a matrix or convex analysis point of view it is interesting to address the question when is this function convex in this paper we prove that the 2 dimensional kantorovich function is convex if and only if the condition number of its matrix is less than or equal to 3 22 thus the convexity of the function with two variables can be completely characterized by the condition number the upper bound 3 22 is turned out to be a necessary condition for the convexity of the kantorovich function in any finite dimensional spaces we also point out that when the condition number of the matrix which can be any dimensional is less than or equal to 5 26 the kantorovich function is convex furthermore we prove that this general sufficient convexity condition can be improved to 2 3 in 3 dimensional space our analysis shows that the convexity of the function is closely related to some modern optimization topics such as the semi infinite linear matrix inequality or robust positive semi definiteness of symmetric matrices in fact our main result for 3 dimensional cases has been proved by finding an explicit solution range to some semi infinite linear matrix inequalities

a converse to the jensen inequality its matrix extensions and inequalities for minors and eigenvalues we obtain a scalar inequality converse to the jensen inequality we also derive an operator converse to the jensen inequality as special cases we obtain inequalities similar to the kantorovich one as well as some operator generalizations of them using some exterior algebra we prove a generalization of the sylvester determinant theorem we also deduce some determinant analogs of the additive and multiplicative kantorovich inequalities

a method for discovering kantorovich type inequalities and a probabilistic interpretation abstract the kantorovich inequality z az z a 1z 1 n 2 4 1 n can be proved using convexity and elementary geometry this method of proof is generalized so that it can often yield the upper and lower bounds for f z az z bz z cz where a b c are commuting n x n hermitian matrices and z is a unit vector in c n among the examples is the mulholland smith inequality it is also shown that our method finds probability distributions giving f e e e its upper and lower bounds

degenerate regularization of forward backward parabolic equations the regularized problem we study a quasilinear parabolic equation of forward backward type in one space dimension under assumptions on the nonlinearity which hold for a number of important mathematical models for example the one dimensional perona malik equation using a degenerate pseudoparabolic regularization proposed in barenblatt et al siam j math anal 24 1414 1439 1993 which takes time delay effects into account we prove existence and uniqueness of positive solutions of the regularized problem in a space of radon measures we also study qualitative properties of such solutions in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the lebesgue measure in this respect the existence of a family of viscous entropy inequalities plays an important role

new inequalities of the kantorovich type for bounded linear operators in hilbert spaces some new inequalities of the kantorovich type are established they hold for larger classes of operators and subsets of complex numbers than considered before in the literature and provide refinements of the classical results in the case when the involved operator satisfies the usual conditions several new reverse inequalities for the numerical radius of a bounded linear operator are obtained as well

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

passage to the limit over small parameters in the viscous cahn hilliard equation abstract we study singular passage to the limit over different small parameters for the viscous cahn hilliard equation under weak growth assumptions on the nonlinearity a rigorous proof of convergence to solutions of either the cahn hilliard equation or of the allen cahn equation or of the sobolev equation depending on the choice of the parameter is provided we also study the singular limit of the cahn hilliard equation as the parameter in the fourth order term goes to zero in particular we show that a radon measure valued solution of the limiting ill posed problem can arise depending on the behavior of the nonlinearity at infinity

radon measure valued solutions for a class of quasilinear parabolic equations initial value problems for quasilinear parabolic equations having radon measures as initial data have been widely investigated looking for solutions which for positive times take values in some function space in contrast it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the radon measures of the initial data we call such solutions measure valued in contrast to function valued solutionspreviously considered in the literature we first show that there is a natural notion of measure valued solution of problem p below in spite of its nonlinear character a major consequence of our definition is that if the space dimension is greater than one the concentrated part of the solution with respect to the newtonian capacity is constant in time subsequently we prove that there exists exactly one solution of the problem such that the diffuse part with respect to the newtonian capacity of the singular part of the solution with respect to the lebesgue measure is concentrated for almost every positive time on the set where the regular part with respect to the lebesgue measure is large moreover using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time finally the regularity problem is addressed as we give conditions depending on the space dimension the initial data and the rate of convergence at infinity of the nonlinearity to ensure that the measure valued solution of problem p is in fact function valued

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

free vibrations of a simply supported rectangular plate an exact elasticity solution abstract an exact three dimensional solution for the free vibrations of simply supported rectangular plates of arbitrary thickness within the linear theory of elastodynamics is given in this paper the solution obtained in a semi inverse fashion as was the solution of the elastostatic problem for such plates satisfies all of the boundary conditions of the problem in a pointwise manner it is found that there are two types of modes of oscillation possible which are consistent with the kinematic assumptions made to find the semi inverse solution other modes of oscillation may exist in the three dimensional theory of elastodynamics for such plates but our kinematic assumptions would not be consistent with such modes the two types of modes found are analogous to the flexural modes of classical plate theory and the thickness twist modes here called breathing modes of mindlin plate theory some numerical results are given which indicate that the predictions of mindlin plates are uncannily good approximations to the flexural frequencies given by the present three dimensional analysis even for very thick plates however the predictions of mindlin plate theory for the thickness twist or breathing frequencies are not nearly so good these discrepancies are discussed briefly in an appendix

degenerate regularization of forward backward parabolic equations the vanishing viscosity limit we address the vanishing viscosity limit of the regularized problem studied in smarrazzo and tesei arch rat mech anal 2012 in press we show that the limiting points in a suitable topology of the family of solutions of the regularized problem can be regarded as suitably defined weak measure valued solutions of the original problem in general these solutions are the sum of a regular term which is absolutely continuous with respect to the lebesgue measure and a singular term which is a radon measure singular with respect to the other by using a family of entropy inequalities we prove that the singular term is nondecreasing in time we also characterize the disintegration of the young measure associated with the regular term proving that it is a superposition of two dirac masses with support on the branches of the graph of the nonlinearity varphi

operator trigonometry of statistics and econometrics a new and useful geometric point of view for the understanding and analysis of certain matrix methods as they are used in statistics and econometrics is presented applications to statistical efficiency parameter estimation and correlation theory are given in particular we show that worst case relative least squares efficiency although achieved by maximally inefficient regressors is also achieved by maximal covariance matrix turning vectors also we elaborate geometrically a commutator trace efficiency result of p bloomfield and g s watson biometrika 62 1975 121 well established lagrange multiplier methods for constrained optimizations are compared to use of an euler equation from the new geometric theory