vibration analysis of orthotropic mindlin plates with edges elastically restrained against rotation abstract the free vibration analysis of the orthotropic mindlin plates with edges elastically restrained against rotation has been accomplished using the rayleigh ritz method polynomials with properties corresponding to those of the timoshenko beam functions are introduced as trial functions in the spatial representation of the deflection and rotations of cross sections in two directions of the plates the results obtained by using polynomials are nearly the same as the timoshenko beam functions results but were obtained with considerably less numerical efforts

vibrations and elastic stability of thin circular plates with variable profile abstract circular plates of thicknesses varying to the functional relationho 1 r a n wherenis a positive integer are studied in the present paper uniform elastic boundary restraints are considered and the first two natural frequency coefficients corresponding to axisymmetric modes and the buckling in plane pressure are determined using simple polynomial co ordinate functions which identically satisfy the boundary conditions and the classical rayleigh ritz method an independent solution is also obtained by means of the differential quadrature technique

computing eigenvalues of real symmetric matrices with rational filters in real arithmetic powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals best known are the works of sakurai and coauthors and polizzi and coauthors even if the matrices are real symmetric most such methods rely on complex arithmetic leading to expensive linear systems to solve an appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems for which there is no need to leave the real axis investigation shows that using rational interpolation per se suffers from instability however related techniques involving real rational filters can be very effective this article presents a technique of this kind that is related to previous work published in japanese by murakami

fast and stable rational interpolation in roots of unity and chebyshev points a new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented when the interpolation nodes are roots of unity or chebyshev points the algorithm is particularly simple and relies on discrete fourier transform matrices which results in a fast implementation using the fast fourier transform the method is generalized for arbitrary grids which requires the construction of polynomials orthogonal on the set of interpolation nodes the appearance of common factors in the numerator and denominator due to finite precision arithmetic is explained by the behavior of the singular values of the linear system associated with the rational interpolation problem the new algorithm has connections with other methods particularly the work of jacobi and kronecker berrut and mittelmann and egecioglu and koc short matlab codes and numerical experiments are included

a new method for free vibration and buckling analysis of rectangular orthotropic plates a new method for free vibration and buckling analysis of rectangular orthotropic plates is presented the proposed method differs significantly from previously published methods in that it allows obtaining practically exact results for free vibration and buckling of orthotropic plates by enhancing the method of superposition the boundary value problem is essentially reduced to an infinite system of linear algebraic equations of particular significance of this research is that the existence of bounded solution of the infinite system after elimination of the unknowns has been captured in an elegant but quantitative way to converge upon any natural frequency or critical buckling load of the orthotropic plate to any desired accuracy the theory is applied to the particular case of orthotropic plates with clamped edges but it can be extended to other boundary conditions buckling analysis is carried out as a degenerate case of free vibration at zero frequency comparisons of computed results from the present theory with those available in the literature are made wherever possible and some noteworthy conclusions are drawn

3 d free vibration analysis of annular plates on pasternak elastic foundation via p ritz method in this study a three dimensional 3 d free vibration analysis of thick annular plates resting on elastic foundation with different combinations of free soft simply supported hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate is presented on the basis of the polynomials ritz method the elastic foundation is considered as a pasternak model with adding a shear layer to the winkler model the analysis procedure is based on the linear small strain and 3 d elasticity theory in this analysis method a set of orthogonal polynomial series in cylindrical polar coordinate is used to extract eigenvalue equation yielding the natural frequencies and mode shapes for the annular plates the accuracy of these results is verified by appropriate convergence studies and checked with the available literature finite element method fem analysis and the mindlin theory furthermore the effect of the foundation stiffness parameters thickness radius ratio inner outer radius ratio and different combinations of boundary conditions on the ill conditioning of the mass matrix as well as on the vibration behavior of the annular plates is investigated finally the validity and the range of applicability of the results obtained on the basis of the mindlin and classical plate theories for a thin and moderately thick annular plate with different values of the winkler foundation stiffness are graphically presented through comparing them with those obtained by the present 3 d p ritz solution

effect of elastic foundation on asymmetric vibration of polar orthotropic linearly tapered circular plates asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation of winkler type are discussed on the basis of the classical plate theory ritz method has been employed to obtain the natural frequencies of vibration using the functions based upon the static deflection of polar orthotropic plates which has faster rate of convergence as compared to the polynomial co ordinate functions frequency parameter of the plate with elastically restrained edge conditions are presented for various values of taper parameter rigidity ratio and foundation parameter a comparison of the results with those available in the literature obtained by finite element method receptance method and polynomial co ordinate functions shows an excellent agreement

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

dynamic stiffness formulation and free vibration analysis of specially orthotropic mindlin plates with arbitrary boundary conditions abstract a novel dynamic stiffness formulation which includes the effects of shear deformation and rotatory inertia is proposed to carry out the free vibration analysis of thick rectangular orthotropic plates i e the formulation is based on an extension of the mindlin theory to orthotropic plates in a dynamic stiffness context the modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in series form permitting the derivation of an infinite system of linear algebraic equations which connect the fourier coefficients of the kinematic boundary conditions of its four edges the boundary conditions are essentially the deflections and angles of rotations constituting the displacement vector and shear forces and bending moments constituting the force vector the force displacement relationship is then obtained by relating the two vectors via the dynamic stiffness matrix essentially the ensuing infinite system forms the fundamental basis of the paper which defines the dynamic stiffness matrix for an orthotropic mindlin plate in an exact sense numerical results are obtained by using the wittrick williams algorithm as solution technique the theory is first validated using published results and then further illustrated by a series of numerical examples the paper concludes with significant conclusions

the set of unattainable points for the rational hermite interpolation problem abstract we describe geometrically and algebraically the set of unattainable points for the rational hermite interpolation problem i e those points where the problem does not have a solution we show that this set is a union of equidimensional complete intersection varieties of odd codimension the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem each of these equidimensional varieties can be further decomposed as a union of as many rational irreducible varieties as input data points we exhibit algorithms and equations defining all these objects

buckling and vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation abstract axisymmetric vibrations of polar orthotropic circular plates of linearly varying thickness subjected to hydrostatic in plane force and resting on an elastic foundation winkler type are discussed on the basis of the classical plate theory an approximate solution of the problem is obtained by a ritz method frequency parameters of plates with different edge conditions for the first three modes of vibration are presented as computed for various values of the taper parameter the rigidity parameter the foundation parameter the flexibility parameter and the in plane force parameter

asymptotically efficient estimation of the index of regular variation we propose a conditional mle of the index of regular variation when the functional form of a slowly varying function is assumed known in the tail and we study its asymptotic properties we prove asymptotic normality of p theta k n a probability measure whose density is the product of the joint conditional density of the k n largest order statistics from f theta x given z n k the n k th order statistic and a density of z n k with parameter theta based on this result we show that this conditional mle is asymptotically normal and asymptotically efficient in many senses whenever k n is o n we also propose an iterative estimator of theta given only partial knowledge of l theta x this estimator is asymptotically normal asymptotically unbiased and asymptotically efficient

accurate free vibration analysis of point supported mindlin plates by the superposition method the superposition method is exploited to obtain free vibration eigenvalues for thick mindlin plates resting on point supports the mindlin governing differential equations are satisfied exactly and boundary conditions are satisfied to any desired degree of accuracy by taking more terms in the solution series very good agreement is obtained with published results based on thin plate theory when the thickness ratio of the mindlin plates is allowed to take on values characteristic of thin plates eigenvalues are tabulated for a square plate resting on four point supports distributed symmetrically along the plate diagonals significant differences are encountered for higher modes when eigenvalues computed by the mindlin theory are compared with those based on the thin plate theory

a new method for analytical solution of inplane free vibration of rectangular orthotropic plates based on the analysis of infinite systems abstract in this paper a new method for in plane free vibration analysis of rectangular orthotropic plate is presented the cases for completely clamped and completely free plates are considered in detail the boundary value problem is essentially reduced to infinite systems of linear algebraic equations new mathematical theory and associated results are presented for investigating the asymptotic behavior of non trivial solution of quasi regular infinite system by enhancing the theory an effective and accurate algorithm for determining the natural frequencies and modes shapes vibrations is developed the accuracy and computational efficiency of the method are demonstrated by examples

vibrations of mindlin rectangular plates with elastically restrained edges using static timoshenko beam functions with the rayleigh ritz method abstract a set of static timoshenko beam functions is developed as the admissible functions for the free vibration analysis of mindlin rectangular plates with uniform elastic translational and or rotational edge constraints by the use of the rayleigh ritz method this set of beam functions is made up of two parts the transverse deflection functions and rotation functions which are the static solutions of a timoshenko beam under the transverse loads of a fourier sinusoidal series distributed along the length of the beam therefore the effect of shear deformation on the admissible functions is taken into account in fact the beam is considered to be a unit width strip taken from the rectangular plate in a direction parallel to the edge of the plate and has the end supports corresponding to the edge conditions of the particular plate under consideration it can be seen that the method is sound theoretically each of the beam functions is a polynomial function of not more than third order plus a sine or cosine function for all cases and the change of the boundary supports only results in a corresponding change of the coefficients of the low order polynomial therefore a unified program with little computational efforts can be easily developed for the vibration analysis of mindlin rectangular plates with arbitrary boundary conditions and thickness span ratios the accuracy and efficiency of the proposed method are demonstrated by comparison and convergency studies finally sets of reasonably accurate natural frequencies are presented which can serve as a supplement to the database of vibration characteristics of moderately thick plates

free vibration and buckling of symmetric cross ply rectangular laminates the method of superposition is further developed to analyze free vibration and buckling of symmetric crossply laminated rectangular plates the improved plate theory developed by mindlin is implemented a four building block technique is employed that reduces the number of combinations of classical boundary conditions that need to be studied combinations of clamped and simply supported boundary conditions are investigated eigenvalues and critical buckling loads with four digit accuracy are obtained for plates with five distinct combinations of simply supported and clamped boundary conditions a b c

a fast algorithm for rational interpolation via orthogonal polynomials l algorithme propose genere un ensemble de polynomes orthogonaux par la relation de recurrence a trois termes et ensuite utilise l interpolation de newton pour determiner les polynomes au numerateur et au denominateur de la fonction rationnelle d interpolation

computationally efficient applications of the euclidean algorithm to zero location abstract we characterize the inertia of the n n hankel matrix associated with a pair of real polynomials in terms of the quotients generated by the euclidean algorithm applied to these polynomials in this way we solve certain classical zero location problems at the computational cost of o n 2 arithmetic operations which can be reduced to o n log 2 n if fast polynomial arithmetic is used

on the estimation of the extreme value index and large quantile estimation l article se constitue de deux parties on donne une preuve simple de la convergence faible de l estimation de pickand pour le parametre principal d une loi de valeurs extremes on donne d autres conditions naturelles pour la convergence forte et pour la normalite asymptotique de l estimation ensuite une grande quantile d une loi est estimee par une combinaison de statistiques d ordre intermediaires ou extremes ceci mene a un intervalle de confiance asymptotique

rational interpolation via orthogonal polynomials abstract relations between rational interpolants and hankel matrices are investigated a modification of a jacobi like algorithm for rational interpolation proposed in 1 is developed the proposed algorithm eliminates all the auxiliary assumptions required to implement the algorithm of 1 in this way it can be used to construct all existing rational interpolants with the cost of o n 2 arithmetic operations where n 1 is the number of data points

a test for extreme value domain of attraction abstract a simple statistic is proposed to test the hypothesis that a sample comes from a distribution in the domain of attraction of the gumbel distribution it is based on the top k order statistics and is a generalization of the shapiro wilk goodness of fit statistic the critical region of the test and its power against the alternative that the sample comes from a distribution in another domain of attraction are studied theoretically and by simulation the power turns out to be superior to that of other tests previously proposed

global sufficiency of extreme order statistics in location models of weibull type in janssen and reiss 1988 it was shown that in a location model of a weibull type sample with shape parameter 1 a 1 thek n lower extremes are asymptotically local sufficient in the present paper we show that even global sufficiency holds moreover it turns out that convergence of the given statistical experiments in the deficiency metric does not only hold for compact parameter sets but for the whole real line

comparison of location models of weibull type samples and extreme value processes four different location parameter models are compared within the sufficiency and deficiency concept the starting is a location model of a weibull type sample with shape parameter 1 a 1 here our basic inequality concerns the approximate sufficiency of the k lower extremes in addition the lower extremes are approximately equal in distribution to left s m 1 1 a t right m leqq k where s m is the sum of m i i d standard exponential random variables and t is the location parameter the final step leads us to the model of extreme value processes left s m 1 1 a t right m 1 2 3

on the super additivity and estimation biases of quantile contributions sample measures of top centile contributions to the total concentration are downward biased unstable estimators extremely sensitive to sample size and concave in accounting for large deviations it makes them particularly unfit in domains with power law tails especially for low values of the exponent these estimators can vary over time and increase with the population size as shown in this article thus providing the illusion of structural changes in concentration they are also inconsistent under aggregation and mixing distributions as the weighted average of concentration measures for a and b will tend to be lower than that from a b in addition it can be shown that under such fat tails increases in the total sum need to be accompanied by increased sample size of the concentration measurement we examine the estimation superadditivity and bias under homogeneous and mixed distributions

free vibration analysis of stepped circular mindlin plates this paper presents the first known investigation on exact vibration of circular mindlin plates with multiple step wise thickness variations a stepped circular plate is divided into multiple annular and one circular segments along the locations of the step variations the governing differential equations for harmonic vibration of annual and circular segments are derived and an analytical method based on the domain decomposition technique is developed to solve the plate vibration problem exact vibration solutions are presented for circular mindlin plates of different edge support conditions and various combinations of step wise thickness variations the influence of plate boundary conditions plate thickness ratios step location ratios and step thickness ratios on the vibration behaviour of stepped circular mindlin plates are examined the exact vibration results can serve as benchmark values for researchers to validate their numerical methods for such circular plate problems

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

dynamic stiffness matrix of a rectangular plate for the general case the dynamic stiffness matrix of a rectangular plate for the most general case is developed by solving the bi harmonic equation and finally casting the solution in terms of the force displacement relationship of the freely vibrating plate essentially the frequency dependent dynamic stiffness matrix of the plate when all its sides are free is derived making it possible to achieve exact solution for free vibration of plates or plate assemblies with any boundary conditions previous research on the dynamic stiffness formulation of a plate was restricted to the special case when the two opposite sides of the plate are simply supported this restriction is quite severe and made the general purpose application of the dynamic stiffness method impossible the theory developed in this paper overcomes this long lasting restriction the research carried out here is basically fundamental in that the bi harmonic equation which governs the free vibratory motion of a plate in harmonic oscillation is solved in an exact sense leading to the development of the dynamic stiffness method it is significant that the ingeniously sought solution presented in this paper is completely general covering all possible cases of elastic deformations of the plate the wittrick williams algorithm is applied to the ensuing dynamic stiffness matrix to provide solutions for some representative problems a carefully selected sample of mode shapes is also presented

on testing the extreme value index via the pot method n alternatives based on the point processes n of the exceedances among y n i over a sequence of thresholds t it turns out that the random number of n exceedances t n over t is the central sequence for the log likelihood n ratio d l n rd l n yielding its local asymptotic normality lan b n b n n 0 this result implies in particular that t n carries asymptotically all the information about the underlying parameter b which is contained in n n we establish sharp bounds for the rate at which t n becomes asymptoti cally sufficient which show however that this is quite a poor rate these results remain true if we add an unknown scale parameter 1 n a model assumption on the upper tail of the underlying d f such as m is for example indispensable if one is interested in extreme quantities of the underlying d f that is such quantities of f which are usually outside the range of the given data y y a typical example is inference about 1 n y1 4 extreme quantiles f p inf t g r f t g p with p being close to 1 this problem is typically tackled by hydrologists for the prediction of large w

free vibration of polar orthotropic circular plates of variable thickness with elastically restrained edge abstract asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness with elastic rigid support are discussed on the basis of the classical plate theory an approximate solution of the problem is obtained by the rayleigh ritz method using functions based on static deflection of polar orthotropic circular plates this type of approximating functions has a faster rate of convergence as compared to the polynomial co ordinate functions frequency parameters of the plate have been obtained for different values of taper parameter flexibility parameter and rigidity ratio in the first three modes of vibration a comparison of results for special cases of isotropy uniform thickness and of axisymmetric vibrations obtained by other techniques such as the finite element method and the receptance method shows excellent agreement

local asymptotic normality of truncated empirical processes given n iid copies x 1 x n of a random element x in some arbitrary measurable space s we are only interested in those observations that fall into some subset d having but a small probability of occurrence it is assumed that the distribution p x of x belongs on d to a parametric family p x d p v v r d nonlinear regression analysis and the peaks over threshold pot approach in extreme value analysis are prominent examples for the pot approach on s r and p v being a generalized pareto distribution it is known that the complete information about the underlying parameter v 0 is asymptotically contained in the number n of observations in d among x 1 x n but not in their actual values this result is formulated in terms of local asymptotic normality of the log likelihood ratio of the point process of exceedances with n being the central sequence in this paper we establish a necessary and sufficient condition such that n has this property for a general truncated empirical process in an arbitrary sample space and for an arbitrary parametric family the known results are then consequences of this result we can moreover characterize the influence of the actual observations in d on the central sequence if this condition is violated immediate applications are asymptotically optimal tests for testing v 0 and if r asymptotic efficiency of the ml estimator n satisfying p vn d n n where these statistics are based on n only

use of two dimensional orthogonal polynomials for vibration analysis of circular and elliptical plates abstract a general numerical method is developed to estimate the natural frequencies and the mode shapes for transverse vibration of circular and elliptical plates a set of orthogonal plate functions is used as the admissible functions in the rayleigh ritz approach to approximate the eigenvalues and eigenvectors new starting functions are proposed which are used in a recurrence formula to generate the higher terms in the set of orthogonal plate functions to approximate the mode shapes for circular and elliptical plates these functions are general and easily formed for circular and elliptical plates having classical boundary conditions several numerical examples are solved to demonstrate the applicability and the accuracy of the new proposed plate functions

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

buckling and vibrations of polar orthotropic annular plates on elastic foundation subjected to hydrostatic peripheral loading an analysis is presented of the effect of hydrostatic peripheral loading and that of an elastic foundation on the natural frequencies of free axisymmetric vibrations and critical buckling parameter of linearly varying thickness annular plates exhibiting polar orthotropic characteristics on the basis of classical plate theory the numerical solution of the governing differential equation derived by using hamilton s energy principle is obtained by the spline technique method for three different boundary conditions inner and outer both clamped inner clamped and outer simply supported inner free and outer clamped transverse displacements moments in normalized form for a particular set of plate parameters and buckling loads in compression for different values of taper constant foundation modulus and rigidity ratios have been computed for the first two modes of vibration the numerical results are compared with exact values and with the results of other authors the close agreement of the values illustrates the versatility of this simple and elegant method provided the number of nodal points is taken appropriately

treatments of over restrained boundaries for doubly connected plates of arbitrary shape in vibration analysis abstract the problem of the free flexural vibrations of plates of doubly connected arbitrary shape is investigated a hybrid energy approach which combines both the advantages of the rayleigh ritz method and the lagrange multiplier method is developed for analysing the aforementioned plate problems the rayleigh ritz method with a set of pb 2 shape functions is used to formulate the plates with classical boundary conditions as a single continuum element while the lagrange multipliers are imposed at discrete points in the inner plate boundaries for instance when the overrestrained boundary has occurred the admissible pb 2 shape function consists of the product of a two dimensional polynomial and a basic function the basic function is defined by the product of the equations of the prescribed piecewise continuous boundary shape each raised to the power of 0 1 or 2 corresponding respectively to a free simply supported or clamped edge this set of functions automatically satisfies all the kinematic boundary conditions of the plate at the outset numerical results for several examples of doubly connected plates of arbitrary shape are presented to demonstrate the applicability and accuracy of the present method

non linear vibration analysis of arbitrarily laminated thin rectangular plates on elastic foundations abstract solutions to free undamped large amplitude vibrations of arbitrarily laminated thin rectangular plates on elastic foundations based on an assumed mode shape are determined simply supported and clamped boundary conditions with both movable and immovable edges are considered the governing equations of motion of plates on linear winkler pasternak foundations and a non linear winkler foundation are provided using von karman s theory the galerkin s method reduces the von karman equation to a time dependent ordinary differential equation solutions are then obtained and compared using the runge kutta method the direct numerical integration method and the perturbation method the effects on the non linear frequency of several parameters such as amplitude material lamination aspect ratio boundary conditions as relate to movable and immovable edges and the influence of elastic foundation constants are provided

accurate free vibration analysis of the completely free rectangular mindlin plate the superposition method is exploited to obtain accurate solutions for the natural frequencies and mode shapes of the completely free mindlin plate computed eigenvalues are tabulated for a number of plate aspect and thickness ratios steps taken to avoid computational instability are described difficulties associated with choosing mode shape functions particularly when free edges are involved have always hindered researchers utilizing the rayleigh ritz method such difficulties are obviated here to the authors knowledge this represents the first accurate comprehensive solution to this important plate vibration problem

estimating tails of probability distributions we study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold a key idea is the use of pickands generalised pareto distribution and its fitting in most cases by the method of maximum likelihood the results cover all three limiting types of extreme value theory we propose a new estimator for an index of regular variation and show that it often performs better than hill s estimator we give new results for estimating the endpoint of a distribution extending earlier work by hall and by smith and weissman finally we give detailed results for the domain of attraction of exp e x and show that in most cases our proposed estimator is more efficient than two others one based on the exponential distribution and the other due to davis and resnick we also touch briefly on the problem of large deviations from a statistical point of view the results make extensive use of existing work on rates of convergence

free vibrations of a solid circular plate of linearly varying thickness and attached to a winkler foundation abstract the title problem is solved by using linear analysis and the rayleigh schmidt method frequency coefficients corresponding to the lower symmetric and antisymmetric modes of transverse vibration are tabulated since it is assumed that the thin isotropic plate is elastically restrained against translation and rotation the free edge case turns out to be a particular situation of the mechanical system under study a comparison with results obtained by means of the finite element method is also presented very good agreement is achieved for the situations considered

rational interpolation and the euclidean algorithm abstract a general framework leading to a parametrization of all rational functions which interpolate a given set of pairs of points is investigated this framework is based on the euclidean algorithm the resulting parametrization has the property that it keeps track of the complexity of both the numerator and of the denominator polynomials of the interpolating functions the cauchy interpolation problem and the related pade approximation problem can be treated within this framework

free vibration analysis of rectangular mindlin plates with elastic restraints uniformly distributed along the edges abstract free vibration analysis of isotropic mindlin plates with edges elastically restrained against rotation and translation is carried out by a variational method the elastic restraints are assumed to be uniformly distributed along the edges elastically restrained timoshenko beam functions are used as constitutive shape functions and the boundary conditions not satisfied remain as boundary terms in the final variational energy expression the results for reduced problems are compared with those available in the literature and found to be in good agreement results showing the effect of elastic restraints on the natural frequencies of a thick plate indicate that the combined variation of rotational and translational restraint is more significant than their independent variations variations in mode shapes caused by the changes in the boundary conditions are also presented

effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness abstract axisymmetric vibrations of a polar orthotropic circular plate of variable thickness linear as well as parabolic resting on an elastic foundation of winkler type are discussed on the basis of the classical theory of plates an approximate solution of the problem is obtained by the ritz method frequency parameters of plates with different edge conditions for the first three modes of vibration and for various values of the taper parameter the foundation modulus and the rigidity ratio are presented