mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

vibration of arbitrarily shaped triangular plates with elastically restrained edges abstract a solution method is proposed for the vibration analysis of arbitrarily shaped triangular plates with elastically restrained edges an arbitrarily shaped triangular plate is first mapped into a right angled isosceles triangular plate and subsequently stretched to form a square plate by padding to it another right angled isosceles triangular plate of near zero thickness the displacement function is then expressed as a two dimensional fourier cosine series supplemented by several one dimensional series introduced to improve the convergence and accuracy of the displacement solution regardless of the shapes and boundary conditions of the triangular plate the final stiffness and mass matrices can always be calculated analytically which makes the current solution method highly efficient in implementations numerical examples are presented to verify the applicability of the proposed method to triangular plates of different shapes and perhaps more importantly different boundary conditions the current method can be easily extended to plates with heterogeneous material properties variable thickness non uniform elastic restraints etc

a simple formula for predicting resonant frequencies of a rectangular plate with uniformly restrained edges abstract this paper provides empirical formulas for rapidly calculating the resonant frequencies of an orthotropic rectangular plate with its edges constrained by elastic supports in particular the classical boundary condition with guided supports at its edges is considered other boundary conditions such as the guided free edges guided clamped edges and guided free support edges have also been included in the present study simple and closed form empirical formulas have been derived to allow straightforward computations of the modal resonant frequencies the empirical formulas are based on the analytical results obtained from the rayleigh ritz method however the coefficient is determined empirically from the results obtained by other more accurate computational scheme e g finite element method the method is further generalized to predict the resonant frequencies for general boundary conditions of a square orthotropic plate

calculation of the natural frequencies of composite plates by the rayleigh ritz method with orthogonal polynominals orthogonal polynomials are used as admissible functions and the approach has been found to be numerically stable

transverse vibrations of rectangular plates with edges elastically restrained against translation and rotation abstract the fundamental frequency coefficient for a rectangular plate with edges elastically restrained against both translation and rotation is calculated by using polynomial coordinate functions and the rayleigh ritz method the approach is simple and straightforward and allows the solution of a rather difficult elastodynamics problem complicating factors orthotropic properties in plane forces concentrated masses etc can also be taken into account without formal difficulties

free vibration analysis of mindlin plates with uniform elastic edge support by the superposition method abstract utilizing the superposition method a solution is obtained for the free vibration eigenvalues of mindlin plates resting on uniform lateral elastic edge support subsequently it is shown how minor modifications to the eigenvalue matrix permit the incorporation of the additional effects of rotational stiffness into the analysis solutions are based on the mindlin thick plate theory convergence is rapid and eigenvalue curves are plotted for square plates and plates of aspect ratio 1 5 for various lateral and rotational spring stiffnesses digital results are tabulated in order that other analysts will have reliable data against which their findings may be compared

harmonic univalent mappings into a half plane with nonreal vertical slits a class of harmonic univalent mappings is constructed by applying the method of clunie and sheil small these mappings assuming their values in a half plane with a vertical boundary omit two vertical half lines symmetric w r t the real axis several basic properties are proved

simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates abstract the simulation of classical edge conditions by finite values of elastic edge restraints is a problem frequently encountered in experimental vibration analysis the support fixtures utilized in such studies always introduce undesired flexibilities or stiffnesses causing the support conditions to deviate from ideal classical ones there are also some analytical studies in which finite values of flexibilities are used in conjunction with a specific distribution over the edge to represent classically supported edges with point supports thus one may encounter in the simulation of classical edge conditions a small but finite flexibility where it is required to be zero or a large but finite flexibility where it is required to be infinity the present study attempts to answer the question how large can a small value of flexibility be and yet represent zero flexibility and how small can a large value be and yet represent infinite flexibility in the simulation of classical boundary conditions the effects of finite values and distributions of the elastic edge restraint on the natural frequencies of rectangular plates are examined and guidelines are provided in terms of their ranges and limits for their use in representing classical edge conditions based on an analysis of the errors in the edge forces and moments the values of the flexibility coefficients suitable for stress analysis are also identified

strong orthogonal arrays and associated latin hypercubes for computer experiments this paper introduces constructs and studies a new class of arrays called strong orthogonal arrays as suitable designs for computer experiments a strong orthogonal array of strength t enjoys better space filling properties than a comparable orthogonal array in all dimensions lower than t while retaining the space filling properties of the latter in t dimensions latin hypercubes based on strong orthogonal arrays of strength t are more space filling than comparable orthogonal array based latin hypercubes in all g dimensions for any 2 g t 1 copyright 2013 oxford university press

a constructive method for computing the tension parameters in convexity preserving spline in tension interpolation the idea initiated by opfer for constructing conformal mappings from the disk onto starlike domains is generalized for univalent harmonic mappings this is of some interest since such mappings are not characterized by analytic means

on the use of orthogonal polynomials in the rayleigh ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates series composed of recently proposed orthogonal polynomial functions are used in the rayleigh ritz method to generate results for a number of flexural vibration and buckling problems for rectangular isotropic and orthotropic plates it is demonstrated that the approach yields excellent results not only for natural frequencies and buckling loads but also for the more sensitive mode shape bending moment and shear force distribution problem

on meromorphically harmonic starlike functions with respect to symmetric conjugate points abstract in a previous paper m p chen z r wu and z z zou m p chen z r wu z z zou on functions starlike with respect to symmetric conjugate points j math anal appl 201 1996 25 34 developed a method using some operators to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc then the same method is employed to functions meromorphic by z z zou and z r wu zhong zhu zou zhuo ren wu on meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points j math anal appl 261 2001 17 27 now the method can be employed to functions meromorphic harmonic in the punctured disc 0 z 1 especially a sharp coefficient estimate and a structural representation of such functions are obtained

free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints abstract an analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges and elastically coupled together at an arbitrary angle the interactions of all four wave groups bending waves out of plane shearing waves in plane longitudinal waves and in plane shearing waves have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses each of the transverse and in plane displacement functions is expressed as the superposition of a two dimensional 2 d fourier cosine series and several supplementary functions which are introduced to ensure and improve the convergence of the series representation by removing the discontinuities that the original displacement and its derivatives will potentially exhibit at the edges when they are periodically expanded onto the entire x y plane as mathematically implied by a 2 d fourier series the unknown expansions coefficients are calculated using the rayleigh ritz procedure which is actually equivalent to solving the governing equation and the boundary and coupling conditions directly when the assumed solutions are sufficiently smooth over the solution domains numerical examples are presented for several different coupling configurations a good comparison is observed between the current results and the fea models although this study is specifically focused on the coupling of two plates the proposed method can be directly extended to structures consisting of any number of plates

vibrations of rectangular plates with arbitrary non uniform elastic edge restraints abstract arbitrary non uniform elastic edge restraints represent the most general class of boundary conditions for plate problems and are encountered in many real world applications the vibrations of plates with this kind of boundary conditions however are rarely studied in the literature perhaps because there is a lack of suitable analytical or numerical techniques in this investigation a general analytical method is derived for the vibration analysis of rectangular plates with elastic edge restraints of varying stiffness both rotational and translational restraints can be arbitrarily applied to an edge and their stiffness distributions are generally described in terms of a set of invariants cosine functions the displacement solution is sought simply as a linear combination of several one and two dimensional fourier cosine series expansions all the unknown fourier coefficients are treated equally as a set of independent generalized coordinates and solved directly from the rayleigh ritz formulation unlike the existing techniques the current method does not require any special procedures or schemes to deal with different boundary conditions a few classical problems involving non uniform rotational restraints are first solved and used to check the current solution against some of the existing techniques the modal results are also presented for plates with more complicated boundary conditions in which an edge is no longer completely restrained in the translational direction the accuracy and reliability of the current method are repeatedly demonstrated through all these examples

natural frequencies of rectangular plates using a set of static beam functions in rayleigh ritz method abstract a fast converging series consisting of a set of static beam functions which is a combination of sine series and polynomials is developed and these functions are used as the basis functions in the rayleigh ritz method to study the vibrational characteristics of thin isotropic rectangular plates in applications a relatively small number of functions are required and consequently a modest amount of computation is necessary the approach is very simple and straightforward some numerical results are given for rectangular plates with various aspects ratios and boundary conditions good accuracy and fast convergence of the method are demonstrated by comparison with available results complicating factors orthotropic properties in plane forces etc can also be taken into account without formal difficulties

bending of a rectangular plate with rotationally restrained edges under a concentrated force the bending problem of a rectangular plate with rotationally restrained edges under a concentrated force is studied an emphasis is placed on the determination of the corner forces and deflection the problem is solved by superposition of classical navier s and levy s solutions analytic solutions in terms of series are obtained for rsss rrss and rsrs plates respectively where r and s stand for rotational restraint and simple support respectively some important design parameters such as maximum deflection and corner force are evaluated the effect of edge restraint on these parameters is illustrated

space filling properties of good lattice point sets we study space filling properties of good lattice point sets and obtain some general theoretical results we show that linear level permutation does not decrease the minimum distance for good lattice point sets and we identify several classes of such sets with large minimum distance based on good lattice point sets some maximin distance designs are also constructed

on harmonic univalent mappings with nonzero pole abstract in this article we consider the class a h p of all sense preserving harmonic functions f in the open unit disc d having a simple pole at z p 0 1 with the normalizations f 0 f z 0 1 0 we first derive a sufficient condition for univalence of such functions next we consider the class s h 0 p which consists of all functions belonging to a h p and are univalent with the additional normalization f z 0 0 we study the class s h 0 p in the geometric function theoretic viewpoint as a byproduct of our investigation we see that consideration of nonzero pole yields nontrivial lower bounds for the taylor coefficients of such functions we also prove a growth result for a non vanishing subclass associated with s h 0 p

vibrations of rectangular plates reinforced by any number of beams of arbitrary lengths and placement angles this paper presents an analytical method for the vibration analysis of plates reinforced by any number of beams of arbitrary lengths and placement angles both the plate and stiffening beams are generally modeled as three dimensional 3 d structures having six displacement components at a point and the coupling at an interface is generically described by a set of distributed elastic springs each of the displacement functions is here invariably expressed as a modified fourier series which consists of a standard fourier cosine series plus several supplementary series functions used to ensure and improve uniform convergence of the series representation unlike most existing techniques the current method offers a unified solution to the vibration problems for a wide spectrum of stiffened plates regardless of their boundary conditions coupling conditions and reinforcement configurations several numerical examples are presented to validate the methodology and demonstrate the effect on modal parameters for a stiffened plate with various boundary conditions coupling conditions and reinforcement configurations

construction of 2 m4 n designs via a grouping scheme on developpe une methode pour grouper les 2 k 1 effets factoriels dans un plan factoriel a deux niveaux dans des ensembles disjoints mutuellement s t st ou st est l interaction generalisee des effets s et t comme application on construit des tableaux orthogonaux oa 2 k 2 m 4 n 2 de taille 2 k m contraintes avec 2 niveaux et n contraintes avec 4 niveaux satisfaisant m 3n 2 k 1 et de puissance 2 le nombre maximum de contraintes avec 4 niveaux dans la construction ne peut plus etre ameliore dans ce sens le schema de groupement qu on donne est optimal on discute les avantages de la presente approche par rapport a d autres methodes de construction

an exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports abstract an analytical method is developed for the vibration analysis of rectangular plates with elastically restrained edges the displacement solution is expressed as a two dimensional fourier series supplemented with several one dimensional fourier series mathematically such a series expansion is capable of representing any function including the exact displacement solution whose third order partial derivatives are required to be continuous over the area of the plate since the discontinuities or jumps potentially related to the partial derivatives at the edges when they are periodically extended onto the entire x y plane as implied by a two dimensional fourier series expansion have been explicitly absorbed by the supplementary terms all the series expansions for up to the fourth order derivatives can be directly obtained through term by term differentiations of the displacement series thus an exact solution can be obtained by letting the series simultaneously satisfy the governing differential equation and the boundary conditions on a point wise basis because the series solution has to be truncated numerically the exact solution should be understood as a solution with arbitrary precision several numerical examples are presented to illustrate the excellent accuracy of the current solution the proposed method can be directly extended to other more complicated boundary conditions involving non uniform elastic restraints point supports partial supports and their combinations

support points of families of analytic functions described by subordination we determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination the families we study include the functions with a positive real part the typically real functions and the functions which are subordinate to a given majorant if the majorant f is univalent then each support point has the form f 0 where p is a finite blaschke product and o 0 o this completely characterizes the set of support points when f is convex the set of support points is found for some specific majorants including f z 1 z l z p where p 1 let k and st denote the set of normalized convex and starlike mappings respectively we find the support points of the families k and st defined by the property of being subordinate to some member of k or st respectively introduction let a z e c i z i 1 and let e denote the set of functions analytic in a then 6 is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of la by a continuous linear functional on e we mean a complex valued functional defined on 6f that is linear and continuous in other words if j is such a functional then j af bg aj f bj g whenever a and b belong to c and f and g belong to b also if fn e 6b n 1 2 and fn f then j fn j f each continuous linear funcn tional j on ef is given by a sequence bn n 0 1 which satisfies limn n i n i 1 and

free vibration and buckling of in plane loaded plates with rotational elastic edge support the superposition method is employed to obtain buckling loads and free vibration frequencies for a family of elastically supported rectangular plates subjected to one directional uniform in plane loading two edges running in the direction of the in plane loading are free lateral displacement is forbidden along the other two edges which are given uniform elastic rotational support accurate buckling loads are tabulated for a fairly broad range of plate geometries and edge support stiffnesses computed free vibration eigenvalues are also tabulated for square plates with typical in plane loading and a range of stiffnesses in the edge supports

vibration analysis of rectangular plates with general elastic boundary supports abstract in this investigation the rayleigh ritz method is used to determine the modal characteristics of a rectangular plate with general elastic supports alone its edges each of the admissible functions here is composed of a trigonometric function and an arbitrary continuous function that is introduced to ensure the sufficient smoothness of the so called residual displacement function at the edges as a result a drastic improvement of the convergence can be expected of the solution expressed as a series expansion in terms of the admissible functions perhaps more importantly this study has developed a general approach for deriving a complete set of admissible functions that can be universally applied to various boundary conditions several numerical examples are given to demonstrate the accuracy and convergence of the current solution

arbitrary precision frequencies of a free rectangular thin plate abstract a variational method developed by the authors and named whole element method wem is used to find the arbitrary precision frequencies of a rectangular thin plate within the germain lagrange theory having its four borders free of constraint wem consists is proposing an adequate functional and a sequence representing the plate transversal displacementw x y such a sequence is made of a linear combination of functions belonging to a complete set inl2 the sequence and not each co ordinate function is required to satisfy the essential or geometric conditions the sequence generation is systematic and no analysis of the classical natural modes of the plate is needed in particular trigonometric functions which a priori belong to a complete set in the domain are used in the present analysis the solving equations involving very simple sums arise from the minimization of the functional wem is based on theorems which show the ultimate exactness of the eigenvalues and the uniform convergence of the essential functions of the problem to the authors knowledge this problem has no classical solution

free vibration analysis of corner supported rectangular plates with symmetrically distributed edge beams abstract analytical type solutions are obtained for the free vibration frequencies and mode shapes of thin corner supported rectangular plates with symmetrically distributed reinforcing beams or strips attached to the plate edges the method of superposition is employed equations governing reactions at plate beam interfaces are developed in dimensionless form the approach is comprehensive in that both lateral and rotational stiffness and inertia of the beam are incorporated into the analysis for illustrative purposes computed eigenvalues and mode shapes are presented for two plate beam systems of realistic geometries it is shown that the method is easily extended to cover the case where the edge beams do not have a symmetrical distribution this appears to be the first comprehensive analytical study of this problem of industrial interest

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

univalent functions in h _ h d etude des fonctions univalentes dans h h d solutions d une equation aux derivees partielles non lineaire elliptique

orthogonal and nearly orthogonal designs for computer experiments we introduce a method for constructing a rich class of designs that are suitable for use in computer experiments the designs include latin hypercube designs and two level fractional factorial designs as special cases and fill the vast vacuum between these two familiar classes of designs the basic construction method is simple building a series of larger designs based on a given small design if the base design is orthogonal the resulting designs are orthogonal likewise if the base design is nearly orthogonal the resulting designs are nearly orthogonal we present two generalizations of our basic construction method the first generalization improves the projection properties of the basic method the second generalization gives rise to designs that have smaller correlations sample constructions are presented and properties of these designs are discussed

the nitsche conjecture the nitsche conjecture is deeply rooted in the theory of doubly connected minimal surfaces however it is commonly formulated in slightly greater generality as a question of existence of a harmonic homeomorphism between circular annuli h a a r r onto a r r a in the early 1960s while attempting to describe all doubly connected minimal graphs over a given annulus a j c c nitsche observed that their conformal modulus cannot be too large then he conjectured in terms of isothermal coordinates even more a harmonic homeomorphism h a onto a exists if an only if r r 1 2 r r r r this fascinating and engaging problem remained open for almost a half of a century in the present paper we provide among further generalizations an affirmative answer to his conjecture

strong orthogonal arrays of strength two plus strong orthogonal arrays were recently introduced and studied in he and tang biometrika100 2013 254 260 as a class of space filling designs for computer experiments to enjoy the benefits of better space filling properties when compared to ordinary orthogonal arrays strong orthogonal arrays need to have strength three or higher which may require run sizes that are too large for experimenters to afford to address this problem we introduce a new class of arrays called strong orthogonal arrays of strength two plus these arrays while being more economical than strong orthogonal arrays of strength three still enjoy the better two dimensional space filling property of the latter among the many results we have obtained on the characterizations and constructions of strong orthogonal arrays of strength two plus worth special mention is their intimate connection with second order saturated designs

vibration of orthotropic plates discussion on the completeness of the solutions used in direct methods fil rosales marta beatriz consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet bahia blanca argentina universidad nacional del sur departamento de ingenieria argentina

vibrations of rectangular plates with elastically restrained edges abstract the rayleigh ritz method is used to determined natural frequencies in transverse vibration of rectangular plates with elastically restrained edges by treating an elastically restrained edge as intermediate between an appropriate pair of classical boundary conditions and using the corresponding vibration mode shapes of beams with classical boundary conditions as assumed functions a relatively small number of functions is required consequently only a modest quantity of computation is necessary the good accuracy of the method is demonstrated by solving test problems the method can be applied to a wide range of elastic restraint conditions any aspect ratio and for higher modes in addition to the fundamental the usefulness and accuracy of existing simplified approaches to the problem are assessed the effect of in plane forces on the natural frequencies and the determination of critical loads for plates with these restraint conditions are considered also

free vibrations of beams with general boundary conditions a simple and unified approach is presented for the vibration analysis of a generally supported beam the flexural displacement of the beam is sought as the linear combination of a fourier series and an auxiliary polynomial function the polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives as a result not only is it always possible to expand the displacement in a fourier series for beams with any boundary conditions but also the solution converges at a much faster speed the reliability and robustness of the proposed technique are demonstrated through numerical examples

free vibration solution for clamped orthotropic plates using lagrangian multiplier technique using the lagrangian multiplier technique an analysis was developed to compute the natural frequencies and mode shapes of clamped orthotropic laminated plates transverse shear deformation effects were included using a higher order plate theory predictions made with the developed analysis demonstrated excellent agreement with the available experimental test results and other analytical solutions the validated free vibration analysis has also been used to develop an analysis to predict the dynamic response of clamped orthotropic plates subjected to low velocity impact by a hard object

optimization of natural frequencies of a plate structure by modifying boundary conditions a combined approach based on finite element method and genetic algorithm fem ga is proposed for optimizing the natural frequencies of plate structures this approach can identify the optimal boundary conditions so that the plate s natural frequencies can be adjusted simultaneously to their corresponding target values in this approach the natural frequencies of plates with arbitrary boundary conditions are calculated by fem while ga is employed for searching the optimal solutions of the multiple objective optimization problem the fem is validated by comparing with previous results the proposed approach is illustrated by numerical examples the results demonstrate the effectiveness of this approach

stability and accuracy of three fourier expansion based strong form finite elements for the free vibration analysis of laminated composite plates summary in the present paper strong form finite elements are employed for the free vibration study of laminated arbitrarily shaped plates in particular the stability and accuracy of three different fourier expansion based differential quadrature techniques are shown these techniques are used to solve the partial differential system of equations inside each computational element the three approaches are called harmonic differential quadrature fourier differential quadrature and improved fourier expansion based differential quadrature methods the improved fourier expansion based differential quadrature method implements auxiliary functions in order to approximate functional derivatives up to the fourth order with respect to the fourier differential quadrature method that has a basis made of sines and cosines all the present applications are related to literature comparisons and the presentation of new results for further investigation within the same topic a study of such kind has never been proposed in the literature and it could be useful as a reference for future investigation in this matter copyright 2016 john wiley sons ltd

a unified approach for predicting sound radiation from baffled rectangular plates with arbitrary boundary conditions this paper discusses sound radiation from a baffled rectangular plate with each of its edges arbitrarily supported in the form of elastic restraints the plate displacement function is universally expressed as a 2 d fourier cosine series supplemented by several 1 d series the unknown fourier expansion coefficients are then determined by using the rayleigh ritz procedure once the vibration field is solved the displacement function is further simplified to a single standard 2 d fourier cosine series in the subsequent acoustic analysis thus the sound radiation from a rectangular plate can always be obtained from the radiation resistance matrix for an invariant set of cosine functions regardless of its actual dimensions and boundary conditions further this radiation resistance matrix unlike the traditional ones for modal functions only needs to be calculated once for all plates with the same aspect ratio in order to determine the radiation resistance matrix effectively an analytical formula is derived in the form of a power series of the non dimensional acoustic wavenumber the formula is mathematically valid and accurate for any wavenumber several numerical examples are presented to validate the formulations and show the effect of the boundary conditions on the radiation behavior of planar sources

the minimum support stiffness required to raise the fundamental natural frequency of plate structures the minimum stiffness and optimum position of a flexible point support is calculated that raises the fundamental natural frequency of plate structures for a single support the maximum fundamental natural frequency of the supported structure is equal to the second natural frequency of the unsupported structure the structure is modelled using finite element analysis allowing a wide range of applications and boundary conditions if a support is positioned within a finite element then the shape functions are used to calculate the contribution of the support to the system stiffness and also the slope of the mode at the support efficient methods are used to calculate the minimum support stiffness required numerical results demonstrate that a system with flexible supports may be designed with the same fundamental natural frequency as that of a system with rigid supports examples of plate structures with one or two point supports are analysed and show that the method is efficient in these cases

an exact series solution for calculating the eigenfrequencies of orthotropic plates with completely free boundary abstract an analysis for solving boundary value problems in structural mechanics which was proposed by wang and lin 1996 journal of sound and vibration 196 285 293 1999 journal of applied mechanics 66 380 387 1 2 has been extended to the calculation of the eigenfrequencies of an orthotropic plate under all free boundary conditions the convergence of the series solution is assured and the procedure leads to pointwise exact solutions the calculated eigenfrequencies have been verified by a different approach and indicate that the present method is simple and effective

a variational formulation for vibro acoustic analysis of a panel backed by an irregularly bounded cavity abstract a weak form variational based method is developed to study the vibro acoustic responses of coupled structural acoustic system consisting of an irregular acoustic cavity with general wall impedance and a flexible panel subjected to arbitrary edge supporting conditions the structural and acoustical models of the coupled system are formulated on the basis of a modified variational method combined with multi segment partitioning strategy meanwhile the continuity constraints on the sub segment interfaces are further incorporated into the system stiffness matrix by means of least squares weighted residual method orthogonal polynomials such as chebyshev polynomials of the first kind are employed as the wholly admissible unknown displacement and sound pressure field variables functions for separate components without meshing and hence mapping the irregular physical domain into a square spectral domain is necessary the effects of weighted parameter together with the number of truncated polynomial terms and divided partitions on the accuracy of present theoretical solutions are investigated it is observed that applying this methodology accurate and efficient predictions can be obtained for various types of coupled panel cavity problems and in weak or strong coupling cases for a panel surrounded by a light or heavy fluid the inherent principle of velocity continuity on the panel cavity contacting interface can all be handled satisfactorily key parametric studies concerning the influences of the geometrical properties as well as impedance boundary are performed finally by performing the vibro acoustic analyses of 3d car like coupled miniature we demonstrate that the present method seems to be an excellent way to obtain accurate mid frequency solution with an acceptable cpu time

dynamic analysis of generally supported beams using fourier series the applicability of fourier series to the dynamic analysis of beams having arbitrary boundary conditions is explored for the simplicity of presenting the analysis procedure beams having zero end displacement are used in the discussion numerical results for free vibration and dynamic response based on the present analysis procedure are compared with existing results for natural frequencies and dynamic behavior obtained by the conventional modal analysis these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is seemingly simpler and more straightforward than the conventional modal analysis

univalent harmonic functions several families of complex valued univalent harmonic functions are studied from the point of view of geometric function theory one class consists of mappings of a simply connected domain onto an infinite horizontal strip with a normalization at the origin extreme points and support points are determined as well as sharp estimates for fourier coefficients and distortion theorems next mappings in izi 1 are considered that leave infinity fixed some coefficient estimates distortion theorems and covering properties are obtained for such mappings with real boundary values many extremal problems are solved explicitly

an analytical method for the in plane vibration analysis of rectangular plates with elastically restrained edges in this investigation the in plane vibration problems are solved for plates with general elastically restrained boundary conditions under the current framework all the classical homogeneous boundary condition for in plane displacements can be easily simulated by simply setting the stiffnesses of the restraining springs to either infinite or zero the vibration problems are solved using an improved fourier series method in which the in plane displacements are expressed as the superposition of a double fourier cosine series and four supplementary functions in the form of the product of a polynomial function and a single cosine series expansion the use of these supplementary functions is to overcome the discontinuity problems which the original displacement functions will potentially encounter along the edges when they are viewed as a periodic function defined over the entire x y plane the excellent accuracy and convergence of the current solution are demonstrated through numerical examples to the best of authors knowledge this work represents the first time that an analytical solution has been obtained for the in plane vibrations of a rectangular plate with elastically restrained edges

on the symplectic superposition method for new analytic free vibration solutions of side cracked rectangular thin plates abstract in this study a first attempt is made to develop an up to date symplectic superposition method for some new analytic free vibration solutions of side cracked rectangular thin plates that were not obtained by conventional semi inverse methods in contrast with the classical lagrangian system and euclidean space the present method is implemented within the hamiltonian system framework and symplectic space the solution procedure involves expressing the problems in the hamiltonian system and dividing a side cracked plate into several sub plates that are analytically solved by the symplectic superposition method where the imposed quantities are determined by the plate boundary conditions free edge conditions along the crack and interfacial continuity conditions between the sub plates in the analytic solution of a sub plate specifically the symplectic eigenvalue problems are formulated followed by the symplectic eigen expansion the integration of the solutions of the sub plates yields the final solution of a side cracked plate the rigorous mathematical techniques without predetermination of solution forms qualify the present method as an unusual approach for exploring more analytic solutions comprehensive natural frequency and mode shape solutions of the side cracked plates under three representative boundary constraints are provided and well validated by other methods the new analytic solutions obtained may serve as benchmarks for other potential solution methods