on elliptic curves of prime power conductor over imaginary quadratic fields with class number 1 the main result of this paper is to extend from q to each of the nine imaginary quadratic fields of class number one a result of serre 1987 and mestre oesterle 1989 namely that if e is an elliptic curve of prime conductor then either e or a 2 3 or 5 isogenous curve has prime discriminant for four of the nine fields the theorem holds with no change while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime the proof is conditional in two ways first that the curves are modular so are associated to suitable bianchi newforms and second that a certain level lowering conjecture holds for bianchi newforms we also classify all elliptic curves of prime power conductor and non trivial torsion over each of the nine fields in the case of 2 torsion we find that such curves either have cm or with a small finite number of exceptions arise from a family analogous to the setzer neumann family over q

hasse schmidt derivations and coefficient fields in positive characteristics abstract we show how to express any hasse schmidt derivation of an algebra in terms of a finite number of them under natural hypothesis as an application we obtain coefficient fields of the completion of a regular local ring of positive characteristic in terms of hasse schmidt derivations

free vibration analysis of functionally graded plates using the element free kp ritz method a free vibration analysis of metal and ceramic functionally graded plates that uses the element free kp ritz method is presented the material properties of the plates are assumed to vary continuously through their thickness according to a power law distribution of the volume fractions of the plate constituents the first order shear deformation plate theory is employed to account for the transverse shear strain and rotary inertia and mesh free kernel particle functions are used to approximate the two dimensional displacement fields the eigen equation is obtained by applying the ritz procedure to the energy functional of the system convergence studies are performed to examine the stability of the proposed method and comparisons of the solutions derived with those reported in the literature are provided to verify its accuracy four types of functionally graded rectangular and skew plates al al2o3 al zro2 ti 6al 4v aluminum oxide and sus304 si3n4 are included in the study and the effects of the volume fraction boundary conditions and length to thickness ratio on their frequency characteristics are discussed in detail

free vibration of skew sandwich plates with laminated facings this paper is concerned with the free vibration of skew sandwich plates composed of an orthotropic core and laminated facings the p ritz method has been adopted for the analysis the ritz functions are formed from the product of mathematically complete polynomials and boundary equations raised to appropriate integer powers depending on the boundary conditions the boundary equations ensure the satisfaction of the geometric boundary conditions a priori and facilitate the handling of any type of boundary conditions for generality better accuracy and ease in imposition of geometric boundary condition for the oblique edges the ritz formulation was non dimensionalized and cast in the skew co ordinates system since no vibration solutions are available for such skew sandwich plates the validity convergence and accuracy of the ritz formulation were established by comparing with other researchers vibration frequencies for various subset plate problems involving rectangular sandwich plates and skew laminated plates the paper features extensive generic vibration frequencies of these skew sandwich plates for various aspect ratios and boundary conditions lamination designs of facing material properties of core and facings

three dimensional free vibrations of thick skewed cantilevered plates abstract the natural frequencies of skewed cantilevered thick plates are determined by using the ritz method the present work is the first known three dimensional study of the problem assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly and which are mathematically complete accurate natural frequencies are calculated for skewed thick plates having arbitrary degrees of skewness detailed numerical studies reveal interesting trends concerning the variation of frequencies with increasing skew angle results obtained by using the present method are compared with those obtained by using three dimensional finite elements

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

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

natural vibration of pre twisted shear deformable beam systems subject to multiple kinds of initial stresses free vibration and buckling of pre twisted beams exhibit interesting coupling phenomena between compression moments and torque and have been the subject of extensive research due to their importance as models of wind turbines and helicopter rotor blades the paper investigates the influence of multiple kinds of initial stresses due to compression shears moments and torque on the natural vibration of pre twisted straight beam based on the timoshenko theory the derivation begins with the three dimensional green strain tensor the nonlinear part of the strain tensor is expressed as a product of displacement gradient to derive the strain energy due to initial stresses the frenet formulae in differential geometry are employed to treat the pre twist the strain energy due to elasticity and the linear kinetic energy are obtained in classical sense from the variational principle the governing equations and the associated natural boundary conditions are derived it is noted that the first mode increases together with the pre twisted angle but the second decreases seeming to close the first two modes together for natural frequencies and compressions the gaps close monotonically as the angle of twist increases for natural frequencies and buckling compressions however unlike natural frequencies and compressions the closeness is not monotonic for buckling shears moments and torques

weak form quadrature element method for accurate free vibration analysis of thin skew plates in this paper a novel weak form quadrature element method qem is proposed for accurate free transverse vibration analysis of thin isotropic skew plates with general boundary conditions in the formulation of the stiffness matrix the first and second order derivatives of the shape functions at integration points are computed explicitly by using the differential quadrature rule this leads to a great simplicity in formulating an n n node quadrature plate element and a large reduction of programming effort different from the existing weak form quadrature element method or differential quadrature finite element method the element nodes can be either the same or different from the integration points convergence studies are performed free vibration of skew thin plates with various skew angles and different combinations of boundary conditions is analyzed it is shown that although the assumed displacement field does not explicitly consider the bending moment singularities at the obtuse angles the proposed qem can yield accurate frequencies even for the thin isotropic skew plate with large skew angles

tight closure and differential simplicity the behavior of the hasse schmidt algebra under etale extension is used to show that the hasse schmidt algebra of a smooth algebra of finite type over a field equals the ring of differential operators these techniques show that the formation of hasse schmidt derivations does not commute with localization providing a counterexample to a question of brown and kuan their conjecture is reformulated in terms of the hasse schmidt algebra these techniques also imply that a smooth domain r is differentially simple tight closure is used to show that the test ideal is hasse schmidt stable indeed differentially simple rings of prime characteristic are strongly f regular

application of the spline element method to analyze vibration of skew mindlin plates with varying thickness in one direction this paper presents an application of the spline element method based on the mindlin plate theory to analyze the vibration of thick skew plates with varying thickness in the longitudinal direction to demonstrate the convergence and accuracy of the present method several examples are solved and results are compared with those obtained by other numerical methods good accuracy is obtained frequencies of skew mindlin plates with varying cross sections having arbitrary boundary conditions are analyzed for different thickness ratios aspect ratios ratios of the width to thickness and skew angles

free vibration analysis of a flat plate using the hierarchical finite element method abstract the hierarchical finite element method is used to determined the natural frequencies and modes of a flat rectangular plate ten different boundary conditions including free edges and point supports are considered in this paper extensive results are presented for each case including the variation of frequency with the aspect ratio and the poisson ratio and these are shown to be in very good agreement with the work of other investigators this confirms both the applicability and accuracy of solution of the hfem to problem of this type

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

a triangular fourier p element for the analysis of membrane vibrations abstract a triangular fourier p element for the analysis of membrane vibrations is presented the element s transverse displacement is written in terms of dimensionless area co ordinates and is described by three linear shape functions plus a variable number of trigonometric shape functions the three nodal displacements and the amplitudes of the trigonometric functions on the three edges and in the interior of the element respectively are used as generalized co ordinates inter element compatibility is achieved by matching the generalized co ordinates at the three nodes and the three edges results are obtained for a right isosceles triangular and a square membranes and comparisons are made with exact and linear triangular finite element solutions the results of both membranes confirm that the solutions converge very fast from above to the exact values as the number of trigonometric terms is increased and highly accurate values are obtained with the use of a very few terms the results also show that the triangular fourier p element produces a much higher accuracy than the linear triangular finite element with fewer system degrees of freedom

on the efficacy of the wavelet decomposition for high frequency vibration analyses abstract this paper reports the extraordinary ability of the wavelet decomposition for vibration analyses under the framework of rayleigh ritz method using a beam as an example daubechies wavelet scale functions are used as admissible functions for decomposing the flexural displacement of the structure along with the artificial springs at the boundary to predict vibration of an euler bernoulli beam in an extremely large frequency range it is shown that the use of wavelet basis allows reaching very high frequencies typically covering more than 1000 modes using conventional computational facility within the available numerical dynamics of the computers with no particular care needed for round off errors as a side benefit the use of spring boundary also allows handling any elastic boundary conditions through a dynamic contribution in the hamiltonian of the beam the wavelet decomposed approach combines the flexibility of the global methods and the accuracy of local methods by inheriting the versatility of the rayleigh ritz approach and the superior fitting ability of the wavelets numerical results on both free and forced vibrations are given in excellent agreement with predictions of classical methods

free vibration of skew mindlin plates by p version of f e m abstract accurate natural frequencies and mode shapes of skew plates with and without cutouts are determined by p version finite element method using integrals of legendre polynomials for p 1 14 the hierarchical plate element is formulated based on mindlin s plate theory including rotatory inertia effects and based on a skew co ordinate system non dimensional frequency parameter and mode shapes are presented for a range of skew angle aspect ratio a b thickness width ratio h b cutout dimensions and different boundary conditions the results were verified by comparison with those available in the open literature

optimal quotients of modular jacobians let n be a positive integer and f a normalized newform of weight two on 0 n attached to f is an optimal quotient a f of the jacobian j 0 n we prove two theorems concerning such optimal quotients a let denote the algebra of endomorphisms of j 0 n generated by the hecke operators and let f denote the ideal of fusion of f in if f denotes the dual abelian variety to a f then the canonical polarization of j 0 n induces a polarization f f a f we show that there is an embedding ker f j 0 n f whose cokernel is supported at maximal ideals of for which j 0 n is not two dimensional b if n is prime let c denote the subgroup of j 0 n generated by the divisor 0 on x 0 n mazur has shown that c is equal to the full torsion subgroup of j 0 n and that specialization modulo n induces an isomorphism of c with the group of connected components j 0 n of the characteristic n fibre of the neron model of j 0 n we prove that analogous results hold for every optimal quotient of prime conductor thereby generalizing results of mestre and oesterle who treated the case of strong weil curves and confirming william stein s refined eisenstein conjecture the key idea in the proof of these two theorems is encapsulated in corollary 2 5 below which allows us to apply multiplicity one results in a novel way to the study of optimal quotients

stability and vibration of shear deformable plates first order and higher order analyses abstract this work presents the highly accurate numerical calculation of the natural frequencies and buckling loads for thick elastic rectangular plates with various combinations of boundary conditions the reissener mindlin first order shear deformation plate theory and the higher order shear deformation plate theory of reddy have been applied to the plate s analysis the governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the total energy the solution is obtained by the extended kantorovich method this approach is combined with the exact element method for the vibration and stability analysis of compressed members which provides for the derivation of the exact dynamic stiffness matrix including the effect of in plane and inertia forces the large number of numerical examples demonstrates the applicability and versatility of the present method the results obtained by both shear deformation theories are compared with those obtained by the classical thin plate s theory and with published results many new results are given too

the application of symbolic computing to the hierarchical finite element method one difficulty associated with the hierarchical finite element method is that of accurately determining the integrals necessary to establish the stiffness and mass matrices of arbitrarily high order elements this paper describes how such integrals can be calculated exactly using symbolic computing results are presented which permit the construction of high order hierarchical finite elements applicable to a variety of problems involving two dimensional elasticity and plate and shell bending three dimensional solid or brick elements are not considered

free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first order shear deformation theory in this study free vibration of laminated skew plates was investigated discrete singular convolution dsc method is used for numerical solution of vibration problems the straight sided quadrilateral domain is mapped into a square domain in the computational space using a four node element by using the geometric transformation typical results are presented for different geometric parameters and boundary conditions it is concluded from the results that the skew angle have considerable influence on the variations of the frequencies with fibre orientation angle and number of layers in the laminate the results obtained by dsc method are compared with those obtained by analytical and numerical approaches it is shown that reasonable accurate results are obtained present work also indicates that the method of dsc is a promising and potential approach for computational mechanics copyright 2009 john wiley sons ltd

discrete singular convolution for the prediction of high frequency vibration of plates theoretical analysis of high frequency vibrations is indispensable in a variety of engineering designs much effort has been made on this subject in the past few decades however there is no single technique which can be applied with confidence to various engineering structures for high frequency predictions at present this paper introduces a novel computational approach the discrete singular convolution dsc algorithm for high frequency vibration analysis of plate structures square plates with six distinct boundary conditions are considered to validate the proposed method a completely independent approach the levy method is employed to provide exact solutions for a comparison the proposed method is also validated by convergence studies we demonstrate the ability of the dsc algorithm for high frequency vibration analysis by providing extremely accurate frequency parameters for plates vibrating in the first 5000 modes

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

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

vibration of skew plates resting on point supports abstract this paper deals with the vibrations of skew plates resting on point supports the spline element method is used to demonstrate the accuracy of the method several examples are solved and results are compared with those obtained by other approximate methods the effects of skew angles and locations of point supports on natural frequencies of isotropic skew plates are also investigated

an efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates abstract in this paper a simple and efficient mixed ritz differential quadrature dq method is presented for free vibration and buckling analysis of orthotropic rectangular plates the mixed scheme combines the simplicity of the ritz method and high accuracy and efficiency of the dq method the accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature it is shown that highly accurate results can be obtained using a small number of ritz terms and dq sample points the proposed method is suitable for the problem considered due to its simplicity and potential for further development

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

high accuracy mixed finite element ritz formulation for free vibration analysis of plates with general boundary conditions abstract a high order accurate mixed finite element ritz method is introduced and developed to study the vibration problem of plates with general boundary conditions the finite element method fem with higher order interpolation functions is first used to discretize the spatial partial derivatives with respect to a co ordinate direction of the plate the ritz method is then employed to analogize the resulting system of ordinary differential equations a novel technique is also presented to exactly satisfy the mixed natural boundary conditions the proposed method is applied here to solve some benchmark vibration problems of plates including isotropic and anisotropic rectangular plates variable thickness rectangular plates multi span rectangular plates and skew plates comparisons with existing numerical and analytical solutions show that the proposed mixed method can yield highly accurate results for vibration problem of plates involving free edges free corners and irregular boundaries using a small number of finite elements and ritz terms

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

first order shear deformation p version finite element for laminated plate nonlinear vibrations a p version hierarchical finite element for moderately thick composite laminated plates is presented where the effects of the rotary inertia transverse shear and geometrical nonlinearity are taken into account the time domain free vibration equations of motion are obtained by applying the principle of virtual work those equations are mapped into the frequency domain by the harmonic balance method and solved by a predictor corrector procedure the linear natural frequencies of vibration of several plates are determined and the convergence properties of the element are investigated it is shown that the element is not prone to shear locking and that a moderate number of degrees of freedom is sufficient for accuracy the influences of the plate s thickness of the width to length ratio and of the fiber orientation on nonlinear free vibrations are investigated

an efficient differential quadrature methodology for free vibration analysis of arbitrary straight sided quadrilateral thin plates in this paper a new differential quadrature dq methodology is employed to study free vibration of irregular quadrilateral straight sided thin plates a four nodded super element is used to map the irregular physical domain into a square domain in the computational domain second order transformation schemes with relative ease and less computation are employed to transform the fourth order governing equation of thin plates between the two domains the only degree of freedom within the domain is the displacement whereas along the boundaries the displacement as well as the second order derivative of the displacement with respect to associated normal co ordinate variable in computational domain are the two degrees of freedom implementing the method the formulation for the dq method for the free vibration analysis of plates of straight sided shapes was presented together with the implementation procedure for the different boundary conditions to demonstrate the accuracy convergency and stability of the new methodology detail studies are made on isotropic plates at acute angles with different geometries boundary and loading conditions including dq free edge boundary condition implementations accurate results even with fewer degrees of freedom than for those of comparable numerical algorithms were achieved

hierarchical finite element analyses of geometrically non linear vibration of beams and plane frames geometrically non linear vibrations of beams and plane frameworks are analyzed by the hierarchical finite element method hfem two main points are of interest the first is to compare polynomials trigonometric functions and beam eigenfunctions as displacement shape functions for beam hierarchical finite elements the second is to examine the suitability of the hfem for time domain non linear analyses

on the use of three dimensional h and p version finite elements in solving vibration response problems abstract the potential for reliably solving vibration response problems by using h and p versions of the finite element fe method is investigated each fe model is based on full three dimensional 3 d displacement based continuum theory special attention is given to the ability to handle thin and or nearly incompressible elastic and viscoelastic materials steady state time harmonic problems in the low and mid frequency ranges are treated convergence studies are performed on plate like model problems with simply supported boundary conditions using a series elasto dynamic navier solution extended to the viscoelasto dynamic case by means of the elasto viscoelastic correspondence principle the importance of eliminating the discretisation error when using numerical solutions to estimate the frequency dependent viscoelastic material parameters from experiments is stressed the superior efficiency of p enrichments compared to h refinements for resolving regular elastostatic solutions is observed in elasto viscoelasto dynamics as well requirements on p enrichments to avoid shear and volumetric locking are given the frequently used polynomial interpolation functions employed are shown to produce a mass contribution to the dynamic stiffness whose condition number deteriorates with increasing polynomial order the practical implications and limitations of this observation are outlined the recommended approach is corroborated using the measured frequency response on a laminate consisting of two aluminium plates and a constrained viscoelastic polymer damping treatment

vibrations of cantilevered skewed plates with corner stress singularities in conjunction with classical thin plate theory the ritz trial space of assumed transverse displacement functions consists of algebraic polynomials which are mathematically complete and corner functions which account for the singular behaviour of bending stresses at the reentrant corner

on the behavior of modules of m integrable derivations in the sense of hasse schmidt under base change abstract we study the behavior of modules of m integrable derivations of a commutative finitely generated algebra in the sense of hasse schmidt under base change we focus on the case of separable ring extensions over a field of positive characteristic and on the case where the extension is a polynomial ring in an arbitrary number of variables

on the modules of m integrable derivations in non zero characteristic abstract let k be a commutative ring and a a commutative k algebra given a positive integer m or m we say that a k linear derivation of a is m integrable if it extends up to a hasse schmidt derivation d id d 1 d 2 d m of a over k of length m this condition is automatically satisfied for any m under one of the following orthogonal hypotheses 1 k contains the rational numbers and a is arbitrary since we can take d i i i 2 k is arbitrary and a is a smooth k algebra the set of m integrable derivations of a over k is an a module which will be denoted by ider k a m in this paper we prove that if a is a finitely presented k algebra and m is a positive integer then a k linear derivation of a is m integrable if and only if the induced derivation p a p a p is m integrable for each prime ideal p a in particular for any locally finitely presented morphism of schemes f x s and any positive integer m the s derivations of x which are locally m integrable form a quasi coherent submodule ider s o x m der s o x such that for any affine open sets u spec a x and v spec k s with f u v we have u ider s o x m ider k a m and ider s o x m p ider o s f p o x p m for each p x we also give for each positive integer m an algorithm to decide whether all derivations are m integrable or not

a novel approach for the analysis of high frequency vibrations abstract despite much effort in the past few decades the numerical prediction of high frequency vibrations remains a challenging task to the engineering and scientific communities due to the numerical instability of existing computational methods however such prediction is of crucial importance to certain problems of pressing practical concern as pointed out by langley and bardell 1998 the aeronautical journal102 287 297 this paper introduces the discrete singular convolution dsc algorithm for the prediction and analysis of high frequency vibration of structures both a beam and two span plates are employed as test examples to demonstrate the capability of the dsc algorithm for high frequency vibration analysis a completely independent approach the levy method is employed to provide exact solutions for a cross validation of the proposed method the reliability of the dsc results is also validated by convergence studies remarkably extremely accurate and stable results are obtained in this work e g the relative dsc errors for the first 7100 modes of the beam and the first 4500 modes of the two span plates are all

periodic geometrically nonlinear free vibrations of circular plates abstract the geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi degree of freedom model which is based on thin plate theory and on von karman s nonlinear strain displacement relations the middle plane in plane displacements are included in the formulation and the common axisymmetry restriction is not imposed the equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in plane and transverse displacement fields are given by weighted series of spatial functions these spatial functions are based on hierarchical sets of polynomials which have been successfully used in p version finite elements for beams and rectangular plates and on trigonometric functions employing the harmonic balance method the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method convergence with the number of shape functions and of harmonics is analysed the numerical results obtained are presented and compared with available published results it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom internal resonances are found and the ensuing multimodal oscillations are described

on the inclusion principle for the hierarchical finite element method the inclusion principle provides a qualitative characterization of the eigenvalues of a matrix the principle has been shown to apply to systems described by a single hermitian matrix the most important of which being the real symmetric matrix self adjoint distributed systems when discretized by either the classical rayleigh ritz method or by the finite element method lead to algebraic eigenvalue problems described in terms of two real symmetric matrices the algebraic eigenvalues problem derived by the classical rayleigh ritz method possesses the embedding feature required by the inclusion principle but that derived by the finite element method in general does not this paper demonstrates that the inclusion principle can be extended to discretized systems derived by the hierarchical finite element method

a sector fourier p element applied to free vibration analysis of sectorial plates abstract a sector fourier p element is presented and applied to free vibration analysis of sectorial plates an important feature of this element is that it can describe the geometry of a sectorial plate exactly and is therefore suitable for this type of plate the element is formulated in terms of a fixed number of cubic polynomial shape functions plus a variable number of trigonometric hierarchical shape functions the cubic polynomial shape functions are used to describe the element s nodal d o f and the trigonometric hierarchical shape functions are used to give additional freedom to the edges and the interior of the element results are obtained for a number of sectorial plates with various boundary conditions and comparisons are made with exact and 16 d o f sector finite element solutions the results show that the solutions converge very quickly from above to the exact values as the number of trigonometric terms is increased and highly accurate values are obtained with the use of very few terms the results also show that the sector fourier p element gives a much higher accuracy than the 16 d o f sector finite element with far fewer system d o f

vibration and buckling of flat free free plates under non uniform in plane thermal stresses detailed consideration is given to the modes and frequencies of a free rectangular kirchoff plate subjected to in plane stresses generated by prescribed non uniform surface temperature distributions which are doubly symmetrical about the plate central axes physical understanding is sought of phenomena observed by previous investigators stress distributions corresponding to three different temperature distributions have first been studied and incorporated in a rayleigh ritz analysis to find natural frequencies and modes all frequencies change as the temperature changes some much more than others all eventually vanish one after the other as the temperature reaches certain critical positive and negative values at which the plate goes into statically unstable buckling modes whether the frequencies rise or fall with rising temperature at the plate centre depends on the relative magnitudes of pairs of positive and negative critical temperatures the modes of buckling at each pair of critical temperatures may differ greatly from one another and also from the vibration modes at zero temperature the relationship between the square of the frequency and the temperature is then no longer approximately linear although it is exactly so for certain simple in plane stress distributions conditions have nevertheless been identified under which it is a very good approximation to the actual frequencies of the heated plate over wide temperature ranges

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

a perturbation approach for the eigenfrequency analysis of separable kirchhoff thin plate problems on a rectangle we introduce a method for the estimation of the eigenfrequencies for separable kirchhoff thin plate problems on a rectangle particularly those involving the free boundary condition we believe that this paper is the first to give such a treatment for the low end of the spectrum the method is an adaptation of an asymptotic perturbation method used for the treatment of various beam problems and in the setting of the plate is shown to be a generalization and refinement of the asymptotic methods of bolotin and keller and rubinow the wave propagation method we compare our results with those of our own legendre tau approximation and where available with numerical results extant in the literature excellent agreement is found in each case

natural frequencies of rectangular plates with free edges abstract this note presents vibration analysis of isotropic rectangular plates with free edges by the rayleigh ritz method with b spline functions to show the accuracy of the present method the results are compared with existing results based on other numerical methods and found to be in good agreement accurate frequencies of rectangular plates are analyzed for different aspect ratios and boundary conditions the effects of poisson s ratio on natural frequencies of square plates with free edges are also investigated

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

a differential quadrature hierarchical finite element method using fekete points for triangles and tetrahedrons and its applications to structural vibration abstract a differential quadrature hierarchical finite element method dqhfem using fekete points was formulated for triangles and tetrahedrons and applied to structural vibration analyses first orthogonal polynomials on triangles and tetrahedrons that can be used as bases of the hierarchical finite element method hfem were derived and simple formulas of transforming one dimensional non uniform nodes to simplexes were presented then the non uniform nodes were used as initial guesses to solve the fekete points on simplexes through newton raphson s method together with the orthogonal polynomials new differential quadrature dq rules on simplexes were formulated using the hfem bases and the fekete points the numbers of nodes or bases on different edges and faces and inside the body of the new dq elements do not relate with each other like the hfem that can freely assign different numbers of bases on different edges and faces and inside the body so the new dq method was named as a differential quadrature hierarchical dqh method that uses either interpolation functions or orthogonal polynomials as bases inside the element its weak form was named as the dqhfem besides the dqh method and its weak form a simple method of generating high quality linear and high order triangular and tetrahedral meshes from a single nurbs patch was presented numerical tests of the dqhfem through structural vibration analyses showed that high accuracy results can be obtained using only a few nodes even on curvilinear domains using the dqh bases on both physical and geometric fields it was concluded that wide applications of the dqh method and the dqhfem to science and engineering are possible and commercial codes based on them are deserved to be developed