a new code for electrostatic simulation by numerical integration of the vlasov and ampere equations using maccormack s method abstract we present a new simulation code for electrostatic waves in one dimension which uses the vlasov equation to integrate the distribution function and ampere s equation to integrate the electric field forward in time previous vlasov codes used the vlasov and poisson equations using ampere s equation has two advantages first boundary conditions do not have to be set on the electric field second it forms a logical basis for an electromagnetic code since the time integration of the electric and magnetic fields is treated in a similar way maccormack s method is used to integrate the vlasov equation which was found to be easy to implement and reliable a stability analysis is presented for the maccormack scheme applied to the vlasov equation conditions for stability are more stringent than the simple courant friedrich s lewy cfl conditions for the spatial and velocity grids we provide a simple linear function which when combined with the cfl conditions should ensure stability simulation results for landau damping are in excellent agreement with numerical solutions of the linear dispersion relation for a wide range of wavelengths the code is also able to retain phase memory as demonstrated by the recurrence effect and reproduce the effects of particle trapping the use of ampere s equation enables standing and traveling waves to be produced depending on whether the current is zero or non zero respectively in simulations where the initial current is non zero and maxwell s equations are satisfied initially additional standing waves may be set up which could be important when the coupling of wave fields from a transmitter to a plasma is considered

discrete singular convolution and its application to the analysis of plates with internal supports part 2 applications part 2 of this series of two papers presents the applications of the discrete singular convolution dsc algorithm the main purpose of this paper is to explore the utility test the accuracy and examine the convergence of the proposed approach for the vibration analysis of rectangular plates with internal supports both partial internal line supports and complex internal supports are considered for 21 square plates of various combinations of edge support conditions the effects of different size shape and topology of the internal supports and different boundary conditions on the vibration response of plates are investigated the partial internal line supports may vary from a central point support to a full range of cross or diagonal line supports several closed loop supports such as ring square and rhombus and their combinations are studied for complex internal supports convergence and comparison studies are carried out to establish the correctness and accuracy of the dsc algorithm the dsc results are compared with those in the available literature obtained by using other methods numerical results indicate that the dsc algorithm exhibits controllable accuracy for plate analysis and shows excellent flexibility in handling complex geometries boundary conditions and support conditions copyright 2002 john wiley sons ltd

an analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports abstract a general analytical solution is obtained for the free vibration of rectangular plates supported on their lateral surface by symmetrically distributed point supports highly accurate eigenvalues are provided for a square plate supported at various points along the diagonal results are compared with earlier findings obtained by a finite element method it is shown that the method may be extended to solve various other problems the tabulation of eigenvalues appears to be the most comprehensive yet made for a square plate with point supports on the diagonals

dynamic stability of rotating blades beams eccentrically clamped to a shaft with fluctuating speed abstract in plane bending vibrations of a beam rotating with a periodically fluctuating speed are considered the partial differential equation of motion is discretisized via galerkin s method and a set of mathieu hill equations is obtained the constant speed rotation problem is reviewed and its reflections onto the fluctuating speed problem are underlined dynamic stability analysis is performed via a monodromy matrix method and a generalized bolotin method examples of stability charts are worked out reflecting stability s dependence on various pairs of system parameters

the influence of concentrated masses and pasternak soil on the free vibrations of euler beams exact solution abstract the free vibration frequencies of a beam with flexible ends resting on pasternak soil are determined in the presence of a concentrated mass at an arbitrary intermediate abscissa the differential equation of motion is deduced and solved and the resulting frequency equation gives the exact frequencies of the system some numerical examples and comparisons end the paper

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

conservative numerical schemes for the vlasov equation abstract a new scheme for solving the vlasov equation using a phase space grid is proposed the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity several numerical results are presented in two and four dimensional phase space and the scheme is compared with the semilagrangian method this method is almost as accurate as the semilagrangian one and the local reconstruction technique is well suited for parallel computation

natural frequencies of rectangular plates using characteristic orthogonal polynomials in rayleigh ritz method natural frequencies of rectangular plates are obtained by employing a set of beam characteristic orthogonal polynomials in the rayleigh ritz method the orthogonal polynomials are generated by using a gram schmidt process after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods the method yields superior results for lower modes particularly when plates have some of the edges free

transverse vibration of simply supported elliptical and circular plates using boundary characteristic orthogonal polynomials in two variables abstract the first four frequencies and mode shapes have been computed for elliptical and circular plates of uniform thickness with simply supported edges by employing orthogonal polynomials in the rayleigh ritz method in general this is found to give better results than traditional methods successive approximations have been worked out to ensure convergence comparisons have been made with existing results in some cases calculations have been done for several values of b a i e the ratio of the semi major and semi minor axes mode shapes have been used to obtain three dimensional plots of plates in displaced configurations

flexural vibration of axially loaded beams with linear or parabolic taper abstract curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections previous curves have only included axial forces in a very limited number of cases whereas here axial forces within the range of greatest practical importance can always be allowed for the present curves cover 11 combinations of end conditions three types of taper all taper ratios between 0 2 and 1 and all axial forces between p c and 0 6 p c where p c is the critical buckling load for pure compression and can be found from curves provided a list of principal nomenclature is given in appendix 2 the cross sections covered include thin circular ones of constant thickness and thin walled cross sections consisting of constant thickness flat plates which all have their breadths tapering in the same way truncated wedges of varying depth and constant width are also covered the taper is linear between the ends or linear between a maximum value at the centre and equal values at the ends or parabolic with a maximum value at the centre simple examples show how the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1 the curves also illustrate how natural frequencies and buckling loads of thin walled members are altered by changing the amount and type of a member s taper while keeping its wall thickness and mass constant finally it is shown that for the ranges covered by the curves frequencies which separate the first five natural frequencies can be found a priori the theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy exact bernoulli euler dynamic stiffnesses which allowed for axial force effects were used for these uniform components in appendix 1 the effects of using exact timoshenko stiffnesses instead of the bernoulli euler ones are illustrated

free vibration analysis of a cylindrical shell circular plate system with general coupling and various boundary conditions abstract the free vibration of a structure consisting of a finite circular cylindrical shell closed at one end by a circular plate is analyzed in this paper emphasis is given to the characterization of structural coupling and boundary conditions these are incorporated into the model by means of continuous distributions of springs along the shell and the plate interface a general formulation based on a variational principle is used this formulation allows a wide spectrum of boundary conditions and coupling conditions between the shell and the plate an issue the importance of which is clearly shown by a literature review very good accuracy of the method is demonstrated by solving test problems for lower order modes of the plate and of the shell for which some results are available in the literature comparisons are also made using finite element analysis on a plate ended shell showing that the proposed approach is a convenient effecient and accurate one for determining the modal behavior of such a complex structural system other numerical results are then presented for a shell rigidly attached to a plate to illustrate the coupling phenomena between the shell and the plate it is shown that there exist three different types of modes for this combined structure they are termed plate controlled modes shell controlled modes and strongly coupled modes respectively it is also shown that each type is closely related to the modal character of each of uncoupled elements

dynamics of multi span continuous straight bridges subject to multi degrees of freedom moving vehicle excitation abstract the paper presents an analytical approach to the problem of vehicle bridge dynamic interaction starting from early studies based on a simply supported beam interacting with a lumped mass moving at constant speed in recent years researchers have improved the models of both the bridge and the vehicle on this basis the bridge is modelled here as a multi span continuous isotropic plate its response to external loads is defined by applying the mode superposition principle and takes into account both flexural and torsional mode shapes the latter being usually neglected in the literature the plate is considered proportionally damped and its modes are computed by means of the rayleigh ritz method the scheme adopted for the vehicle consists of a seven degrees of freedom system moving at constant speed over the isotropic rough bridge surface the numerical investigation based on these analytical models refers to a three span bridge and includes the importance of torsional mode shapes of road surface irregularities and of vehicle speed

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

minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by rayleigh ritz method abstract the rayleigh ritz method is employed to determine the minimum stiffness location of the elastic point support for raising the fundamental natural frequency of a rectangular plate to the second frequency of the unsupported plate which usually is the upper limit of the first frequency for a single support based on the optimal design of the support position the minimum stiffness can be obtained numerically by solving a characteristic eigenvalue sub problem in the rayleigh ritz procedure the boundary characteristic orthogonal polynomials are used for the admissible functions several typical examples of plate structures with the additional point supports are analyzed in detail and the results prove that the proposed method is very effective in the solution to the optimal design of the supports it will be shown that elastic supports can be designed like rigid ones for the purpose of increasing the fundamental natural frequency of a rectangular plate

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

the free vibration of a type of tapered beams this paper studies the vibrational characteristics of tapered beams with continuously varying rectangular cross section of depth and breadth proportional to xs and xt respectively where both s and t are arbitrary real numbers for a truncated beam and arbitrary positive numbers for a sharp ended beam and x is the axial co ordinate measured from the sharp end of the beam the bernoulli euler theory of bending is used to describe the motion of the beam a new set of beam functions are developed as the admissible functions which are the complete solution of the tapered beam under an arbitrary static load expanded into a taylor series the eigenfrequency equation is obtained by the rayleigh ritz method the accuracy is assured from the convergency and comparison studies the effect of the location of the taylor series expanding point on the convergency is discussed the analysis shows that the present approach is convergent for arbitrary truncation factor by taking the midpoint of the beam as the expanding point of the taylor series numerical results are tabulated for three different tapered beams with various boundary conditions and truncation factors it is shown that the eigenfrequencies can be obtained with high accuracy by using only a small number of terms of the static beam functions 2000 elsevier science s a all rights reserved

arbitrarily high order convected scheme solution of the vlasov poisson system abstract the convected scheme cs is a forward trajectory semi lagrangian method for solution of transport equations which has been most often applied to the kinetic description of plasmas and rarefied neutral gases in its simplest form the cs propagates the solution forward in time by advecting the so called moving cells along their characteristic trajectories and by remapping them on the mesh at the end of the time step the cs is conservative positivity preserving simple to implement and it is not subject to time step restriction to maintain stability recently guclu and hitchon 2012 1 a new methodology was introduced for reducing numerical diffusion based on a modified equation analysis the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping while the spatial accuracy was increased from 2nd to 4th order the new scheme retained the important properties of the original method and was shown to be extremely simple and efficient for constant advection problems here the cs is applied to the solution of the vlasov poisson system which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal coulomb interactions the vlasov equation is split into two constant advection equations one in configuration space and one in velocity space and high order time accuracy is achieved by proper composition of the operators the splitting procedure enables us to use the constant advection solver which we extend to arbitrarily high order of accuracy in time and space a new improved procedure is given which makes the calculation of the corrections straightforward focusing on periodic domains we describe a spectrally accurate scheme based on the fast fourier transform the proposed implementation is strictly conservative and positivity preserving the ability to correctly reproduce the system dynamics as well as resolving small scale features in the solution is shown in classical 1d 1v test cases both in the linear and the non linear regimes

free vibration analysis of a type of tapered beams by using adomian decomposition method abstract the adomian decomposition method adm is employed in this paper to investigate the free vibrations of tapered euler bernoulli beams with a continuously exponential variation of width and a constant thickness along the length under various boundary conditions based on adm the governing differential equation for the tapered beam becomes a recursive algebraic equation by using the boundary condition equations the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously the computed results for different boundary conditions and non uniformity ratios are presented the accuracy is assured from the convergence and comparison published results

in plane free vibration of circular annular disks abstract this paper presents a generalized formulation for the in plane modal characteristics of circular annular disks under combinations of all possible classical boundary conditions the in plane free vibration of an elastic and isotropic disk is studied on the basis of the two dimensional linear plane stress theory of elasticity the boundary characteristic orthogonal polynomials are employed in the rayleigh ritz method to obtain the natural frequencies and associated mode shapes two approaches have been used to represent a clamped boundary condition the first approach assumes a polynomial expression that satisfies the clamped conditions while the second approach uses a disk with free boundary supported on artificial springs with stiffness tending to infinity the natural frequencies are tabulated and compared with data available in the literature mode shapes are presented to illustrate the free vibration behavior of the disk

a high order time splitting method based on integral deferred correction for semi lagrangian vlasov simulations semi lagrangian schemes with various splitting methods and with different reconstruction interpolation strategies have been applied to kinetic simulations for example the order of spatial accuracy of the algorithms proposed in qiu and christlieb 2010 29 is very high as high as ninth order however the temporal error is dominated by the operator splitting error which is second order for strang splitting it is therefore important to overcome such low order splitting error in order to have numerical algorithms that achieve higher orders of accuracy in both space and time in this paper we propose to use the integral deferred correction idc method to reduce the splitting error specifically the temporal order accuracy is increased by r with each correction loop in the idc framework where r 1 2r 1 2 for coupling the first order splitting and the strang splitting respectively the proposed algorithm is applied to the vlasov poisson system the guiding center model and two dimensional incompressible flow simulations in the vorticity stream function formulation we show numerically that the idc procedure can automatically increase the order of accuracy in time we also investigate numerical stability of the proposed algorithm via performing fourier analysis to a linear model problem

free vibration analysis of a uniform cantilever beam with point masses by an analytical and numerical combined method abstract the natural frequencies and mode shapes of a uniform cantilever beam carrying any number of concentrated masses were determined by using an analytical and numerical combined method anc method the eigenvalue equation was first derived analytically by using the expansion theorem and then the eigenvalues and eigenvectors were calculated numerically in comparison with the general finite element method fem the anc method has the advantage that there is no necessity to derive the property matrices of each beam element and then to develop the overall ones of the entire beam in comparison with the transfer matrix method tmm the anc method requires no tedious matrix multiplication so that some computer time may be saved the anc method is also better than the pure analytical closed form method since instead of a few special cases a uniform beam carrying any number of point masses of various magnitudes or distributions along the beam length can be easily solved besides by using the anc method there is no difficulty in taking more modes in the mode superposition equation and hence more higher mode natural frequencies of better accuracy may be obtained

axisymmetric vibration of circular plates and its analog in elliptical plates using characteristic orthogonal polynomials the axisymmetric vibration of circular plates and its analog in elliptic plates with clamped simply supported and free boundary conditions are investigated using boundary characteristic orthogonal polynomials in the rayleigh ritz method modified polar coordinates are employed to analyze the vibration of elliptical plates with circular plates as a special case axisymmetric vibration of circular plates and its analog in elliptical plates with concentric nodal elipses require only one dimensional shape functions in the rayleigh ritz method the first six natural frequencies and the parameters associated with nodal ellipses are tabulated

a new conservative unsplit method for the solution of the vlasov equation we have developed a new conservative method for solving the vlasov equation without using any splitting technique our goal is to maintain the positivity of the distribution function and to avoid un physical oscillations which might lead to numerical instabilities based on a finite volume conservative discretization of the conservative form of the vlasov equation we implemented a highly accurate second order upwind scheme in order to avoid un physical oscillations and their possible numerical instability we apply a flux limiter in the second order we validate our new vlasov solver by considering standard cases of one dimensional current driven ion acoustic instabilities solving a vlasov ampere set of equations

a newtonian procedure for the solution of the kron characteristic value problem abstract in this paper a quadratically convergent newtonian procedure is presented to determine the latent roots of the kron matrix the roots are found in ascending order with the certainty that none has been missed since the procedure is allied to a sturm sequence algorithm the procedure has been programmed and it performs at least 50 faster than the bisection scanning method available for the kron matrix examples of successful use of the procedure to solve for the natural frequencies of frames with large numbers of degrees of freedom are presented

free vibration of rectangular plates with internal column supports a direct method is proposed to derive the exact solution for the free vibration of a thin rectangular plate with two opposite edges simply supported and with internal column supports considering the compatibility of displacements and rotations between the plate and the columns the coupled vibration of a plate column system is derived the accuracy and correctness of the present method are demonstrated through the comparison of the results obtained from the proposed method and from finite element analysis then the effects of column sizes and column models on the natural frequencies of a plate column system are investigated in detail a parametric study is focused on fully simply supported rectangular plates with a single internal column it is shown that the effect of column flexibility in plate column structures should be considered the solution provided in the paper is general and includes several particular solutions for fixed point supports pinned point supports and elastic point supports results with high accuracy have been obtained and these can be used as the benchmark for the further investigation and for other approximate methods

free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method abstract starting from the governing differential equations of motion in free vibration the dynamic stiffness matrix of a uniform rotating bernoulli euler beam is derived using the frobenius method of solution in power series the derivation includes the presence of an axial force at the outboard end of the beam in addition to the existence of the usual centrifugal force arising from the rotational motion this makes the general assembly of dynamic stiffness matrices of several elements possible so that a non uniform or tapered rotating beam can be analyzed for its free vibration characteristics by idealizing it as an assemblage of many uniform rotating beams the application of the derived dynamic stiffness matrix is demonstrated by investigating the free vibration characteristics of uniform and non uniform tapered rotating beams with particular reference to the wittrick williams algorithm the results from the present theory are compared with published results it is shown that the proposed dynamic stiffness method offers an accurate and effective method of free vibration analysis of rotating beams

a finite beam element for vibration analysis of rotating tapered timoshenko beams a finite beam element for vibration analysis of a rotating tapered beam including shear deformations and rotary inertia is derived in this paper the finite element has four degrees of freedom and accounts for linear tapering in two planes this formulation permits any combination of taper ratios as well as unequal element lengths explicit expressions for the finite element mass and stiffness matrices are derived using the consistent mass approach while accounting for the centrifugal stiffness effects the generalized eigenvalue problem is defined and numerical solutions are generated for a wide range of rotational speed and taper ratios results obtained include the first ten free vibrational modes for both fixed and hinged end conditions comparisons are made whenever possible with exact solutions and with numerical results available in the literature the results display high accuracy when compared with other numerical results

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

further flexural vibration curves for axially loaded beams with linear or parabolic taper curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections these cross sections complement those of an earlier paper such that the curves in the two papers cover all of the most common types of tapered member the curves presented cover eleven combinations of end conditions three types of taper all taper ratios between 0 3 and 1 and all axial forces between pc and 0 6pc where pc is the critical buckling load for pure compression and can be found from curves provided the results cover truncated wedges which have their depth and width varying identically so that all cross sections are similar in the sense that similar triangles are the cross sections covered also include thin circular ones with thickness proportional to diameter and thin walled cross sections consisting of flat plates which all have their breadths and thicknesses tapering identically the taper is linear between the ends or linear between a maximum value at the centre and equal values at the ends or parabolic with a maximum value at the centre the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1 the curves also illustrate how natural frequencies and buckling loads are altered by changing the amount and type of a member s taper while keeping its mass constant finally it is hown that for the ranges covered by the curves frequencies which separate the first five natural frequencies can be found a priori the theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy exact bernoulli euler dynamic stiffnesses which allowed for axial force effects were used for these uniform components illustrative results show the effects of using exact timoshenko stiffnesses instead of the bernoulli euler ones

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

a note on the vibrations of restrained beams and rods with point masses abstract this paper is concerned with the approximate determination of the fundamental frequency and first mode shape of a beam or rod to which springs and point masses are attached after formulating the general eigenvalue problem which forms the basis of the analysis the theory is applied to three examples and it is shown that the results are quite satisfactory

analytical and experimental investigation on vibrating circular plates with stepped thickness over a concentric circular region transverse vibrations of the structural elements described in the title are studied for the case where the edges are elastically restrained against translation and rotation since finding an exact solution is a difficult task it was considered convenient to approximate the response of the plate in the case of free axisymmetric vibrations by means of a summation of simple polynomial coordinate functions that satisfy the governing boundary conditions the ritz method is used in order to generate the frequency equation the natural frequency coefficients are optimized by minimizing each eigenvalue with respect to an undetermined exponential parameter included in each coordinate function

free vibration analysis by the superposition method of rectangular mindlin plates with internal columns resting on uniform elastic edge supports free vibration analysis of rectangular mindlin plates is carried out via the superposition method and is shown to produce accurate results the analytical method proposed by the authors in this paper solves the problem for the case where the plate has simultaneous elastic edge and internal supports the conditions of edge supports are uniform lateral rotational and torsional elastic supports whereas the internal supports are column supports with finite area compatibility between the plate and column is achieved by requiring that the column and plate rotations be equal the results presented herein are verified through comparison with results presented by others numerical examples presented in this study confirm that the analytical method is able to model the property of the elastic edge supports while simultaneously considering the effect of column restraint an effect which increases with column number and area

arbitrary order time accurate semi lagrangian spectral approximations of the vlasov poisson system abstract the vlasov poisson system modeling the evolution of non collisional plasmas in the electrostatic limit is approximated by a semi lagrangian technique spectral methods of periodic type are implemented through a collocation approach groups of particles are represented by the fourier lagrangian basis and evolve for a single timestep along a high order accurate representation of the local characteristic lines the time advancing technique is based on truncated taylor series that can be in principle of any order of accuracy a variant is obtained by coupling the phase space discretization with high order accurate backward differentiation formulas bdf at each timestep particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs thus these techniques combine the excellent features of spectral approximations with high order time integration the resulting method has excellent conservation properties indeed it can be proven that the total number of particles proportional to the total mass and charge is conserved up to the machine precision series of numerical experiments are performed in order to assess the real performance in particular comparisons with standard benchmarks are examined

inference based on ranks for the multiple design multivariate linear model abstract multivariate models more general than the standard multivariate linear model have received considerable attention in both the statistical and econometric literature see srivastava 1966 1967 1968 and kleinbaum 1973 the multiple design multivariate mdm linear model generalizes the standard multivariate linear model in the sense that a different design matrix is used for each response dependent variable important applications of the mdm linear model include changing covariates in repeated measurement designs and p variate regression systems p 1 with different regressors for each of the p response variables the latter situation has been described as seemingly unrelated regression equations in the econometric literature zellner 1962 in this article an approach to statistical inference is presented under the mdm linear model based on multivariate rank and aligned rank statistics in section 1 estimation and hypothesis testing for the mdm linear model are reviewed under parametric

stiff string approximations in rayleigh ritz method for rotating beams the governing differential equation of the rotating beam reduces to that of a stiff string when the centrifugal force is assumed as constant the solution of the static homogeneous part of this equation is enhanced with a polynomial term and used in the rayleighs method numerical experiments show better agreement with converged finite element solutions compared to polynomials using this as an estimate for the first mode shape higher mode shape approximations are obtained using gram schmidt orthogonalization estimates for the first five natural frequencies of uniform and tapered beams are obtained accurately using a very low order rayleigh ritz approximation

the rayleigh ritz method still competitive abstract an extended rayleigh ritz method for computing two sided eigenvalue bounds of the one dimensional schrodinger equation is presented the method is based on the b spline approximation over the truncated domain for the low lying eigenvalues and problems with rapidly increasing spectrum the corresponding lower bounds are much more accurate than the known temple bounds numerical results for problems exhibiting different kinds of the eigenfunction behaviour are presented

vibration of complex structures the modal constraint method abstract a method is described for establishing the natural frequencies of an arbitrary structure with arbitrary supports the method is based on the modal constraint technique described in a previous paper 1 as shown in the present paper weinstein s theory for the intermediate problem can be regarded as equivalent to the lagrangian multiplier method i e both methods result in the same eigenvalue equations weinstein s theory deals with modifications of base differential operators whereas the lagrangian multiplier method deals with modifications of base energy functionals the modal constraint technique is an extension of weinstein s theory or in energy terms the generalized fourier expansion of the lagrangian multiplier the merits of this method lie in the fact that the eigenvalues and eigenfunctions of structures are used as base structures the coupling of these structures are taken into account by lagrangian generalized forces of the constraint acting on the base structures some examples are given and the results compared with known solutions

generalized power law exponent based shear deformation theory for free vibration of functionally graded beams free vibration of functionally graded fg beams is presented based on a newly proposed shear deformation theory pesdbt the material properties of the beam vary along the thickness direction according to the power law form rayleigh ritz method is used in the modeling to obtain the eigenfrequency equation relationships of different shear deformation theories with pesdbt have also been noticed and incorporated here results in special cases are compared and new results are also reported present study deals with free vibration of functionally graded fg beams subject to different sets of boundary conditions the analysis is carried out on the basis of a newly proposed generalized power law exponent based shear deformation beam theory pesdbt the proposed theory exactly satisfies the transverse stress boundary conditions on the bottom and top surfaces of the beam material properties of the beam vary continuously in the thickness direction according to the power law exponent form the displacement components of the cross sections of the beam are expressed in simple algebraic polynomials rayleigh ritz method is used to estimate frequency parameters in order to handle various sets of boundary conditions at the edges the objective is to study the effects of constituent volume fractions slenderness ratios and the beam theories on the natural frequencies new results for frequency parameters are also incorporated after checking the convergence pattern and validation of results computed for previously published shear deformation beam theories

free vibration of rotating tapered beams using the dynamic stiffness method abstract the free bending vibration of rotating tapered beams is investigated by using the dynamic stiffness method the range of problems considered includes beams for which the depth and or width of the cross section vary linearly along the length first the governing differential equation of motion of the rotating tapered beam in free flap bending vibration is derived for the most general case using hamilton s principle allowing for the effects of centrifugal stiffening an arbitrary outboard force and the hub radius term for harmonic oscillation the differential equation is solved for bending displacement by applying the frobenius method of series solution the expressions for bending rotation shear force and bending moment at any cross section of the beam are also obtained in explicit analytical form then the dynamic stiffness matrix is developed by relating the amplitudes of forces and moments to those of the displacements and rotations at the ends of the harmonically vibrating tapered beam next the wittrick williams algorithm is used as a solution technique to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of some illustrative examples a parametric study is carried out to demonstrate the effects of rotational speed taper ratio and hub radius on the results which are discussed and compared with the published ones finally some conclusions are drawn

discontinuous galerkin methods for the vlasov maxwell equations discontinuous galerkin methods are developed for solving the vlasov maxwell system methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy such properties in general are hard to achieve within other numerical method frameworks for simulating the vlasov maxwell system the proposed scheme employs discontinuous galerkin discretizations for both the vlasov and the maxwell equations resulting in a consistent description of the distribution function and electromagnetic fields it is proven up to some boundary effects that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the maxwell equations and the underlying approximation spaces error estimates are established for several flux choices the scheme is tested on the streaming weibel instability the order of accuracy and conservation properties of the proposed method are verified

analysis of repeated measures designs with changing covariates in clinical trials summary a multivariate model is proposed for analysing repeated measures designs with chang ing or within subject covariates under the assumption of multivariate normality the likelihood ratio tests are derived for testing certain linear hypotheses some special designs are discussed and a numerical example is given to illustrate the method for analysing a two period crossover design during a long term clinical trial environmental factors can affect a disease condition in different ways air pollution is likely to influence the frequency of asthma attacks in patients with chronic bronchial asthma the effectiveness of a therapeutic drug in the treatment of atherosclerosis is probably influenced by factors such as diet exercise and smoking response measurements obtained periodically during a trial are affected jointly by the treatment being received and by certain environmental factors if the influencing factors are quantified over the periods between successive clinic visits a repeated measures model can be constructed to accommodate such information in a between patient i e parallel group design each patient receives the assigned treatment throughout the trial in a within patient design each patient receives some or all of the treatments in an assigned order crossover and incomplete block designs are examples of a within patient design a fixed or between patient covariate remains unchanged during the trial a changing or within patient covariate varies from one visit to the next for the same patient the covariates associated with environmental factors fall into this category in a within patient design the baseline obtained just before the start of each treatment period also falls into this category a similar situation occurs in a between patient design when the responses at successive visits are evaluated relative to the corresponding baseline performances for example in a parallel group study designed to evaluate once a day short acting therapies in the treatment of chronic stable angina treadmill walking time is recorded just before the administration of a dose and then at some post dose time at each clinic visit the effectiveness of the visit doses is evaluated relative to the corresponding baseline walking times however there must be a strong evidence prior to study initiation that the baseline values are independent of the assigned treatments whenever baseline is to be used as a changing covariate the situations described above are special cases of the multiple design multivariate linear model studied by roy srivastava 1963 and srivastava 1967 this model in

on the analysis of laterally vibrating slender beams subject to various complicating effects the rayleigh ritz method with orthogonally generated polynomials as admissible functions is used to obtain a simple analysis of the free vibration problem of slender beams subject to various complicating effects these effects are the presence of concentrated masses and or concentrated moments of inertia intermediate supports axial loading and non uniformity of cross section such may exist separately or in any combination numerical results are presented for several example beams illustrating the accuracy and versatility of the approach

on the flexural vibration of rectangular plates approached by using simple polynomials in the rayleigh ritz method abstract sets of simple polynomials are proposed for use as admissible functions in the rayleigh ritz method for the study of the flexural vibration of rectangular plates the plates may be subject to a number of different complicating factors including the presence of in plane forces concentrated masses point and line spring supports and special orthotropy of the plate material the sets of polynomials are shown to be equivalent to the orthogonally generated polynomials used very successfully in earlier studies but are more easily generated and the subsequent evaluation of the integrals in the resulting eigenvalue problem are more simply performed numerical results are presented for several problems illustrating the applicability of the method and its accuracy

a perturbation approach for evaluating natural frequencies of moderately thick elliptic plates natural frequencies for moderately thick elliptic plates are calculated by perturbing initial values corresponding to the kirchhoff classical theory of plate bending the proposed approach utilizes a universal algebraic equation for perturbed eigenvalues previously derived by the authors for elliptic plates of a small eccentricity a dual procedure is developed to evaluate the sought for natural frequencies starting from those for a circular thin plate a comparison with finite element computations is presented an importance of the perturbation techniques in question for interpreting of finite element data is emphasized

free vibration analysis of functionally graded size dependent nanobeams abstract this paper presents free vibration analysis of functionally graded fg size dependent nanobeams using finite element method the size dependent fg nanobeam is investigated on the basis of the nonlocal continuum model the nonlocal elastic behavior is described by the differential constitutive model of eringen which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators the material properties of fg nanobeams are assumed to vary through the thickness according to a power law the nanobeam is modeled according to euler bernoulli beam theory and its equations of motion are derived using hamilton s principle the finite element method is used to discretize the model and obtain a numerical approximation of the equation of motion the model is validated by comparing the obtained results with benchmark results numerical results are presented to show the significance of the material distribution profile nonlocal effect and boundary conditions on the dynamic characteristics of nanobeams

numerical study of the two species vlasov ampere system in this paper we propose energy conserving eulerian solvers for the two species vlasov ampere va system and apply the methods to simulate current driven ion acoustic instability the two species va systems are of practical importance in applications and they conserve many physical quantities including the particle number of each species and the total energy that is comprised of kinetic energy for both species and the electric energy the main goal of this paper is to generalize our previous work for the single species va system 9 and vlasov maxwell vm system 8 to the two species case the methodologies proposed involve careful design of temporal discretization and the use of the discontinuous galerkin dg spatial discretizations we show that the energy conserving time discretizations for single species equations 9 8 can also work for the two species case if extended properly compared to other high order schemes we emphasize that our schemes can preserve the total particle number and total energy on the fully discrete level regardless of mesh size making them very attractive for long time simulations we benchmark our algorithms on a test example to check the one species limit and the current driven ion acoustic instability to simulate the current driven ion acoustic instability a slight modification for the implicit method is necessary to fully decouple the split equations this is achieved by a gauss seidel type iteration technique numerical results verified the conservation and performance of our methods finally we remark that the schemes in this paper can be readily extended to applications when the models take more general form such as the multi species vm equations

on the free transverse vibration of annular and circular thin sectorial plates subject to certain complicating effects abstract the rayleigh ritz method with orthogonally generated polynomials as admissible functions is used to obtain an eigenvalue equation for the free transverse vibration of thin annular sectorial plates circular sectorial plates are treated with this equation by permitting the inner radius to become negligibly small the plates may be of isotropic or polar orthotropic material of uniform or radially and or circumferentially varying thickness and may include interior circumferential or radial line supports numerical values for the natural frequency parameters of several different plates are presented and compared with previously published values where available

exact bernoulli euler dynamic stiffness matrix for a range of tapered beams bernoulli euler theory and bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial torsional and flexural vibrations of any beam which is tapered such that a varies as yn and gj and i both vary as y n 2 where a gj and i have their usual meanings y cx l 1 c is a constant such that c 1 l is the length of the beam and x is the distance from one end of the beam numerical checks give better than seven figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements a procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped this enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members

transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the rayleigh ritz method abstract natural frequencies and mode shapes of a rotating uniform cantilever beam with a tip mass are studied by using beam characteristic orthogonal polynomials in the rayleigh ritz method the set of orthogonal polynomials which satisfy the geometrical boundary conditions are generated by using the gram schmidt process the results are compared with those obtained by the myklestad method the extended galerkin method and finite element methods the variation of natural frequencies with the speed of rotation is plotted for several parameter combinations such as setting angle tip mass moment of inertia of tip mass etc mode shapes at different rotational speeds are also plotted use of orthogonal polynomials for the deflection shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any numerical difficulties which is not the case when arbitrary polynomial expressions are used

modal analysis of cantilever plates undergoing accelerated in plane motion a modeling method for the modal analysis of cantilever plates undergoing accelerated in plane motion is presented in this paper von karman strain measures are employed to derive the in plane and the lateral equations of motion in plane strain measures of the accelerated plates are obtained from the in plane equations and substituted into the lateral equations to obtain the linear equations for modal analysis the resulting equations are transformed into dimensionless forms in which dimensionless parameters are identified the effects of dimensionless parameter variations on the modal characteristics of the accelerated plates are investigated through numerical studies

vibration frequencies of a rotating tapered beam with end mass abstract the natural frequencies of vibration of a rotating tapered beam with tip mass are investigated explicit expressions for the finite element mass and stiffness matrices are derived by using a consistent mass formulation the beam is assumed to be linearly tapered in two planes the generalized eigenvalue problem is defined and numerical solutions are generated for a wide range of rotational speed and tip mass variations both fixed and hinged end conditions are considered comparisons are made whenever possible with exact solutions and numerical results available in the literature the results display high accuracy when compared with the exact solutions

a note on transverse vibrations of continuous beams subject to an axial force and carrying concentrated masses abstract polynomial co ordinate functions are used in order to obtain an approximate solution to the title problem the algorithmic procedure is quite simple leads to adequate engineering accuracy and can be easily extended to the case of several intermediate supports

on the lateral vibration of thin annular and circular composite plates subject to certain complicating effects abstract series comprising orthogonally generated polynomial functions are used in the rayleigh ritz method to obtain an eigenvalue equation for the flexural vibration of thin flat annular and circular plates the plates may be of isotropic or polar orthotropic material of uniform or radially varying thickness and may pass over intermediate concentric ring supports numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

analysis of repeated measures designs with changing covariates summary a model proposed by patel 1986 is re examined it is shown that the model can be written as seemingly unrelated regressions and estimation can be carried out simply estimation under constraints given by the general linear hypothesis is also discussed and a two stage procedure proposed connexions with seemingly unrelated regressions with constraints between the regression parameters and the sum of profiles model of verbyla venables 1988 are also given

vibration of an euler bernoulli beam of constant depth and with linearly varying breadth abstract the transverse vibration of a homogeneous euler bernoulli beam of constant depth and linearly varying breadth is investigated in this paper a direct analytical solution is presented for the mode shape equation the constraints imposed by the boundary conditions yield the frequency equation the four ideal boundary conditions considered are clamped pinned sliding and free for a complete beam sharp ended it is a necessary condition that the sharp end remain free and as a result the frequency equations for clamped pinned sliding or free boundary conditions at the large end are the determinant of a 2 2 matrix equated to zero a truncated beam is considered as part of a hypothetical complete beam the frequency equation for truncated beam is the determinant of a 4 4 matrix equated to zero the roots of the frequency equations are determined by an iterative procedure the first four natural frequencies of complete beam and the first three frequencies of truncated beams for 16 combinations of the ideal boundary conditions for various truncation factors are presented the results are benchmarks and are therefore presented in tabular form to preserve the accuracy to six figures

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

modelling the covariance structure of repeated measurements models for repeated measurements used by wilson hebel and sherwin 1981 biometrics 37 553565 and mansour nordheim and rutledge 1985 biometrics 41 287 294 are considered these two models are generalized by the a p e k model obtained by introducing an extra parameter k a more general class of models described by kenward unpublished ph d thesis university of reading 1981 is briefly discussed simulation results are presented in which model parameters are estimated by maximum likelihood and likelihood ratios are used as criteria for choosing between models a set of blood pressure data is used to illustrate model fitting

free vibration analysis of rotating beams by a variable order finite element method the free vibration of rotating beams is analyzed by means of a finite element method of variable order this method entails displacement functions that are a complete power series of a variable number of terms the terms are arranged so that the generalized coordinates are composed of displacements and slopes at the element extremities and additionally displacements at certain points within the element the displacement is assumed to be analytic within an element and thus can be approximated to any degree of accuracy desired by a complete power series numerical results are presented for uniform beams with zero and nonzero hub radii tapered beams and a nonuniform beam with discontinuities since the present method reduces to a conventional beam finite element method for a cubic displacement function the results are compared and found to be superior to the conventional results in terms of accuracy for a given number of degrees of freedom indeed essentially exact eigenvalues and eigenvectors are obtained with this technique which is far more rapidly convergent than other approaches in the literature

hierarchical finite element method for rotating beams abstract the free vibration characteristics of rotating beams are analyzed by means of a finite element method of hierarchical form in which the displacement function for the element is taken to be a polynomial of variable degree the first four terms of the polynomial are expressed in terms of the displacements and slopes at the extremities of the elements and the rest of the terms are considered as internal degrees of freedom numerical results are presented for cantilever beams with uniform non uniform and discontinuous sectional properties a comparison of the present results with those in the published literature indicates that the present relatively simple formulation is as efficient as the other well established techniques finally some clarification is also provided of certain subtle issues concerning the accuracy of the finite element solution and monotonic convergence characteristics and their dependence on the arithmetic precision of the computer employed is indicated

flux corrected transport this paper describes a class of explicit eulerian finite difference algorithms for solving the continuity equation which are built around a technique called flux correction these flux corrected transport fct algorithms are of indeterminate order but yield realistic accurate results in addition to the mass conserving property of most conventional algorithms the fct algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well this first paper concentrates on a simple one dimensional version of fct utilizing shasta a new transport algorithm for the continuity equation which is described in detail

the flexural vibration of rectangular plates with point supports abstract orthogonally generated polynomial functions are used in the lagrangian multiplier method to study the free flexural vibration problem of point supported thin flat rectangular plates the analysis applies to isotropic and specially orthotropic plates having any combination of clamped simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges numerical results are presented for a number of specific problems illustrating the accuracy and versatility of the approach and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports

transverse vibrations of a clamped elliptical plate carrying a concentrated mass at an arbitrary position free vibrations of clamped thin elliptical plates carrying a concentrated mass at an arbitrary position are studied in order to analyze the mass effect on the natural frequencies and mode shapes of the plate a variational approach the well known ritz method is used where the displacement amplitude is approximated by polynomial expressions in the cartesian coordinates the present proposal exhibits excellent agreement with particular cases of the problem available in the literature and also with an independent solution obtained by means of the finite element method some new results are presented for the natural frequencies and modes of clamped elliptical plates with a concentrated mass

a numerical method for simulating transverse vibrations of an axially moving string a modified finite difference method is presented to simulate transverse vibrations of an axially moving string by discretizing the governing equation and the stress strain relation at different frictional knots two linear sparse finite difference equations are obtained which can be computed alternatively the numerical method makes the non linear model easier to deal with and of small truncation errors it also shows stable for small initial values so it can be used in analyzing the non linear vibration of viscoelastic moving string efficiently a practical way of testing the precise of the numerical results is given by using a conservative quantity numerical examples are presented and dynamical analysis is given by using the numerical results

analysis of weak solution of euler bernoulli beam with axial force in this paper we discuss about the existence and uniqueness of the weak form of the non uniform cantilever euler bernoulli beam equation with variable axial tensile and compressive force we investigate the reason of the buckling from the coercivity analysis the frequencies of the beam with tensile force are found by the galerkin method in the sobolev space h2 with proper norm using this method a system of ordinary differential equations in time variable is formed and the corresponding mass and stiffness matrices are constructed a very general form of these matrices which is very simple and suitable for calculations is derived here with a standard basis numerical results for rotating beams with polynomial stiffness and mass variation typical of wind turbine and helicopter rotor blades are obtained these results match well with the published literature a new polynomial generating set is found using two elements of this set a formula to find the eigenfrequencies is derived the proposed approach is easy to implement in symbolic computing software

free vibration analysis of rectangular plates on elastic intermediate supports a procedure using the finite strip element method in combination with a spring system is proposed to treat the free vibration analysis of plates on elastic intermediate supports results indicate that the spring system can successfully simulate elastic intermediate supports such as point supports line supports local uniformly distributed supports and mixed edge supports from the results it is also evident that support stiffness and support areas have significant influence on the free vibration response of plates on line supports and local uniformly distributed supports

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

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

the role of non linear theories in transient dynamic analysis of flexible structures abstract transient dynamic analysis of flexible structures undergoing large motions is considered for rotating structures it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non linear at least second order beam theory use of a first order linearized linear beam theory results in a spurious loss of bending stiffness for a rotating plane beam a set of linear partial differential equations of motion that includes all inertia effects coriolis centrifugal acceleration of revolution and coupling between extensional and flexural deformations is derived from the fully non linear beam theory by consistent linearization the analysis is subsequently extended to the more general case of a plate accomodating shear deformation and undergoing a general three dimensional rotating motion the discretization process of the resulting linear equations of motion for the beam and the plate is also discussed

free vibration analysis of rectangular plates with variable thickness and point supports a discrete method is developed for analyzing the free vibration problem of rectangular plates with point supports the fundamental differential equations involving dirac s delta function are established for the bending problem of the plate with point supports by transforming these differential equations into integral equations and using numerical integration the solution of these equations is obtained and used as green function to obtain the characteristic equation of the free vibration the effects of the point support the boundary condition the variable thickness and aspect ratio on the frequencies are considered by comparing the numerical results obtained by the present method with those previously published the efficiency and accuracy of the present method are investigated

an extension of the growth curve model summary an extension of the growth curve model of potthoff roy 1964 is presented the extension arises naturally when parallel profiles are required or profiles are concurrent at some known time point and may be of use for the analysis of longitudinal data from designed experiments linearization of a nonlinear model also results in this model under certain conditions transformation of the model leads to seemingly unrelated multivariate regressions in this case questions of identifiability and estimation are handled easily an analysis of data on the growth of lambs is presented to illustrate the approach in this paper we consider the analysis of repeated measurements on experimental units which may be grouped according to some experimental design usually successive measurements are taken over time but this need not be the case we restrict ourselves to the case where the same number of observations at identical time points are made on all the experimental units this is likely to be the case in designed experiments though not perhaps in observational studies the approach given here extends that of potthoff roy 1964 and we show below that the potential applications are quite widespread we emphasize that in this paper that we are attempting to present what we regard as a useful and practical extension to the classical growth curve model it is likely to be useful if the aim of the analysis is the estimation of an explicit model for interpretative or predictive purposes this is rather in contrast to the traditional approach to the analysis of such data where the aim is more directed towards testing for effects of treatments or covariates if testing for effects is the primary purpose of analysis then the simpler traditional methods as given initially by wishart 1938 and subsequently by many authors such as rowell walters 1976 may well suffice in this approach selected linear functions of the response over time are calculated and treatment effects are assessed using the analysis of variance since this approach concentrates on individual linear functions seriatim and generally but not always ignores correlations between such linear functions we would regard this as essentially a univariate technique however if testing is the main concern the loss of efficiency in ignoring the multivariate character of the data may not be severe our approach is more complex but the aim is more comprehensive we aim to provide an explicit model for the mean response over time if the model is then used to answer questions that could have been considered by separate tests the answers will of course usually be essentially the same as those obtained by the simpler traditional approach

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