natural frequencies of arbitrarily shaped plates using the rayleigh ritz method together with natural co ordinate regions and normalized characteristic orthogonal polynomials abstract a rayleigh ritz method mapping arbitrarily shaped plates on to a natural co ordinate plane using serendipity functions is presented the displacement function uses normalized beam characteristic orthogonal polynomials the kinetic and strain energy equations are mapped from cartesian co ordinates to natural co ordinates and the rayleigh coefficient is minimized flexural free vibration analyses of various shaped plates is performed results are in excellent agreement with those found in the literature this paper corrects the theory previously presented on this topic

transverse vibration of a rectangular plate with bidirectional thickness variation abstract the transverse vibrations of a rectangular plate of variable thickness have been investigated with different combinations of boundary conditions at the four edges the thickness variation is bidrectional and is the cartesian product of linear variations along two concurrent edges of the plate successive approximations have been worked out by the rayleigh ritz method using basis functions satisfying essential boundary conditions convergence is ensured by taking a sufficient number of approximations extensive numerical work has been carried out to obtain the first three frequencies for some selected cases the mode shapes have also been plotted comparison is made with results available in the literature for speacial cases

transverse vibrations of non homogeneous orthotropic rectangular plates of variable thickness a spline technique an analysis and numerical results are presented for free transverse vibrations of non homogeneous orthotropic rectangular plates of non uniform thickness and resting on an elastic foundation of winkler type on the basis of classical plate theory the non homogeneity of the plate material is assumed to arise due to the exponential variation in young s moduli and density along one direction following levy approach i e the two parallel edges are simply supported the fourth order differential equation governing the motion of such plates of exponentially varying thickness in one direction has been solved by using the quintic splines interpolation technique for three different combinations of clamped simply supported and free boundary conditions at the other two edges effect of the non homogeneity and elastic foundation together with other plate parameters such as orthotropy aspect ratio and thickness variation on the natural frequencies of vibration is illustrated for the first three modes of vibration normalized displacements are presented for specified plates for all the three boundary conditions

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

vibrations of axially moving viscoelastic plate with parabolically varying thickness based on the two dimensional viscoelastic differential constitutive relation the differential equation of motion of the axially moving viscoelastic rectangular plate constituted by the kelvin voigt model with parabolically varying thickness in the y direction is derived the dimensionless complex frequencies of axially moving viscoelastic plate with different boundary conditions versus the dimensionless moving speed for various aspect ratio thickness parameter and the dimensionless delay time are analyzed by the differential quadrature method the effects of various parameters such as aspect ratio thickness parameter the dimensionless moving speed and the dimensionless delay time on the vibration characteristics of the axially moving viscoelastic rectangular plate with parabolically varying thickness are discussed

vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices this work provides a general formulation to solve vibration problems for continuous systems with damping effects including modal transient harmonic and spectrum response analyses in modal analysis the system eigenvalues and corresponding eigenfunctions can be determined the orthogonal relations of eigenfunctions are shown for transient harmonic and spectrum analyses the generic force actuator functions and response sensing operators are introduced respectively and used to derive the system response the time domain response is obtained for transient analysis the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis the solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force actuator and response sensor mode shape functions in particular one dimensional beam and two dimensional plate vibration analyses are illustrated by following the developed generic formulation this work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well

multidimensional thermodynamic uncertainty relations we extend a class of recently derived thermodynamic uncertainty relations to vector valued observables in contrast to the scalar valued observables examined previously this multidimensional thermodynamic uncertainty relation provides a natural way to study currents in high dimensional systems and to obtain relations between different observables our proof is based on the generalized cramer rao inequality which we interpret as a relation between physical observables and the fisher information this allows us to develop high dimensional versions of both the original steady state uncertainty relation and the more recently obtained generalized uncertainty relation for time periodic systems we apply the multidimensional uncertainty relation to obtain a new constraint on the performance of steady state heat engines which is tighter than previous bounds and reveals the role of heat work correlations as a second application we show that the uncertainty relation is connected to a bound on the differential mobility as a result of this connection we find that a necessary condition for equality in the uncertainty relation is that the system obeys the equilibrium fluctuation dissipation relation

free vibration analysis of rectangular plates with variously shaped holes an approximate method for analyzing the free vibration of rectangular plates with a hole of different shapes is proposed the shapes of the holes are circular semi circular elliptic square rectangular triangular rhombic etc these rectangular plates with a hole can be considered ultimately as a kind of rectangular plates with non uniform thickness a hole in a plate can be considered as an extremely thin part of the plate therefore the free vibration problem of a plate with a hole can be translated into the free vibration problem of the equivalent rectangular plate with non uniform thickness for some plates with different shaped holes the convergency and accuracy of the numerical solutions calculated by the proposed method are investigated

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

buckling of a conical thin shell with variable thickness under a dynamic loading this study considers the buckling of an elastic truncated conical shell having a meridional thickness expressed by an arbitrary function subject to a uniform external pressure which is a power function of time at first the fundamental relations and donnell type dynamic buckling equation of an elastic conical shell with variable thickness have been obtained then employing galerkin s method those equations have been reduced to a time dependent differential equation with variable coefficients finally applying the ritz type variational method the critical static and dynamic loads the corresponding wave numbers dynamic factor and critical stress impulse have been found analytically using the results thus obtained the effects of the thickness variations with a power or an exponential function the variation of the semi vertex angle and the variation of the power of time in the external pressure expression are studied through pertinent computations it is observed that these factors have appreciable effects on the critical parameters of the problem in the heading

a generalization of the thermodynamic uncertainty relation to periodically driven systems the thermodynamic uncertainty relation expresses a universal trade off between precision and entropy production which applies in its original formulation to current observables in steady state systems we generalize this relation to periodically time dependent systems and relatedly to a larger class of inherently time dependent current observables in the context of heat engines or molecular machines our generalization applies not only to the work performed by constant driving forces but also to the work performed while changing energy levels the entropic term entering the generalized uncertainty relation is the sum of local rates of entropy production which are modified by a factor that refers to an effective time independent probability distribution the conventional form of the thermodynamic uncertainty relation is recovered for a time independently driven steady state and additionally in the limit of fast driving we illustrate our results for a simple model of a heat engine with two energy levels

a dqem for transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions in this paper a differential quadrature element method dqem for free transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions is proposed governing equations the compatibility conditions at the damaged cross sections and implementation of the external boundary conditions are derived and formulated by the differential quadrature analogue the accuracy convergence and versatility of the proposed method are confirmed by the exact solution of the uniform beam which has been presented by other authors and 2d finite element method fem numerical results for non uniform beam after the validation of the presented method the effect of quantity depth and location of the cracks on the frequency values of vibrations are investigated the achieved results show that the existence of the crack leads to a decrease in the frequencies of the vibrations through decrease in the stiffness of the beam meanwhile the compatibility conditions at the damaged section is considered as a discontinuity in slope and vertical displacement where the effect of the discontinuity in the slope is more considerable as many authors have neglected the discontinuity in the vertical displacement as it will be shown consideration of the discontinuity in the vertical displacement causes more decrease in the frequencies

free vibration analysis of right triangular plates with variable thickness an approximate method for analyzing the free vibration of right triangular plates with arbitrary variable thickness and various boundary conditions is proposed in this paper a right triangular plate is considered as a kind of rectangular plate with non uniform thickness therefore the free vibration characteristics of any right triangular plate are obtained by analyzing the equivalent rectangular plate with non uniform thickness the approximate method is based on the green function of an equivalent rectangular plate the green function of a rectangular plate with arbitrary variable thickness is obtained as a discrete form solution for deflection of the plate with a concentrated load by applying the green function the free vibration problem of plate is translated into the eigenvalue problem of matrix

vibratory characteristics of general laminates i symmetric trapezoids abstract this paper presents part i of investigations of free flexural vibration of multi layered symmetric and unsymmetric composite laminates with symmetric trapezoidal planform of arbitrary combinations of edge conditions the ritz procedure withpb 2shape functions as the admissible functions is used to arrive at the governing eivenvalue equation the set ofpb 2shape functions is intrinsically a product of a class of orthogonally generated two dimensional polynomial functions and a basic function which ensures the satisfaction of geometric boundary conditions at the outset different boundary conditions with various combinations of in plane constraints are considered the effects of wide ranges of chord ratios angles of fibre orientation and stacking sequences on the vibration response are investigated with the aim of increasing the existing database sets of previously unavailable vibration frequencies and mode shapes are presented as benchmarks for future reference

finite time generalization of the thermodynamic uncertainty relation for fluctuating currents in nonequilibrium steady states the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving we show that this relation holds not only for the long time limit of fluctuations as described by large deviation theory but also for fluctuations on arbitrary finite time scales this generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments for which infinite time scales are not accessible importantly often this finite time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long time limit we illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle

free vibration analysis of orthotropic rectangular plates with variable thickness and general boundary conditions a discrete method is developed for analyzing the free vibration problem of orthotropic rectangular plates with variable thickness the green function which is obtained by transforming the differential equations into integral equations and using numerical integration is used to establish the characteristic equation of the free vibration the effects of the aspect ratios boundary conditions the variation of the thickness on the frequencies are considered by comparing the numerical results obtained by the present method with the those previously published the efficiency and accuracy of the present method are investigated the frequency parameters are obtained for the orthotropic plates with general boundary conditions and variable thickness in one or two directions the model shapes are given for some of the square plates with three kinds of thickness variations

vibration analysis of simply supported rectangular plates with unidirectionally arbitrarily varying thickness in this paper a practical analytical method for the free vibration analysis of a simply supported rectangular plate with unidirectional arbitrary thickness variation is proposed first the plate is divided into a number of regions of which the values of thickness are assumed to be constant and are given by the known thickness function of the plate the close form frequency function that yields the eigenvalues of the plate is extracted by considering the condition of continuity in displacement and slope between the regions and by considering the simply supported boundary condition of the plate it is shown by several case studies that the proposed method has good convergence characteristics and yields accurate eigenvalues and mode shapes compared with other analytical methods including fem ansys

the free vibrations of tapered rectangular plates using a new set of beam functions with the rayleigh ritz method abstract in19991999 academic pressacademic pressthis paper the free vibrations of a wide range of non uniform rectangular plates in one or two directions are considered the domain of the plate is bounded by copyright x 1a a and y 1b b in rectangular co ordinates the thickness of the plate is continuously varying and proportional to the power functionxsyt a variety of tapered rectangular plates can be described by giving the taper factors s and t different values s and t may be given arbitrary real numbers if both 1 0 and 1 0 or arbitrary non negative numbers if 1 0 or 1 0 the uniform rectangular plate is a special case by letting both s and t equal to zero a new set of admissible functions which are the static solutions of the tapered beam or a strip taken from the tapered rectangular plate under an arbitrary static load expanded into a taylor series is developed unlike conventional admissible functions the set of static beam functions will vary appropriately with the thickness variation of the plate the eigenfrequency equation is obtained by the use of the rayleigh ritz method a general computer program has been compiled and some numerical results are tabulated on the basis of comparison with available results in the literature it is shown that the first few eigenfrequencies can be obtained with good accuracy by using only a small number of terms of the static beam functions

fluctuation theorem uncertainty relation the fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations such as the second law of thermodynamics and the jarzynski equality recently the thermodynamic uncertainty relation was discovered which states that the fluctuation of observables is lower bounded by the entropy production in the present letter we derive a thermodynamic uncertainty relation from the fluctuation theorem we refer to the obtained relation as the fluctuation theorem uncertainty relation and it is valid for arbitrary dynamics stochastic as well as deterministic and for arbitrary antisymmetric observables for which a fluctuation theorem holds we apply the fluctuation theorem uncertainty relation to an overdamped langevin dynamics for an antisymmetric observable we demonstrate that the antisymmetric observable satisfies the fluctuation theorem uncertainty relation but does not satisfy the relation reported for current type observables in continuous time markov chains moreover we show that the fluctuation theorem uncertainty relation can handle systems controlled by time symmetric external protocols in which the lower bound is given by the work exerted on the systems

eigenfrequencies of tapered rectangular plates with intermediate line supports the free vibrations of a wide range of tapered rectangular plates with an arbitrary number of intermediate line supports in one or two directions are investigated the domain of the plate is bounded by x a a 0 1 and y b b 0 1 in the rectangular co ordinates the thickness of the plate is continuously varying and proportional to a power function xsyt a variety of tapered rectangular plates can be described by giving the taper factors s and t various values the intermediate line supports run parallel to the edges of the plate a new set of admissible functions which are the static solutions of the tapered beam with intermediate point supports or a strip taken from the plate structure in one or the other direction under a taylor series of loads is developed consistent convergency independent of the truncation factors and of the plate can be obtained by taking the midpoint of the beam as the expanding point of the taylor series unlike conventional admissible functions this set of static beam functions can appropriately vary with the thickness variation of the plate the eigenfrequency equation of the plate is derived by the rayleigh ritz approach a general computer program has been compiled it can be seen that the convergency of the numerical computation is very rapid and that the first few eigenfrequencies can be obtained with good accuracy by using only a small number of terms of the static beam functions sets of first time reported eigenfrequency data are included for future reference

natural frequencies of a functionally graded cracked beam using the differential quadrature method the present work is concerned with the free vibration analysis of an elastically supported cracked beam the beam is made of a functionally graded material and rested on a winkler pasternak foundation the line spring model is employed to formulate the problem the method of differential quadrature is applied to solve it the obtained results agreed with the previous similar ones further a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the problem on the values of natural frequencies and mode shape functions

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

distribution of work in isothermal nonequilibrium processes diffusive motion in an externally driven potential is considered it is shown that the distribution of work required to drive the system from an initial equilibrium state to another is gaussian for slow but finite driving our result is obtained by projection method techniques exploiting a small parameter defined as the switching rate between the two states of the system the exact solution for a simple model system shows that such an expansion may fail in higher orders since the mean and the variance following from the exact distribution show nonanalytic behavior

on the equivalence of generalized differential quadrature and highest order finite difference scheme abstract it is shown in this paper that the global method of generalized differential quadrature gdq is equivalent to the highest order finite difference scheme which is the extension of low order finite difference schemes the effect of grid point distribution on the accuracy of numerical results is studied by two numerical examples the results showed that the ability to alter the grid point distribution in the gdq method is an important feature to obtain accurate numerical solutions in certain problems

abstract time dependent transport equations abstract we consider the abstract time dependent linear transport equation as an initial boundary value evolution problem in the banach spaces lp 1 p

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

vibration of mindlin plates using boundary characteristic orthogonal polynomials abstract the vibration analysis of shear deformable plates formulated on the basis of first order mindlin theory is presented the displacement and rotational functions of the plates are approximated by sets of boundary characteristic orthogonal polynomials the ease of generation and manipulation of these polynomial functions greatly enhances the computational efficiency of the numerical method the energy functional of the shear deformable plates derived from the mindlin plate theory is minimized in the ritz procedure to arrive at the governing eigenvalue equation corresponding natural frequencies and mode shapes can be obtained by solving the eigenvalue equation computed frequency results for various boundary conditions aspect ratios a b and thickness ratios t b are presented to demonstrate the effects of each factor on the vibration frequencies of moderately thick plates vibration mode shapes in the form of contour plots are presented for a thickness ratio of t b 0 1 these plots are believed to be the first known in the open literature

on the performance of three iterative methods for solution of gdq algebraic equations abstract in this paper the three conventional iterative methods namely j oe or jacobi gauss seidel g s and sor methods are extended to solve the gdq algebraic equations the performances of these three iterative methods for the gdq and second order finite difference fd algebraic equations are comparatively studied by their application to solve the two dimensional poisson equation it was found that sor method gives the fastest convergence rate for both gdq and fd algebraic equations however the effect of relaxation factor for gdq algebraic equations is reduced the high efficiency and accuracy of gdq over second order fd scheme are also shown in the paper furthermore the performance of j oe and sor methods was validated by their application to a complicated problem that is numerical simulation of natural convection in the annulus between two concentric cylinders it was found that for this complicated case jacobi and gauss seidel methods cannot get converged solution and the efficiency of sor method is reduced to be around three times faster than the j oe method

thermodynamic uncertainty relation for arbitrary initial states the thermodynamic uncertainty relation tur describes a trade off relation between nonequilibrium currents and entropy production and serves as a fundamental principle of nonequilibrium thermodynamics however currently known turs presuppose either specific initial states or an infinite time average which severely limits the range of applicability here we derive a finite time tur valid for arbitrary initial states from the cram er rao inequality we find that the variance of an accumulated current is bounded from below by the instantaneous current at the final time which suggests that the boundary is constrained by the bulk we apply our results to feedback controlled processes and successfully explain a recent experiment which reports a violation of a modified tur with feedback control we also derive a tur that is linear in the total entropy production and valid for discrete time markov chains with nonsteady initial states the obtained bound exponentially improves the existing bounds in a discrete time regime

transverse vibration of triangular plates with variable thickness transverse vibration of a triangular plate with thickness varying as a linear function of co ordinates in the plane of the plate has been solved by working out several approximations in the rayleigh ritz method the basis functions are chosen so that the essential boundary conditions are satisfied three parameters control the three types of boundary conditions at the edges two parameters control the thickness variation and two parameters control the shape of the plate a general procedure has been laid down to compute the frequencies and the associated mode shapes for any set of these parameters convergence has been ensured by working out a sufficiently large number of approximations so that at least the first three frequencies converge to three significant digits the results are tabulated for various values of the parameters and comparisons have been made with known results in special cases three dimensional plots of the mode shapes have also been given in some cases

free vibration analysis of rectangular plates with variable thickness abstract an approximate method for analyzing the free vibration of thin and moderately thick rectangular plates with arbitrary variable thickness is proposed the approximate method is based on the green function of a rectangular plate the green function of a rectangular plate with arbitrary variable thickness is obtained as a discrete form solution for deflection of the plate with a concentrated load the discrete form solution is obtained at each discrete point equally distributed on the plate it is shown that the numerical solution for the green function has good convergency and accuracy by applying the green function the free vibration problem of the plate is translated into the eigenvalue problem of the matrix the convergency and accuracy of the numerical solutions for the natural frequency parameter calculated by the proposed method are investigated and the frequency parameters and their modes of free vibration are shown for some rectangular plates

variational asymptotic analysis for plates of variable thickness abstract it is a common assumption in plate theory that the effect of varying thickness on plate elastic constants can be simply accounted for by incorporating the thickness distribution in the formulae obtained for a plate of uniform thickness for plates whose thickness varies linearly in the global directions the variation described by a set of taper parameters a plate theory has been developed using the variational asymptotic method in a first of its kind development both the 8 8 stiffness matrix and the 3d stress and strain recovery are shown to contain these taper parameters explicitly as opposed to their merely being present by virtue of the varying plate thickness furthermore terms such as bending shear and twist shear coupling are shown to be present which are typically disregarded in published works on tapered plates it is later clearly demonstrated using the full 3d finite element solutions from a test case that an analysis of this kind is needed to accurately determine the complete stress distribution in conclusion merely changing the thickness distributions in results for uniform plates may work for some problems but it is clearly not sufficient in general nor can it guarantee good agreement with all aspects of plate analysis this is because these kinds of approximations are not in tune with the mechanics of such a problem as they disregard the tilt of the outward directed normals in the boundary conditions for top and bottom surfaces furthermore as always a theory developed using the variational asymptotic method does not make any assumptions regarding the deformation and is generally well suited for capturing such effects without fundamental changes in the procedure

free vibration analysis of tapered rectangular plates by differential quadrature method a semi analytical approach abstract the particular type of thin rectangular plate configurations considered in the present work are the ones having two opposite edges simply supported and are of varying thickness between the other two edges the boundary conditions at these two edges may be quite general the plates may isotropic or specially orthotropic for the numerical solution to evaluate the free vibration characteristics of such plates a semi analytical approach is adopted first by using the trigonometric sine functions for the mode shape between the simply supported edges the eigenvalue partial differential equation is reduced to an ordinary differential equation the solution of the reduced equation is then carried out by the differential quadrature method on the basis of comparison with the available results in the published literature it is believed that this solution method is capable of yielding highly accurate results moreover the very small computation times required for the evaluation of the vibration characteristics demonstrate the high computational efficiency of the method and point to its possible use in the real time analysis and design of vibrating plate systems

natural vibration study on rectangular plates with a line hinge and various boundary conditions a discrete method is proposed for analyzing the natural vibration problem of rectangular plates with a line hinge the fundamental differential equations and the solutions of these equations are derived for two parts of the plate which are obtained by dividing the plate along the line hinge by transforming these equations into integral equations and using numerical integration and the continuous conditions along the line hinge the solutions of the whole plate can be expressed by the unknown quantities on the boundary and the quantities of the rotation along the hinge the green function of the deflection problem is used to obtain the characteristic equation of the free vibration the effects of the position of the line hinge the aspect ratio the thickness ratio and the boundary condition on the natural frequency parameters 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

extension of hencky bar net model for vibration analysis of rectangular plates with rectangular cutouts abstract comprising rigid bars and spring systems at the joints and in the cells the hencky bar net model hbm has been shown to be a physical structural representation of the finite difference plate model fdm in this paper the hbm is extended for the vibration analysis of rectangular plates with rectangular cutouts this extension addresses the rotationally elastic and transverse spring stiffnesses for the hbm at the cutout corners after verifying the hbm model by comparing the vibration results with existing solutions for some plate problems the model is used to obtain some new free vibration solutions for plates having various boundary conditions including cracked corners modelled by free small rectangular cutouts

tightening the uncertainty principle for stochastic currents we connect two recent advances in the stochastic analysis of nonequilibrium systems the loose uncertainty principle for the currents which states that statistical errors are bounded by thermodynamic dissipation and the analysis of thermodynamic consistency of the currents in the light of symmetries employing the large deviation techniques presented by gingrich et al phys rev lett 116 120601 2016 prltao0031 900710 1103 physrevlett 116 120601 and pietzonka barato and seifert phys rev e 93 052145 2016 2470 004510 1103 physreve 93 052145 we provide a short proof of the loose uncertainty principle and prove a tighter uncertainty relation for a class of thermodynamically consistent currents j our bound involves a measure of partial entropy production that we interpret as the least amount of entropy that a system sustaining current j can possibly produce at a given steady state we provide a complete mathematical discussion of quadratic bounds which allows one to determine which are optimal and finally we argue that the relationship for the fano factor of the entropy production rate var mean 2 is the most significant realization of the loose bound we base our analysis both on the formalism of diffusions and of markov jump processes in the light of schnakenberg s cycle analysis

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 semi analytical solution of the flexural vibration of orthotropic plates of variable thickness a semi analytical method based on the finite strip technique in conjunction with the transition matrix is applied to investigate the flexural vibration of orthotropic rectangular plates of variable thickness in one direction the effect of orthotropy aspect ratio and taper ratio on the natural frequencies of the vibration of such plates is investigated the efficiency of the proposed technique is illustrated by comparing the results with those available in the literature for several examples the results show excellent agreement and stable convergence for all plate configurations considered in this paper

free vibration of laminated composite plates with cutout the present investigation is concerned with free vibration analysis of composite plates in the presence of cutouts undergoing large amplitude oscillations the ritz finite element model using a nine nodedc0continuity isoparametric quadrilateral element along with a higher order displacement theory which accounts for parabolic variation of transverse shear stresses is used to predict the dynamic behavior results have been obtained for laminated plates with various cutout geometries such as square rectangle circle and ellipse in the large amplitude range backbone curves are drawn for various boundary conditions and aspect ratios of the cutout

thermodynamic uncertainty relations including measurement and feedback thermodynamic uncertainty relations quantify how the signal to noise ratio of a given observable is constrained by dissipation fluctuation relations generalize the second law of thermodynamics to stochastic processes we show that any fluctuation relation directly implies a thermodynamic uncertainty relation considerably increasing their range of applicability in particular we extend thermodynamic uncertainty relations to scenarios which include measurement and feedback since feedback generally breaks time reversal invariance the uncertainty relations involve quantities averaged over the forward and the backward experiment defined by the associated fluctuation relation this implies that the signal to noise ratio of a given experiment can in principle become arbitrarily large as long as the corresponding backward experiment compensates e g by being sufficiently noisy we illustrate our results with the szilard engine as well as work extraction by free energy reduction in a quantum dot

universal bounds on current fluctuations for current fluctuations in nonequilibrium steady states of markovian processes we derive four different universal bounds valid beyond the gaussian regime different variants of these bounds apply to either the entropy change or any individual current e g the rate of substrate consumption in a chemical reaction or the electron current in an electronic device the bounds vary with respect to their degree of universality and tightness a universal parabolic bound on the generating function of an arbitrary current depends solely on the average entropy production a second stronger bound requires knowledge both of the thermodynamic forces that drive the system and of the topology of the network of states these two bounds are conjectures based on extensive numerics an exponential bound that depends only on the average entropy production and the average number of transitions per time is rigorously proved this bound has no obvious relation to the parabolic bound but it is typically tighter further away from equilibrium an asymptotic bound that depends on the specific transition rates and becomes tight for large fluctuations is also derived this bound allows for the prediction of the asymptotic growth of the generating function even though our results are restricted to networks with a finite number of states we show that the parabolic bound is also valid for three paradigmatic examples of driven diffusive systems for which the generating function can be calculated using the additivity principle our bounds provide a general class of constraints for nonequilibrium systems

statical and dynamical behaviour of thin fibre reinforced composite laminates with different shapes abstract based on the classical laminated plate theory a variational approach for the study of the statical and dynamical behaviour of arbitrary quadrilateral anisotropic plates with various boundary conditions is developed the analytical formulation uses the ritz method in conjunction with natural coordinates to express the geometry of general plates in a simple form the deflection of the plate is approximated by a set of beam characteristic orthogonal polynomials generated using the gram schmidt procedure the algorithm developed is quite general and can be used to study fibre reinforced composite laminates with symmetric lay ups which may have general anisotropy and any combinations of clamped simply supported and free edge support conditions various numerical applications are presented and some results are compared with existing values in the literature to demonstrate the accuracy and flexibility of the present method new results were also determined for plates with different geometrical shapes combinations of boundary conditions several stacking sequences and various angles of fibre orientation

a semi analytical dqem for free vibration analysis of thick plates with two opposite edges simply supported this paper is introducing a semi analytical differential quadrature element method dqem based on first order shear deformation theory for free vibration analysis of thick plates by decomposing the plate into a series of sub domains or elements any discontinuity in geometry material properties and elastic foundations in one direction can be handled conveniently classical as well as non classical boundary conditions can be imposed one benefits of this semi analytical dqem is reducing the storage requirements and the computational efforts needed to assemble the global equations the method would also increase the computational efficiency in comparison with the finite element method fem or differential quadrature method dqm analysis of such problems the accuracy versatility stability and convergence of the algorithm are certified

transverse vibrations of an orthotropic rectangular plate of linearly varying thickness and with a free edge fil bambill diana virginia consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet bahia blanca argentina universidad nacional del sur departamento de ingenieria argentina

on the frequency response of moderately thick simply supported rectangular plates with arbitrary lamination a here to fore unavailable analytical solution to the boundary value problem of free vibration response of arbitrarily laminated moderately thick rectangular plates subjected to an admissible boundary condition is presented a novel boundary continuous solution approach is employed to solve the highly coupled governing partial differential equations that arise from the implementation of the yang norris stavsky 1966 theory incorporating first order shear deformation and rotary and in plane inertias the numerical results presented for various parametric effects to study first lowest seven eigenvalues and corresponding eigenvectors can be capitalized as bench marks for a future comparison an eight node isoparametrics shear flexible element is considered to validate its performance with respect to the present solutions

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

quintic splines in the study of transverse vibrations of non uniform orthotropic rectangular plates an analysis and numerical results are presented for free transverse vibrations of orthotropic rectangular plates of linearly varying thickness along one direction and resting on an elastic foundation of the winkler type on the basis of classical plate theory following the levy approach i e two parallel edges being simply supported the fourth order differential equation governing the motion of such plates has been solved by using the quintic splines interpolation technique for three different combinations of clamped simply supported and free boundary conditions at the other two edges the effect of the elastic foundation together with the orthotropy aspect ratio and thickness variation on the natural frequencies of vibration is illustrated for the first three modes of vibration normalized displacements are presented for two different values of the taper constant keeping other plate parameters fixed for all the three boundary conditions a comparison of the results with those available in literature is presented

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

application of a new differential quadrature methodology for free vibration analysis of plates a new methodology is introduced in the differential quadrature dq analysis of plate problems the proposed approach is distinct from other dq methods by employing the multiple boundary conditions in a different manner for structural and plate problems the methodology employs the displacement within the domain as the only degree of freedom whereas along the boundaries the displacements as well as the second derivatives of the displacements with respect to the co ordinate variable normal to the boundary in the computational domain are considered as the degrees of freedom for the problem employing such a procedure would facilitate the boundary conditions to be implemented exactly and conveniently in order to demonstrate the capability of the new methodology all cases of free vibration analysis of rectangular isotropic plates in which the conventional dq methods have had some sort of difficulty to arrive at a converged or accurate solution are carried out excellent convergence behaviour and accuracy in comparison with exact results and or results obtained by other approximate methods were obtained the analogous dq formulation for a general rectangular plate is derived and for each individual boundary condition the general format for imposing the given conditions is devised it must be emphasized that the computational efforts of this new methodology are not more than for the conventional differential quadrature methods copyright 2002 john wiley sons ltd

vibration of elastically restrained cross ply laminated plates with variable thickness abstract the natural frequencies of symmetrically laminated plates of variable thickness are analyzed using the finite strip transition matrix technique in this paper the natural frequencies of such plates are determined for edges with being elastically restrained against both rotation and transition or both a successive conjunction of the classical finite strip method and the transition matrix method is applied to develop a new modification of the finite strip method to reduce the complexity of the problem the displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction using the new transition matrix after necessary simplification and the satisfaction of the boundary conditions yields a set of simultaneous equations that leads to the characteristic matrix of vibration the mode shapes and the frequency parameters for different combinations of elastic or translational restraint coefficients have been presented and compared with those available from other methods in the literature also the effect of the tapered ratio and the aspect ratio on the natural frequencies and the mode shapes of the plates are presented the good agreement with other methods demonstrates the validity and the reliability of the proposed method

vibration analysis of edge cracked beam on elastic foundation with axial loading using the differential quadrature method abstract the eigenvalue problems of clamped free and hinged hinged bernoulli euler beams on elastic foundation with a single edge crack axial loading and excitation force were numerically formulated using the differential quadrature method dqm appropriate boundary conditions accompanied the dqm to transform the partial differential equation of a bernoulli euler beam with a single edge crack into a discrete eigenvalue problem the dqm results for the natural frequencies of cracked beams agree well with other literature values the sampling point number effect the location of the crack effect and the depth of the crack effect on the accuracy variation of calculated natural frequencies are presented by using two elements in this work the effects of axial loading foundation stiffness opening crack and closing crack are also studied

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

free vibration analysis of rectangular plate with a hole by means of independent coordinate coupling method this paper is concerned with the vibration analysis of a rectangular plate with a rectangular hole or a circular hole in the previous researches the effect of the hole is taken into account by subtracting the energies of the hole domain from the total energies of the whole plate in doing so the integrals of the functions consisting of the admissible functions based on the plate domain over hole domain lead to complex numerical computations in this paper a new methodology called independent coordinate coupling method iccm was developed in which the energies corresponding to the rectangular plate domain and the hole domain were derived independently and the two independent coordinates were coupled by imposing kinematic relations by matching the deflection condition imposed on the expressions the relationship between the global axes and the local axes can be easily derived and then used to construct the mass and stiffness matrices the numerical results show the efficacy of the proposed method

harmonic differential quadrature finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation abstract the geometrically nonlinear static and dynamic analysis of thin rectangular plates resting on elastic foundation has been studied winkler pasternak foundation model is considered dynamic analogues von karman equations are used the governing nonlinear partial differential equations of the plate are discretized in space and time domains using the harmonic differential quadrature hdq and finite differences fd methods respectively the analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges various types of dynamic loading namely a step function a sinusoidal pulse and an n wave are investigated and the results are presented graphically the accuracy of the proposed hdq fd coupled methodology is demonstrated by the numerical examples

flexural vibration of polygonal plates treatments of sharp re entrant corners abstract the free vibrations of a class of polygonal plates with sharp re entrant corners are investigated a highly versatile and accurate domain decomposition method ddm is proposed for solutions frequency parameters and modes shapes of thin isotropic plates having l t and c shapes planforms are computed the l shaped plate in particular is employed as the basic building element for the other configurations in the preparatory stage of the solution process this l shaped element is subdivided into three smaller rectangular subdomains each subdomain is assigned with appropriate polynomial based displacement shape function continuities in displacement slope and higher derivatives between adjacent subdomains are enforced at the interconnecting edges this results in a set of continuity matrices which are used to couple the respective strain and kinetic energies of each subdomain a global energy functional which is the aggregate of the respective energy contributions from each subdomain is derived and minimized according to the ritz principle to arrive at the linear eigenvalue equation for the complete plate domain a comprehensive range of mode shapes is presented in contour plots also results of extensive parametric studies to investigate the effects of changing geometric parameters on the natural frequencies of the complex plates are presented

universal trade off relation between power and efficiency for heat engines for a general thermodynamic system described as a markov process we prove a general lower bound for dissipation in terms of the square of the heat current thus establishing that nonvanishing current inevitably implies dissipation this leads to a universal trade off relation between efficiency and power with which we rigorously prove that a heat engine with nonvanishing power never attains the carnot efficiency our theory applies to systems arbitrarily far from equilibrium and does not assume any specific symmetry of the model

a new differential quadrature methodology for beam analysis and the associated differential quadrature element method this paper presents a computationally efficient and an accurate new methodology in differential quadrature analysis of beam elements the methodology would overcome the difficulties in boundary conditions implementations of fourth order differential equations encountered in such problems the methodology benefited from defining the second order derivatives along the boundaries as independent degrees of freedom would enable the differential quadrature method to exactly satisfy some types of boundary conditions where by most other conventional algorithms have to be satisfied approximately the weighting coefficients employed are not exclusive and any accurate and efficient method such as the generalized differential quadrature method may be used to produce the methods weighting coefficients by solving some typical stability deflection and frequency analysis beam problems and by comparing the results with those of exact solutions and or those of other methodologies accuracy convergency and efficiency of the methodology is asserted in order to generalize its application to large scale beam structures and to the cases with the discontinuity in loading conditions and geometry a new one dimensional differential quadrature element method formulation is presented and implemented

proof of the finite time thermodynamic uncertainty relation for steady state currents this paper presents the finite time version of the thermodynamic uncertainty relation for nonequilibrium steady states which sets a bound on the magnitude of current fluctuations the authors use a methodology that is similar to the proof for the long time limit but with large deviation theory applied to an ensemble of many copies

vibratory characteristics of cantilevered rectangular shallow shells of variable thickness in this paper a comprehensive study on the modeling of vibratory response of variable thickness cantilevered shallow cylindrical shells of rectangular planform is carried out through the principle of minimum total energy the equations of stretching and bending strain energies kinetic energy and the associated boundary conditions of the shells are derived the present model is developed based on the ritz formulation with the assumption of sets of mathematicall y complete admissible two dimensional polynomials to approximate the in plane and transverse displacement amplitude functions a basic function is introduced in the approximate method to enforce automatic satisfaction of the kinematic boundary conditions sets of comprehensive reasonably accurate vibration frequencies of the shallow shells are presented for wide ranges of aspect ratio thickness variation ratios and shallowness ratio these results where possible are verified by comparing with other established experimental and theoretical solutions few selected contour plots for the first known mode shapes of these cylindrical shells of variable thickness are included

rayleigh s optimization concept and the use of sinusoidal co ordinate functions the present note reports some numerical experiments performed on the determination of the fundamental frequency vibration of a rectangular plate with three simply supported edges while the fourth is free the optimization parameter is contained in the argument of the sinusoidal terms of a truncated fourier series

vibration of shear deformable plates with variable thickness first order and higher order analyses this work presents accurate numerical calculations of the natural frequencies for elastic rectangular plates of variable thickness with various combinations of boundary conditions the thickness variation in one or two directions of the plate is taken in polynomial form the first order shear deformation plate theory of mindlin and the higher order shear deformation plate theory of reddy have been applied to the plate analysis the governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the lagrangian function the solution is obtained by the extended kantorovich method this approach is combined with the exact element method for the vibration analysis of members with variable flexural rigidity which provides for the derivation of the exact dynamic stiffness matrix of varying cross sections strips 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 theory and with published results

free vibration of orthotropic square plates with a square hole an approximate method is extended for analyzing the free vibration problem of orthotropic square plate with a square hole in this paper a square plate with a square hole is transformed into an equivalent square plate with non uniform thickness by considering the hole as an extremely thin part of the equivalent plate therefore the dynamic characteristics of a plate with a hole can be obtained by analyzing the equivalent plate the green function which is the discrete solution for the deflection of the equivalent plate is used to obtain the characteristic equation of the free vibration the effects of the side to thickness ratio hole side to plate side ratio and the variation of the thickness on the frequencies are considered some numerical analyses are carried out for the simply supported orthotropic square plate with a square hole the efficiency and accuracy of the numerical solutions by the present method are investigated

numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non uniform thickness abstract a comparison of natural frequencies of transverse vibration of thin rectangular plates of non uniform thickness with different combinations of boundary conditions is presented in this paper as obtained by using the following methodologies 1 the rayleigh ritz method with characteristic orthogonal polynomial shape functions 2 the rayleigh ritz method with a shape function which includes two exponents that are to be determined by minimizing the fundamental frequency coefficient 3 the optimized kantorovich method which has been proposed rather recently 4 the finite element method the results are presented in tabular form and discussed

propagation in a kinetic reaction transport equation travelling waves and accelerating fronts in this paper we study the existence and stability of travelling wave solutions of a kinetic reaction transport equation the model describes particles moving according to a velocity jump process and proliferating thanks to a reaction term of monostable type the boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves the minimal speed of propagation of waves is obtained from an explicit dispersion relation we construct the waves using a technique of sub and supersolutions and prove their weak stability in a weighted l2 space in case of an unbounded velocity set we prove a superlinear spreading it appears that the rate of spreading depends on the decay at infinity of the velocity distribution in the case of a gaussian distribution we prove that the front spreads as t3 2

high accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain summary based on the differential quadrature dq rule the gauss lobatto quadrature rule and the variational principle a dq finite element method dqfem is proposed for the free vibration analysis of thin plates the dqfem is a highly accurate and rapidly converging approach and is distinct from the differential quadrature element method dqem and the quadrature element method qem by employing the function values themselves in the trial function for the title problem the dqfem without using shape functions essentially combines the high accuracy of the differential quadrature method dqm with the generality of the standard finite element formulation and has superior accuracy to the standard fem and fdm and superior efficiency to the p version fem and qem in calculating the stiffness and mass matrices by incorporating the reformulated dq rules for general curvilinear quadrilaterals domains into the dqfem a curvilinear quadrilateral dq finite plate element is also proposed the inter element compatibility conditions as well as multiple boundary conditions can be implemented simply and conveniently as in fem through modifying the nodal parameters when required at boundary grid points using the dq rules thus the dqfem is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter element compatibilities a series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the dqfem copyright q 2009 john wiley sons ltd

a semi analytical solution for vibration of rectangular plates with abrupt thickness variation a semi analytical analysis of free vibration of plates with cross sectional discontinuities due to abrupt changes in thickness is presented a basic square element divided into suitable subdomains dependent upon the positions of these abrupt changes is used as the basic building element admissible functions that satisfy the essential or geometric boundary conditions are used to define the transverse deflection of each subdomain continuities in the displacement slope moment and higher derivatives between adjacent subdomains are enforced at the interconnecting edges the resulting global energy functional from the proper assembly of the coupled strain and kinetic energy contributions of each subdomain is then minimized via the ritz procedure to extract the frequencies and mode shapes contour plots of a range of new mode shapes are presented for the enhancement of understanding the dynamic behavior of this class of plates

simple bounds on fluctuations and uncertainty relations for first passage times of counting observables recent large deviation results have provided general lower bounds for the fluctuations of time integrated currents in the steady state of stochastic systems a corollary are so called thermodynamic uncertainty relations connecting precision of estimation to average dissipation here we consider this problem but for counting observables i e trajectory observables which in contrast to currents are non negative and nondecreasing in time and possibly symmetric under time reversal in the steady state their fluctuations to all orders are bound from below by a conway maxwell poisson distribution dependent only on the averages of the observable and of the dynamical activity we show how to obtain the corresponding bounds for first passage times times when a certain value of the counting variable is first reached and their uncertainty relations just like entropy production does for currents dynamical activity controls the bounds on fluctuations of counting observables

limit sets for multidimensional nonlinear transport equations abstract we propose a hyperbolic generalisation of the well known reaction diffusion equation and study the structure of its limit sets under a dissipativity condition on the nonlinearity we show quasi convergence of the flow saying that the limit set of any orbit is non empty compact connected and contained in the set of equilibrium solutions

free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution dsc approach algorithm and verification this study presents a detailed procedure for the implementation of a discrete singular convolution dsc approach to the free vibration analysis of composite plates based on classical laminated plate theory clpt the approach performs a numerical solution of differential equation of motion by using a grid discretization based on distribution theory and wavelets in the paper firstly computational algorithm of the dsc method is presented then the accuracy of the computer code developed is verified by comparing dsc solutions with the exact results of simply supported isotropic thin beams fully simply supported one layer isotropic and specially orthotropic plates and also some symmetrically laminated thin composite plates orientated to become specially orthotropic besides dsc predictions for laminated composite plates with different boundary conditions and ply numbers for which there is no analytical solution are compared with those of several distinguished works available in the literature it is noteworthy that dsc results completely match with the exact solutions and are in perfect agreement with those of compared studies

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

a global continuum ritz formulation for flexural vibration of pretwisted trapezoidal plates with one edge built in abstract the development of a continuum ritz model for twisted plate vibration on the basis of minimization of the energy functional is presented this method involves the use of a set of orthogonally generated pb 2 shape functions to approximate the transverse and in plane deflections in this paper a two dimensional recurrence formula is proposed to generate this set of functions during the orthogonalization process a basic function is introduced to each transverse and in plane displacement to ensure the automatic satisfaction of the essential boundary conditions the whole process can be easily implemented numerically and the accuracy of the method can be achieved by increasing the number of terms used in the polynomial functions following the above procedures a governing eigenvalue equation is derived this equation is solved to determine the vibration response of trapezoidal plates with initial twist in order to verify the validity and reliability of this method in addition to the convergence study the present results are compared with the available values from the literature where possible

transverse vibration of trapezoidal plates of variable thickness unsymmetric trapezoids abstract a set of simple two dimensional polynomial functions is employed as the admissible displacement function in the rayleigh ritz energy approach for the free transverse vibration analysis of symmetric trapezoidal plates with linearly varying thickness the admissible function consists the product of i a two dimensional polynomial function and ii a basic function the basic function is defined by the product of the equations of the prescribed continuous piecewise boundary shape each raised to the power of 0 1 or 2 corresponding to a free simply supported or clamped edge respectively the set of functions generated ensures the satisfaction of all the kinematic boundary conditions at the outset the proposed method is applied to solve several symmetric trapezoidal plates with different combinations of boundary conditions and variable thickness the results for some cases are compared with the available published values from the open literature these new results may serve as benchmark data for the development of other numerical methods