exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates the dimensionless equations of motion are derived based on the mindlin plate theory to study the transverse vibration of thick rectangular plates without further usage of any approximate method the exact closed form characteristic equations are given within the validity of the mindlin plate theory for plates having two opposite sides simply supported the six distinct cases considered involve all possible combinations of classical boundary conditions at the other two sides of rectangular plates accurate eigenfrequency parameters are presented for a wide range of aspect ratio and thickness ratio for each case the three dimensional deformed mode shapes together with their associated contour plots obtained from the exact closed form eigenfunctions are also presented finally the effect of boundary conditions aspect ratios and thickness ratios on the eigenfrequency parameters and vibratory behavior of each distinct cases are studied in detail it is believed that in the present work the exact closed form characteristic equations and their associated eigenfunctions except for the plates with four edges simply supported for the rest of considered six cases are obtained for the first time

a differential quadrature procedure for three dimensional buckling analysis of rectangular plates abstract this paper presents an application of the differential quadrature dq method for three dimensional buckling analysis of rectangular plates the governing equations of the plate model are first presented in terms of displacement stress displacement relationship and boundary conditions with three dimensional flexibility these equations are then normalised and discretised using the dq procedure example problems pertaining to the buckling of rectangular plates with generic boundary conditions are selected to illustrate the efficiency and simplicity of implementing the dq procedure the convergence characteristics of the method are first conducted based on numerical studies the dq solutions are then compared where possible with exact or approximate solutions it is found that the differential quadrature method yields accurate results for the plate problems under the current investigation in addition to the above some parametric studies are carried out by varying the plates aspect ratio boundary conditions and thickness to width ratio under axial and biaxial loading

modeling via differential quadrature method three dimensional solutions for rectangular plates this paper presents the formulation and numerical analysis of the three dimensional plate model using the differential quadrature dq method the equilibrium equations stress displacement relations and boundary conditions for the three dimensional plate model are first presented these equations are then normalised and discretised using the dq procedure example problems for rectangular plates with generic boundary conditions are selected to illustrate the computational efficiency and applicability of the dq procedure the convergence characteristics of the method are first conducted based on numerical studies the solutions are then compared where possible with the analytical or exact solutions it is found that the differential quadrature method yields accurate results for the plate problems under the current investigation

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

an approximate numerical method for the complex eigenproblem in systems characterised by a structural damping matrix this paper presents an efficient numerical method that approximates the complex eigenvalues and eigenvectors in structural systems with viscoelastic damping materials characterised by the complex structural damping matrix this new method begins from the solution of the undamped system and approximates the complex eigenpair by finite increments using the eigenvector derivatives and the rayleigh quotient it is implemented in three different approaches single step incremental and iterative schemes the single step technique is presented for systems with low and medium damping from numerical examples it can be verified that the errors committed by the approximate single step technique with respect to the exact solutions obtained by the iram method are less than 0 2 when the loss factor of the material damping is lower than 1 this is a considerable improvement on other approximate methods for higher damped systems an improvement to the previous approach is proposed by an incremental technique that keeps the accuracy without significantly increasing the computational time the complex eigenproblem for materials whose mechanical properties are dependent on frequency is solved by a fast iterative approach whose validity is proved using a four parameter fractional derivative model

analysis of four parameter fractional derivative model of real solid materials the introduction of fractional derivatives into the constitutive equation of the differential operator type of linear solid materials has led to the development of the so called fractional derivative models one of these models characterized by four parameters has been found usable for describing the variation of dynamics elastic and damping properties in a wide frequency range provided that there is only one loss peak in this paper this four parameter model is theoretically analyzed the effect of the parameters on the frequency curves is demonstrated and it is shown that there is a strict relation between the dispersion of the dynamic modulus the loss peak and the slope of the frequency curves the half value bandwidth of the loss modulus frequency curve is investigated and conditions are developed to establish the applicability of the model for a class of materials moreover it is shown that the model can be used to predict the frequency dependences of dynamic properties for a wide range even if measurements are made in only a narrow frequency range around the loss peak

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

three dimensional vibration analysis of a cantilevered parallelepiped exact and approximate solutions the paper investigates the hypothetical assumption of neglecting transverse normal stress in vibration analysis for cantilevered thick plates and rectangular parallelepiped the analysis solves the three dimensional elasticity energy functional including as well as excluding transverse normal stress and obtains free vibration solutions for a cantilevered parallelepiped although it is widely accepted the omission of transverse normal stress is well justified in kirchhoff love thin plate theory and higher order thick plate models the transverse normal stress effects and thickness extent to which the thick plate models apply as the thickness increases are practically unknown the inconsistency of assuming constant transverse normal displacement through thickness for thick plate models is also addressed the paper concludes that for a rectangular parallelepiped with thickness exceeding a certain limit there is considerable discrepancy if transverse normal stress is neglected

computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed point iteration abstract linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions the free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency dependent damping matrix in the present paper a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non proportional viscoelastic structures is developed the central idea is the construction of two complex valued functions of a complex variable whose fixed points are precisely the eigenvalues this important property allows the use of these functions in a fixed point iterative scheme with help of some results in fixed point theory necessary conditions for global and local convergence are provided it is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping in addition under certain conditions the recursive method can also be used for the calculation of non viscous real eigenvalues in order to validate the mathematical results two numerical examples are analyzed one for single degree of freedom systems and another for multiple ones

asymptotic properties of an adaptive beam former algorithm the asymptotic properties of a recursive adaptive beam former algorithm are studied both decreasing gain and constant gain cases are treated for the case of decreasing gain the mean square convergence result is obtained whereas for constant gain a sharp bound is derived and asymptotic analysis for the normalized error is carried out the analysis provides a clear picture of the local behaviour of the iterates near the optimal value a sequence of scale deviations or normalized errors is shown to converge to a gauss markov diffusion process which satisfies a stochastic differential equation

three dimensional vibration analysis of rectangular thick plates on pasternak foundation the free vibration characteristics of rectangular thick plates resting on elastic foundations have been studied based on the three dimensional linear and small strain elasticity theory the foundation is described by the pasternak two parameter model the ritz method is used to derive the eigenvalue equation of the rectangular plate by augmenting the strain energy of the plate with the potential energy of the elastic foundation the chebyshev polynomials multiplied by a boundary function are selected as the admissible functions of the displacement functions in each direction the approach is suitable for rectangular plates with arbitrary boundary conditions convergence and comparison studies have been performed on square plates on elastic foundations with different boundary conditions it is shown that the present method has a rapid convergent rate stable numerical operation and very high accuracy parametric investigations on the dynamic behaviour of clamped square thick plates on elastic foundations have been carried out in detail with respect to different thickness span ratios and foundation parameters some results found for the first time have been given and some important conclusions have been drawn

three dimensional vibration analysis of thick rectangular plates using chebyshev polynomial and ritz method this paper describes a method for free vibration analysis of rectangular plates with any thicknesses which range from thin moderately thick to very thick plates it utilises admissible functions comprising the chebyshev polynomials multiplied by a boundary function the analysis is based on a linear small strain three dimensional elasticity theory the proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions a very simple and general programme has been compiled for the purpose for a plate with geometric symmetry the vibration modes can be classified into symmetric and antisymmetric ones in that direction in such a case the computational cost can be greatly reduced while maintaining the same level of accuracy convergence studies and comparison have been carried out taking square plates with four simply supported edges as examples it is shown that the present method enables rapid convergence stable numerical operation and very high computational accuracy parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail with respect to different thickness side ratios aspect ratios and poissons ratios these results may serve as benchmark solutions for validating approximate two dimensional theories and new computational techniques in future 2002 elsevier science ltd all rights reserved

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

three dimensional elasticity solutions to some orthotropic plate problems abstract this paper presents a three dimensional analysis of rectangular orthotropic plates by employing the differential quadrature dq method the derivation of the governing equations from the governing equations and the stress strain relationship of the three dimensional elasticity model is detailed the constrained conditions of the orthotropic plate edges are given the governing equations and boundary conditions are first normalized and discretized according to the dq procedure example problems pertaining to the bending buckling and free vibration of orthotropic plates with generic boundary conditions are selected to illustrate the simplicity and applicability of the dq method the convergence characteristics of the dq method are first obtained based on numerical studies after which the dq solutions are compared where possible with exact solutions it is found that the dq method yields accurate results for the plate problems under investigation

comparative accuracy of dq and hdq methods for three dimensional vibration analysis of rectangular plates an accuracy study between the differential quadrature dq and harmonic differential quadrature hdq methods for three dimensional elasticity solutions of free vibration of rectangular plates is carried out the solution capability of the dq and hdq methods is first studied then the numerical performance of both the methods is compared it is found that the dq method displays more superior convergence characteristics over the hdq method for the lower modes of vibration however the hdq method is advantageous over the dq method for computing higher modes of vibration it is also discovered that the dq and hdq methods produce better convergent solutions than the finite element method fem when a similar number of discrete points nodes are used copyright 1999 john wiley sons ltd

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

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

free vibration analysis of three dimensional solids by bem in practical applications it is often necessary to determine the natural frequencies and mode shapes of a solid in addition to the static response under a variety of loading conditions this paper discusses the calculation of natural frequencies and mode shapes of three dimensional solids using the boundary element method the method developed is based on the use of particular integrals of the elastic equilibrium equations and employs only real arithmetic it represents an extension to three dimensions of work previously carried out for two dimensional problems in references 1 and 15 the method has been incorporated for multiregion analysis in the best3d boundary element solution technology three dimensional computer program 11 results of the boundary element calculations are compared with analytical finite element and experimental results

on the three dimensional vibrations of the cantilevered rectangular parallelepiped a solution is presented for the three dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces this problem apparently is previously unsolved in the published literature the ritz method is used with displacements assumed in the form of algebraic polynomials convergence is studied numerical results are given for the first five frequencies of each of the four symmetry classes of vibration for five thick parallelepiped configurations including the cube contour plots are exhibited for the modal displacements of the cube the effects of varying poisson s ratio are also observed

static analysis of thick rectangular laminated plates three dimensional elasticity solutions via differential quadrature element method this article presents the first endeavor to develop the differential quadrature element method for static solution of three dimensional elasticity equations of thick rectangular laminated composite plates the domain decomposition technique is employed to decompose a laminated plate into elements according to material layers the differential quadrature dq method is then applied to each element where the material properties are continuous to form the element weighting coefficient matrix and element force vector the discretized element weighting coefficient matrices and element force vectors are assembled together to form the global weighting coefficient matrix and global force vector for the whole plate using connection conditions the solution for the entire plate is obtained by solving the final algebraic equation system detailed formulations and numerical procedures are presented and the convergence characteristics of the method are investigated the numerical results are then compared where possible with the analytical solutions to verify the present solutions consequently some new numerical results are computed and analyzed using the present numerical method for laminated rectangular plates with different boundary conditions which are not solvable directly by the global dq method

averaging procedures in adaptive filtering an efficient approach an averaging procedure for adaptive filtering is developed in contrast to the traditional approach two sequences x sub n and x sub n are constructed where x sub n is the arithmetic average of x sub n the authors show that the algorithm so designed has optimal rate of convergence therefore the averaging approach is asymptotically efficient

3 d theory versus 2 d approximate theory of free orthotropic isotropic plate and shell vibrations part 1 derivation of governing equations in part 1 of the paper the basic assumptions of the shells with added masses considered are introduced they serve for a derivation of the governing equations which are analyzed in part 2 of the paper

a continuum three dimensional vibration analysis of thick rectangular plates abstract this paper presents a continuum three dimensional ritz formulation for the vibration analysis of homogeneous thick rectangular plates with arbitrary combinations of boundary constraints this model is formulated on the basis of the linear three dimensional small deformation elasticity theory to predict the vibratory responses of these thick rectangular plates the displacement fields in the transverse and in plane directions are expressed by sets of orthogonally generated polynomial functions these shape functions are intrinsically a product of a class of orthogonal polynomial functions and a basic function which are chosen to satisfy the essential geometric boundary conditions at the outset sets of frequency data for plates with various aspect ratios and thickness ratios have been presented these data are used to examine the merits and limitations of the classical plate theory and mindlin plate theory by direct comparisons finally using the three dimensional continuum approach sets of first known deformed mode shapes have been generated thus helping to understand the vibratory motion furthermore these results may also serve as the benchmark to further research into the refined plate theories

second order convergence analysis of stochastic adaptive linear filtering the convergence of an adaptive filtering vector is studied when it is governed by the mean square error gradient algorithm with constant step size we consider the mean square deviation between the optimal filter and the actual one during the steady state this quantity is known to be essentially proportional to the step size of the algorithm however previous analyses were either heuristic or based upon the assumption that successive observations were independent which is far from being realistic actually in most applications two successive observation vectors share a large number of components and thus they are strongly correlated in this work we deal with the case of correlated observations and prove that the mean square deviation is actually of the same order or less than the step size of the algorithm this result is proved without any boundedness or barrier assumption for the algorithm as it has been done previously in the literature to ensure the nondivergence our assumptions are reduced to the finite strong memory assumption and the finite moments assumption for the observation they are satisfied in a very wide class of practical applications

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

numerical aspects for free vibration of thick plates part ii numerical efficiency and vibration frequencies a concise formulation and an efficient method of solution for free vibration of thick shear deformable plates have been presented in part i of this paper an efficient numerical algorithm is developed here and executed on various computational platforms numerical results show that upon classifying the symmetries of vibration modes as much as 40 to 65 of the execution time can be saved for single symmetry classification and as much as 45 to 85 of the execution time can be saved for double symmetry classification while maintaining the same level of accuracy in addition memory requirements can be considerably reduced a set of comprehensive frequency parameters is presented and the frequency trend is discussed the conclusion is reached that the transverse dominant vibration frequencies are dependent on plate thickness while the inplane dominant vibration frequencies are insensitive to variations in thickness

eigensolutions of nonviscously damped systems based on the fixed point iteration abstract in this paper nonviscous nonproportional symmetric vibrating structures are considered nonviscously damped systems present dissipative forces depending on the time history of the response via kernel hereditary functions solutions of the free motion equation leads to a nonlinear eigenvalue problem involving mass stiffness and damping matrices this latter as dependent on frequency viscous damping can be considered as a particular case involving damping forces as function of the instantaneous velocity of the degrees of freedom in this work a new numerical procedure to compute eigensolutions is proposed the method is based on the construction of certain recursive functions which under a iterative scheme allow to reach eigenvalues and eigenvectors simultaneously and avoiding computation of eigensensitivities eigenvalues can be read then as fixed points of those functions a deep analysis of the convergence is carried out focusing specially on relating the convergence conditions and error decay rate to the damping model features such as the nonproportionality and the viscoelasticity the method is validated using two 6 degrees of freedom numerical examples involving both nonviscous and viscous damping and a continuous system with a local nonviscous damper the convergence and the sequences behavior are in agreement with the results foreseen by the theory

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

three dimensional free vibration analysis of isotropic rectangular plates using the b spline ritz method abstract this paper presents three dimensional free vibration analysis of isotropic rectangular plates with any thicknesses and arbitrary boundary conditions using the b spline ritz method based on the theory of elasticity the proposed method is formulated by the ritz procedure with a triplicate series of b spline functions as amplitude displacement components the geometric boundary conditions are numerically satisfied by the method of artificial spring to demonstrate the convergence and accuracy of the present method several examples with various boundary conditions are solved and the results are compared with other published solutions by exact and other numerical methods based on the theory of elasticity and various plate theories rapid stable convergences as well as high accuracy are obtained by the present method the effects of geometric parameters on the vibrational behavior of cantilevered rectangular plates are also investigated the results reported here may serve as benchmark data for finite element solutions and future developments in numerical methods

weak convergence and asymptotic properties of adaptive filters with constant gains the basic adaptive filtering algorithm x n 1 epsilon x n epsilon epsilon y n y n x n epsilon psi n is analyzed using the theory of weak convergence apart from some very special cases the analysis is hard when done for each fixed epsilon 0 but the weak convergence techniques are set up to provide much information for small epsilon the relevant facts from the theory are given define x epsilon cdot by x epsilon t x n epsilon on n epsilon n epsilon epsilon then weak distributional convergence of x epsilon cdot and of x epsilon cdot t epsilon is proved under very weak assumptions where t epsilon rightarrow infty as epsilon rightarrow 0 the normalized errors x n epsilon theta sqrt epsilon are analyzed where theta is a stable point for the mean algorithm the asymptotic properties of a projection algorithm are developed where the x n epsilon are truncated at each iteration if they fall outside of a given set

derivatives of eigenvalues and eigenvectors of a general complex matrix a survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non hermitian matrices is presented in addition a modification of one method based on a better normalizing condition is proposed methods are classified as direct or adjoint and are evaluated for efficiency operation counts are presented in terms of matrix size number of design variables and number of eigenvalues and eigenvectors of interest the effect of the sparsity of the matrix and its derivatives is also considered and typical solution times are given general guidelines are established for the selection of the most efficient method

free vibrations of a simply supported rectangular plate an exact elasticity solution abstract an exact three dimensional solution for the free vibrations of simply supported rectangular plates of arbitrary thickness within the linear theory of elastodynamics is given in this paper the solution obtained in a semi inverse fashion as was the solution of the elastostatic problem for such plates satisfies all of the boundary conditions of the problem in a pointwise manner it is found that there are two types of modes of oscillation possible which are consistent with the kinematic assumptions made to find the semi inverse solution other modes of oscillation may exist in the three dimensional theory of elastodynamics for such plates but our kinematic assumptions would not be consistent with such modes the two types of modes found are analogous to the flexural modes of classical plate theory and the thickness twist modes here called breathing modes of mindlin plate theory some numerical results are given which indicate that the predictions of mindlin plates are uncannily good approximations to the flexural frequencies given by the present three dimensional analysis even for very thick plates however the predictions of mindlin plate theory for the thickness twist or breathing frequencies are not nearly so good these discrepancies are discussed briefly in an appendix

iterative method for calculating derivatives of eigenvectors the modified modal method uses the static solution to account for the contribution of truncated higher eigenvectors which significantly improves the accuracy or effectiveness of the tuncated modal superposition in this paper we extended this method to an iterative procedure

on the three dimensional vibration analysis of simultaneously skewed and twisted cantilevered parallelepipeds the natural frequencies of simultaneously skewed and twisted cantilevered parallelepipeds are determined using the ritz method the present work is the first known three dimensional study of the problem assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly and which are mathematically complete reasonably accurate natural frequencies are calculated for low aspect ratio thick parallelepipeds having arbitrary degrees of twist and skewness detailed numerical studies show that a three dimensional analysis is essential in monitoring complicated coupled mode sensitivities in the variation of non dimensional natural frequencies with increasing skew and twist angles and thickness ratio results obtained using the present method are compared with those obtained in a detailed three dimensional finite element analysis where the suitability and limitations of s version solid element discretizations are clearly brought out

up to lowest 100 frequencies of rectangular plates using jacobi polynomials and tsndt abstract this paper finds up to lowest 100 free vibration frequencies of isotropic and linearly elastic plates with thickness to side length ratio varying from 0 001 to 0 5 by using the ritz method a third order shear and normal deformable plate theory tsndt and weighted jacobi polynomials as admissible functions that are mutually orthogonal and exactly satisfy essential boundary conditions the numerical method is stable even for 18th degree polynomials as basis functions it is shown that this approach requires fewer degrees of freedom than those in the traditional finite element method fem to find converged lowest 100 frequencies and the corresponding mode shapes no shear correction factor is employed in the tsndt the presently computed results agree well with those from either analytical or numerical solutions of the corresponding 3 dimensional linearly elastic problems obtained with the commercial fe software abaqus furthermore results for plates made of an incompressible material can be computed by setting poisson s ratio 0 49

bem analysis of free vibration problems using polynomial particular integrals abstract the development of a boundary element method for free vibration analysis using particular solutions is discussed the formulation presented is based on treating the forcing function in the governing differential equation as an initially unknown distributed body force within the domain the volume integral due to this body force is then eliminated by approximating the inertia force by global interpolation and polynomial function representations and finding particular solutions for the governing inhomogeneous equations the resulting non symmetric system matrix is solved by using a modified arnoldi s algorithm that takes advantage of the special structure of the substructured boundary element method the techniques described here are embedded in a computer program gpbest and the numerical examples arc solved by using this computer code

frequency dependences of complex moduli and complex poisson s ratio of real solid materials abstract the concept of a complex modulus of elasticity is a powerful and widely used tool for characterizing the linear dynamic elastic and damping properties of solid materials in the frequency domain it is shown in this paper that typical characters of frequency dependences of all complex moduli shear young s etc and complex poisson s ratios of real solid materials can be determined by transforming the causal and real relaxation and creep responses respectively from the time domain into the frequency domain even without having to specify the processes of relaxation and creep it is proved that all dynamic moduli monotonically increase and the dynamic poisson s ratio monotonically decreases with increasing frequency and all respective loss factors pass through at least one maximum these frequency dependences are generally valid for any real solid material regardless of the actual damping mechanisms some experimental results are presented and interpreted in the light of the theory the usefulness of theoretical predictions in materials engineering measurements of dynamic properties and in modelling dynamic behaviour is discussed

higher order eigensensitivities based numerical method for the harmonic analysis of viscoelastically damped structures this paper presents an efficient numerical method for the harmonic analysis of viscoelastically damped structural systems characterized by a frequency dependent structural damping matrix making use of the complex mode superposition method departing from the undamped eigensolution the proposed numerical method updates the complex and frequency dependent eigenpair avoiding the solution of a complex eigenproblem for each computational frequency the complex eigenvalues and eigenvectors are updated within the desired tolerance by an adaptive step size control scheme using the first and higher order eigenderivatives the influence on the computation time of the considered number of eigenderivatives and the tolerance is discussed and the efficiency of the proposed numerical method for the harmonic analysis of viscoelastically damped large ordered structural systems is proved finally a practical application is presented where the harmonic response of a constrained layer damping cantilever beam subjected to a base motion is analyzed the complex and frequency dependent eigenvalues and eigenvectors are computed the modal contributions to the total response are determined and the total response is approximated by the complex mode superposition method finally the approximated response is validated with the exact one computed by the direct frequency method and with that experimentally measured copyright 2011 john wiley sons ltd

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