monodromy matrix for linear difference operators with almost constant coefficients a new method for solving the discrete scattering problem for a linear single valued difference operator of arbitrary order with almost constant coefficients is proposed the treatment is concerned with the asymptotic behavior of its eigenfunctions as t the purpose of the paper is to investigate the transition between the asymptotically free states of the underlying system defined in terms of the monodromy operator a rare mathematical tool is used an infinite matrix product it is shown that if the coefficients of the aforementioned operator are suitably behaved the monodromy operator exists in the form of the convergent two sided infinite product of matrices associated with the matrix eigenvalue equation corresponding to the scattering problem the application of the method to the solution of the scattering problem for toda lattice equations is demonstrated however the approach is quite general and should be applicable to other forms of lattice equations

inclusion principle for the rayleigh ritz based substructure synthesis according to this substructure synthesis the motion of every substructure is modeled in terms of quasicomparison functions which are linear combinations of admissible functions capable of satisfying all the boundary conditions

rayleigh ritz based substructure synthesis for flexible multibody systems this paper is concerned with the modeling of flexible multibody systems by a rayleigh ritz based substructure synthesis method so that certain advantages can be accrued by using the variational approach to derive the eigenvalue problem as with the classical rayleigh ritz method if the admissible functions used to represent the motion of the substructures are not chosen properly convergence can suffer this paper presents a new substructure synthesis method with superior convergence characteristics achieved by representing the motion by means of a recently developed class of functions namely the class of quasi compari son functions this improved convergence is shown to be related to improved approximation of both the differential equations and the natural boundary conditions the theory is demonstrated by means of a numerical example i introduction t his investigation is concerned with the modeling of flexible multibody systems many structures such as fixedwing aircraft helicopters flexible spacecraft flexible robots etc can be modeled as assemblages of interacting flexible bodies a method proposed by hurty1 2 in the early 1960s and known as component mode synthesis consists of modeling the motion of the individual substructures referred to as components by means of component modes he uses three types of modes to represent the motion of a substructure rigid body modes constraint modes and normal modes to define constraint modes hurty divides the constraints into two classes statically determinate and redundant the constraint modes are equal in number to the number of redundant constraints and are defined by producing a unit displacement on each redundant constraint in turn with all other constraints fixed the normal modes represent the modes of vibration of the components with all constraints fixed an approach by craig and hampton3 differs from that of hurty 1 2 mainly in the selection of the component modes indeed craig and hampton suggest two types of component modes boundary modes providing for displacements and rotations at points along the substructure boundaries and related to the constraint modes of hurty and substructure modes corresponding to completely restrained boundaries the substructures modeled individually in refs 1 3 are made to act together as a single structure by eliminating the redundant generalized coordinates arising from the fact that displacements at points common to two adjacent substructures are included twice in the overall problem formulation once for each substructure the elimination process is based on the use of constraint equations resulting from the enforcement of compatibility conditions which amounts to saying that the displacement of a boundary point shared by two substructures is the same in another approach to the problem proposed by

vibration of an oscillator with random damping analytical expression for the probability density function this study deals with the impact of probabilistic viscous and hysteretic damping upon the dynamic response of a linear single degree of freedom oscillator assuming that damping is governed by an unspecified probability density function pdf an analytical expression of the output pdf of the oscillator s transfer function is provided in terms of the input damping pdf the instance of a uniform pdf is thoroughly treated for both viscous and hysteretic damping cases the methodology also yields to analytical expressions of all moments of the output law of probability a definition of a new concept of envelopes is then introduced and their analytical expression is derived exact analytical results are compared with monte carlo simulations for an academic case study an identification procedure of a predefined input damping pdf is proposed allowing one to obtain probabilistic parameters from experimental frequency response data

a general solution for the free vibration of rectangular plates resting on uniform elastic edge supports abstract in this paper it is shown how the superposition method is employed to obtain accurate solutions for the free vibration frequencies and mode shapes of rectangular plates resting on elastic edge supports the analysis is completely general in that uniform elastic rotational and translational supports of any stiffness magnitudes are considered to act simultaneously along each of the four edges eight stiffness coefficients being required to define the problem it is shown how all of the classical boundary conditions are approached when the coefficients are allowed to take on appropriate limits a limited number of plots of fundamental mode eigenvalues vs stiffness coefficients are presented for square plate problems of special interest to the author s knowledge this represents the first accurate comprehensive treatment of this problem to appear in the literature

a general procedure for improving substructures representation in dynamic synthesis in the general substructure synthesis method for the dynamic analysis of complex flexible structures developed earlier by the authors the motion of each substructure is represented by a given number of substructure admissible functions and the otherwise disjoint substructures are connected together to form a whole structure by imposing approximate geometric compatibility conditions by means of weighted residuals in this paper a general procedure for improving the admissible functions representing each substructure in the synthesis is developed the procedure is an iterative one and it leads to an improved eigensolution for the intermediate structure i e a fictitious structure defined by the approximate compatibility conditions where the improved eigensolution is obtained while always using the same number of degrees of freedom to represent a given substructure in any iteration the computations associated with each substructure are independent of those for all other substructures and they can be performed in parallel by increasing the number of iterations the eigensolution for the intermediate structure is approached the procedure is applicable to assembled structures that are either positive definite or positive semidefinite moreover the procedure represents an extension of the concept of subspace iteration to structures composed of substructures

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

modal probabilistic analysis in this paper we first recall a method for estimating the numerical probability density function pdf with parzen rosenblatt estimators of experimental data like the sound absorption coefficient with the kundt s tube then we focus on the model of mikki which is used as a predictive tool we are interested then in inversing the model and in identifying the 3 parameters q of mikki s model but we want to go further and to identify from the estimated pdf directly the pdf of the 3 parameters we explain first when it is possible then we give some examples to illustrate the method this work allows for evaluating the robustness of a model from experimental data

on a decomposition of generalized constraint forces an additive decomposition of generalized constraint forces into a component determined by the motion of the system and a constitutively prescribed component is discussed in this paper the former component is shown to be necessary and sufficient to ensure that the motion of the system satisfies the constraint while the prescription of the latter is shown to be at the discretion of the dynamicist the decomposition is also related to previous works on constraint forces in the dynamics of discrete mechanical systems

rigorous implementation of the galerkin method for uniform and stepped columns the aim of this paper is to deal with the often occurring mistake in the implementation of the galerkin method and its correction it is shown that the flexural rigidity should be represented as a

from euler ritz and galerkin to modern computing the so called ritz galerkin method is one of the most fundamental tools of modern computing its origins lie in hilbert s direct approach to the variational calculus of euler lagrange and in the thesis of walther ritz who died 100 years ago at the age of 31 after a long battle with tuberculosis the thesis was submitted in 1902 in gottingen during a period of dramatic developments in physics ritz tried to explain the phenomenon of balmer series in spectroscopy using eigenvalue problems of partial differential equations on rectangular domains while this physical model quickly turned out to be completely obsolete his mathematics later enabled him to solve difficult problems in applied sciences he thereby revolutionized the variational calculus and became one of the fathers of modern computational mathematics we will see in this article that the path leading to modern computational methods and theory involved a long struggle over three centuries requiring the efforts of many great mathematicians

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

a general dynamic synthesis for structures with discrete substructures abstract this paper presents a substructure synthesis method for the dynamic simulation of complex structures where the structures consist of an assemblage of discrete substructures an analogy between distributed and discrete structures is extensively invoked to stimulate the motion of discrete substructures the concept of admissible vectors is introduced where admissible vectors represent the discrete counterpart of admissible functions for distributed substructures the individual substructures are forced to act as a whole structure by imposing certain geometric compatibility on internal boundaries shared by any two substructures a numerical example illustrating the method is presented

finite volume schemes for the biharmonic problem on general meshes finite volume schemes for the approximation of a biharmonic problem with dirichlet boundary conditions are constructed and analyzed first on grids which satisfy an orthogonality condition and then on general possibly non conforming meshes in both cases the piece wise constant approximate solution is shown to converge in l2 to the exact solution similar results are shown for the discrete approximate of the gradient and the discrete approximate of the laplacian of the exact solution error estimates are also derived these results are confirmed by numerical results

a new approach for the representation of a point support in the analysis of plates abstract a new approach for the representation of a point support in the analysis of plates is presented it is based on the use of a flexibility function representing the distribution of a fictitious elastic restraint over the boundary which is such that it has a zero value at the point support location but assumes large values resulting in negligible restraint over the free boundary the application of this flexibility function approach is demonstrated by considering the free vibrations of rectangular plates with two opposite edges simply supported and mid point supports on the other two edges which are otherwise free the results obtained by using this approach are compared with those obtained by using the method based on the representation of the concentrated force at the point support by a fourier series expansion of the impulse function it is shown that the flexibility function approach as compared to the impulse function approach gives a slightly faster convergence and yields results which are generally very accurate it is also shown that the flexibility function approach has the scope to simulate practical point support conditions in which the support reaction is spread over a definite band

optimization of support positions to maximize the fundamental frequency of structures in this paper the position optimization of simple supports is implemented to maximize the fundamental frequency of a beam or plate structure both elastic and rigid supports are taken into account first the frequency sensitivity with respect to the movement of a simple support is derived using the discrete method by means of the shape functions of the finite element method closed form sensitivity formulations are developed straightforwardly then a heuristic approach called evolutionary shift method is presented for optimizing support positions with a fixed grid mesh scheme based on the design sensitivity analysis the support with the highest efficiency is shifted in priority along the elementary edges with the interval step of the elementary size to facilitate the convergence of the process the interpolation technique is employed to evaluate the solution more accurately finally three numerical examples are presented to demonstrate the validity of the sensitivity analysis and the effectiveness of the optimization method copyright 2004 john wiley sons ltd

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

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

on proximity of rayleigh quotients for different vectors and ritz values generated by different trial subspaces abstract the rayleigh quotient is unarguably the most important function used in the analysis and computation of eigenvalues of symmetric matrices the rayleigh ritz method finds the stationary values of the rayleigh quotient called ritz values on a given trial subspace as optimal in some sense approximations to eigenvalues in the present paper we derive upper bounds for proximity of the ritz values in terms of the proximity of the trial subspaces without making an assumption that the trial subspace is close to an invariant subspace the main result is that the absolute value of the perturbations in the ritz values is bounded by a constant times the gap between the original trial subspace and its perturbation the constant is the spread in the matrix spectrum i e the difference between the largest and the smallest eigenvalues of the matrix it s shown that the constant cannot be improved we then generalize this result to arbitrary unitarily invariant norms but we have to increase the constant by a factor of 2 our results demonstrate in particular the stability of the ritz values with respect to a perturbation in the trial subspace

invertibly convergent infinite products of matrices abstract the standard definition of convergence of an infinite product of scalars is extended to the infinite product p n 1 b n of k k matrices that is p is convergent according to the definition given here if and only if there is an integer n such that b n is invertible for n n and p lim n m n n b m is invertible a family of sufficient conditions for this kind of convergence is given along with examples showing that they have nontrivial applications

free vibration and buckling of symmetric cross ply rectangular laminates the method of superposition is further developed to analyze free vibration and buckling of symmetric crossply laminated rectangular plates the improved plate theory developed by mindlin is implemented a four building block technique is employed that reduces the number of combinations of classical boundary conditions that need to be studied combinations of clamped and simply supported boundary conditions are investigated eigenvalues and critical buckling loads with four digit accuracy are obtained for plates with five distinct combinations of simply supported and clamped boundary conditions a b c

optimal support positions for a structure to maximize its fundamental natural frequency abstract a procedure and related theories are developed to find loci of optimal support positions for a structure to maximize its fundamental eigenvalue by increasing the support stiffness the concept of limit eigenvalue which is the upper bound of fundamental eigenvalue achieved by adding supports is introduced a condition is derived on which the fundamental eigenvalue can be reached to its limit eigenvalue a sensitivity formula of eigenvalues with respect to the change of support positions is also derived to set up an optimization problem and to obtain its optimal support positions it is found that the loci of m supports start from the maximum displacement position of the structure s first eigenfunction and end at certain positions on the nodal line of its m 1 th eigenfunction if the fundamental eigenvalue can reach its limit eigenvalue the suggested method is tested to find the loci for a beam and a plate structure

the kosterlitz thouless phase in a hierarchical model the author uses renormalisation group methods to study the long distance behaviour of hierarchical cos phi 2 lattice field theories in the kosterlitz thouless phase

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

a comprehensive overview of a non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics abstract in structural dynamics a predictive model is constructed by developing a mathematical mechanical model of a designed system in order to predict the response of the real system which is the manufactured system realized from the designed system the mathematical mechanical modelling process of the designed system introduces two fundamental types of uncertainties the data uncertainties and the model uncertainties uncertainties have to be taken into account for improving the predictability of the model model uncertainties cannot be modelled by using the usual parametric probabilistic approach recently a general non parametric probabilistic approach of model uncertainties for dynamical systems has been proposed using the random matrix theory this paper gives a comprehensive overview of this approach in developing its foundations in simple terms and in illustrating all the concepts and the tools introduced in the general theory by using a simple example this paper deals with 1 notions of designed systems real systems mean models as predictive models errors and uncertainties 2 the definition of a simple example in linear elastodynamics 3 a comprehensive overview of the non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics 4 a summary of the random matrix ensembles which are necessary for the non parametric modelling of random uncertainties 5 the estimation of the dispersion parameters of the non parametric probabilistic model using experimental data 6 the method to solve the stochastic equation of the dynamical system with non parametric probabilistic model of random uncertainties 7 a numerical simulation and the validation for the simple example

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

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

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

gibbs phenomenon in structural mechanics the concepts of complementary boundary condition and interior condition are discussed guidelines are developed to eliminate the gibbs effect when selecting trial functions for the series expansion

rigidity of the three dimensional hierarchical coulomb gas a random set of points in euclidean space is called rigid or hyperuniform if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i i d random points this phenomenon has received considerable attention in recent years due to its appearance in random matrix theory the theory of coulomb gases and zeros of random analytic functions however most of the published results are in dimensions one and two this paper gives the first proof of hyperuniformity in a coulomb type system in dimension three known as the hierarchical coulomb gas this is a simplified version of the actual 3d coulomb gas the interaction potential in this model inspired by dyson s hierarchical model of the ising ferromagnet has a hierarchical structure and is locally an approximation of the coulomb potential hyperuniformity is proved at both macroscopic and microscopic scales with upper and lower bounds for the order of fluctuations that match up to logarithmic factors the fluctuations have cube root behavior in agreement with a well known prediction for the 3d coulomb gas for completeness analogous results are also proved for the 2d hierarchical coulomb gas and the 1d hierarchical log gas

optimal a priori error bounds for the rayleigh ritz method we derive error bounds for the rayleigh ritz method for the approximation to extremal eigenpairs of a symmetric matrix the bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector we also present a sharp bound

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

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

the equations of lagrange for a continuous deformable body with rigid body degrees of freedom written in a momentum based formulation abstract the present paper is concerned with lagrange s equations applied to a deformable body in the presence of rigid body degrees of freedom the lagrange description of continuum mechanics is used an exact version of the equations is derived first this version which represents an identical extension of the fundamental law of dynamics does involve the idea of virtual motions the virtual motion is described in the framework of the ritz ansatz but our derivation does not make use of d alemberts principle the principle of virtual work or variational principles from the exact version by involving arguments related to the galerkin approximation technique we derive an approximate ritz type version of lagrange s equations this approximate version coincides with the traditional one which is based on the notion of kinetic energy however since our derivation stems from the fundamental law of dynamics we have at our disposal an alternative formulation which is based on the notion of local momentum this momentum based version which is the main topic of the present contribution can be used for the purpose of performing independent checks of the energy based version of lagrange s equations the momentum based version also clarifies that and how certain terms in the energy based version do cancel out the momentum based version is worked out in the framework of the floating frame of reference formulation of multibody dynamics explicit formulas for the single terms of lagrange s equations are derived for the translational rotational and flexible degrees of freedom of the deformable body respectively corresponding lagrange s equations are explained in the light of the relations of balance of total momentum balance of total moment of momentum of the mean stress theorem and the notion of virial of forces an embedding into the literature is given

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

analysis of thick isotropic and cross ply laminated plates by radial basis functions and a unified formulation in this paper we combine carrera s unified formulation and a radial basis function collocation technique for predicting the static deformations and free vibration behavior of thin and thick isotropic and cross ply laminated plates through numerical experiments the capability and efficiency of this collocation technique for static and vibration problems are demonstrated and the numerical accuracy and convergence are thoughtfully examined

the convergence of harmonic ritz values harmonic ritz vectors and refined harmonic ritz vectors this paper concerns a harmonic projection method for computing an approximation to an eigenpair x of a large matrix a given a target point r and a subspace w that contains an approximation to x the harmonic projection method returns an approximation x to x three convergence results are established as the deviation e of x from w approaches zero first the harmonic ritz value r converges to a if a certain rayleigh quotient matrix is uniformly nonsingular second the harmonic ritz vector x converges to x if the rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic ritz values third better error bounds for the convergence of are derived when x converges however we show that the harmonic projection method can fail to find the desired eigenvalue in other words the method can miss if it is very close to r to this end we propose to compute the rayleigh quotient p of a with respect to x and take it as a new approximate eigenvalue p is shown to converge to once x tends to x no matter how r is close to finally we show that if the rayleigh quotient matrix is uniformly nonsingular then the refined harmonic ritz vector or more generally the refined eigenvector approximation introduced by the author converges we construct examples to illustrate our theory

a procedure for improving discrete substructure representation in dynamic synthesis n important problem in the dynamic analysis for flexible structures is that of determining the natural frequencies and associated natural modes of vibration particularly the lower ones complex structures are often represented by mathematical models such as finite element models possessing a very large number of degrees of freedom where the number can reach into the tens of thousands because of the large number of degrees of freedom a direct iterative method for obtaining a partial eigensolution know as subspace iteration was developed the subspace iteration method has a long history an extensive bibliography is given in ref 1 and the current status of the method is discussed in refs 2 4 in the context of structural dynamics even the subspace iteration method however can be overwhelmed by the large number of degrees of freedom in the mathematical model so that a method for prior reduction of the number of degrees of freedom is desirable complex structures are often modeled by breaking the structure into a number of simpler components or substructures the substructure models are then coupled together to form the whole structure model the technique is known as substructure synthesis and its origin can be found in refs 5 and 6 the idea of refs 5 and 6 is to represent the motion of each substructure by a set of substructure normal modes obtained by solving an eigenvalue problem for each substructure then each substructure is represented in the synthesis process by a reduced number of lower substructure modes the synthesis leads to an eigenvalue problem for the assembled structure of a substantially smaller order than that of the original formulation the price paid in reducing the order of the eigenvalue problem is that its solution is only an approximation of the actual eigensolution of the original structure

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

angles between infinite dimensional subspaces with applications to the rayleigh ritz and alternating projectors methods we define angles for infinite dimensional subspaces of hilbert spaces inspired by the work of e j hannan 1961 1962 the angles of dixmier and friedrichs and the gaps are characterized we establish connections between the angles corresponding to orthogonal complements the sensitivity of angles with respect to subspaces is estimated we show that the squared cosines of the angles from one subspace to another can be interpreted as ritz values in the rayleigh ritz method the hausdorff distance between the ritz values corresponding to different trial subspaces is shown to be bounded by a constant times the gap between the subspaces we prove a similar eigenvalue perturbation bound that involves the gap squared an ultimate acceleration of the classical alternating projectors method is proposed its convergence rate is estimated in terms of the angles we illustrate the acceleration for a domain decomposition method with a small overlap for the 1d diffusion equation

invertibly convergent infinite products of matrices with applications to difference equations abstract the standard definition of convergence of an infinite product of scalars is extended to the infinite product p m 1 b m of k k matrices that is p is convergent according to the definition given here if and only if there is an integer n such that bm is invertible for m n and p lim n n m n i a m is invertible sufficient conditions for this kind of convergence are given some of the results seem to be new even for infinite products of scalars the results are derived by considering related systems of difference equations and have implications concerning the asymptotic behavior of solutions of these systems

on the substructure synthesis method substructure synthesis is a model reduction method whereby a complex structure is regarded as an assemblage of substructures the motion of each of the substructures is represented by a series of admissible functions vectors and the substructures are made to act as a single structure by imposing certain geometric compatibility conditions at the boundary between any two adjacent substructures because the series of admissible functions vectors represents the motion of the substructures only approximately and because the geometric compatibility conditions are only approximations of the true conditions the computed eigensolution is only an approximation of the actual one convergence of computed eigenvalues to actual eigenvalues requires two limiting processes one in which the number of admissible functions vectors is increased and the other in which the number of geometric compatibility conditions is increased

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

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

the role of eigenparameter gradients in the detection of perturbations in discrete linear systems abstract continuous mechanical systems can be characterized through analysis of discrete linear models such models can provide approximations for the eigenvalues and eigenvectors collectively eigenparameters although the eigenparameters do not qualify as measures for the state of the system in response to specific loading and boundary conditions they do reflect the identity of the system and this in itself has important applications in particular the eigenparameters can be used to study the sensitivity of a system to perturbations due perhaps to damage incurred by one or more discrete elements these studies can rationalize the choice and weighting of eigenparameters for system identification strategies damage detection algorithms and damage assessment methods to this end this paper develops a set of sensitivity coefficients based on gradients of the eigenparameters sensitivities are normalized with respect to that of the harmonic oscillator and generalized to include the mode vectors through the definition of a figure of merit analytical and numerical examples based on appropriate elements are used to illustrate the utility of the approach

an efficient general approach to modal analysis of frame resonators with applications to support loss in microelectromechanical systems abstract the design of high q resonators such as xylophone bar resonators xbrs capable of being fabricated using micro electro mechanical systems mems processes is of considerable interest in light of the widespread and rapidly growing use of systems dependent on their availability and performance this paper is concerned with vibration analysis and q optimisation of an xbr with the method extending directly to other planar frames and straightforwardly to more complex structures the rayleigh ritz method is discussed in some detail first treating the discrete case followed by developing and applying a kinematical procedure to an l frame structure attention is given to geometric interpretation of the rayleigh ritz procedure and to developing an intuitive understanding the method before turning to the xbr case having developed an approximation for system dynamics the results are used in conjunction with an analytical model of elastic wave propagation in the substrate to obtain an estimate for the support q factor natural frequencies mode shapes and support q values are presented and compared to finite element models of the same problem with excellent agreement observed at substantially lower computational cost for the first time in the literature the geometric impedance tuning principle underlying the xbr design is validated and quantified including sensitivity to manufacturing error

renormalization group flow of a hierarchical sine gordon model by partial differential equations we use a renormalization group differential equation to rigorously control the renormalization group flow in a hierarchical lattice sine gordon field theory in the kosterlitz thouless phase

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

optimization of boundary conditions for maximum fundamental frequency of vibrating structures a sensitivity formula of eigenvalues with respect to the change of boundary conditions is derived using the material derivative concept based on a variational formulation the change of boundary conditions is described with the introduction of the tangential component of the design velocity field used in shape design simply supported and partially welded plates are taken as numerical examples to check the accuracy of the sensitivity formula the sensitivites of the distinct and multiple eigenvalues calculated by the formulas are compared with those calculated by finite differences optimal support locations are then determined by use of a gradient based optimization method it is shown that a crossing of eigenvalues can occur in the solution process

numerical modal analysis of structures based on a revised substructure synthesis approach the classical substructuring approach to the problem of modal analysis of structures suffers from a number of drawbacks which can sometimes lead the analyst to prefer other often more cumbersome techniques a modified rayleigh ritz based approach is presented which circumvents most of the drawbacks and improves the precision of the method while also enhancing its applicability to complex structures the scheme is also particularly well suited to symbolic implementation in view of its systematic problem independent formulation the theoretical results are validated by case studies chosen to highlight various features of the approach viz a beam structure a hinged simple square plate a hinged plate structure of fairly complicated shape and a rectangular plate with two cutouts

pseudo rigid continua basic theory and a geometrical derivation of lagrange s equations pseudorigid bodies are regarded here as globally constrained threedimensional homogeneous continua the constraint reaction stresses play a fundamental role in maintaining the homogeneity of the de

a double sweeping preconditioner for the helmholtz equation a new preconditioner is developed to increase the efficiency of iterative solution of the helmholtz equation the key idea of the proposed preconditioner is to split the domain of interest into smaller subdomains and sequentially approximate the forward and backward components of the solution the sequential solution is facilitated by approximate interface conditions that ignore the effect of multiple reflections the efficiency of the proposed method is tested using various 2 d heterogeneous media we observe that the proposed preconditioner results in good convergence with number of iterations growing very slowly with increasing frequency we also note that the mesh size and number of subdomains do not affect the convergence rate finally we find that the overall computational time is much smaller than that of the sweeping preconditioner

ground states and hyperuniformity of the hierarchical coulomb gas in all dimensions stochastic point processes with coulomb interactions arise in various natural examples of statistical mechanics random matrices and optimization problems often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties that is the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i i d random points a well known conjecture from physics appearing in the works of jancovici lebowitz manificat martin and yalcin see jancovici et al in j stat phys 72 3 4 773 787 1993 lebowitz in phys rev a 27 1491 1494 1983 martin and yalcin in j stat phys 22 4 435 463 1980 states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i i d random points in a recent beautiful work chatterjee in rigidity of the three dimensional hierarchical coulomb gas 2017 arxiv preprint arxiv 1708 01965 to appear in probability theory and related fields chatterjee gave the first proof of such a result in dimension three for a coulomb type system known as the hierarchical coulomb gas inspired by dyson s hierarchical model of the ising ferromagnet dyson in phys rev 92 1331 1338 1953 dyson in commun math phys 12 2 91 107 1969 however the case of dimensions greater than three had remained open in this paper we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three for the hierarchical model using similar methods we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two a key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems and could be of independent interest

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

free and forced flexural vibration analysis of cantilever plates with attached point mass abstract in this paper the method of superposition or the gorman method is employed to obtain an analytical solution for free and forced vibrations of cantilever plates carrying point masses the free lateral vibration analysis is carried out by employing five building blocks four for the cantilever plate and one for the point mass computed eigenvalues show that the proposed scheme is accurate and convergent for free vibration analysis once the eigen analysis is completed the modal summation method is then used to deal with the base induced lateral vibration of cantilever plates carrying a point mass effects of mass ratios and locations of the point mass on eigenvalues and modal participation factors are investigated for square and rectangular plates

convergence of the classical rayleigh ritz method and the finite element method it is demonstrated in this paper that convergence of the classical rayleigh ritz method can be vastly improved by introducing a new class of admissible functions called quasi comparison functions

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

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

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

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

chladni figures and the tacoma bridge motivating pde eigenvalue problems via vibrating plates teaching linear algebra routines for computing eigenvalues of a matrix can be well moti vated for students by using interesting examples we propose in this paper to use vibrating plates for two reasons first they have many interesting applications from which we chose the chladni figures representing sand ornaments which form on a vibrating plate and the tacoma bridge one of the most spectacular bridge failures second the partial differential operator that arises from vibrating plates is the biharmonic operator which one does not encounter often in a first course on numerical partial differential equations and which is more challenging to discretize than the standard laplacian seen in most textbooks in addition the history of vibrating plates is interesting and we will show both spectral discretizations leading to small dense matrix eigenvalue problems and a finite difference discretization leading to large scale sparse matrix eigenvalue problems hence both the qr algorithm and lanczos can be well illustrated

free vibration analysis of doubly curved shallow shells using the superposition galerkin method abstract this paper demonstrates the applicability of the superposition method for free vibration analysis of doubly curved thin shallow shells of rectangular planform with any possible combination of simply supported and clamped edges the same building block yields the natural frequencies for 55 combinations of edge conditions the natural frequency parameters of the shells were obtained using the superposition galerkin method sgm for seven sets of boundary conditions several different curvature ratios and two aspect ratios the sgm uses approximate steady state solutions as building blocks but the method proves to be accurate and efficient it has also been shown that even with approximate building blocks the monotonic nature of convergence of the natural frequencies with respect to the number of driving coefficients holds as long as the number of admissible functions in the steady state solution is kept constant the results for natural frequencies of the seven boundary conditions may be considered as benchmarks