on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

direct and inverse spectral problems for a class of non self adjoint periodic tridiagonal matrices abstract the spectral properties of a class of tridiagonal matrices are investigated the reconstruction of matrices of this special class from given spectral data is also studied necessary and sufficient conditions for that reconstruction are found the obtained results extend some results on the direct and inverse spectral problems for periodic jacobi matrices and for some non self adjoint tridiagonal matrices

analysis of in plane wave propagation in periodic structures with sierpinski carpet unit cells abstract in this study the dispersion relationships and characteristics of directional periodic structure waveguides with sierpinski carpet unit cells were investigated finite element method was used to analyze the sierpinski carpet periodic structures with square pb cylinders inserted in the rubber background the dispersion curves of unit cells with different stages were investigated first then group velocities of periodic structures at given frequencies were calculated and used to analyze the directional waveguide characteristics in the sierpinski carpet periodic structures the dynamic responses of finite periodic structures were investigated to observe the directional features and the results were used to support the fact that sierpinski carpet unit cells with different stages can control the propagation behavior of elastic waves in specific ways the dispersion curves grew compressed as the stage increased multiple band characteristics were observed in the case of the second stage fractal configuration the directional propagation characteristics in sierpinski carpet periodic structures can be utilized to further extend the range of vibration reduction zones the results of this study have significant potential in terms of the application of sierpinski carpet periodic structures for vibration isolation

a new approach for displacement functions of a curved timoshenko beam element in motions normal to its initial plane summary in existing literature either analytical methods or numerical methods the formulations for free vibration analysis of circularly curved beams normal to its initial plane are somewhat complicated particularly if the effects of both shear deformation sd and rotary inertia ri are considered it is hoped that the simple approach presented in this paper may improve the above mentioned drawback of the existing techniques first the three functions for axial or normal to plane displacement and rotational angles about radial and circumferential or tangential axes of a curved beam element were assumed since each function consists of six integration constants one has 18 unknown constants for the three assumed displacement functions next from the last three displacement functions the three force displacement differential equations and the three static equilibrium equations for the arc element one obtained three polynomial expressions equating to zero the coefficients of the terms in each of the last three expressions respectively one obtained 17 simultaneous equations as functions of the 18 unknown constants excluding the five dependent ones among the last 17 equations one obtained 12 independent simultaneous equations solving the last 12 independent equations one obtained a unique solution in terms of six unknown constants finally imposing the six boundary conditions at the two ends of an arc element one determined the last six unknown constants and completely defined the three displacement functions by means of the last displacement functions one may calculate the shape functions stiffness matrix mass matrix and external loading vector for each arc element and then perform the free and forced vibration analyses of the entire curved beam good agreement between the results of this paper and those of the existing literature confirms the reliability of the presented theory copyright 2005 john wiley sons ltd

a mixed stress model for linear elastodynamics of arbitrarily curved beams this work presents a mixed stress finite element for linear elastodynamics of arbitrarily curved beams based on a modified hellinger reissner functional a rational approach to choose the stress approximation is proposed in particular the self equilibrated stress is augmented by some stress modes obtained from the lower order displacement approximation using the equilibrium equations in such a way that the total number of stress modes is equal to the number of strain modes the rationale is to preserve all the interactions among the stresses proper of a curved structure without compromising the flexibility of the element an arbitrarily curved geometry is described using a parametric hermitian interpolation scheme tuned by minimizing the initial curvature of the arch the effectiveness of the present approach is numerically demonstrated copyright 2007 john wiley sons ltd

a force identification method using cubic b spline scaling functions for force identification the solution may differ from the desired force seriously due to the unknown noise included in the measured data as well as the ill posedness of inverse problem in this paper an efficient basis function expansion method based on wavelet multi resolution analysis using cubic b spline scaling functions as basis functions is proposed for identifying force history with high accuracy which can overcome the deficiency of the ill posed problem the unknown force is approximated by a set of translated cubic b spline scaling functions at a certain level and thereby the original governing equation of force identification is reformulated to find the coefficients of scaling functions which yields a well posed problem the proposed method based on wavelet multi resolution analysis has inherent numerical regularization for inverse problem by changing the level of scaling functions the number of basis functions employed to approximate the identified force depends on the level of scaling functions a regularization method for selecting the optimal level of cubic b spline scaling functions by virtue of condition number of matrix is proposed in this paper the validity and applicability of the proposed method are illustrated by two typical examples of volterra fredholm integral equations that are both typical ill posed problems force identification experiments including impact and harmonic forces are conducted on a cantilever beam to compare the accuracy and efficiency of the proposed method with that of the truncated singular value decomposition tsvd technique

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

radiation conditions for a semi infinite elastic strip high frequency vibrations of a semi infinite elastic strip with traction free faces are considered the conditions on end data that are derived do not allow non radiating in sommerfeld s sense of polynomial modes at thickness resonance frequencies these represent a high frequency analogue of the well known decay conditions in statics that agree with the classical saint venant principle the proposed radiation conditions are applied to the construction of boundary conditions in the theories of high frequency long wave vibrations describing slow varying motions in the vicinity of thickness resonance frequencies the derivation is based on the laplace transform technique along with the asymptotic methodology that is typical for thin plates and shells

modelling of wave propagation in composite plates using the time domain spectral element method abstract this paper presents results of numerical simulation of the propagation of transverse elastic waves corresponding to the a0 mode of lamb waves in a composite plate the problem is solved by the spectral element method spectral plate finite elements with 36 nodes defined at gauss lobatto legendre points are used as a consequence of the selection of lagrange polynomials for element shape functions discrete orthogonality is obtained leading to the diagonal form of the element mass matrix this results in a crucial reduction of numerical operations required for the solution of the equation of motion by time integration numerical calculations have been carried out for various orientations and relative volume fractions of reinforcing fibres within the plate the paper shows how the velocities of transverse elastic waves in composite materials depend on the orientation and the relative volume fraction of the reinforcement

a hierarchy of rational timoshenko dispersion relations abstract this paper shows that the conventional timoshenko dispersion relation for bending waves is the 0 1 member of a two parameter family m n of approximations to the exact dispersion relation obtained from linear elasticity higher members of the family are shown to converge rapidly to the exact dispersion relation in particular any number of branches can be captured accurately over their entire length i e up to arbitrarily high frequencies and wavenumbers the theory admits a rigorous accuracy analysis and demonstrates that the conventional timoshenko dispersion relation is a completely rational approximation thus answering some previous doubts about this matter lower members of the two parameter family include the euler bernoulli dispersion relation and a rayleigh type inertia correction especially useful is the timoshenko 1 2 dispersion relation which extends the conventional timoshenko 0 1 dispersion relation by capturing the first four branches of the exact dispersion relation rather than merely the first two and is accurate over a wide range an inertia corrected timoshenko dispersion relation is also derived and is shown to include many previous approximations as special cases

on the rational derivation of a hierarchy of dynamic equations for a homogeneous isotropic elastic plate abstract flexural equations of motion for a homogeneous isotropic elastic plate are derived by an antisymmetric expansion in the thickness coordinate of the displacement components all but the lowest order expansion functions are eliminated with the help of the three dimensional equations of motion and are plugged into the boundary conditions eliminating between these an equation is obtained for the mean plane vertical displacement which also includes arbitrary loading on the plate surface this equation can be truncated to any order in the thickness and it is in particular noted that the corresponding dispersion relation seems to correspond to a power series expansion of the exact rayleigh lamb dispersion relation to all orders various truncations of the equation are discussed and are compared numerically with each other the exact three dimensional solution and mindlin s plate equation both the dispersion relation and the corresponding displacement components as well as an excitation problem are used for the comparisons the theories are reasonably close to each other and in order to be on the safe side none of them should in fact be used for frequencies above the cutoff of the second flexural mode

classification of extremal vertex operator algebras with two simple modules in recent work wang and tener defined a class of extremal vertex operator algebras voas consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge in this article we show that there are exactly 15 character vectors of extremal voas with two simple modules all but one of the 15 character vectors are realized by a previously known voa the last character vector is realized by a new voa with central charge 33

dynamic equivalence self equilibrated excitation and saint venant s principle for an elastic strip abstract the distinction between the near and far fields for a semi infinite elastic strip has been exploited to derive conditions under which different dynamic excitations can be considered as equivalent these different excitations are equivalent in the sense that they produce the same displacement field far from the excited end it is shown that dynamically equivalent excitations degenerate to statically equivalent loads in the limit of a vanishing frequency the no radiation condition is derived and its relation to self equilibrated dynamic and static loads is presented dynamically equivalent excitations are utilized to formulate a dynamic version of saint venant s principle for symmetric excitations with frequencies below the first cut off frequency of a strip it has been shown that the requirement of self equilibrium of a load for the decay of end effects for static fields can be deduced from the requirement of zero average power for the dynamic fields

vibration of non uniform rods and beams we show that for some non uniform rods and beams the equation of motion can be transformed into the equation of motion for a uniform rod or beam then when the ends are completely fixed the eigenvalues of the non uniform continuum are the same as those of uniform rods or beams for other end support conditions exact solutions are obtained an efficient procedure is used to analyze the free vibration of non uniform beams with general shape and arbitrary boundary conditions simple formulas are presented for predicting the fundamental natural frequency of non uniform beams with various end support conditions

first order solutions for the buckling loads of weakened timoshenko columns in this work closed form expressions for the buckling loads of weakened timoshenko columns with different boundary conditions and shear force approaches proportional to the bending rotation or to the total slope are presented the crack model used promotes discontinuities in both transversal displacements and rotation due to bending to solve the buckling problem the perturbation method is used considering that the solutions for both the cracked and the uncracked columns are slightly different this procedure leads to first order closed form expressions for the buckling loads of the timoshenko cracked column which were compared with those found by directly solving the corresponding eigenvalue problem establishing validity limits for these solutions

the dynamic analysis of a cracked timoshenko beam by the spectral element method the aim of this paper is to introduce a new finite spectral element of a cracked timoshenko beam for modal and elastic wave propagation analysis the proposed approach deals with the spectral element method this method is suitable for analyzing wave propagation problems as well as for calculating modal parameters of the structure in the paper the results of the change in modal parameters due to crack appearance are presented the influence of the crack parameters especially of the changing location of the crack on the wave propagation is examined responses obtained at different points of the beam are presented proper analysis of these responses allows one to indicate the crack location in a very precise way this fact is very promising for the future work in the damage detection field

a modified method for reconstructing periodic jacobi matrices in this note we discuss the reconstruction of periodic jacobi matrices from spectral data the method combines ideas and techniques from the algorithms given by boley and golub 1 2 and ferguson 3 resulting in a numerically stable algorithm applicable to a larger class of problems the number of initial data items needed for this method equals the number of items in the resulting matrix namely 2n

modal and characteristics based approaches for modeling elastic waves induced by time dependent boundary conditions in this paper we present a characteristics based approach for solving elastic wave problems with time dependent traction boundary conditions a generalized mathematical model for this important class of problems is expressed as a set of first order linear hyperbolic partial differential equations we analyze the mathematical structure of this first order linear system verify its hyperbolicity derive its characteristic form and deduce its eigenvalues eigenvectors and riemann invariants the eigenvalues correspond to the wave speeds while the riemann invariants are used to construct a solution by the method of characteristics we benchmark the method of characteristics against several popular modal approaches two of these which we refer to as the concentrated body force method cbfm and the homogeneous eigenfunction expansion method heem were developed to simplify the well established but tedious mindlin goodman method to homogenize the boundary conditions and enable modal analysis the cbfm and heem forgo the usual formalism of linear transformations a la mindlin goodman in favor of intuitive modeling assumptions and postulated solution structures we find however that these approaches introduce an artificial stress discontinuity at the forced boundary in their reformulated problems when these reformulated problems are solved by modal analysis spurious oscillations and significant overshoot similar to the gibbs phenomenon emerge in the stress profile at the artificial discontinuity we demonstrate that these oscillations and overshoot are physical manifestations of a series solution for stress obtained from term by term differentiation that is not uniformly convergent as required by the formalism of mathematical analysis the method of characteristics solution on the other hand is exact to within machine precision yielding no artificial discontinuities spurious oscillations or unphysical overshoot unlike the modal approaches the method of characteristics solves the first order problem with time dependent boundary conditions as is without any reformulation restructuring or postulated solution structures further its solutions require no post processing no a posteriori solution treatment like l hopital s rule to accommodate resonance resonant behavior is inherently captured without the emergence of singularities no term by term differentiation to deduce stress stress is primitive in a first order velocity stress formulation of elastodynamics and no convergence tests

on solving first kind integral equations using wavelets on a bounded interval the conventional method of moments mom when applied directly to integral equations leads to a dense matrix which often becomes computationally intractable to overcome the difficulties wavelet bases have been used previously which lead to a sparse matrix the authors refer to mom with wavelet bases as wavelet mom there have been three different ways of applying the wavelet techniques to boundary integral equations 1 wavelets on the entire real line which requires the boundary conditions to be enforced explicitly 2 wavelet bases for the bounded interval obtained by periodizing the wavelets on the real line and 3 wavelet like basis functions furthermore only orthonormal on bases have been considered the present authors propose the use of compactly supported semi orthogonal so spline wavelets specially constructed for the bounded interval in solving first kind integral equations they apply this technique to analyze a problem involving 2d em scattering from metallic cylinders it is shown that the number of unknowns in the case of wavelet mom increases by m 1 as compared to conventional mom where m is the order of the spline function results for linear m 2 and cubic m 4 splines are presented along with their comparisons to conventional mom results it is observed that the use of cubic spline wavelets almost diagonalizes the matrix while maintaining less than 1 5 of relative normed error the authors also present the explicit closed form polynomial representation of the scaling functions and wavelets

hermitian plane wavelet finite element method wave propagation and load identification abstract the two dimensional hermitian interpolation wavelet is constructed by using the tensor product of the modified hermitian wavelets expanded at each coordinate then the two dimensional hermitian interpolation wavelet is substituted into finite element formulations to address the wave propagation and load identification problems hermitian wavelet finite element can be used to describe the wave propagation and to reveal the rule of the wave propagation in plane the wave propagation response is used to solve the load identification inverse problem results show that the identified load value is similar to the applied load when the location of the response node is close to the applied load position the proposed method can accurately identify the location waveform and amplitude of the applied load

a closed form solution of a longitudinal bar with a viscous boundary condition abstract a closed form solution of a longitudinal bar with a viscous boundary condition subjected to point loading is developed in this paper a new series solution is formulated that allows the time and space modes of the beam to decouple this expansion yields explicit eigenvalues and eigenvectors a frequency domain example is presented and the results are compared with finite element solutions of the same problem it is shown that the closed form solution is computationally more efficient than a finite element solution additionally the truncation error at lower frequencies is shown to be extremely small the method is easily implemented and can provide time and frequency domain solutions to this class of problems

wave motion analysis in arch structures via wavelet finite element method abstract the application of b spline wavelet on interval bswi finite element method for wave motion analysis in arch structures is presented in this paper instead of traditional polynomial interpolation scaling functions at certain scales have been adopted to form the shape functions and construct wavelet based elements different from other wavelet numerical methods adding wavelets directly the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix the energy functional of the arch is obtained by the generalized shell theory and the finite element model for wave motion analysis is constructed according to hamilton s principle and the central difference method in time domain taking the practical application into account damaged arch waveguides are also investigated proper analysis of the responses from structure damages allows one to indicate the location very precisely this paper mainly focuses on the crack in structures based on castigliano s theorem and the pairs equation the local flexibility of crack is formulated for bswi element numerical experiments are performed to study the effect of wave propagations in arch waveguides that is frequency dispersion and mode spilt in the arch the responses of the arch with cracks are simulated under the broad band narrow band and chirp excitations in order to estimate the spatial time and frequency concentrations of responses the reciprocal length time frequency transform and correlation coefficient are introduced in this investigation

free and forced vibrations of stepped rods and coupled systems abstract analytical and closed form frequency responses of the longitudinal vibration of n stepped rods as well as the coupled system of multi stepped rods and lumped elements with non classical boundary conditions are proposed in this paper the closed form frequency equation for these systems is also obtained in this article a new two way state flow model based on the dynamics of the rod is developed in addition to the state flow model developed for lumped elements concentrated mass and spring and damper the exact frequency response of both stepped rod and the coupled system can be directly written out according to the graph theory since the results are analytical and closed form they can be directly implemented for analytical and numerical analysis without resorting to complex operations such as inversion recursive calculation or iteration computation moreover the derived equations are feasible not only for the multi stepped and multi stage rods but also for the coupled system of multi stepped rods and lumped elements finally the results were checked using multi stepped rods with classical and non classical boundary conditions as well as a combination of stepped rods and lumped elements to demonstrate the efficiency of this method

wave propagation modelling in 1d structures using spectral finite elements abstract the application of spectral finite elements sfe to one dimensional 1d elastic wave propagation problems is presented travelling waves in an isotropic rod and timoshenko beam have been investigated the rod has been modelled using 1d sfes while the beam has been modelled using 1d and 2d sfes numerical results have been compared to those obtained from the classical finite element approach this comparison highlighted the efficiency of the sfe method the numerical results have been also verified experimentally a high degree of accuracy has been observed

the finite product method in the theory of waves and stability this paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental the method involves replacing each transcendental term in the dispersion relation by a finite product polynomial the finite products chosen must be consistent with the low frequency low wavenumber limit but the method is nevertheless accurate up to high frequencies and high wavenumbers full details of the method are presented for a non trivial example that of anti symmetric elastic waves in a layer the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers it is proved that the method is accurate because certain gamma function expressions which occur as ratios of transcendental terms to finite products largely cancel out nullifying runge s phenomenon the polynomial approximations which are unrelated to taylor series introduce no spurious branches into the dispersion relation and are ideal for numerical computation the method is potentially useful for a very wide range of problems in wave theory and stability theory

discontinuity location on a timoshenko beam by using a novel wave reflection technique abstract based on the physical phenomenon that some propagating waves are partially reflected by discontinuities in waveguide structures this paper proposes a novel discontinuity locating method which is potentially able to detect both the added and embedded discontinuities such as supports and cracks in this method the reflection coefficients of a discontinuity can be obtained experimentally by using a generalized discontinuity approach without the information of the discontinuity position on the other hand these reflection coefficients can be derived from a reflection coefficient prediction model in terms of the discontinuity position by minimizing the error between the reflection coefficients from measurements and those from the prediction model in the frequency bands in which the guided waves are sensitive to the discontinuity and can propagate with low decay rate the discontinuity position parameter in the prediction model can then be identified two numerical cases of locating an added discontinuity and an embedded discontinuity on a beam are proposed to introduce the effectiveness of the method simulation results show that these two discontinuities can be located accurately with zero error in practice the accuracy of discontinuity locating may be influenced by some factors 1 boundary conditions of the beam 2 sizes and locations of the discontinuity and 3 measurement noise the investigation of these factors in simulation shows that this locating method is insensitive to the boundary conditions of the beam effective to small size of the discontinuity and robust to the measurement noise finally three experiments were developed to locate an added discontinuity on a finite beam a crack in an infinite beam with depth ratio of 1 2 and a crack in an infinite beam with depth ratio of 1 6 respectively the corresponding locating errors of these three experiments are 0 percent 0 077 percet and 0 154 percent

a unified solution for longitudinal wave propagation in an elastic rod this paper proposes a unified analytical solution of the wave equation governing the propagation of the longitudinal stress wave in an elastic rod within a single formula derived by using the laplace transform and inverse transform the solution covers the contributions of the external excitations the nonzero initial conditions and the inhomogeneous boundary conditions altogether including such boundary conditions that the dependent variables at the ends of the rod are restricted with an equation the proposed formula particularly suitable for transient problems could be regarded as an exact interpolation function in the time domain provided the rod works as a component in a complex system etc four examples are presented to show the applications of the solution

meromorphic c 24 conformal field theories modular invariant conformal field theories with just one primary field and central charge c 24 are considered it has been shown previously that if the chiral algebra of such a theory contains spin 1 currents it is either the leech lattice cft or it contains a kac moody sub algebra with total central charge 24 in this paper all meromorphic modular invariant combinations of the allowed kac moody combinations are obtained the result suggests the existence of 71 meromorphic c 24 theories including the 41 that were already known

structured matrices continued fractions and root localization of polynomials we give a detailed account of various connections between several classes of objects hankel hurwitz toeplitz vandermonde and other structured matrices stietjes and jacobi type continued fractions cauchy indices moment problems total positivity and root localization of univariate polynomials along with a survey of many classical facts we provide a number of new results

continued fractions and periodic jacobi matrices abstract we describe an algorithm based on a continued fraction expansion to reconstruct a periodic jacobi matrix from spectral data we also compare various different sets of spectral data which characterize a periodic jacobi matrix

a three noded shear flexible curved beam element based on coupled displacement field interpolations an efficient shear flexible three noded curved beam element is proposed herein the shear flexibility is based on timoshenko beam theory and the element has three degrees of freedom viz tangential displacement u radial displacement w and the section rotation a quartic polynomial interpolation for flexural rotation is assumed a priori making use of the physical composition of in terms of and u a novel way of deriving the polynomial interpolations for u and w is presented by solving force moment and moment shear equilibrium equations simultaneously the field interpolation for is then constructed from that of and u the procedure leads to high order polynomial field interpolations which share some of the generalized degrees of freedom by means of coefficients involving material and geometric properties of the element when applied to a straight euler bernoulli beam all the coupled coefficients vanish and the formulation reduces to classical quintic in w and quadratic in u element with u w and w x as degrees of freedom the element is totally devoid of membrane and shear locking phenomena the formulation presents an efficient utilization of the nine generalized degrees of freedom available for the polynomial interpolation of field variables for a three noded curved beam element numerical examples on static and free vibration analyses demonstrate the efficacy and locking free property of the element copyright 2001 john wiley sons ltd

analysis of wave motion in one dimensional structures through fast fourier transform based wavelet finite element method abstract this paper presents a hybrid method that combines the b spline wavelet on the interval bswi finite element method and spectral analysis based on fast fourier transform fft to study wave propagation in one dimensional 1d structures bswi scaling functions are utilized to approximate the theoretical wave solution in the spatial domain and construct a high accuracy dynamic stiffness matrix dynamic reduction on element level is applied to eliminate the interior degrees of freedom of bswi elements and substantially reduce the size of the system matrix the dynamic equations of the system are then transformed and solved in the frequency domain through fft based spectral analysis which is especially suitable for parallel computation a comparative analysis of four different finite element methods is conducted to demonstrate the validity and efficiency of the proposed method when utilized in high frequency wave problems other numerical examples are utilized to simulate the influence of crack and delamination on wave propagation in 1d rods and beams finally the errors caused by fft and their corresponding solutions are presented

free vibration and lateral buckling of a cantilever slender beam with an edge crack experimental and numerical studies the effects of crack ratios and positions on the fundamental frequencies and buckling loads of slender cantilever euler beams with a single edge crack are investigated both experimentally and numerically using the finite element method based on energy approach the governing matrix equations are derived from the standard and cracked beam elements combined with the local flexibility concept the experiments are conducted using specimens having edge cracks of different depths at different positions to validate the numerical results obtained the numerical results are shown to be in good agreement with the experimental results for the considered crack ratios

on classification of extremal non holomorphic conformal field theories rational chiral conformal field theories are organized according to their genus which consists of a modular tensor category and a central charge c a long term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories we conjecture that for any unitary modular tensor category there exists a unitary chiral conformal field theory so that its modular tensor category is in this paper we initiate a mathematical program in and around this conjecture we define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge and non trivial representation theory we show that there are finitely many different characters of extremal vertex operator algebras possessing at most three different irreducible modules moreover we list all of the possible characters for such vertex operator algebras with

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

longitudinal wave propagation part i comparison of rod theories abstract the paper introduces new four spectral elements for analysis of longitudinal wave propagation in structures the developed elements are based on the elementary love mindlin herrmann and three mode theories certain differences in wave propagation behaviour are observed for the analysed models for lower excitation frequencies the results obtained for all models are similar and from a practical point of view the spectral element based on the love theory is adequate for high frequency excitation these differences are considerable and only the mindlin herrmann or three mode models give correct results

a higher order hybrid mixed harmonic shell of revolution element abstract an accurate higher order hybrid mixed c 0 harmonic shell of revolution element is proposed unlike existing hybrid mixed shell of revolution elements the present element introduces additional nodeless degrees only for displacement field interpolation in order to enhance the solution convergence rate in the course of developing the present element we address some fundamental issues such as the effect of the nodeless degrees and the role of the stress field approximation consistent with the displacement field through several numerical examples the use of the additional nodeless degrees is shown to yield very accurate results only with a few elements

a hierarchy of high order theories for symmetric modes in an elastic layer abstract a hierarchy of high order theories for symmetric modes in an infinitely long elastic layer of the constant thickness is derived by means of the inertia corrected polynomial approximations for each member of the hierarchy boundary conditions for layers of the finite length are formulated the forcing problems at several approximation levels are solved with the use of the bi orthogonality conditions accuracy of these approximations is assessed by comparison of results with the exact solution of the rayleigh lamb problem special attention is paid to the power flow analysis in alternative modal excitation cases and to the applicability of the saint venant s principle in stationary elasto dynamics

the green s matrix and the boundary integral equations for analysis of time harmonic dynamics of elastic helical springs helical springs serve as vibration isolators in virtually any suspension system various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions the method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time harmonic dynamics of elastic springs in the framework of bernoulli euler beam theory in this paper the derivations of the green s matrix of the somigliana s identities and of the boundary integral equations are presented the vibrational power transmission in an infinitely long spring is analyzed by means of the green s matrix the eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations in the course of analysis the essential features and advantages of the method of boundary integral equations are highlighted the reported analytical results may be used to study the time harmonic motion i

modeling of wave propagation in plate structures using three dimensional spectral element method for damage detection abstract this paper presents the application of three dimensional 3 d spectral element method sem to wave propagation problems in plate structures for the purpose of damage detection the excellent characteristic of the sem is that the mass matrix is diagonal because of the choice of lagrange interpolation function supported on the gauss lobatto legendre gll points in conjunction with the gll integration rule therefore numerical calculation can be significantly efficient in comparison with the classical finite element method fem by taking advantage of this characteristic a 3 d spectral finite elements sfes based model is developed to simulate the wave propagation in plate structures lamb waves propagating in aluminum plates with and without a crack are investigated responses from 3 d sfes and 2 d sfes based models are compared different lamb wave modes are generated using different excitation approaches and subsequently characteristics of those modes are analyzed the results demonstrate that the proposed model can offer efficient and realistic simulation for lamb wave propagation in plate structures so as to detect damages in those structures

on propagation of plane symmetric waves in a periodically corrugated straight elastic layer abstarct the classical bernoulli euler model is employed for analysis of the plane wave propagation in an infinitely long elastic layer with a periodically varying thickness the floquet theory is applied to derive asymptotic formulas defining location and broadness of frequency stop bands for several corrugation shapes with different levels of discontinuity for a layer with the perfectly smooth periodic corrugation the equation of the axial wave motion is solved by the method of multiple scales in vicinity of critical frequencies then it is transformed to the canonical mathieu equation and its stability diagram is compared with predictions of the floquet theory the qualitative differences between the shapes of stop bands are discussed the wentzel kramers brillouin approximation is employed to solve the bernoulli euler equation for the layer with perfectly smooth periodic corrugation and the results are discussed in view of existence of pass and stop bands and in view of the levels of approximation used in alternative asymptotic methods

structural damage detection of space truss structures using best achievable eigenvectors a method is presented by which measured modes and frequencies from a modal test can be used to determine the location and magnitude of damage in a space truss structure the damage is located by computing the euclid ean distances between the measured mode shapes and the best achievable eigenvectors the best achievable eigenvectors are the projection of the measured mode shapes onto the subspace defined by the refined analytical model of the structure and the measured frequencies loss of both stiffness and mass properties can be located and quantified to examine the performance of the method when experimentally measured modes are employed various damage detection studies using a laboratory eight bay truss structure were conducted the method per forms well even though the measurement errors inevitably make the damage location more difficult uture structures in space will be orders of magnitude larger and more complex than their predecessors structures such as space station freedom will typically be built around a large flexi ble frame and consist of habitation and experiment modules flexi ble and articulating appendages and numerous utility trays and pay loads along with the need to design the spacecraft to be light weight strong and modular for ease of expansion repair and modification the complexity and size of these structures require that the integrity of the structures be monitored periodically several researchers have studied the possibility of detecting damage or structural anomalies to large space trusses and locating the site of this damage based on changes in the natural frequencies and mode shapes of the structures references 1 and 2 provide a comprehensive overview of many of the proposed methods in this respect and the issues involved with on orbit modal identification and structural verification there are several distinct classes of model improvement damage location methods each of these methods begins with an initial finite element representation of the structure in the case of model improvement the goal is to use test data of the actual structure to modify this initial finite element model so that it may serve as an accurate representation of the un damaged structure in the case of damage location the goal is to use test data from the damaged structure and the updated corre lated finite element model of the undamaged structure to deter mine changes to the stiffness and or mass matrices that would ac count for the observed differences in the measured modes and frequencies the class of update methods that has been explored most widely is known as optimal matrix modification the goal of the optimal matrix modification is to find an updated matrix stiffness and or mass that is as close as possible to the original matrix assumed to be almost correct that accurately produces the measured modes

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

a note on curved cracked beams abstract this paper is concerned with evaluating stress intensity factors for a curved cracked beam of small to moderate curvature using a simple engineering method which allows us to estimate the strain energy release rate based on elementary beam theory the basic methodology is briefly discussed and an example application to a cracked circular beam gives results which are in reasonable agreement with more accurate numerical calculations in the literature