flexural wave propagation in beams with periodically attached vibration absorbers band gap behavior and band formation mechanisms abstract this paper is concerned with flexural wave propagation and vibration transmission in beams with periodically attached vibration absorbers such periodic systems feature unique wave filtering characteristics that can find applications in the control of wave propagation in flexural beam structures the study is performed by using an exact analytical approach based on a combination of the spectral element method and periodic structure theory both infinite and finite periodic structures are considered an explicit expression is provided for the calculation of propagation constants and thus the complex band structures and it is further developed to examine the effects of various system parameters on the band gap behavior including the position width and wave attenuation performance of all the band gaps the band formation mechanisms of such periodic systems are explained via both derivations and physical models yielding explicit equations to enable the prediction of all the band edge frequencies in an exact manner without the need to calculate propagation constants based on these equations explicit formulas are further derived to determine the conditions for the transition and near coupling between local resonance and bragg scattering each being a unique band gap opening mechanism

calculating the forced response of two dimensional homogeneous media using the wave and finite element method abstract the forced response of two dimensional infinite homogenous media subjected to time harmonic loading is treated the approach starts with the wave and the finite element wfe method where a small segment of a homogeneous medium is modelled using commercial or in house finite element fe packages the approach is equally applicable to periodic structures with a periodic cell being modelled this relatively small model is then used along with periodicity conditions to formulate an eigenvalue problem whose solution yields the wave characteristics of the whole medium the eigenvalue problem involves the excitation frequency and the wavenumbers or propagation constants in the two directions the wave characteristics of the medium are then used to obtain the response of the medium to a convected harmonic pressure chp since the fourier transform of a general two dimensional excitation is a linear combination of chps the response to a general excitation is a linear combination of the responses to chps thus the response of a two dimensional medium to a general excitation can be obtained by evaluating an inverse fourier transform this is a double integral one of which is evaluated analytically using contour integration and the residue theorem the other integral can be evaluated numerically hence the approach presented herein enables the response of an infinite two dimensional or periodic medium to an arbitrary load to be computed via a modelling a small segment of the medium using standard fe methods and post processing its model to obtain the wave characteristics b formulating the fourier transform of the response to a general loading and c computing the inverse of the fourier transform semi analytically via contour integration and the residue theorem followed by a numerical integration to find the response at any point in the medium numerical examples are presented to illustrate the approach

orthomodularity and quadratic transformations in probabilistic theories of physics a characterization of orthomodularity of a set of effects is given in terms of a triangle closedness of this set this leads to a simple probabilistic characterization of observables with a boolean range quadratic transformations on expectation functionals of observables are studied in order to investigate the derivability of the heisenberg inequality and the form of the lower bound of such an inequality for a pair of observables

a symplectic analytical wave based method for the wave propagation and steady state forced vibration of rectangular thin plates abstract a semi analytical method is used to investigate the wave propagation characteristics and steady state forced vibration response for rectangular thin plates by way of a rigorous but simple derivation the governing differential equations for transverse vibration of rectangular thin plates are first converted into hamiltonian canonical equations following the method of separation of variables a symplectic eigenproblem is formed whose solution gives analytically the dispersion equation and the wave mode shape using the wave modes i e the wave propagation parameters and wave shapes and combining the directly excited waves the wave propagation within the plate and the wave reflection at the boundary the forced response of the plate can be computed in the wave domain with high precision and high efficiency the present method is based on the basic elasticity equations of the plate and can give the symplectic analytical solutions for the wave modes for any combination of simple boundary conditions along the plate edges the present method eliminates the limitation of the traditional analytical wave propagation method which can only obtain wave modes for plates with two opposite edges simply supported in contrast to numerical wave propagation methods the present method provides symplectic analytical solutions for the wave modes and hence the computations are of high precision and well conditioned also continuously distributed external forces can be easily considered in the numerical examples the wave propagation characteristics are analyzed for plates with three different combinations of boundary conditions i e with two opposite edges either both simply supported or both clamped or one simply supported and the other clamped the steady state forced responses are also computed for plates excited by point forces as well as for line and area distributed forces for the three combinations of boundary conditions comparison of the present results with analytical results wave finite element and finite element results validates the high accuracy and high efficiency of the present method

a spectral super element for modelling of plate vibration part 1 general theory the dynamic response of vibrating structures is studied with a proposed merger of the standard finite element method with the more computationally efficient spectral finite element method first a plate structure is modelled with a newly developed spectral super element then this element is coupled to other parts that can have a more complex geometry and are modelled entirely with conventional finite elements some numerical examples are given to illustrate and validate the developed method and studies of numerical stability are also presented in an accompanying paper the predicted and measured response of a turbulence excited aircraft panel are compared

wave finite elements in guided elastodynamics with internal fluid abstract this paper presents a general formulation which addresses the issue of wave propagation in guided elastodynamic structures filled with acoustic fluid the formulation is based on a finite element description of periodic systems it leads to a general spectral problem whose eigenvalues and eigenvectors are related to the free propagating wave properties the formulation incorporates many simplified elastodynamic models of an analytical nature here the formulation is stated for a fluid structure guided medium the free and forced frequency response of such elasto acoustic system is fully formulated in the wave space reduction of the wave basis is discussed numerical simulations and comparisons with classic simplified theories clearly shows the pertinence of the proposed formulation moreover it is clearly shown that unlike most of the existing theories the proposed formulation is not low frequency limited

computation of propagative waves in free rail using a finite element technique abstract a method is presented for the numerical computation of the wavenumbers and associated modes of the cross section of solid waveguides the method based on the finite element technique is well suited to the computation of both the propagative and the evanescent waves in a straight waveguide with an arbitrary cross section the solution is obtained by factorization with the cross section of the waveguide being modelled by numerical discretization the dispersion curves of propagative waves in a free rail are computed by using triangular and quadrilateral finite elements of the cross section the evolution of cross section modes as the frequency increases is evaluated and discussed

the treatment of forces in bloch analysis abstract in periodic lattice structures wave propagation on the infinite domain can be greatly simplified by invoking the floquet bloch theorem the theorem allows a system s degrees of freedom to be reduced to a small subset contained in a repeating unit cell the equations of motion governing this subset contain internal force terms which must be eliminated before establishing the eigenvalue problem for the dispersion relationships there are subtle issues with regard to the elimination of these forces which we address in this paper we demonstrate that for any two or three dimensional periodic lattice the internal forces vanish when acted upon by the linear transformation engendered by the degree of freedom reduction

coupled tension torsion vibration of repetitive beam like structures equivalent continuum stiffness properties derived from the eigenanalysis of a single cell of a planar beam like repetitive structure have previously been employed within well known dynamic theories such as euler bernoulli and timoshenko for flexural vibration suitably modified to predict natural frequencies of vibration here the approach is applied to two structure types that exhibit tension torsion coupling the first is modelled on a nasa deployable structure with a cross section of equilateral triangular form but with asymmetric triangulation on the faces the second is a related more symmetric structure but with pre twist the simplest tension torsion dynamic theory due to di prima is employed and this is extended to more general end conditions this combined periodic structure substitute continuum approach provides excellent agreement with predictions from the finite element method especially for the lower modes of vibration typically agreement is within 1 for the lowest 8 10 natural frequencies for the longer 30 cell structures considered here the majority of these being torsional modes and within 1 for the lowest 4 5 modes for the shorter ten cell structures this level of accuracy is attainable so long as a single wavelength spans 2 3 cells of the repetitive structure

internal sets and internal functions in colombeau theory this paper is devoted to the study of generalized functions as pointwise functions so called internal functions on certain sets of generalized points so called internal sets we treat the case of the colombeau algebras of generalized functions for which these notions have turned out to constitute a fundamental technical tool we provide general foundations for the notion of internal functions and internal sets and prove a saturation principle various applications to colombeau algebras are given

free wave propagation in periodically ring stiffened cylindrical shells abstract the free wave propagation characteristics of several periodically ring stiffened cylindrical shells have been investigated using a previously developed finite element based methodology results indicate that predicted stop bands that are associated primarily with shell flexure can be classified into three types the most important stop band type for structurally efficient cylinders is one in which the structural half wavelength or integer multiples there of approaches the stiffener spacing for the cylinders investigated these stop bands are broadband and can be easily tuned to a particular frequency band cylinder designs utilizing double curvature are shown to have significantly more coupling between flexural and longitudinal wave modes this conclusion is reinforced by results from an analytical investigation of a curved beam an axisymmetric formulation for the prediction of propagation constants is also identified

finite element prediction of wave motion in structural waveguides a method is presented by which the wavenumbers for a one dimensional waveguide can be predicted from a finite element fe model the method involves postprocessing a conventional but low order fe model the mass and stiffness matrices of which are typically found using a conventional fe package this is in contrast to the most popular previous waveguide fe approach sometimes termed the spectral finite element approach which requires new spectral element matrices to be developed in the approach described here a section of the waveguide is modeled using conventional fe software and the dynamic stiffness matrix formed a periodicity condition is applied the wavenumbers following from the eigensolution of the resulting transfer matrix the method is described estimation of wavenumbers energy and group velocity discussed and numerical examples presented these concern wave propagation in a beam and a simply supported plate strip for which analytical solutions exist and the more complex case of a viscoelastic laminate which involves postprocessing an ansys fe model the method is seen to yield accurate results for the wavenumbers and group velocities of both propagating and evanescent waves

numerical investigation of dispersion relations for helical waveguides using the scaled boundary finite element method abstract in this paper the dispersion properties of elastic waves in helical waveguides are investigated the formulation is based on the scaled boundary finite element method sbefm with a set of orthogonal unit basis introduced as the contravariant basis the helical coordinate is firstly considered where components of tensor retain the dimension of original quantity based on the strain displacement relation the eigenvalue matrix is obtained about wavenumbers and frequencies the cross section of the waveguides is discretized by using high order spectral elements moreover the formulated linear matrix is utilized to design efficient and accurate algorithms to compute the eigenvalues of helical waveguides compared with the pochhammer chree curves the convergence and accuracy of the sbfem are discussed finally we give some dispersion curves for a wide range of lay angles and analyze in detail properties of cut off frequency mode separation and mode transition for elastic wave propagation in the helical waveguides

finite element analysis of the vibrations of waveguides and periodic structures many structural components can be regarded as waveguides they are uniform in one direction so that the cross section of the waveguide has the same physical and geometric properties at all points along the axis of the waveguide in this paper a method is presented to calculate the forced response of such a structure using a combination of wave and finite element fe approaches the method involves post processing a conventional but low order fe model in which the mass and stiffness matrices are typically found using a conventional fe package a section of the waveguide is meshed and the eigenvalues and eigenvectors of the resulting transfer matrix found the eigenvectors form a set of basis functions for the analysis of the structure as a whole allowing the global dynamic stiffness matrix to be built easily and then the forced response to be calculated very efficiently the main advantage of the approach over the alternative waveguide fe approach often termed the spectral fe method is that conventional fe packages can be used to form the stiffness and mass matrices so that structures with complex geometries or material distributions can be analysed with relative ease to demonstrate the efficacy of the method examples of the forced response for a finite beam and plate strip are presented

wave propagation in one dimensional waveguides with slowly varying random spatially correlated variability this paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation more specifically it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium this variability plays a significant role in the so called mid frequency region for dynamics and vibration but wave based methods are typically only applicable to homogeneous and uniform waveguides the wkb approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion an alternative wave formulation approximation with piecewise constant properties is also derived and included so that the internal reflections are taken into account but this requires a discretization of the waveguide moreover a fourier like series the karhunen loeve expansion is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam it is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the wkb approach is given

factorization of complex canonical transformations we investigate the lie series representation of the canonical transformations in a finite dimensional complex phase space it is shown that any transformation of this type can be factorized into a product of three factors associated with a pure imaginary generating function a holomorphic function and an element of the cyclic group c4 the imaginary function can be considered as an observable in the sense of classical mechanics some hints are given which suggest that the holomorphic function can be connected with the notion of the state of a physical system moreover a special kind of mappings is studied which provides a link between entropy action and state functions the occurrence of these important physical quantities shows that the mathematical structure goes beyond a formal analogy to quantum physics at least in the finite dimensional case

vibration suppression for strings with distributed loading using spatial cross section modulation abstract a problem of vibration suppression in any preassigned region of a bounded structure subjected to action of an external time periodic load which is distributed over its domain is considered a passive control is applied in which continuous spatially periodic modulations of structural parameters are used as a means for vibration suppression as an example stationary vibrations of a string under action of a distributed time periodic load are studied this system in a simplified form models such processes as interaction between membranes and colloids oscillations of transmission lines under action of rain and wind and dynamics of suspension bridges and stay cables suppression of vibration in predefined regions of the string is performed by continuous spatial modulation of its cross section for analyzing the problem considered a novel approach named the method of varying amplitudes is employed this approach is applicable for solving differential equations without a small parameter and may be considered as a natural continuation of the classical methods of harmonic balance and averaging as a result optimal parameters for the string cross sectional area modulation are determined for the cases of harmonically uniformly and arbitrarily distributed load which allows for completely suppressing or considerably reducing vibration in the prescribed part of the string compared to the case without modulation

repetitive beam like structures distributed loading and intermediate support theory is developed for the elastostatic transfer matrix analysis of a repetitive beam like structure subject to point wise distributed loading with and without an intermediate support this complements previous work in which loading is applied at one end only and reacted at the other state vectors are expressed in terms of participation coefficients of the eigen and principal vectors leading to the simplest description of the evolution of nodal displacement and force components as one moves along the structure in terms of powers of the jordan canonical form inaccuracy due to ill conditioning is explained in terms of powers of greater than unity eigenvalues

general absorbing boundary for acoustic and elastic waves in the finite element computation of unbounded acoustic problems the domains must be of finite size and artificial absorbing conditions have to be introduced in order to avoid reflections at the truncated boundary this communication proposes a new method in the frequency domain to generate efficient absorbing boundary conditions without the need to consider high order derivatives on the boundary moreover the approach is quite general and can consider media with mean flow or be applied to structural vibrations it needs the knowledge of the dynamic stiffness matrix of only one element which can be obtained from any finite element software the final result is an impedance condition linking the forces and displacements on the boundary which is determined numerically from a wave analysis of one element of the media using this impedance condition a finite element problem can be solved by standard methods this is then applied to study some examples for 2d standard acoustic acoustic problems with mean flo

transfer matrix analysis of the elastostatics of one dimensional repetitive structures transfer matrices are used widely for the dynamic analysis of engineering structures increasingly so for static analysis and are particularly useful in the treatment of repetitive structures for which in general the behaviour of a complete structure can be determined through the analysis of a single repeating cell together with boundary conditions if the structure is not of infinite extent for elastostatic analyses non unity eigenvalues of the transfer matrix of a repeating cell are the rates of decay of selfequilibrated loading as anticipated by saint venant s principle multiple unity eigenvalues pertain to the transmission of load e g tension or bending moment and equivalent homogenized continuum properties such as cross sectional area second moment of area and poisson s ratio can be determined from the associated eigen and principal vectors various disparate results the majority new others drawn from diverse sources are presented these include calculation of principal vectors using the moore penrose inverse bi and symplectic orthogonality and relationship with the reciprocal theorem restrictions on complex unity eigenvalues effect of cell left to right symmetry on both the stiffness and transfer matrices eigenvalue veering in the absence of translational symmetry and limitations on possible jordan canonical forms it is shown that only a repeating unity eigenvalue can lead to a non trivial jordan block form so degenerate decay modes cannot exist the present elastostatic analysis complements langley s langley 1996 proc r soc a 452 1631 1648 transfer matrix analysis of wave motion energetics

a multiplication of distributions if denotes an open subset of rn n 1 2 we define an algebra g which contains the space d of all distributions on and such that c is a subalgebra of g the elements of g may be considered as generalized functions on and they admit partial derivatives at any order that generalize exactly the derivation of distributions the multiplication in g gives therefore a natural meaning to any product of distributions and we explain how these results agree with remarks of schwartz on difficulties concerning a multiplication of distributions more generally if q 1 2 and om r2q a classical schwartz notation for any g1 gq g we define naturally an element g1 gq g these results are applied to some differential equations and extended to the vector valued case which allows the multiplication of vector valued distributions of physics

resonant frequency primitive waveforms and star waves in lattices for square and triangular lattices we have found a new line localized primitive waveform lpw existing at a resonant frequency in two dimensional 2d case the lpw represents a line of oscillating particles while the lattice outside this line remains at rest we show that a a single lpw does not conduct energy however a band consisting of two or more neighboring lpws is a conductor with the phase shift dependent energy flux velocity b any canonical sinusoidal wave consists of lpws in turn the lpw can be represented by a superposition of the sinusoidal waves these two types of waves are connected by the discrete fourier transform c there are two three lpw orientations for the square triangular lattice and this is why the sinusoidal wave group velocity orientation is piecewise constant at this frequency it coincides with the nearest lpw orientation d lpw can also exist at a lower frequency being localized at the lattice halfplane boundary further for 3d lattices plane localized waveforms are found to exist in a frequency region finally for the point harmonic excitation of 2d lattices we show that starlike waves develop with the rays in the lpw directions

wave propagation in continuous periodic structures research contributions from southampton 1964 1995 abstract after brief reference to some early studies by other investigators this paper focuses mainly on methods developed at the university of southampton since 1964 to analyze and predict the free and forced wave motion in continuous periodic engineering structures beginning with receptance methods which have been applied to periodic beams and rib skin structures it continues with a method of direct solution of the wave equation this uses floquet s principle and has been applied to beams and quasi one dimensional periodic plates and cylindrical shells sample curves of the propagation and attenuation constants pertaining to these structures are presented a limited discussion of the transfer matrix then follows after which the method of space harmonics is introduced as the method best suited to the prediction of sound radiated from a vibrating periodic structure reviewed next are some theorems and variational principles relating to periodic structures which have been developed at southampton and which form a basis for finding natural frequencies of finite structures or for computing free and forced wave notion by energy methods this has led to the finite element method in its standard and hierarchical forms being used to study wave motion in genuine two dimensional and three dimensional structures examples of this work are shown the method of phased array receptance functions is then introduced as possibly the easiest way of setting up exact equations for the propagation constants of uniform quasi one dimensional periodic structures a summary is finally presented of the limited and early work performed at southampton on simple disordered periodic structures

generalized solutions of a nonlinear parabolic equation with generalized functions as initial data abstract in h brezis a friedman nonlinear parabolic equations involving measures as initial conditions j math pure appl 9 1983 73 97 brezis and friedman prove that certain nonlinear parabolic equations with the measure as initial data have no solution however in j f colombeau m langlais generalized solutions of nonlinear parabolic equations with distributions as initial conditions j math anal appl 1990 186 196 colombeau and langlais prove that these equations have a unique solution even if the measure is substituted by any colombeau generalized function of compact support here we generalize colombeau and langlais result proving that we may take any generalized function as the initial data our approach relies on recent algebraic and topological developments of the theory of colombeau generalized functions and results from j aragona colombeau generalized functions on quasi regular sets publ math debrecen 2006 371 399

passive vibration suppression of flexible space structures via optimal geometric redesign acomputationalframework is presented for the design of large flexible space structures with nonperiodicgeometries to achieve passive vibration suppression the present system combines an approximationmodel management framework ammf developed for evolutionary optimization algorithms eas with reduced basis approximate dynamic reanalysis techniques a coevolutionary genetic search strategy is employed to ensure that design changes during the optimization iterations lead to low rank perturbations of the structural system matrices for which thereduced basis methods give high quality approximations the k means algorithm is employed for cluster analysis of the population of designs to determine design points at which exact analysis should be carried out the fitness of the designs in an ea generation is then approximated using reduced basis models constructed around the points where exact analysis is carried out results are presented for the optimal design of a two dimensional cantilevered space structure to achieve passive vibration suppression it is shown that significant vibration isolation of the order of 50 db over a 100 hz bandwidth can be achieved further it is demonstrated that the ammf can potentially arrive at a better design compared to conventional approaches when a constraint is imposed on the computational budget available for optimization

generators for nonlinear canonical transformations canonical transformations whose generators are linear in the basic operators pj and arbitrary in the canonically conjugate operators qj are explicitly constructed it is shown that they correspond to gauge transformations and changes of variables some applications are mentioned in the one dimensional case

eigenanalysis and continuum modelling of pre twisted repetitive beam like structures a repetitive pin jointed pre twisted structure is analysed using a state variable transfer matrix technique within a global coordinate system the transfer matrix is periodic but introduction of a local coordinate system rotating with nodal cross sections results in an autonomous transfer matrix for this floquet system eigenanalysis reveals four real unity eigenvalues indicating tension torsion coupling and equivalent continuum properties such as poisson s ratio cross sectional area torsion constant and the tension torsion coupling coefficient are determined a variety of real and complex near diagonal jordan decompositions are possible for the multiple eight complex unity eigenvalues and these are discussed in some detail analysis of the associated principal vectors shows that a bending moment produces curvature in the plane of the moment together with shear deformation in the perpendicular plane but no bending bending coupling the choice of a structure having an equilateral triangular cross section is thought responsible for this unexpected behaviour as the equivalent continuum second moments of area are equal about all cross sectional axes in addition an asymmetric stiffness matrix is obtained for bending moment and shearing force coupling and possible causes are discussed

hilbert ℂ modules structural properties and applications to variational problems abstract we develop a theory of hilbert c over tilde modules which forms the core of a new functional analytic approach to algebras of generalized functions particular attention is given to nitely generated submodules projection operators representation theorems for c over tilde linear functionals and c over tilde sesquilinear forms we establish a generalized lax milgram theorem and use it to prove existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form

wave properties in poroelastic media using a wave finite element method abstract the application of the bidirectional wave finite element method wfe to biot allard s theory of poroelasticity is investigated this method has been successfully used in previous elastodynamics studies in the case of porous media the rigidity of the layer is very low leading to very small wavelengths and a high dissipation rate occurs within the pores these differences with the elastic case justify a study of their consequences on numerical results in this paper it is shown that despite these difficulties the wfe provides an efficient tool to compute the waves propagating through poroelastic media the influence of boundary conditions on wave propagation is discussed as well as the convergence of the numerical results depending on the spatial discretization the order of shape functions and the choice of the formulation finally the wavenumbers predicted with this method are compared with some simplified models such as equivalent fluid models or equivalent plate models it is shown that the wfe can be used to validate the assumptions made by the simplified models

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides the wave and finite element wfe method is a numerical approach to the calculation of the wave properties of structures of arbitrary complexity the method starts from a finite element fe model of only a short segment of the structure typically by using existing element libraries and commercial fe packages the dynamic stiffness matrix of the segment is obtained a periodicity condition applied and an eigenvalue problem formed whose solution gives the dispersion equations and wave mode shapes these define a wave basis from which the forced response can be found straightforwardly although straightforward in application the wfe method is prone to numerical difficulties these are discussed in this paper and methods to avoid or remove them described attention is focused on 1 dimensional waveguide structures for which numerical problems are most severe three ways of phrasing the eigenvalue problem for free wave propagation are presented and a method based on singular value decomposition is proposed to determine eigenvectors at low frequencies discretisation errors are seen to occur if the segment is too large while round off errors occur if the segment is too small this can be overcome by forming a super segment from the concatenation of two or more segments the forced response is then considered the use of a reduced wave basis removes many problems direct calculation of the waves excited by a point force is very prone to poor numerical conditioning but can be circumvented by exploiting the orthogonality of the left and right eigenvectors numerical examples are presented

higher order effective modeling of periodic heterogeneous beams ii derivation of the proper boundary conditions for the interior asymptotic solution abstract this paper constitutes the second part of a study devoted to the 1d modeling of beam like structures of arbitrary cross section with periodic properties along the axis in the first part the asymptotic expansion method has been described and the equations involving the successive terms of the outer solution have been given in particular it has been shown that neither the leading term nor the full outer solution are able to satisfy arbitrary prescribed edge data a specific study is thus necessary and it is presented in this paper the method used is the decay analysis initially proposed for homogeneous plates by gregory and wan j elast 14 1984 27 this approach consists in deriving the appropriate set of boundary conditions for the outer expansion solution for any arbitrary edge data stress displacement or mixed data these boundary conditions are obtained from the solution of canonical beam problems without computing any inner solution the method is illustrated through numerical examples on a lattice structure the results obtained prove the importance of a correct treatment of the edge effects in order that the inclusion of higher order terms in the beam model serves actually to increase the accuracy of the outer expansion solution

dynamic modelling of railway track a periodic model based on a generalized beam formulation abstract dynamic modelling of railway track is a vital step in view of traffic noise reduction considering the track as a periodic structure the classical properties and methods associated with one dimensional periodic structures are recalled then applied to the problem within the framework of the euler bernouilli and timoshenko beam theories these methods turn out to be insufficient when compared with experimental data especially concerning the lateral receptance spectrum therefore an extended beam model is developed essentially based on the notion of generalized cross section displacements finally the application of this theory to railway track results in a significant improvement in the computed spectrum and suggests directions for further investigations

quantization scheme based on the extension of phase space a new quantization scheme is proposed the similarity of geometric and covariant quantization of a certain class of constrained dynamical systems is shown a method to calculate anomalies is developed

model reduction and perturbation analysis of wave finite element formulations for computing the forced response of coupled elastic systems involving junctions with uncertain eigenfrequencies abstract the wave finite element method is investigated for computing the low and mid frequency forced response of coupled elastic systems involving straight structures with junctions the relevance of the method is discussed when a component mode synthesis procedure is used for modeling the junctions a norm wise selection criterion is proposed so as to reduce efficiently the number of junction modes retained in the wave based formulations component wise perturbation bounds of the wave based displacement force solutions are also derived to address slight uncertainties for the junction eigenfrequencies numerical comparisons with standard finite element solutions as well as monte carlo simulations highlight the relevance of the formulation

harmonic green s functions for flexural waves in semi infinite plates with arbitrary boundary conditions and high frequency approximation for convex polygonal plates abstract this paper provides a method for obtaining the harmonic green s function for flexural waves in semi infinite plates with arbitrary boundary conditions and a high frequency approximation of the green s function in the case of convex polygonal plates by using a generalised image source method the classical image source method consists in describing the response of a point driven polygonal plate as a superposition of contributions from the original source and virtual sources located outside of the plate which represent successive reflections on the boundaries the proposed approach extends the image source method to plates including boundaries that induce coupling between propagating and evanescent components of the field and on which reflection depends on the angle of incidence this is achieved by writing the original source as a fourier transform representing a continuous sum of propagating and evanescent plane waves incident on the boundaries thus the image source contributions arise as continuous sums of reflected plane waves for semi infinite plates the exact green s function is obtained for an arbitrary set of boundary conditions for polygonal plates a high frequency approximation of the green s function is obtained by neglecting evanescent waves for the second and subsequent reflections on the edges the method is compared to exact and finite element solutions and evaluated in terms of its frequency range of applicability

optimum design of band gap beam structures abstract the design of band gap structures receives increasing attention for many applications in mitigation of undesirable vibration and noise emission levels a band gap structure usually consists of a periodic distribution of elastic materials or segments where the propagation of waves is impeded or significantly suppressed for a range of external excitation frequencies maximization of the band gap is therefore an obvious objective for optimum design this problem is sometimes formulated by optimizing a parameterized design model which assumes multiple periodicity in the design however it is shown in the present paper that such an a priori assumption is not necessary since in general just the maximization of the gap between two consecutive natural frequencies leads to significant design periodicity the aim of this paper is to maximize frequency gaps by shape optimization of transversely vibrating bernoulli euler beams subjected to free standing wave vibration or forced time harmonic wave propagation and to study the associated creation of periodicity of the optimized beam designs the beams are assumed to have variable cross sectional area given total volume and length and to be made of a single linearly elastic material without damping numerical results are presented for different combinations of classical boundary conditions prescribed orders of the upper and lower natural frequencies of maximized natural frequency gaps and a given minimum constraint value for the beam cross sectional area to study the band gap for travelling waves a repeated inner segment of the optimized beams is analyzed using floquet theory and the waveguide finite element wfe method finally the frequency response is computed for the optimized beams when these are subjected to an external time harmonic loading with different excitation frequencies in order to investigate the attenuation levels in prescribed frequency band gaps the results demonstrate that there is almost perfect correlation between the band gap size location of the emerging band structure and the size location of the corresponding natural frequency gap in the finite structure

bloch theorem for isogeometric analysis of periodic problems governed by high order partial differential equations abstract with the emergence of isogeometric analysis iga numerical rotation free solutions to high order partial differential equations pdes are facilitated indeed iga allows the use of high order continuous basis functions through the domain for periodic systems the bloch theorem is useful for analyzing wave propagation restricting the analysis to a single unit cell and applying periodic boundary conditions however in the literature boundary conditions are only accounting for c 0 continuity and we propose in the present paper to enforce the g 1 continuity as well or to maintain the angle in case of unit cells meeting at kinks while g 1 continuity is enforced by maintaining the collinearity between adjacent control points in the presence of kinks variations of the field derived with respect to opposite in plane edge normals are linked by the rodrigues rotation matrix this method applied at collocation points belonging to the unit cell edges can be viewed as a novel approach of coupling multi patch geometries with kinks avoiding the use of nonlinear multipliers or additional bending strip elements both transfer matrix and indirect methods of the bloch theorem are investigated for respectively quasi one and quasi two dimensional wave propagation in periodic structures discretized by nurbs based iga dispersion curves of a set of numerical problems treating isogeometric rotation free extensible elastica beams and kirchhoff love shells are then analyzed

bloch theorem with revised boundary conditions applied to glide screw and rotational symmetric structures abstract wave propagation in complex periodic systems is often addressed with the bloch theorem and consists in applying periodic boundary conditions to a discretized unit cell while this method has been developed for structures periodic by translation in a recent work for quasi one dimensional wave propagation it has been shown that screw translation plus rotation and glide translation plus reflection periodicities can be accounted for as well keeping the cartesian coordinate system but revisiting the periodic boundary conditions the goal of the present paper is to generalize this concept to quasi two dimensional wave propagation two dimensional waves propagating in three dimensional structures dispersion relations for a set of reduced problems are then compared to results from the classical method when available by considering a smaller periodicity the computational cost is decreased and the number of folding curves and non interacting intersecting curves is reduced improving their interpretability while the size of a unit cell is divided by a factor two when glide symmetries are considered this ratio is significantly increased for screw or rotational symmetries moreover the proposed revisited bloch method is applicable to screw symmetric structures that do not possess purely translational symmetries and for which the classical method cannot be used e g chiral nanotubes longitudinally wrinkled helicoids

high frequency dynamics of an isotropic timoshenko periodic beam by the use of the time domain spectral finite element method in this work results of numerical simulations and experimental measurements related to the high frequency dynamics of an aluminium timoshenko periodic beam are presented it was assumed by the authors that the source of beam structural periodicity comes from periodical alterations to its geometry due to the presence of appropriately arranged drill holes as a consequence of these alterations dynamic characteristics of the beam are changed revealing a set of frequency band gaps the presence of the frequency band gaps can help in the design process of effective sound filters or sound barriers that can selectively attenuate propagating wave signals of certain frequency contents in order to achieve this a combination of three numerical techniques were employed by the authors they comprise the application of the time domain spectral finite element method in the case of analysis of finite and semi infinite computational domains damage modelling in the case of analysis of drill hole influence as well as the bloch reduction in the case of analysis of periodic computational domains as an experimental technique the scanning laser doppler vibrometry was chosen a combined application of all these numerical and experimental techniques appears as new for this purpose and not reported in the literature available

on saint venant s principle in pin jointed frameworks numerical experiments performed by hoff 50 years ago on the application of saint venant s principle to frameworks are extended and his conclusions qualified guided by these results a general treatment is formulated employing a state variable approach this leads directly to an eigenvalue problem in which the decay rates are the eigenvalues of the cell transfer matrix unity eigenvalues correspond to the transmitting modes of rigid body displacement and the stress resultants of tension bending moment shearing force and twisting moment consideration of the associated eigenvectors and principal vectors gives exact values for the equivalent continuum beam properties of the framework such as equivalent cross sectional area poisson s ratio second moment of area shear coefficient and torsion constant

structural waveguide behaviour of a beam plate system structures which have a constant cross section normal to a longitudinal axis can be considered as waveguides in which vibration can propagate in the form of various waves in the longitudinal direction the dynamic behaviour of such systems can be found by using a fourier transform approach in terms of wavenumbers in the longitudinal direction analytical solutions are available for simple infinitely long waveguides whereas for more complex waveguides numerical approaches have been developed using finite element techniques to describe the cross section in the present paper an analytical fourier transform approach is used to find the dynamic behaviour of a system consisting of two parallel beams coupled by a plate when a point force is applied to one of the beams multiple waves occur in the longitudinal direction the number of waves depending on the number of modes of the equivalent cross section however the motion of the driven beam is shown to be dominated by the contribution from only one or two waves at each frequency these having wavenumbers closest to that of the uncoupled beam the motion of the plate is also shown to be dominated by these wavenumbers for excitation on the beam experimental results are obtained on beam plate beam systems with identical and non identical beams which show good agreement with the predictions in particular these confirm that the plate response is dominated by waves with wavenumbers in the beam direction that follow those of the excited beam

transverse tilting and cross coupling stiffness of cylindrical rubber isolators in the audible frequency range the wave guide solution audio frequency wave guide models for antisymmetric dynamic stiffness of arbitrary long elastomer cylinders are presented the locally non mixed boundary conditions at the lateral and radial surfaces are simultaneously satisfied by using the modes corresponding to the dispersion relation for axial waves in cylinders satisfying the stress free boundary conditions at the curved radial boundaries while the displacement conditions on the flat cylinder ends are satisfied by mode matching the elastomer is modelled as nearly incompressible with deviatoric visco elasticity based on a fractional derivative standard linear solid embodying a mittag leffler relaxation kernel the main advantage being the minimum parameter number required to successfully model the material properties over a broad frequency band the stiffness is found to depend strongly on frequency displaying resonances and anti resonances the method is compared with and verified against finite element models in addition comparison to thin beam theories i e euler and timoschenko theory and a simple shear model is presented illustrating the limitations of these models

a finite element based method for the analysis of free wave propagation in stiffened cylinders abstract a finite element based method is developed to evaluate elastic wave propagation characteristics in orthogonally stiffened cylindrical shells the method is general and allows for detailed modelling of the shell and stiffners it is assumed that identical stiffeners are spaced at regular intervals in both the axial and circumferential directions so that the structure is periodic in these two dimensions wave propagation techniques are applied to a finite element model of a single periodic unit to determine propagation constants the shell variables are expanded in a fourier series in the circumferential direction and the axial propagation constants are presented for different values of the circumferential wavenumber as a function of frequency the method is verified for unstiffened cylinders by comparing results with exact solutions

calculating the forced response of cylinders and cylindrical shells using the wave and finite element method abstract the dynamic response of circular cylinders can be obtained analytically in very few and simple cases for complicated thick or anisotropic circular cylinders researchers often resort to the finite element fe method this can lead to large models especially at higher frequencies which translates into high computational costs and memory requirements in this paper the response of axially homogenous circular cylinders that can be arbitrarily complex through the thickness is obtained using the wave and finite element wfe method here the homogeneity of the cylinder around the circumference and along the axis are exploited to post process the fe model of a small rectangular segment of the cylinder using periodic structure theory and obtain the wave characteristics of the cylinder the full power of fe methods can be utilised to obtain the fe model of the small segment then the forced response of the cylinder is posed as an inverse fourier transform however since there are an integer number of wavelengths around the circumference of a closed circular cylinder one of the integrals in the inverse fourier transform becomes a simple summation whereas the other can be resolved analytically using contour integration and the residue theorem the result is a computationally efficient technique for obtaining the response to time harmonic arbitrarily distributed loads of axially homogenous circular cylinders with arbitrary complexity across the thickness

a recursive approach for the finite element computation of waveguides abstract the finite element computation of structures such as waveguides can lead to heavy computations when the length of the structure is large compared to the wavelength such waveguides can in fact be seen as one dimensional periodic structures in this paper a simple recursive method is presented to compute the global dynamic stiffness matrix of finite periodic structures this allows to get frequency response functions with a small amount of computations examples are presented to show that the computing time is of order log 2 n where n is the number of periods of the waveguide

the saint venant problem and principle in elasticity abstract the traditional semi inverse solution method of the saint venant problem and the saint venant principle which were described in the euclidian space under the lagrange system formulation are updated to be solved in the symplectic space under the conservative hamiltonian system thus the saint venant problem and the saint venant principle have been unified by the direct method it is proved in the present paper that all the saint venant solutions can be obtained directly via the zero eigenvalue solutions and all their jordan normal forms of the corresponding hamiltonian operator matrix and the saint venant principle corresponds to neglect of all the non zero eigenvalue solutions in which the non zero eigenvalue gives the decay rate

power flow in two and three dimensional frame structures abstract a general matrix method is presented for the analysis of power flow in two and three dimensional frames using a wave scattering approach axial torsional and flexural wave modes are included since the method is intended to be used for mid and high frequency analysis shear deformation and rotatory inertia are included in the flexural wave modes a general formulation is presented to determine wave scattering at structural junctions and boundaries although the finite element method can be used to calculate power flow a large number of elements is required at mid and high frequencies to obtain accurate results in the current method exact wave mode expressions are used and therefore only one element is required for each structural member at any frequency power flow results using the wave scattering approach are compared to the conventional finite element method for two and three dimensional frames

simplified description of out of plane waves in thin annular elastic plates abstract dispersion relations are derived for the out of plane wave propagation in planar elastic plates with constant curvature using the classical kirchhoff thin plate theory the dispersion diagrams and the mode shapes are compared with their counterparts for a straight plate strip and the role of curvature is assessed for plates with unconstrained edges elementary bernoulli euler theory for a beam of rectangular cross section with the circular shape of its axis is also employed to analyze the wave guide properties of this structure in its out of plane deformation the applicability range of the elementary beam theory is validated the wave finite element method in the formulation of the three dimensional elasticity theory is used to ensure that the comparison of dispersion diagrams is performed in the frequency range where the classical thin plate theory is valid thus the paper summarizes the effects brought to the propagation of out of plane waves in thin elastic plates by their constant curvature and the models of these plates

modelling wave propagation in two dimensional structures using finite element analysis a method is described by which the dispersion relations for a two dimensional structural component can be predicted from a finite element fe model the structure is homogeneous in two dimensions but the properties might vary through the thickness this wave finite element wfe method involves post processing the mass and stiffness matrices found using conventional fe methods of a segment of the structure this is typically a 4 noded rectangular segment although other elements can be used periodicity conditions are applied to relate the nodal degrees of freedom and forces the wavenumbers real imaginary or complex and the frequencies then follow from various resulting eigenproblems the form of the eigenproblem depends on the nature of the solution sought and may be a linear quadratic polynomial or transcendental eigenproblem numerical issues are discussed examples of a thin plate an asymmetric laminated plate and a laminated foam cored sandwich panel are presented for the last two examples developing an analytical model is a formidable task at best the method is seen to give accurate predictions at very little computational cost furthermore since the element matrices are typically found using a commercial fe package the meshing capabilities and the wealth of existing element libraries can be exploited

a new approach to the analysis of oscillations of one dimensional spatially periodic structures abstract a new approach to the analysis of oscillations of spatially periodic structures which is based on the method of direct separation of motions mdsm is proposed in the paper as an example of the approach application the study of oscillations of a string with variable cross section is conducted as the result analytical expressions for the eigenmodes and the eigenfrequencies of the system are obtained it is shown that modulation of the string cross section leads to a change of the eigenfrequencies as compared with their non modulated values and to the emergence of a spectrum of additional high eigenfrequencies which correspond to large wave lengths it is noted that application of the multiple scales method and other classical asymptotic methods for solving the considered equations is rather cumbersome and leads to erroneous results because an unknown parameter is present in them the fundamental difference between the mdsm and these methods is revealed in particular it is shown that the mdsm may be employed for solving equations without small parameter thereby the applicability range of this method turns out to be broader than the one of the classical asymptotic methods to illustrate the validity of this statement a classical problem about the stability of a pendulum with vibrating suspension axis is considered

locally convex topological algebras of generalized functions compactness and nuclearity in a nonlinear context in this paper we introduce hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions these topologies are strong duals of frechet schwartz space topologies and even strong duals of nuclear frechet space topologies in particular any bounded set is relatively compact and one benefits from all deep properties of nuclearity these algebras of generalized functions contain most of the classical irregular functions and distributions they are obtained by replacing the mathematical tool of c functions in the original version of nonlinear generalized functions by the far more evolved tool of holomorphic functions this paper continues the nonlinear theory of generalized functions in which such locally convex topological properties were strongly lacking up to now

stronger uncertainty relations for all incompatible observables the heisenberg robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator notably it does not capture the concept of incompatible observables because it can be trivial i e the lower bound can be null even for two noncompatible observables here we give two stronger uncertainty relations relating to the sum of variances whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system

the image source method for calculating the vibrations of simply supported convex polygonal plates the aim of this paper is to show that the image source method ism can be used both for analytically calculating green s functions of particular simply supported convex polygonal plates and for predicting medium and high frequency vibrations of arbitrarily shaped simply supported convex polygonal plates with controllable precision in the first part of the paper the method for obtaining the green s function of a polygonal plate by ism is developed examples for plates of different geometries rectangle isosceles right triangle half equilateral triangle and equilateral triangle are given in the second part of the paper the pertinence of ism for predicting medium and high frequency vibrations of arbitrarily shaped simply supported convex polygonal plates is investigated an approximation based on the exclusion of image sources beyond a certain distance from the receiver is used in order to take advantage of the dissipation of vibrational energy through wave propagation we investigate the influence of structural damping and truncation distance on the accuracy of such approximation the computed responses are in good agreement with reference solutions which are analytically known or obtained by the finite element method

on the direct solution of wave propagation for repetitive structures abstract the direct method of solving the wave constants for a repetitive structure with given frequency is developed in this paper the analogy between structural mechanics and optimal control theory is applied the dynamic stiffness matrix approach is used to solve for the eigensolutions of wave propagation the weighted adjoint symplectic orthonormality relationship and the eigenvector expansion theorem are also established for this approach the symplectic eigenproblem is derived for skew symmetric matrices a cell triangular decomposition for skew symmetric matrices is introduced to reduce the generalized eigenproblem and the symplectic householder transformation is introduced to cell tri diagonalize the skew symmetric matrix to solve the eigenequation the problem of a wave impinging on the boundary is considered by the eigenvector expansion method which could also be applied to a wave scattering when passing over an abnormal part numerical examples show that resonance will occur for the localized vibration case

entropy of ideal quantum gases as a lie function we investigate the lie series representation of the canonical transformations in a complex phase space it is shown that any canonical mapping in the complex domain can be labelled by two different functions one of these functions corresponds to an observable in the sense of classical mechanics the second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics in particular we show that the entropy of ideal quantum gases generates a special canonical transformation and moreover the entropy itself can be represented as a lie function formed by a special characteristic function

spectral finite element analysis of the vibration of straight fluid filled pipes with flanges spectral finite element analysis of the vibration of straight fluid filled pipes with flanges

wave propagation along transversely periodic structures the dispersion curves for guided waves have been of constant interest in the last decades because they constitute the starting point for nde ultrasonic applications this paper presents an evolution of the semianalytical finite element method and gives examples that illustrate new improvements and their importance for studying the propagation of waves along periodic structures of infinite width periodic boundary conditions are in fact used to model the infinite periodicity of the geometry in the direction normal to the direction of propagation this method allows a complete investigation of the dispersion curves and of displacement stress fields for guided modes in anisotropic and absorbing periodic structures among other examples that of a grooved aluminum plate is theoretically and experimentally investigated indicating the presence of specific and original guided modes

propagation of decaying waves in periodic and piecewise periodic structures of finite length abstract a new method is described for the analysis of long and complicated structures which are composed of spatially periodic units or sections of spatially periodic units the response of such a structure to external excitations is treated as a superposition of wave motions with account taken of the effects of wave reflection due to change in the construction pattern along the structure and boundary conditions since damping usually exists in a real structure a wave motion always decays as it propagates along the structure however this can cause numerical difficulties if the structure is rather long because a decaying wave appears as a growing wave in the opposite direction and since propagations in both directions must be accounted for in the analysis computational errors associated with the apparent growing waves also grow and the numerical results become meaningless in the proposed new scheme use is made of wave reflection and transmission matrices making it possible for all computations to proceed in the direction of wave propagation so that the results are always stable numerically furthermore the required computing time is greatly reduced in comparison with that of an all encompassing finite element analysis for the entire structure application of the new method is illustrated by examples

on wave propagation in repetitive structures two forms of transfer matrix abstract two forms of dynamic transfer matrix are derived for a one dimensional beam like repetitive pin jointed structure with point masses located at nodal cross sections the displacement force transfer matrix g and the displacement displacement transfer matrix h similarity matrices are introduced to relate g and h together with their respective metrics symplectic orthogonality relationships for the eigenvectors of both g and h are derived together with relationships between their respective sets of eigenvectors new expressions for the group velocity are derived for repetitive structures of finite length natural frequency equations are derived employing both g and h including phase closure and the direct application of boundary end conditions besides an exposition of the theory some familiar but much new the focus of the present paper is on the relationships between the two forms of transfer matrix including their respective dis advantages numerical results together with further theory necessary for interpretation are presented in companion papers

an asymptotic decoupling method for waves in layered media this paper presents a technique asymptotic decoup ling for analysing wave propagation in multi layered media the technique leads to a hierarchy of approximations to the exact dispersion relatio

the computation of dispersion relations for three dimensional elastic waveguides using the scaled boundary finite element method abstract in this paper a numerical approach for the computation of dispersion relations for three dimensional waveguides with arbitrary cross section is proposed the formulation is based on the scaled boundary finite element method sbfem it is an extension of the approach previously derived for plate structures it is shown that the wavenumbers of guided waves in a waveguide can be obtained as the eigenvalues of the z matrix which is well known in the sbfem the hamiltonian properties of this matrix are utilized to derive an efficient way to compute the group velocities of propagating waves as eigenvalue derivatives the cross section of the waveguide is discretized using higher order spectral elements it is discussed in detail how symmetry axes can be utilized to reduce computational costs in order to sort the solutions at different frequencies a mode tracking algorithm is proposed based on the pade expansion

on the forced response of waveguides using the wave and finite element method the forced response of waveguides subjected to time harmonic loading is treated the approach starts with the wave and finite element wfe method where a segment of the waveguide is modeled using traditional finite element methods the mass and stiffness matrices of the segment are used to formulate an eigenvalue problem whose solution yields the wave properties of the waveguide the wfe formulation is used to obtain the response of the waveguide to a convected harmonic pressure chp since the fourier transform of the response to a general excitation is a linear combination of the responses to chps the response to a general excitation can be obtained via an inverse fourier transform process this is evaluated analytically using contour integration and the residue theorem hence the approach presented herein enables the response of a waveguide to general loading to be found by a modeling a segment of the waveguide using finite element methods and post processing it to obtain the wave characteristics b using fourier transform and contour integration to obtain the wave amplitudes and c using the wave amplitudes to find the response at any point in the waveguide numerical examples are presented

heisenberg inequality and the complex field in quantum mechanics observables a and b satisfy the heisenberg inequality if the product of their variances has a positive lower bound independent of the state of the system in the hilbert space formulation of quantum mechanics it is a consequence of the schwarz inequality that the heisenberg type inequality var a var b 1 4 a b b a 2 holds for any pair of observables a and b represented as self adjoint operators and for any vector state represented as a unit vector if inf a b b a dom a dom b 0 then a and b satisfy the heisenberg inequality in the present paper the derivability of the heisenberg type inequality is analyzed within the general theoretical frame of a sum logic it is shown that any real valued non negative function a f a of observables a and of states which has a symmetry property f a b f a b 2f a 2f b with respect to observables satisfies the heisenberg type inequality f a f b 1 4 f a b f a f b 2 for all observables a and

stochastic wave finite element for random periodic media through first order perturbation abstract wave propagation features in random guided elastic media are considered in this paper a formulation named stochastic wave finite element swfe is offered this formulation allows wave characteristics definition by means of a stochastic finite element model the case of spatially homogeneous random properties is dealt with through a parametric probabilistic technique numerical experiments show the effectiveness of the proposed approach to predict the statistics of guided wave propagation characteristics means and standard deviations when compared to monte carlo calculations

sensitivity analysis of generalised eigenproblems and application to wave and finite element models abstract the first and second order sensitivity analysis of the eigenvalue problem of generalised non symmetric matrices using perturbation theory is developed these results are then applied to sensitivity analysis of wave propagation in structures modelled using the wave and finite element wfe method three formulations of the wfe eigenvalue problem are considered the transfer matrix method the projection method and zhong s method the sensitivities with respect to system parameters of wavenumbers and wave mode shapes are derived expressions for the group velocity are presented numerical results for a thin beam a foam core panel and a cross laminated timber panel are used to demonstrate the proposed approach it is shown that sensitivities can be calculated at negligible computational cost

finite element computation of absorbing boundary conditions for time harmonic wave problems this paper proposes a new method in the frequency domain to define absorbing boundary conditions for general two dimensional problems the main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general it is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software boundary conditions at various orders of accuracy can be obtained in a simple way this is then applied to study some examples for which analytical or numerical results are available good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method

the forced vibration of one dimensional multi coupled periodic structures an application to finite element analysis abstract a general theory for the forced vibration of multi coupled one dimensional periodic structures is presented as a sequel to a much earlier general theory for free vibration starting from the dynamic stiffness matrix of a single multi coupled periodic element it derives matrix equations for the magnitudes of the characteristic free waves excited in the whole structure by prescribed harmonic forces and or displacements acting at a single periodic junction the semi infinite periodic system excited at its end is first analysed to provide the basis for analysing doubly infinite and finite periodic systems in each case total responses are found by considering just one periodic element an already known method of reducing the size of the computational problem is reexamined expanded and extended in detail involving reduction of the dynamic stiffness matrix of the periodic element through a wave coordinate transformation use of the theory is illustrated in a combined periodic structure finite element analysis of the forced harmonic in plane motion of a uniform flat plate excellent agreement between the computed low frequency responses and those predicted by simple engineering theories validates the detailed formulations of the paper the primary purpose of the paper is not towards a specific application but to present a systematic and coherent forced vibration theory carefully linked with the existing free wave theory