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

an inverse method to determine the dispersion curves of periodic structures based on wave superposition abstract phononic crystals and acoustic metamaterials have unique properties such as the existence of band gaps which give them huge potential in many applications such as vibration isolation acoustic cloaking acoustic lensing and more many methods have been proposed to determine the band structure of these materials but almost all require a model of the structure in this paper an inverse method to calculate the band structure of one dimensional periodic structures based on bloch wave boundary conditions and wave superposition is introduced the proposed method only requires the frequency responses measured at a small number of points within the structure this allows the band structures to be determined experimentally using simple equipment like a shaker and accelerometers the band structure of a simple bi material beam was calculated in this study as a demonstration of the method and the results were found to be in agreement with calculations made using the transfer matrix method the proposed method was then extended to predict the response of a finite periodic bi material beam with arbitrary boundary conditions using only the band structure and components of the eigenvectors some resonance peaks were observed within the band gaps and these were found to be caused by the reflection of the waves at the boundaries the effects of the number of unit cells on the transmissibility of a beam were investigated it was found that the transmissibilities within the band gaps can be estimated to be directly proportional to the number of unit cells lastly an attempt was made to extend the method to two and three dimensional periodic structures and the wave superposition method was found to be able to measure a portion of the dispersion surface of two dimensional structures with a fair degree of accuracy especially at lower bands errors and scatter are present at high frequencies caused by more waves significantly affecting the responses of the system this issue can be alleviated by taking measurements further away from the boundaries or increasing the number of waves considered however the key limitation of the method for two and three dimensional periodic structure is that it can only measure a portion of the dispersion surface or volume

generalized weighted hardy cesaro operators and their commutators on weighted morrey spaces abstract let 0 1 0 s 0 1 r be measurable functions and be a parameter curve in r n given by t x 0 1 r n s t x s t x in this paper we study a new weighted hardy cesaro operator defined by u s f x 0 1 f s t x t d t for measurable complex valued functions f on r n under certain conditions on s t and on an absolutely homogeneous weight function we characterize the weight function such that u s is bounded on weighted morrey spaces l p and then compute the corresponding operator norm of u s we also give a necessary and sufficient condition on the function which ensures the boundedness of the commutator of the operator u s on l p with symbols in bmo

on modifications of newton s second law and linear continuum elastodynamics in this paper we suggest a new perspective where newton s second law of motion is replaced by a more general law which is a better approximation for describing the motion of seemingly rigid macroscopic bodies we confirm a finding of willis that the density of a body at a given frequency of oscillation can be anisotropic the relation between the force and the acceleration is non local but causal in time conversely for every response function satisfying these properties and having the appropriate high frequency limit there is a model which realizes that response function in many circumstances the differences between newton s second law and the new law are small but there are circumstances such as in specially designed composite materials where the difference is enormous for bodies which are not seemingly rigid the continuum equations of elastodynamics govern behaviour and also need to be modified the modified versions of these equations presented here are a generalization of the equations proposed by willis to describe elastodynamics in composite materials it is argued that these new sets of equations may apply to all physical materials not just composites the willis equations govern the behaviour of the average displacement field whereas one set of new equations governs the behaviour of the averageweighted displacement field where the weighted displacement field may attach zero weight to hidden areas in the material where the displacement may be unobservable or not defined from knowledge of the average weighted displacement field one obtains an approximate formula for the ensemble averaged energy density two other sets of new equations govern the behaviour when the microstructure has microinertia i e where there are internal spinning masses below the chosen scale of continuum modelling in the first set the average displacement field is assumed to be observable while in the second set an average weighted displacement field is assumed to be observable

lp and bmo bounds of weighted hardy littlewood averages abstract this paper characterizes those nonnegative functions defined on 0 1 for which the weighted hardy littlewood average u f 0 1 f t t dt is bounded on either l p r n p 1 or bmo r n meanwhile the corresponding operator norms are worked out

the norm inequalities for the weighted cesaro mean operators this paper gives some necessary and sufficient conditions for the weighted cesaro mean operators to be bounded on herz spaces the corresponding new operator norm inequalities are obtained

free vibration of an orthogonally stiffened flat plate abstract a flat plate reinforced by a regular orthogonal array of uniform beams is analyzed by using techniques developed for studying wave propagation in two dimensional periodic structures a plane wave type of motion is considered which may be characterized by different propagation phase constants in the x and y directions the hierarchical finite element method is used to set up the governing equations of free wave motion and these are then solved as an eigenvalue problem for the frequencies at which particular waves will propagate plots of phase constant surfaces vs frequency are presented for a number of different plate beam configurations excellent agreement is found between some of these and the results of earlier investigators when the plate is supported by flexible beams in both directions wave propagation is found to commerce at zero frequency at higher frequencies alternating and overlapping attenuation and propagation bands occur the nature and explanations of these are discussed wave speed surfaces vs frequency are also presented and these give insight into the critical coincidence frequencies of the plate under acoustic excitation

invariant representation of propagation properties for bi coupled periodic structures general bi coupled periodic systems are dealt with by means of transfer matrices of single units the solutions of the associated characteristic equation are discussed in terms of invariant quantities by exploiting the well known reversibility of its coefficients an exhaustive description of the free wave propagation patterns is given on the invariant plane where propagation domains with qualitatively different character are identified the asymptotic behavior of the roots of the characteristic equation when the invariants tend to infinity is analyzed the contour plot of the real part of the propagation constants responsible for the amount of attenuation of the characteristic waves is illustrated on the invariants plane next several models of bi coupled periodic structures made up of beams resting on elastic supports are considered a non linear mapping from the invariants plane to the physical parameters plane provides a concise representation of the pattern of the propagation domains a mechanical interpretation associated with the boundaries of these regions is given finally the proper selection of the physical parameters governing the propagation modes is discussed

continuity of hardy type operators on rectangularly defined spaces abstract we introduce a generalization of rectangular herz spaces defined in 3 and prove boundedness of a version of the n dimensional hardy operator on them we also study the action of the hausdorff operator on some subspaces of bmo

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

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

dispersive elastodynamics of 1d banded materials and structures analysis abstract the elastodynamics of 1d periodic materials and finite structures comprising these materials are studied with particular emphasis on correlating their frequency dependent characteristics and on elucidating their pass band and stop band behaviors dispersion relations are derived for periodic materials and are employed in a novel manner for computing both pass band and stop band complex mode shapes through simulations of harmonically induced wave motion within a finite number of unit cells conformity of the frequency band structure between infinite and finite periodic systems is shown in particular only one or two unit cells of a periodic material could be sufficient for frequency bandedness to carry over from the infinite periodic case and only three to four unit cells are necessary for the decay in normalized transmission within a stop band to practically saturate with an increase in the number of cells dominant speeds in the scattered wave field within the same finite set of unit cells are observed to match those of phase and group velocities of the infinite periodic material within the most active pass band dynamic response due to impulse excitation also is shown to capture the infinite periodic material dynamical characteristics finally steady state vibration analyses are conducted on a finite fully periodic structure revealing a conformity in the natural frequency spread to the frequency band layout of the infinite periodic material the steady state forced response is observed to exhibit mode localization patterns that resemble those of the infinite periodic medium and it is shown that the maximum localized response under stop band conditions could be significantly less than in an equivalent homogenous structure and the converse is true for pass band conditions

almost global existence in the plane wave nonlinear dielectric interaction problem abstract the problem of the time evolution of the electromagnetic field in a plane wave propagating into a nonlinear dielectric with nontrivial conduction vector governed by a nonlinear ohm s law is considered for the relevant associated inhomogeneous quasilinear hyperbolic system of balance laws with associated periodic initial data we establish the following result on almost global existence in time for solutions of the initial value problem either for sufficiently small initial data the maximal time of existence t of a c 1 solution is infinitely large or for finite no matter how small but finite the initial data is taken for all initial data which are sufficiently small t is bounded below by a function of the initial data which increases without bound as the magnitude of the data goes to zero

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

wave propagation in two dimensional periodic lattices plane wave propagation in infinite two dimensional periodic lattices is investigated using floquet bloch principles frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies hexagonal honeycomb kagome lattice triangular honeycomb and the square honeycomb these topologies exhibit dramatic differences in their long wavelength deformation properties long wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices the slenderness ratio of the constituent beams of the lattice or relative density has a significant influence on the band structure the techniques developed in this work can be used to design lattices with a desired band structure the observed spatial filtering effects due to anisotropy at high frequencies short wavelengths of wave propagation are consistent with the lattice symmetries

free vibrations of a mono coupled periodic system abstract vibration problems of periodic systems can be analyzed efficiently by means of the transfer matrix method the frequency equation for the whole system is shown to be obtained in terms of the eigenvalues or their natural logarithms which are often called propagation constants of the transfer matrix for a single periodic subsystem in case of a mono coupled system this frequency equation may be solved graphically by using the propagation constant curve thereby saving a great deal of computational effort two types of mono coupled systems are considered as numerical examples a spring mass oscillating system and a continuous timoshenko beam resting on regularly spaced knife edge supports depending on whether the transfer matrix is derived by an analytical procedure or by the finite element method the numerical solutions become either exact or approximate

solving second order non linear hyperbolic pdes using generalized finite difference method gfdm abstract the generalized finite difference method gfdm has been proved to be a good meshless method to solve several linear partial differential equations pdes wave propagation advection diffusion plates beams etc the gfdm allows us to use irregular clouds of nodes that can be of interest for modelling non linear parabolic pdes this paper illustrates that the gfd explicit formulae developed to obtain the different derivatives of the pde s are based in the existence of a positive definite matrix that it is obtained using moving least squares approximation and taylor series development criteria for convergence of fully explicit method using gfdm for different non linear parabolic pdes are given this paper shows the application of the gfdm to solving different non linear problems including applications to heat transfer acoustics and problems of mass transfer

free vibrations of finite periodic structures in pass and stop bands of the counterpart infinite waveguides abstract the existence of frequency stop bands in which transmission of the vibro acoustic energy is impossible suggests that the periodic structures may be used for vibro isolation in any technical application however only a finite segment of such a structure can be used this paper is concerned with comparison of the eigenfrequency spectra of finite periodic structures with location of stop bands for their infinite counterparts a hierarchy of four mathematical models is considered in each case special attention is paid to eigenfrequencies and eigenmodes of a single periodicity cell with appropriate boundary conditions the influence of the amount of periodicity cells in a finite compound structure on its eigenfrequency spectrum is analyzed several features common for the considered models are found and discussed in the context of the existing knowledge on the subject

analysis of in plane wave propagation in hexagonal and re entrant lattices plane wave propagation in hexagonal and re entrant lattices is investigated through the application of bloch s theorem two dimensional dispersion relations are estimated and analyzed to illustrate peculiar properties of re entrant configurations and investigate the directional behavior of hexagonal lattices for varying construction angles special attention is devoted to the determination of phase and group wave velocities velocity plots are obtained for varying frequency and direction of wave propagation to analyze the dispersion characteristics of the lattices and further investigate their directional characteristics

the p adic weighted hardy cesaro operator and an application to discrete hardy inequalities abstract this paper aims to investigate the boundedness of the p adic analog of the weighted hardy cesaro operator u s f z p f s t t d t on weighted lebesgue spaces and weighted bmo spaces in each case we obtain the corresponding operator norms u s in particular these results have a surprising relevance to discrete hardy inequalities on the real field we prove a reverse bmo hardy inequality and give a necessary condition on so that the commutator of u s is bounded on l r q p n with symbols in bmo q p n

a new method of analyzing wave propagation in periodic structures applications to periodic timoshenko beams and stiffened plates a response function is found for an infinite uniform one dimensional structure which is subjected to an array of harmonic forces or moments spaced equidistantly and which have a constant phase or ratio between any adjacent pair receptance functions are derived for these phased arrays they are used to set up a general determinantal equation for the propagation constants of the infinite structure when it is made periodic by the addition of an infinite set of regular constraints they are also used to set up equations for the response of the structure to a convected harmonic pressure field the method enables the equations for the propagation constants and for the response to convected loading to be set up with much greater facility than by earlier methods it only requires a knowledge of the response function of the infinite uninterrupted structure under a single point harmonic force or moment the general equation for the propagation constants is used to study a a simply supported periodic timoshenko beam and b a parallel plate with periodic beam type stiffeners some calculated propagation constants are presented and discussed the periodic plate results are relevant to integrally stiffened skins of the type used in aeroplanes

hardy type theorems on fourier transforms revised abstract we obtain new conditions on periodic integrable functions so that their transformed fourier series belong to l p this improves the classical hardy and bellman results a counterpart for the fourier transforms is also established our main tool is a new extension of the hausdorff young paley inequality for fourier transforms

the response of two dimensional periodic structures to point harmonic forcing abstract this analysis is concerned with the response of an infinite two dimensional periodic structure to point harmonic loading initially a damped finite system is considered and the system size is then allowed to tend to infinity it is shown that the response in the far field can be expressed in terms of the properties of the phase constant surfaces which arise in the analysis of plane wave motion through the system within a pass band it is found that the physical nature of the response is strongly affected by the presence or otherwise of a caustic in the absence of a caustic the response amplitude has a relatively smooth spatial distribution if a caustic is present then the response amplitude has a complex spatial pattern and a dead region of very low response occurs the theory is applied to a two dimensional mass spring periodic system and a comparison is made with an exact calculation for the response of a large finite system excellent agreement is obtained between the two sets of results the work has application to all types of two dimensional periodic structures including orthogonally stiffened plates and shells and it raises the possibility of designing a periodic structure to act as a spatial filter to isolate sensitive equipment from a localized excitation source

an exact modal analysis approach to vibration analysis of structures with mass spring subsystems and rotational joints abstract a novel exact modal analysis approach is presented for vibration analysis of plane continuous structures which are coupled with discrete mass spring subsystems and include elastic rotational joints modelling local flexibility using the theory of generalised functions to handle the discontinuities of the response variables every continuous member with any number of mass spring subsystems and joints is treated as a two node element for which a 6 6 exact dynamic stiffness matrix is obtained in closed form as a result the global dynamic stiffness matrix is built by a standard finite element assembling procedure with size depending only on the number of nonzero nodal degrees of freedom of member to member nodes upon deriving pertinent orthogonality conditions for the modes the system response under arbitrary loads is obtained by modal impulse and modal frequency response functions under the assumption of proportional damping the solutions are exact and can be used as benchmark for classical finite element solutions the approach is formulated for various mass spring subsystems acting in transverse and axial directions relative to every member

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

formation of local resonance band gaps in finite acoustic metamaterials a closed form transfer function model abstract the objective of this paper is to use transfer functions to comprehend the formation of band gaps in locally resonant acoustic metamaterials identifying a recursive approach for any number of serially arranged locally resonant mass in mass cells a closed form expression for the transfer function is derived analysis of the end to end transfer function helps identify the fundamental mechanism for the band gap formation in a finite metamaterial this mechanism includes a repeated complex conjugate zeros located at the natural frequency of the individual local resonators b the presence of two poles which flank the band gap and c the absence of poles in the band gap analysis of the finite cell dynamics are compared to the bloch wave analysis of infinitely long metamaterials to confirm the theoretical limits of the band gap estimated by the transfer function modeling the analysis also explains how the band gap evolves as the number of cells in the metamaterial chain increases and highlights how the response varies depending on the chosen sensing location along the length of the metamaterial the proposed transfer function approach to compute and evaluate band gaps in locally resonant structures provides a framework for the exploitation of control techniques to modify and tune band gaps in finite metamaterial realizations

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

systems of nonlinear hyperbolic equations associated with problems of classical electromagnetic theory abstract a broad survey of the influence of nonlinearity on diverse phenomena in classical electromagnetic theory is presented with the focus being on the plane wave nonlinear dielectric interaction problem and shock formation in distributed transmission lines connections with recent work in the are of quasilinear hyperbolic systems of balance laws are emphasized

generalized poincare embeddings and weighted hardy operator on qpα q spaces abstract the poincare embeddings w 1 n r n bmo r n and the john nirenberg inequality in bmo r n are classical results in modern analysis and partial differential equations this paper studies the generalized poincare embeddings and the john nirenberg inequality in q p q r n spaces furthermore for the weighted hardy operator u and its adjoint v this paper gives sufficient and necessary conditions on weight function to ensure the boundedness of them on q p q r n spaces

a condition implying the global existence of c1 solutions for a class of nonlinear transmission lines abstract for a distributed parameter nonlinear transmission line with constant self inductance and resistance but voltage dependent capacitance and leakage conductance a condition on the growth in time of the leakage conductance is proposed which implies the existence of global c 1 solutions to the inhomogeneous damped quasilinear hyperbolic system with periodic initial data which governs the evolution of the charge and current in the line for a transmission line in which the leakage conductance may be controllable in time the analysis suggests a direct means of preventing the formation of shocks in the line

vibration reduction in piecewise bi coupled periodic structures the problem of minimizing transmitted vibrations through finitely long periodic structures is addressed bi coupled periodic element properties and arrangement are tailored to localize the response around the excitation source within any assigned frequency range bi dimensional analytical maps of the single unit free wave propagation domains stop pass and complex domains provide the optimal choice of the cell properties and ordering moreover the amount of vibration suppression along the periodic structure is also controlled as it can be described through iso attenuation curves representing the contour plot of the real part of the propagation constants applications to both undamped and damped beams resting on elastic supports are illustrated the response of the periodic structures to harmonic excitations is expressed through the wave vector method taking into account the effects of wave reflection due to changes in the cell properties along the structure and boundary conditions such computational schemes enables one to overcome numerical difficulties arising in the transfer matrix formulation for structures with a large number of periodic units

estimates of weighted hardy littlewood averages on the p adic vector space abstract in the p adic vector space q p n we characterize those non negative functions defined on z p w q p 0 w p 1 for which the weighted hardy littlewood average u f z p f t t d t is bounded on l r q p n 1 r and on bmo q p n also in each case we find the corresponding operator norm u

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