the ensemble statistics of the vibrational energy density of a random system subjected to single point harmonic excitation this paper is concerned with the ensemble statistics of the energy density of a random system subjected to a harmonic point load both the mean and variance of the reverberant energy density i e the total response minus the direct field are investigated it is shown that the ensemble average of the reverberant energy density is not spatially homogeneous but rather the value at the drive point is between two and three times the spatially averaged value depending on the modal overlap factor this result is closely analogous to established results regarding the case of transient excitation expressions are also derived for the relative variance of the reverberant energy density both remote from the drive point and at the drive point and a number of anomalies are found in existing results a comparison is made with simulation results for a randomized plate and this comparison highlights the importance of the ensemble size considered in the simulations the present analysis is based on a random point process model of the system natural frequencies and both gaussian orthogonal ensemble goe statistics and poisson statistics are considered

ensemble averages of power flow in randomly excited coupled beams the transmission of transverse vibrational energy between two coupled beams subjected to random excitation is considered extensive numerical results are given for the frequency decomposition of power flow between the beams the power flow is a function of the three non dimensional parameters describing frequency 1 where 1 is the fundamental resonance frequency of beam one length ratio l 2 l 1 and damping or modal overlap ratio single frequency and octave band averages are shown for various values of l 2 l 1 and means and variances are calculated for the ensemble of systems corresponding to a uniform distribution of 1 l 2 l 1 2 various approximate results based on modal analysis and on sea are compared with the exact results

a graph theory approach to identify resonant and non resonant transmission paths in statistical modal energy distribution analysis statistical modal energy distribution analysis smeda extends classical statistical energy analysis sea to the mid frequency range by establishing power balance equations between modes in different subsystems this circumvents the sea requirement of modal energy equipartition and enables applying smeda to the cases of low modal overlap locally excited subsystems and to deal with complex heterogeneous subsystems as well yet widening the range of application of sea is done at a price with large models because the number of modes per subsystem can become considerable when the frequency increases therefore it would be worthwhile to have at one s disposal tools for a quick identification and ranking of the resonant and non resonant paths involved in modal energy transmission between subsystems it will be shown that previously developed graph theory algorithms for transmission path analysis tpa in sea can be adapted to smeda and prove useful for that purpose the case of airborne transmission between two cavities separated apart by homogeneous and ribbed plates will be first addressed to illustrate the potential of the graph approach a more complex case representing transmission between non contiguous cavities in a shipbuilding structure will be also presented

analysis of beam and plate vibrations by using the wave equation abstract it is shown that the vibrations of beams and plates may be analyzed in the frequency domain by using the wave equation rather than the conventional differential equations of motion provided that certain assumptions are made regarding the response of the system in the vicinity of a structural discontinuity the wave equation being of second order is more amenable to both numerical and analytical solution than the standard fourth order differential equations offering particular advantages in the study of the high frequency response of beam systems and in the free vibration analysis of plate systems the accuracy and robustness of the method is investigated by application to a range of example systems and a comparison is made with the vibrational conductivity approach to high frequency dynamics which is based on the heat conduction equation

the postulations a la d alembert and a la cauchy for higher gradient continuum theories are equivalent a review of existing results in order to found continuum mechanics two different postulations have been used the first introduced by lagrange and piola starts by postulating how the work expended by internal interactions in a body depends on the virtual velocity field and its gradients then by using the divergence theorem a representation theorem is found for the volume and contact interactions which can be exerted at the boundary of the considered body this method assumes an a priori notion of internal work regards stress tensors as dual of virtual displacements and their gradients deduces the concept of contact interactions and produces their representation in terms of stresses using integration by parts the second method conceived by cauchy and based on the celebrated tetrahedron argument starts by postulating the type of contact interactions which can be exerted on the boundary of every suitably regular part of a body then it proceeds by proving the existence of stress tensors from a balance type postulate in this paper we review some relevant literature on the subject discussing how the two postulations can be reconciled in the case of higher gradient theories finally we underline the importance of the concept of contact surface edge and wedge s order forces

characterization of all copulas associated with non continuous random variables we introduce a constructive method by means of a doubly stochastic measure to describe all the copulas that in view of sklar s theorem are able to connect a bivariate distribution to its marginals we use this to give the lower and upper optimal bounds for all the copulas that extend a given subcopula

reformulation of the stochastic bem to improve the computational efficiency in the prediction of the vibro acoustic behaviour of structures with uncertainties abstract non determinism plays a crucial role in the prediction of the vibro acoustic behaviour of structures the stochastic boundary element method sbem allows computing the average response of systems with geometrical uncertainties the method consists in including such perturbations in the standard bem formulation and adding a set of equations representative of the energetic behaviour due to the presence of unknown cross products the addition of a set of auxiliary equation is necessary to solve the problem as a consequence the system dimension becomes very large and the application of the methodology is limited to small models this paper deals with a reformulation of the sbem approach such that the system dimension is drastically decreased and the methodology can be applied to more complex cases moreover the approach is extended to predict the variance of the field variable at no additional computational cost after a theoretical introduction the sbem is applied to two dimensional examples with pressure and velocity boundary conditions the comparison with monte carlo simulations shows a good agreement of the results and a reduction in the computational load is achieved

equipartition and mean square responses in large undamped structures it is shown that the smoothed spectral energy density at one point in a large complex structure may be approximated in a simple manner as proportional to the frequency smoothed admittance at the receiver and at the source and inversely proportional to the global modal density thus the mean square response may be estimated with little computational burden knowing only the local properties at source and receiver and the size of the system the approximate predictions are compared with the results of direct numerical simulations and found to be accurate except in the presence of anderson localization work supported by nsf

hydrofoil vibration induced by a random flow a stochastic perturbation approach the problem of an elastic lifting hydrofoil in a randomly perturbed flow is considered it appears that in these conditions the phenomenon of hydroelastic induced vibrations is controlled by a stochastic differential operator by using the theory of stochastic perturbation a technique of solution for this class of problems is proposed leading to an effective numerical solution the fundamentals of the method are given and it is applied to the general problem of hydroelastic vibrations a numerical application to the case of an elastic control surface for a prototype high speed marine vehicle is presented comparisons between the results obtained by the stochastic perturbation method spm and those provided by standard monte carlo simulations mcs show the accuracy of the proposed method and a useful saving in computational time a method is given for comparing the computational time required by the two methods for a given statistical accuracy

hybrid deterministic statistical analysis of vibro acoustic systems with domain couplings on statistical components abstract recently a hybrid analysis method has been developed in which the various components of a complex vibro acoustic system can be modeled either deterministically or statistically the coupling between these two types of component is effected by using a diffuse field reciprocity relation which relates the cross spectrum of the forces at the boundary of a statistical component to the vibrational energy level of the component in practical applications components may be coupled over a domain rather than just at the component boundaries for example the coupling between a plate and an acoustic volume involves the surface of the plate it is shown in the present work that the reciprocity relation and the hybrid method can be extended to this situation so that for example it is possible to couple a statistical model of a plate to a finite element model of an acoustical cavity or to a statistical acoustical cavity the coupling of statistical acoustical and structural components is not new but rather is done routinely within statistical energy analysis sea this part of the present work demonstrates that the relevant coupling loss factors can be found by using the hybrid method the analysis is illustrated by various numerical examples

the effect of generalised force correlations on the response statistics of a harmonically driven random system if the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the gaussian orthogonal ensemble goe of random matrices previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation the derivation of these expressions employed random point process theory and in the theoretical development it was assumed that the modal generalised forces were uncorrelated although this assumption is often valid there are cases in which correlations between the generalised forces can significantly affect the response variance and in the present work the existing theory is extended to include correlations of this type the extended theory is applicable to both single frequency responses and to band average responses and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure

ensemble energy average and energy flow relationships for nonstationary vibrating systems abstract this paper attempts to introduce a new point of view on energy analysis in structural dynamics with particular emphasis to its link with uncertainty and complexity a linear elastic system undergoing free vibrations is considered the system is subdivided into two subsystems and their respective energies together with the shared energy flow are analysed first the ensemble energy average of the two subsystems assuming uncertain the natural frequencies is investigated it is shown how the energy averages follow a simple law when observing the long term response of the system obtained by a suitable asymptotic expansion the second part of the analysis shows how the ensemble energy average of a set of random samples is representative even of the single case if the system is complex enough the two previous points combined produce a result that applies to the energy sharing between two subsystems even independently of uncertainty for complex systems a simple energy sharing law is indeed stated moreover in the case of absence of damping a nonlinear relation between the energy flow and the energy weighted difference between the two subsystems is derived on the other hand when damping is present this relationship becomes linear including two terms one is proportional to the energy weighted difference between the two subsystems the other being proportional to its time derivative therefore the approach suggests a way for deriving a general approach to energy sharing in vibration with results that in some cases are reminiscent of those met in statistical energy analysis finally computational experiments performed on systems of increasing complexity validate the theoretical results

on the applicability of the lognormal distribution in random dynamical systems this paper is concerned with the probability density function of the energy of a random dynamical system subjected to harmonic excitation it is shown that if the natural frequencies and mode shapes of the system conform to the gaussian orthogonal ensemble then under common types of loading the distribution of the energy of the response is approximately lognormal providing the modal overlap factor is high typically greater than two in contrast it is shown that the response of a system with poisson natural frequencies is not approximately lognormal numerical simulations are conducted on a plate system to validate the theoretical findings and good agreement is obtained simulations are also conducted on a system made from two plates connected with rotational springs to demonstrate that the theoretical findings can be extended to a built up system the work provides a theoretical justification of the commonly used empirical practice of assuming that the energy response of a random system is lognormal

shuffles of copulas we show that every copula that is a shuffle of min is a special push forward of the doubly stochastic measure induced by the copula m this fact allows to generalize the notion of shuffle by replacing the measure induced by m with an arbitrary doubly stochastic measure and hence the copula m by any copula c

random point excitation of coupled beams numerical results are presented for the power flow between two coupled beams one of which is excited at a point by a random force frequency spatial over the excitation point and ensemble by varying the relative lengths of the beams averages are compared with each other and with the results from simple statistical models such as sea the mean values agree well and the standard deviations are surprisingly small

the variational theory of complex rays an answer to the resolution of mid frequency 3d engineering problems the variational theory of complex rays vtrc is an approach to the simulation of mid frequency phenomena whose wavelengths are relatively small compared to the dimensions of the domain this is a wave based computational technique which involves a nonclassical variational formulation this paper focuses on the development of the approach for 3d engineering problems and shows that it is a mature technique illustrations are given to show the capabilities of this method

confinement of vibration by one dimensional disorder ii a numerical experiment on different ensemble averages abstract when producing theoretical estimates of the effect of irregularity on the vibrational behaviour of a structure as has been described previously 1 3 one aims to predict the behaviour of a typical irregular structure within the set considered it might be thought that this would correspond to the ensemble averaged behaviour over the whole set but this need not be the case in fact there are several different possible ways of taking the ensemble average which can produce remarkably different results some of these do indeed correspond to the typical member of the ensemble but some of them certainly do not since rare configurations giving very large contributions to the average distort the answer this problem is investigated here via a numerical experiment on a very simple system the object is to complement the discussion in a companion paper 4 which gives theoretical analysis the aims here are threefold to know which average is the most appropriate in a given context to understand why the other averages are different and to identify the anomalous configurations making them different so that these can be avoided in designing real structures a by product is a better understanding of the averaging procedures used in the past it has not always been recognized that these various averages can give radically different answers and one can find instances where what was used was perhaps the easiest approach rather than the most appropriate one in addition the numerical experiment described here gives a good illustration of the basic phenomenon of anderson localization in an acoustic context

structurally nonlinear fluttering airfoil in turbulent flow the dynamics and response of a structurally nonlinear two dimensional airfoil in turbulent flow are investigated numerically the solution is analyzed in terms of probability density function pdf response mean square power spectral density and lyapunov exponent both the longitudinal and vertical components of turbulence are considered it is shown that turbulent flow due to its longitudinal component diminishes the stability of the aeroelastic system by advancing the flutter point and decreasing the damping thus confirming results from previous work where only the longitudinal component was modeled furthermore the shift in flutter airspeed is strongly dependent on the random stiffness terms it is also observed that turbulence may advance the airspeed at which the pitch angle marginal pdf changes from uni to bimodal compared with the nonexcited deterministic case whereas the heave transition appears to be postponed this observation is attributed to the nonlinearity considered a cubic stiffness in torsion finally in terms of response mean square the system appears to he more sensitive to the presence of the longitudinal excitation component at pre rather than at postinstability airspeeds

insights into the energy equipartition principle in large undamped structures the question of energy sharing among complex engineering vibrating systems is still an open problem on the basis of some recent investigations this paper is addressed to the prediction of the equilibrium energies of interacting conservative resonators and to a better understanding of the principle of energy equipartition for large undamped systems the analysis explores the field of linear and nonlinear vibrations the effects of inhomogeneity the weak or the strong coupling as well as the effect of the initial energy distribution among the subsystems the principle of energy equipartition is shown to react in a different fashion to these factors some general arguments supported by the results of numerical experiments enlightening promoting or inhibiting factors to the reaching of energy equipartition conditions are given

classical wave experiments on chaotic scattering we review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems scattering theory has to be applied for a quantitative interpretation of the measurements most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation in all studied examples a quantitative agreement between experiment and theory is achieved to this end it is necessary however to take absorption and imperfect coupling into account concepts that were ignored in most previous theoretical investigations classical phase space signatures of scattering are being examined in a small number of experiments

accurate extreme value based frequency response bounding for structures with a small number of highly random parameters abstract a modified extreme value based methodology is discussed for computing statistical bounds associated with the magnitude of the frequency response of a specified number of structures with high levels of random parameter uncertainty the methodology intended for small numbers of uncertain parameters is capable of constructing accurate statistical bounds in terms of quantiles associated with the extreme value distribution quantiles can be constructed for an ensemble of structural responses across the entire frequency range without using monte carlo simulation to test the methodology statistical bounds for the energy of an l shaped structure with low and high levels of uniformly distributed length and thickness variability are obtained i via direct integration using an ansys finite element model and ii via statistical energy analysis sea comparisons are shown with bounds obtained using monte carlo simulation the merit of direct integration for computing bounds associated with the responses of an ensemble of structures with high levels of random parameter uncertainty is demonstrated by its simplicity high accuracy and absence of statistical scatter

a note on the copulas invariant with respect to a b transformation abstract this paper answers the problem of copulas invariant with respect to a b transformations recently proposed by horanska and kolesarova 2017 8 using tools from ergodic theory and iterated function systems we prove that the conjecture formulated therein is not true in general and for every a b 0 1 2 we construct a one parametric class of nontrivial copulas invariant with respect to a b transformation

energy flow models from finite element analysis computationally efficient methods are described by which the results of a finite element analysis of a system may be post processed to form energy flow models yielding time and perhaps frequency average subsystem energies as well as input and dissipated powers the methods are particularly efficient for excitation which is spatially distributed or broadband e g rain on the roof or if the frequency average response is required first a method based on a global finite element analysis is presented this involves a global modal decomposition and a reordering of the subsequent numerical calculations the properties of the distribution of the excitation and the system s mass and stiffness lead to subsystem force distribution mass distribution and stiffness distribution matrices the response is given by a sum of terms involving the interaction of a pair of global modes the contribution of each pair depending on the appropriate elements of the distribution matrices frequency averaging is performed by separating the resulting frequency dependent terms and integrating in most practical cases this integration can be done analytically next an alternative method involving component mode synthesis is described in this individual finite element analyses are performed for each subsystem using here the fixed interface method these are then assembled to perform a global modal analysis with the order of the model being much reduced the consequent results are then post processed in the same way finally a system comprising three coupled plates is presented as a numerical example

a general derivation of the statistical energy analysis equations for coupled dynamic systems abstract the response of a multi coupled dynamic system to random excitation is considered with the aim of providing a theoretical derivation of the statistical energy analysis sea approach to vibration prediction it is shown that the general form of the principal sea equation is widely applicable provided that the coupling between the subsystems is conservative and a suitable definition of the subsystem energy is used under the assumption of weak coupling the standard sea equations are recovered expressions for the coupling loss factors are obtained in terms of the frequency and space averaged green functions of the coupled system

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

an efficient probabilistic approach to vibro acoustic analysis based on the gaussian orthogonal ensemble vibro acoustic analysis of complex systems at higher frequencies faces two challenges how to compute the response without using an excessive number of degrees of freedom dofs and how to quantify the uncertainty of the response due to small spatial variations in geometry material properties and boundary conditions which have a wave scattering effect in this study a general method of analysis is presented that provides an answer to both questions while overcoming most limitations of statistical energy analysis the fundamental idea is to numerically compute an artificial ensemble of realizations for the components of the built up system that are highly sensitive to small random wave scatterers this can be efficiently performed because their eigenvalue spacings and mode shapes conform to gaussian orthogonal ensemble spacings and gaussian random fields respectively the dofs of the overall system are therefore limited to those of the deterministic components and the interface dofs of the random components the method is extensively validated by application to plate structures good agreement between the predicted response probability distributions and the results of detailed parametric probabilistic models is obtained also for cases of low modal overlap single point loading and strong subsystem coupling

power flow between two continuous one dimensional subsystems a wave solution abstract the power flow between two point coupled wave bearing subsystems is determined by using a wave approach the subsystems are one dimensional in that each supports just one wave mode and are described by a reflection coefficient from which in certain circumstances the modal overlap and modal density can be defined as continuous functions of frequency the coupling element is described by a reflection coefficient and attention is focused on the case of conservative coupling time harmonic point forces or rain on the roof excitations may act on the subsystems uncertainty exists in the properties of the subsystems which are assumed to be drawn from ensembles of similar subsystems the exact power flow and the ensemble statistics are seen to depend on the magnitudes of the subsystem and coupling reflection coefficients for some particular cases the maximum and minimum power flows are examined together with the ensemble mean and geometric mean the cumulative probability distribution and the probability density function for a given power incident upon the coupling the ensemble mean power flow through the coupling is always less than that expected from a normal sea wave approach due to power re radiation the differences are largest when the damping is light and the coupling strong for a given level of excitation power re radiation effects are also apparent for the ensemble mean power flow from an infinite subsystem to a finite subsystem in the opposite case the ensemble mean power flow from a finite to an infinite subsystem is greater than that between two infinite subsystems due to power re injection the differences are largest for light damping and for weak coupling finally the case where there is uncertainty in one subsystem only is considered

an efficient wave based prediction technique for plate bending vibrations the finite element method is a commonly used prediction method for dynamic simulations of plate bending problems a major disadvantage of this method is its practical frequency limitation as its application involves prohibitively large computational efforts at higher frequencies a newly developed wave based prediction technique pushes up the frequency limit through an enhanced computational efficiency this paper discusses the principles of the wave based method for convex plate problems the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples which include the most commonly encountered boundary conditions

mixed function method for obtaining exact solution of nonlinear differential difference equations and coupled nddes in this paper we construct a new mixed function method for the first time by using this new method we study the two nonlinear differential difference equations named the generalized hybrid lattice and two component volterra lattice equations some new exact solutions of mixed function type such as discrete solitary wave solutions discrete kink and anti kink wave solutions and discrete breather solutions with kink and anti kink character are obtained and their dynamic properties are also discussed by using software mathematica we show their profiles

some results on homeomorphisms between fractal supports of copulas abstract we consider parametric classes t r r 0 1 2 of so called transformation matrices and their induced families a r r 0 1 2 and r r 0 1 2 of two dimensional copulas and doubly stochastic measures with fractal support respectively by using tools from symbolic dynamics we show that for each pair r r 0 1 2 with r r there exists a homeomorphism h r r between the supports of r and r mapping a borel set of r measure one to a set of r measure zero differentiability properties of these homeomorphisms are studied and hausdorff dimensions of related sets are calculated several examples and graphics illustrate the main results

an algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations an algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations compared with most existing tanh methods the jacobi function expansion method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters the solutions obtained in this paper include a polynomial solutions b exponential solutions c rational solutions d triangular periodic wave solutions e hyperbolic and solitary wave solutions and f jacobi and weierstrass doubly periodic wave solutions the efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper new 2 1 dimensional calogero kdv equation 3 1 dimensional jimbo miwa equation symmetric regular long wave equation drinfel d sokolov wilson equation 2 1 dimensional generalized dispersive long wave equation double sine gordon equation calogero degasperis fokas equation and coupled schrodinger boussinesq equation in addition the links among our proposed method the tanh method the extended method and the jacobi function expansion method are also clarified generally

an approach to the theoretical background of statistical energy analysis applied to structural vibration rayleigh s classical approach to the study of vibration of systems having a finite number of degrees of freedom is applied to the problem of coupling of subsystems in a complicated structure in order to probe the regions of applicability of the approach to vibration analysis usually known as statistical energy analysis sea the classical method has advantages of simplicity and rigor over previous approaches to the background of sea in certain cases and provides extensions and simplifications in several areas of the theory it also suggests modifications to sea modeling strategy depending on the type of coupling involved even when that coupling is weak so that earlier analyses might be thought to apply

statistical energy analysis and structural fuzzy the concepts and derivations of the effects of a minor or secondary structure on the dynamics of a well defined major or primary structure have been recently presented as a fuzzy structure interacting with a deterministic one on the face of it this appears like a halfway sea system in that one of the two systems is well defined while the other is probabilistic in the sea sense it is therefore interesting to see if a full sea treatment of the system would lead to difference in at least some of the quoted results the second and more important purpose of this paper is to relate the dynamics of potential minor structures to the properties of the fuzzy system since these properties have been presented in some of the fuzzy literature in rather general terms

on a strong metric on the space of copulas and its induced dependence measure abstract using the one to one correspondence between copulas and markov operators on l 1 0 1 and expressing the markov operators in terms of regular conditional distributions markov kernels allows to define a metric d 1 on the space of copulas c that is a metrization of the strong operator topology of the corresponding markov operators it is shown that the resulting metric space c d 1 is complete and separable and that the induced dependence measure 1 defined as a scalar times the d 1 distance to the product copula has various good properties in particular the class of copulas that have maximum d 1 distance to the product copula is exactly the class of completely dependent copulas i e copulas induced by lebesgue measure preserving transformations on 0 1 hence in contrast to the uniform distance d cannot be approximated arbitrarily well by completely dependent copulas with respect to d 1 the interrelation between d 1 and the so called convergence by mikusinski and taylor as well as the interrelation between 1 and the mutual dependence measure by siburg and stoimenov is analyzed 1 is calculated for some well known parametric families of copulas and an application to singular copulas induced by certain iterated functions systems is given

uncertainty propagation in sea for structural acoustic coupled systems with non deterministic parameters abstract considering limited available information on uncertainties in structural acoustic coupled systems two methods namely the vertex method and the legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the statistical energy analysis sea approach for the vertex method an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded which is well suited for an interval input vector in arbitrary dimension instead of calculating the extremum of the response of interest a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the legendre orthogonal polynomial approximation following the theoretical analysis of the accuracy and efficiency of the proposed methods their validation is performed by one numerical example and two applications

comparison of statistical and interval analysis for the energy flow uncertainties in structural vibrating systems in this paper the energy flow confidence between two structural multimodal systems coupled by a joint with uncertain parameters is computed by two different methods the first one assumes that the joint parameters perturbed randomly the statistical moments of the energy flow are calculated by an analytical procedure the second one uses interval analysis the joint parameters are considered interval variables and the interval of the energy flow is determined the properties of the statistical and interval solution are investigated and compared

a model and numerical method in the medium frequency range for vibroacoustic predictions using the theory of structural fuzzy in linear dynamical analysis of complex mechanical systems the structural fuzzy is defined as the set of minor subsystems that are connected to the master structure but are not accessible by classical modeling the notion of master structure is presently extended to others elements such as an external dense compressible fluid strongly coupled with the primary structure for the low frequency lf dynamical analysis the modeling of the structural fuzzy is commonly made with a system of masses if the lf modeling of the structural fuzzy is applied in the medium frequency mf domain there are some large differences between calculations and experiment it is therefore necessary to take into account internal degrees of freedom of the structural fuzzy a global probabilistic modeling of the structural fuzzy is proposed to improve the calculated estimates of the mf vibrations into the master structure and of the far field radiated by itself in or out of context of the acoustic scattering this paper reviews the author s previous work and introduces the type ii probabilistic constitutive law in this paper 1 a probabilistic modeling of the structural fuzzy is presented 2 two probabilistic constitutive laws of the structural fuzzy are constructed 3 the modeling in the mf range for vibroacoustic predictions using theory of structural fuzzy is developed 4 the fuzzy solution in the mf range is studied and 5 numerical simulations on standard structures and on a submerged complex industrial structure are shown

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

a quantitative criterion validating coupling power proportionality in statistical energy analysis the response of two general spring coupled elements is investigated to develop a unifying approach to the weak coupling criterion in statistical energy analysis sea first the coupled determinis

response variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea of high frequency noise and vibration a complex engineering structure is represented as an assembly of subsystems the response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem and these energies are found by employing the principle of power balance strictly the computed energy is an average taken over an ensemble of random structures and for many years there has been interest in extending the sea prediction to the variance of the energy a variance prediction method for a general built up structure is presented here closed form expressions for the variance are obtained in terms of the standard sea parameters and an additional set of parameters alpha k that describe the nature of the power input to each subsystem k and alpha ks that describe the nature of the coupling between subsystems k and s the theory is validated by comparison with monte carlo simulations of plate networks and structural acoustic systems

gaussian orthogonal ensemble spacing statistics and the statistical overlap factor applied to dynamic systems abstract this paper examines the extent to which gaussian orthogonal ensemble goe statistics is applicable to the natural frequencies of a dynamic system the natural frequencies of a simply supported plate or a rectangular room tend to have an exponential spacing distribution however any disruption of the system symmetries has been shown to promote goe statistics for which the modal spacing distribution is rayleigh in this paper the effect of a range of uncertainties on the modal statistics of structures is numerically characterised this is achieved by examining the modal statistics of mass and or spring loaded plates and plates coupled by springs the natural frequencies of the aforementioned structures have been derived using the lagrange rayleigh ritz technique the degree of uncertainty required to effect the transition from an exponential to a rayleigh distribution and to achieve universality of the statistical properties is investigated a further measure of the randomness required to produce goe statistics can be obtained by examining the amount of mixing and veering between the modes of a dynamic system the statistical overlap factor is a non dimensional parameter related to the random variation in an individual natural frequency from its mean value and is useful to quantify the frequency beyond which the resonant behaviour of individual modes no longer dominates the response statistics using a first order perturbation analysis an approximate expression for the statistical overlap factor has been developed for the randomised plates to estimate the modal range for the occurrence of goe statistics

the statistics of power flow between two continuous one dimensional subsystems abstract the power flow between two continuous subsystems is examined using a wave solution 1 one of the subsystems is excited by time harmonic forces and both subsystems can support only one power transmitting wave mode uncertainty exists in the subsystems which are assumed to be drawn at random from ensembles those ensembles members that have a natural frequency at the excitation frequency are determined the particular members that are resonant are seen to depend on the strength of coupling the statistics of power flow are considered in particular the maximum minimum ensemble arithmetic and geometric means the variance and the cumulative probability distribution of the power flow the maximum power flow undergoes a qualitative change at a particular coupling strength for weak coupling the peak power flows occur at the uncoupled natural frequencies while for stronger coupling they occur at the coupled frequencies the ensemble average power flow may be significantly less than that expected from a normal sea approach due to power re radiation especially for a lightly damped receiving subsystem and strong coupling conversely it may be much greater due to power re injection which occurs particularly for a lightly damped excited subsystem and weak coupling the power flow variance is largest for small damping and weak coupling under which circumstances the cumulative probability distribution has an infrequent but dominant tail of very large power flows

random matrix theory and non parametric model of random uncertainties in vibration analysis recently a new approach called a non parametric model of random uncertainties has been introduced for modelling random uncertainties in linear and non linear elastodynamics in the low frequency range this non parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive definite real random matrices this ensemble differs from the gaussian orthogonal ensemble goe and from the other known ensembles of the random matrix theory the present paper has three main objectives the first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the goe the second one is to compare this new ensemble of random matrices with the goe in the context of the non parametric approach of random uncertainties in structural dynamics for the low frequency range the last objective is to give a new validation for the non parametric model of random uncertainties in structural dynamics in comparing in the low frequency range the dynamical response of a simple system having random uncertainties modelled by the parametric and the non parametric methods these three objectives will allow us to conclude about the validity of the different theories

exact travelling wave solutions of the discrete sine gordon equation obtained via the exp function method abstract in this paper we generalize the exp function method which was used to find new exact travelling wave solutions of nonlinear partial differential equations npdes or coupled nonlinear partial differential equations to nonlinear differential difference equations nddes as an illustration two series of exact travelling wave solutions of the discrete sine gordon equation are obtained by means of the exp function method as some special examples these new exact travelling wave solutions can degenerate into the kink type solitary wave solutions reported in the open literature

a derivation of the coupling loss factors used in statistical energy analysis abstract the conditions under which an exact theory concerning the energy power flow relationships for a complex dynamic system may be reduced to the standard statistical energy analysis sea equations are considered a definition of weak coupling is introduced which encompasses a number of previous definitions and it is shown that the presence of weak coupling does not guarantee the validity of the standard sea assumption that the coupling loss factor clf between two subsystems is zero unless the subsystems are directly connected the standard assumption is likely to be reasonable for systems which are both weakly coupled and reverberant attention is then turned to the clf between two connected subsystems and a general result is obtained which involves a space and frequency averaged green function it is shown that under the appropriate assumptions a number of existing results may be derived from this expression including those produced by both the wave and the modal approaches to sea the exact theory may be used as a basis for developing improved models of the system dynamics

statistical energy analysis with fuzzy parameters to handle populations of structures statistical modal energy distribution analysis smeda approach was developed to enlarge the application field of statistical energy analysis sea when equipartition of modal energies is not achieved smeda gives more precise results than standard sea when compared to exact energy response of a deterministic system in the case of low modal overlap heterogeneous systems or point excitation the present paper was initiated by this question when considering a population of similar structures each of them being described by smeda do the ensemble averaged energies of subsystem and injected power tend to satisfy sea equations in other terms despite the non equipartition of energy observed on each element of the population of structures does the ensemble averaging leads to sea equation where equipartition of energy is assumed the response to that question that rises from this paper is yes if the terms of the sea equation are fuzzy numbers it results that the energy response given by the model can be interpreted using fuzzy numbers theory

the level crossing rates and associated statistical properties of a random frequency response function abstract this work is concerned with the statistical properties of the frequency response function of the energy of a random system earlier studies have considered the statistical distribution of the function at a single frequency or alternatively the statistics of a band average of the function in contrast the present analysis considers the statistical fluctuations over a frequency band and results are obtained for the mean rate at which the function crosses a specified level or equivalently the average number of times the level is crossed within the band results are also obtained for the probability of crossing a specified level at least once the mean rate of occurrence of peaks and the mean trough to peak height the analysis is based on the assumption that the natural frequencies and mode shapes of the system have statistical properties that are governed by the gaussian orthogonal ensemble goe and the validity of this assumption is demonstrated by comparison with numerical simulations for a random plate the work has application to the assessment of the performance of dynamic systems that are sensitive to random imperfections

transient energy exchange between a primary structure and a set of oscillators return time and apparent damping in this paper we examine the conditions that influence the return time the time it takes before energy returns from a set of satellite oscillators attached to a primary structure two methods are presented to estimate the return time one estimate is based on an analysis of the reaction force on a rigid base by a finite number of oscillators as compared with an infinite number of continuously distributed oscillators the result gives a lower bound estimate for the return time a more accurate estimation results from considering the dynamic behavior of a set of oscillators as waves in a waveguide such an analogy explains energy flow between a primary structure and the oscillators in terms of pseudowaves and shows that a nonlinear frequency distribution of the oscillators leads to pseudodispersive waves the resulting approximate expressions show the influence of the natural frequency distribution within the set of oscillators and of their number on the return time as compared with the asymptotic case of a continuous set with infinite oscillators in the paper we also introduce a new method based on a hilbert envelope to estimate the apparent damping loss factor of the primary structure during the return time considering transient energy flow from the primary structure before any energy reflects back from the attached oscillators the expressions developed for return time and damping factor show close agreement with direct numerical simulations the paper concludes with a discussion of the return time and its relation to apparent damping and optimum frequency distribution within a set of oscillators that maximize these quantities

wave component analysis of energy flow in complex structures part ii ensemble statistics a wave based method is presented for the analysis of high frequency vibrations in complex structures the response of the structure to external forcing is described in terms of generalised energy bearing wave components and the structure is represented by global subsystem and junction wave component scattering matrices s and t uncertainty in the properties of the structure is taken into account by assuming that the structure is drawn from an ensemble of structures that differ randomly in detail a scalar random phase ensemble is defined in terms of random eigenvalues of the product st of the scattering matrices and analytical expressions are derived for the average and variance of the energy responses over this ensemble the scalar random phase ensemble is thought to be a reasonable approximation to many practical ensembles and the approach provides a means for estimating response statistics at relatively low computational cost

copulas and associated fractal sets abstract the problem of constructing copulas whose supports are fractals has been studied by fredricks nelsen and rodriguez lallena g a fredricks r b nelsen j a rodriguez lallena copulas with fractal supports insurance math econom 37 1 2005 42 48 in this paper we continue on the path traced by these authors we provide different types of families of self similar copulas using techniques from probability and ergodic theory to give properties on subsets of their fractal supports in particular we give new examples for those copulas and we analyze related topics with mutual singularity of the associated measures hausdorff dimension and the connectedness of their supports

the sine gordon equation on time scales abstract we formulate and discuss integrable analogue of the sine gordon equation on arbitrary time scales this unification contains the sine gordon equation discrete sine gordon equation and the hirota equation doubly discrete sine gordon equation as special cases we present the lax pair check compatibility conditions and construct the darboux backlund transformation finally we obtain a soliton solution on arbitrary time scale the solution is expressed by the so called cayley exponential function

on the statistical properties of random causal frequency response functions abstract in nuclear physics it is known that under broad restrictions a random scattering matrix element h satisfies a condition known as the analyticity ergodicity ae requirement which states that e f h f e h where f is some function of h and e represents the ensemble average a scattering matrix element is directly analogous to a vibrational frequency response function and it is of significant interest to consider whether the ae requirement is also applicable to random engineering systems the proof of the ae condition in nuclear physics rests on the assumptions that h is causal and ergodic causality implies that a lorentzian frequency average satisfies the ae equation and ergodicity implies that lorentzian frequency averages are equal to ensemble averages in vibrational systems it is readily shown that a typical frequency response function is non stationary and non ergodic so that the lorentzian and ensemble averages can differ significantly and this means that the standard proof of the ae requirement breaks down the question then arises as to whether the ae requirement might nonetheless apply to vibrational systems it is shown in the present paper that the requirement does apply providing that the random point process representing the system natural frequencies is at least locally stationary which is a much weaker condition than local stationarity of the frequency response function and a number of the implications of this result are explored

moments and associated measures of copulas with fractal support copulas are closely related to the study of distributions and the dependence between random variables in this paper we develop a recurrence formula for the moments of a measure associated with a copula a bivariate distribution function with uniform one dimensional marginals in the case that its support is a fractal set we do the same for its principal and secondary diagonals we also study certain measures of dependence or association for these copulas with fractal supports

numerical and experimental validation of variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea approach to vibration modeling a complex system is represented as an assembly of coupled subsystems and the method leads to the prediction of the vibrational energy level of each subsystem the averaging procedures implicit in the technique imply that the predicted energy is the mean value taken over an ensemble of random structures such as a set of vehicles leaving a production line recently a new method has been developed to allow the ensemble variance in addition to the mean to be predicted within the context of sea and the present paper concerns further extension and validation of this work the theoretical extension concerns the variance of the energy density at a single point in any of the subsystems and the validation includes both simulation and experimental studies the simulation results concern plate assemblies while experimental results are presented both for a single plate and for a cylinder plate structure in each case an ensemble of random structures has been generated by adding small point masses at random locations on the structure in general good agreement between the predictions and the validation results is observed

wave component analysis of energy flow in complex structures part iii two coupled plates abstract a statistical wave based approach to the analysis of energy flow in structures is applied to example structures comprising two regularly or irregularly coupled rectangular plates the approach which is developed in two companion papers is based on the expression of the response in terms of energy bearing wave components and a description of the structure in terms of subsystem and junction wave component scattering matrices s and t uncertainty in the properties of the structure is taken into account by assuming that the structure is drawn from an ensemble of structures which vary randomly in detail a scalar random phase ensemble is defined in terms of a random distribution of the eigenvalues of the global scattering matrix product st analytical expressions enable the ensemble mean and variance of energy responses over this ensemble to be found at low computational cost scalar random phase ensemble based estimates of these statistics for regularly and irregularly coupled plate structures are found to be in good agreement with the results of monte carlo simulations

response variance prediction for uncertain vibro acoustic systems using a hybrid deterministic statistical method imperfections during the manufacturing process can cause significant variations in the noise and vibration levels exhibited by nominally identical structures any response calculations employed during the design process should ideally take account of these uncertainties and predict the expected range in performance recently a hybrid method has been developed to predict the ensemble average response of a built up system by combining a deterministic model of parts of the system with a statistical model of other components shorter p j and langley r s 2005 j sound vib 288 669 700 in this paper the method is extended to predict the ensemble variance of the response expressions are derived for the variance of the vibrational energies in the statistical components and for the variance of the cross spectrum of the response of the deterministic components which augment the mean values of these quantities predicted by the original theory the method employs a nonparametric model of uncertainty in the sense that the statistical components are taken to carry diffuse wave fields and this obviates the requirement for a detailed description of the system uncertainties the method is validated by application to a range of coupled plate structures and good agreement with detailed monte carlo simulations is found

on the diffuse field reciprocity relationship and vibrational energy variance in a random subsystem at high frequencies a recent paper has shown that under certain conditions the cross spectral matrix of the forces exerted by a vibrational or acoustic wave field on its surrounding boundaries can be expressed in terms of i the energy of the wave field and ii the direct field dynamic stiffness matrix of the boundary this diffuse field reciprocity relation was derived using wave mechanics and it is not immediately clear how the required wave field properties translate to conditions on the vibrational modes of the system or the applied forcing this issue is addressed here by deriving an extended version of the reciprocity relation using modal methods and the conditions required for the extended version to reduce to the existing relation are delineated it is shown that the existing diffuse field reciprocity relation leads to an anomalous result when used to predict the energy variance of a subsystem and that this anomaly is resolved by using the present extended version of the relation a supplementary result arising from the analysis is that for systems with a sufficient degree of randomness the ensemble average of the dynamic stiffness matrix of a random subsystem is equal to the inverse of the ensemble average of the receptance matrix

resonant beam vibration a wave evolution analysis abstract in an attempt to verify the concept of wave mode duality and to understand the wave train closure principle in resonant beam vibration an elastically supported beam subjected to a swept sinusoidal force is investigated through a simulation of wave evolution from initial to final steady state response wave evolution is simulated in the superposition and the degenerate states according to the nature of wave reflection at the boundaries of the beam from the results resonance condition is established and the natural frequencies and mode shapes are calculated using a simply supported beam and a cantilever beam as examples the concept of wave mode duality at resonance is verified an examination of the wave train closure principle shows that although the frequency equations obtained by using the principle are identical to those derived from the wave evolution results the principle does not describe the realistic process of the formulation of vibration modes in general the waves described in the wave train closure principle are not physical but virtual waves that represent final steady state waves which can only be asymptotically approached in reality the principle can be used as a convenient technique to obtain vibration modes from the point of view of wave propagation

copulas with continuous strictly increasing singular conditional distribution functions abstract using iterated function systems induced by so called modifiable transformation matrices t and tools from symbolic dynamical systems we first construct mutually singular copulas a t with identical possibly fractal or full support that are at the same time singular with respect to the lebesgue measure 2 on 0 1 2 afterwards the established results are utilized for a simple proof of the existence of singular copulas a t with full support for which all conditional distribution functions y f x a t y are continuous strictly increasing and have derivative zero almost everywhere this result underlines the fact that conditional distribution functions of copulas may exhibit surprisingly irregular analytic behavior finally we extend the notion of empirical copula to the case of non i i d data and prove uniform convergence of the empirical copula e n corresponding to almost all orbits of a markov process usually referred to as chaos game to the singular copula a t several examples and graphics illustrate both the chosen approach and the main results

validity diagrams of statistical energy analysis abstract this paper is concerned with the validity domain of statistical energy analysis sea which is defined in terms of four criteria the mode count n and the modal overlap m must be high the normalized attenuation factor m and the coupling strength must be small the application of dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of sea must be delimited this domain is discussed on the basis of geometry of the surfaces delimiting it the diagrams of validity of sea are introduced and discussed a numerical simulation on a couple of rectangular plates coupled along one edge illustrates the theoretical approach

statistical energy analysis modelling of complex structures as coupled sets of oscillators ensemble mean and variance of energy expressions are derived for the ensemble means and variances of the subsystem energies of built up systems comprising two subsystems the approach is based on the statistical energy analysis of two spring coupled oscillators and sets of oscillators or coupled continuous subsystems described by mace and ji the statistical energy analysis of coupled sets of oscillators proceedings of the royal society a 1824 2007 the paper focuses on spring coupling although similar results hold for more general forms of conservative coupling randomness is introduced into the system by assuming that the natural frequency spacings in each subsystem conform to certain statistical distributions a coupling coefficient parameter is introduced which together with the coupling strength parameter defined by mace and ji 2007 accounts for the statistics of the coupling stiffness various approximations and assumptions are made it is seen that the variance of the excited subsystem depends primarily on the variance of the input power which in turn depends on the variance of the number of modes of the excited subsystem in the frequency band of excitation and their mode shapes the variance of the undriven subsystem on the other hand depends primarily on the variance of the intermodal coupling coefficients which in turn depend on the variances of the number of in band modes of both subsystems and their mode shapes the cases of poisson and gaussian orthogonal ensemble natural frequency spacing statistics are considered numerical examples of two plates coupled by one or a number of springs are presented

analytical solution for the power exchange between strongly coupled plates under random excitation a test of statistical energy analysis concepts the response of the structural components of a complex structure subjected to random loads is often estimated by approximate methods based on the concepts of statistical energy analysis sea to examine the fundamental sea postulates and assumptions the problem of two coupped plates is rigorously formulated and solved in terms of usual sea variables for randomly distributed loads analytical results are presented for the power flow and plate vibrational energies the formulation allows the extraction of apparent quantifies that correspond to the coupling loss factor and modal density ratio of statistical energy analysis it is demonstrated that this model can yield an understanding of the implications of basic sea assumptions results for simply supported plates clearly show the effects of internal damping and frequency on the coupling parameters

statistical energy analysis of coupled plate systems with low modal density and low modal overlap abstract finite element methods experimental statistical energy analysis esea and monte carlo methods have been used to determine coupling loss factors for use in statistical energy analysis sea the aim was to use the concept of an esea ensemble to facilitate the use of sea with plate subsystems that have low modal density and low modal overlap an advantage of the esea ensemble approach was that when the matrix inversion failed for a single deterministic analysis the majority of ensemble members did not encounter problems failure of the matrix inversion for a single deterministic analysis may incorrectly lead to the conclusion that sea is not appropriate however when the majority of the esea ensemble members have positive coupling loss factors this provides sufficient motivation to attempt an sea model the ensembles were created using the normal distribution to introduce variation into the plate dimensions for plate systems with low modal density and low modal overlap it was found that the resulting probability distribution function for the linear coupling loss factor could be considered as lognormal this allowed statistical confidence limits to be determined for the coupling loss factor the sea permutation method was then used to calculate the expected range of the response using these confidence limits in the sea matrix solution for plate systems with low modal density and low modal overlap relatively small variation uncertainty in the physical properties caused large differences in the coupling parameters for this reason a single deterministic analysis is of minimal use therefore the ability to determine both the ensemble average and the expected range with sea is crucial in allowing a robust assessment of vibration transmission between plate systems with low modal density and low modal overlap

a non poisson model for the vibration analysis of uncertain dynamic systems the response of a dynamic system to medium or high frequency excitation can be very sensitive to small changes in the physical properties of the system previous work has investigated this issue by adopting a poisson model of the system natural frequencies other studies have performed numerical simulations that reveal the effect of parameters that cannot be included in the poisson model the present work considers a point process model of the natural frequencies that encompasses the poisson model as a limiting case but also allows the influence of key parameters such as the statistical overlap factor to be considered a number of general results are derived and these are then applied to the special case of gaussian natural frequencies a closed form solution for the variance of a frequency response function is derived that depends upon a single parameter this parameter is a combination of the statistical overlap factor the modal overlap factor and the degree of correlation between the natural frequencies attention is also directed at the statistical distribution of the natural frequency spacing and various results are derived in terms of the cumulant functions of the point process model the analytical results obtained are confirmed by comparison with simulation results