a regularizing lanczos iteration method for underdetermined linear systems this paper is concerned with the solution of underdetermined linear systems of equations with a very ill conditioned matrix a whose dimensions are so large to make solution by direct methods impractical or infeasible image reconstruction from projections often gives rise to such systems in order to facilitate the computation of a meaningful approximate solution we regularize the linear system i e we replace it by a nearby system that is better conditioned the amount of regularization is determined by a regularization parameter its optimal value is in most applications not known a priori we present a new iterative method based on the lanczos algorithm for determining a suitable value of the regularization parameter by the discrepancy principle and an approximate solution of the regularized system of equations

uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one point dirichlet data abstract we consider an inverse problem arising in laser induced thermotherapy a minimally invasive method for cancer treatment in which cancer tissues is destroyed by coagulation for the dosage planning quantitatively reliable numerical simulation are indispensable to this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance mathematically this problem is a nonlinear and nonlocal parabolic inverse heat source problem we show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one point measurement

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

numerical inversions for a nonlinear source term in the heat equation by optimal perturbation algorithm this paper deals with an inverse problem of identifying a nonlinear source term g g u in the heat equation u t u x x a x g u by data compatibility analysis the forward problem is proved to have a unique positive solution with a maximum of m 0 with which an optimal perturbation algorithm is applied to determine the source function g u on u 0 m numerical inversions are carried out for g u with functional forms of polynomial trigonometric and index functions the inversion reconstruction sources basically coincide with the true source solution showing that the optimal perturbation algorithm is efficient to the inverse source problem here by the computations we find that the inversion results are better for polynomial sources than those of trigonometric and index sources the inversion algorithm seems to be very sharp if the solution s maximum m of the forward problem is relatively small otherwise the deviations in the source solutions become large especially near the endpoint of u m

tikhonov regularization and the l curve for large discrete ill posed problems discretization of linear inverse problems generally gives rise to very ill conditioned linear systems of algebraic equations typically the linear systems obtained have to be regularized to make the computation of a meaningful approximate solution possible tikhonov regularization is one of the most popular regularization methods a regularization parameter specifies the amount of regularization and in general an appropriate value of this parameter is not known a priori we review available iterative methods and present new ones for the determination of a suitable value of the regularization parameter by the l curve criterion and the solution of regularized systems of algebraic equations

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

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

application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire abstract the first order harmonic balance method via the first fourier coefficient is used to construct two approximate frequency amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form this system corresponds to the motion of a mass attached to a stretched wire two procedures are used to approximately solve the nonlinear differential equation in the first the differential equation is rewritten in a form that does not contain the square root expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that in the second procedure application of the harmonic balance method produces an infinite set of harmonics while in the first procedure only two harmonics are produced both approximate frequencies are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed the discrepancy between the second approximate frequency and the exact one never exceeds 2 2

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

a computational procedure to estimate the stochastic dynamic response of large non linear fe models abstract an algorithm for the computation of the stochastic non stationary non linear response of large fe models is presented in stochastic analysis the response is described by the mean and covariance function and possibly by the probability distribution of the non linear response to estimate the covariance matrix the method of equivalent statistical linearization is applied for linearizing all non linear elements the large fully populated and symmetric covariance matrix of dimension 2 n is described by the so called karhunen loeve expansion representation which allows one to employ a feasible description in this context an efficient procedure is suggested to determine the minimal number of karhunen loeve vectors necessary to assure a sufficiently accurate representation this method in fact allows one to employ any available deterministic integration scheme to compute the karhunen loeve vectors to increase the computational efficiency modal coordinates are used to represent the linear sub system special attention is given to the effects of mode truncation which are generally not negligible for large fe systems moreover the suggested approach has no limitation w r t the size of the model in terms of degrees of freedom dofs or the number of non linear elements the feasibility of the proposed procedure is demonstrated in the numerical examples where the methodology is applied to an office building with approximately 25 50 and 100 thousand dofs containing hysteretic damping elements

monotonicity and uniqueness results in identifying an unknown coefficient in a nonlinear diffusion equation we consider the problem of identifying an unknown diffusion coefficient in a nonlinear diffusion equation from overspecified data measured on the boundary conditions ensuring compatibility of the data are derived and it is shown that the mapping which carries the diffusion coefficient into the data is isotonic this result is then used to derive a uniqueness result for the inverse problem and for an associated identification problem some results of numerical experiments are presented to illustrate the various results

harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable abstract the second order harmonic balance method is used to construct three approximate frequency amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable two procedures are used to solve the nonlinear differential equation approximately in the first the differential equation is rewritten in a form that does not contain the y 1 expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first one and the discrepancy between the approximate frequency and the exact one is lower than 1 28

a fuzzy finite element procedure for the calculation of uncertain frequency response functions of damped structures part 2 numerical case studies this work introduces a numerical algorithm to calculate frequency response functions of damped finite element models with fuzzy uncertain parameters part 1 of this paper focusses on the numerical procedure for the solution of the underlying interval finite element problem based on the undamped procedure and the principle of rayleigh damping part 2 of this paper illustrates the applicability of the methodology through four case studies the concepts of the interval and the fuzzy finite element frequency response function analysis are illustrated for different types of uncertainties the obtained results are compared with the results of monte carlo simulations

derivation of statistical energy analysis from radiative exchanges abstract this paper is concerned with the derivation of sea equations from structural ray equations rays are assumed to be uncorrelated leading to the additivity of energy inside all subsystems the energy density is the sum of a direct field from driving forces a reflected field from the boundary and a transmitted field from adjacent subsystems assuming a rain on the roof excitation and a compact shape for subsystems actual and fictitious sources on the boundary are found to be constant furthermore if the attenuation of rays during a mean free path normalized attenuation factor is light the field becomes diffuse i e homogeneous and isotropic the net exchanged power between two adjacent subsystems is then proportional to the difference of energy densities and therefore to the difference of modal energies the derived proportionality coefficient is consistent with the well known formula for coupling loss factor in terms of transmission factors these results are illustrated by a numerical simulation for a multi plate system finally the validity domains of sea and the ray theory are discussed and particularly the diffuse field assumption

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

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

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

numerical solution of the inverse problem of determining an unknown source term in a two dimensional heat equation a numerical procedure for an inverse problem of determination of unknown source term in two dimensional parabolic equation is presented the approach of the proposed method is to approximate unknown function by a piecewise linear function whose coefficients are determined from the solution of minimisation problem based on the overspecified data some numerical examples are presented

nonlinear free vibration analysis of composite plates with material uncertainties a monte carlo simulation approach abstract in this paper the effects of dispersion in material properties on free vibration response of composite plates with geometric nonlinearity in von karman sense are investigated the higher order shear deformation theory is employed for the study reported here an efficient c0 finite element formulation is developed for the analysis using monte carlo simulation the second order statistics i e mean and standard deviation of the nonlinear free vibration response of the composite laminates are obtained for different thickness and amplitude ratios the input statistics mean and variance of the material property are assumed to be known as a priori the computed results demonstrate the influence of the variations in the material properties and amplitudes on the nonlinear free vibration response of the composite plates the results are compared with those available in the literature

a method for solving an inverse biharmonic problem this paper deals with the problem of determining of an unknown coefficient in an inverse boundary value problem using a nonconstant overspecified data it has been shown that the solution to this inverse problem exists and is unique

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

source term identification for an axisymmetric inverse heat conduction problem we consider an inverse heat source problem of determining the heat source term from the final temperature history of a cylinder this problem is ill posed a simplified tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one with a logarithmic type error estimate

efficient exceedance probability computation for randomly uncertain nonlinear structures with periodic loading abstract a method is developed to compute low level response amplitude exceedance probabilities associated with uncertain nonlinear structures with random parameters and deterministic periodic forcing emphasis is focused on accurate and efficient computation in the tails of the exceedance probability distribution function associated with the largest possible response of one displacement variable for unspecified forcing frequency and normally distributed parameters this gives a measure of system reliability when a large amplitude response exceedance of a specified threshold is designated as the mode of failure the method exploits the first order reliability method form in which the failure surface is constructed via the harmonic balance method hbm this combined approach is tested on a duffing oscillator with harmonic forcing and up to three uncertain parameters for which the frequency of multiple solution maximum amplitude is found directly and the probability computed via the hasofer lind reliability index the accuracy of the proposed hbm form in the tails of the amplitude exceedance probability is shown for the duffing example to be acceptably accurate whereas the efficiency is shown to be around 1000 times faster than direct integration and around 200 times faster than monte carlo simulation

a comparison of classical and high dimensional harmonic balance approaches for a duffing oscillator the present study focuses on a novel harmonic balance formulation which is much easier to implement than the standard classical harmonic balance method for complex nonlinear mathematical models and algorithms both harmonic balance approaches are applied to duffing s oscillator to demonstrate the advantages and disadvantages of the two approaches a fundamental understanding of the difference between these two methods is achieved and the properties of each method are analyzed in detail

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

on the steady state response and stability of non linear oscillators using harmonic balance abstract the harmonic balance hb method is used to study the steady state periodic response as well as its stability of a softening duffing oscillator the objectives of the work are 1 to show how the qualitative nature of the hb solution changes if one uses two harmonics rather than one in the approximate solution and 2 to show that the analysis of the mathieu equation which governs local stability of the steady state response must be based on an approximate rather than an exact analysis of this mathieu equation thus floquet theory incorrectly describes the stability of the approximate solution

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

a regularizing l curve lanczos method for underdetermined linear systems many real applications give rise to the solution of underdetermined linear systems of equations with a very ill conditioned matrix a whose dimensions are so large as to make solution by direct methods impractical or infeasible image reconstruction from projections is a well known example of such systems in order to facilitate the computation of a meaningful approximate solution we regularize the linear system i e we replace it by a nearby system that is better conditioned the amount of regularization is determined by a regularization parameter its optimal value is in most applications not known a priori a well known method to determine it is given by the l curve approach we present an iterative method based on the lanczos algorithm for inexpensively evaluating an approximation of the points on the l curve and then determine the value of the optimal regularization parameter which lets us compute an approximate solution of the regularized system of equations

stable determination of a semilinear term in a parabolic equation in 5 the flrst and third authors establish holder type stability estimates for the inverse problem consisting in the determination of a semilinear term of a parabolic equation from a single boundary measurement when the domain is a rectangle in the present paper we extend the results in 5 for a general smooth domain

unicity in an inverse problem for an unknown reaction term in a reaction diffusion equation in this paper we consider an inverse problem in which we seek to deter mine an unknown source or reaction term in a reaction diffusion equation from overspecified data measured on the boundary of the spatial region where the equation applies the analysis is based on the observation that the overspecified data depends monotonically on the unknown source term in the equation here we use this monotonicity to establish a type of unicity result for the inverse problem the attractiveness of the monotonicity methods illustrated here lies in their versatility surveying the rapidly expanding literature devoted to the topic of inverse problems one finds a considerable variety of ad hoc approaches methods which apply to a general class of problems seem to be few monotonicity methods however apply to a variety of inverse problems involving partial differential equations of parabolic or elliptic type this paper is organized as follows in section 1 we formulate the direct initial boundary value problem ibvp and state assumptions on the data and a priori assumptions on the unknown source term several properties of the solution to the direct problem are derived in section 2 the principal monotonicity estimates are established and in section 3 these are applied to the inverse problem to prove a type of unicity result for the solution to the inverse problem in particular it is a corollary of theorem 3 2 that in the class of analytic source terms the solution of the inverse problem is unique the problem of identifying an unknown source term in a heat equation 155

hp finite element discretisation of the electrical impedance tomography problem abstract this paper is concerned with the accurate recovery of the permittivity and conductivity from the solution of the electrical impedance tomography eit inverse problem given the knowledge of either the neumann to dirichlet or the dirichlet to neumann map the linearised eit inverse problem for the fourier regime is discretised by hp finite elements which allows the accurate calculation of the sensitivity matrix entries the variational algorithm of knowles 1998 31 is extended to allow the recovery of both material parameters by solution of the eit problem in the laplace regime the associated forward problem is again discretised by hp finite elements a series of numerical results are included to demonstrate the effectiveness of the proposed techniques

assessment of uncertainty on structural dynamic responses with the short transformation method abstract the predictive capability of finite element models is limited by their deterministic nature typically not all model parameters are exactly known while even small deviations may have significant effects on the predicted response parameter uncertainty should therefore be taken into account e g with fuzzy arithmetic the absence of fuzzy solvers led to interval arithmetic as a numerical alternative the transformation method tm presented by m hanss replaces interval arithmetic with a set of deterministic computations for each interval all parameter extrema are combined in every possible way in a design of experiments terminology the tm is a so called full factorial design the tm is applicable if the output is monotonic in the inputs unlike interval arithmetic it does not overestimate the response uncertainty as only parameter combinations are evaluated that actually occur in this paper the tm has been applied to visualise uncertain frequency response functions frfs obtained with modal superposition this yields accurate results when validated against monte carlo data but the computation time is rather high the short transformation method stm is proposed as an attractive alternative to the original tm a full set of deterministic computations combining all interval extrema is only performed at the lowest interval for higher levels a smaller set is evaluated this allows reconstructing the fuzzy frf from a much lower number of deterministic computations with only a small reduction in the accuracy of frfs both methods are demonstrated on a clamped plate and a car front cradle with uncertain design parameters

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

non probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty effects of uncertainties on the dynamic response of the nonlinear vibration systems with general form are investigated based on interval mathematics modeling the uncertain parameters as interval numbers a non probabilistic interval analysis method which estimates the range of the nonlinear dynamic response with the help of taylor series expansion is presented where the partial derivatives of the dynamic response with respect to uncertain parameters are considered to be interval numbers the sensitivity matrices of dynamic response with the uncertain parameters are derived for the presented method only the bounds on uncertain parameters are needed instead of probabilistic density distribution or statistical quantities numerical examples are used to illustrate the validity and feasibility of the presented method

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

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

calculating frequency response functions for uncertain systems using complex affine analysis this paper is concerned with the effect of parametric uncertainty upon the response of mechanical systems two uncertainty propagation techniques are employed and compared against a benchmark based upon randomly scanning input parameter ranges the first technique complex interval analysis is computationally very efficient but suffers from the issue of dependency which may result in large overestimation of the system response the second technique namely complex affine analysis is presented here for the first time and it keeps track of variable dependencies throughout the calculation thus limiting overestimation albeit at some computational expense the methods are demonstrated on the problem of calculation of the frequency response functions of a simple lumped mass system with uncertain parameters for purposes of transparency of the presented ideas although these ideas are equally applicable to more complex systems it transpired that the complex affine analysis performed significantly better than its interval counterpart

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

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

monte carlo analysis of nonlinear vibration of rectangular plates with random geometric imperfections abstract the effect of random initial geometric imperfections on the vibration behavior of rectangular plates is investigated in this paper using a statistical method the random initial geometric imperfections of plates are described by gaussian random fields and simulated numerically using elishakoff s method lindstedt poincare s perturbation technique is employed to solve duffing s equation with an additional quadratic spring term derived in the vibration analysis of imperfect rectangular plates a monte carlo analysis for simply supported plates is carried out in detail to illustrate the proposed approach it is shown that the effect of random geometric imperfections on the vibration behavior of the plates can be described quantitatively in terms of the frequency reliability function and the hardening type probability

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

stochastic dynamic sensitivity of uncertain structures subjected to random earthquake loading the present study involves computation of stochastic sensitivity of structures with uncertain structural parameters subjected to random earthquake loading the formulations are provided in frequency domain a strong earthquake induced ground motion is considered as a random process defined by respective power spectral density function the uncertain structural parameters are modelled as homogeneous gaussian stochastic field and discretized by the local averaging method the discretized stochastic field is simulated by the cholesky decomposition of respective co variance matrix by expanding the dynamic stiffness matrix about its reference value the advantage of neumann expansion technique is explored within the framework of monte carlo simulation to compute responses as well as sensitivity of response quantities this approach involves only a single decomposition of the dynamic stiffness matrix for the entire simulated structure and the facility that several stochastic fields can be tackled simultaneously are basic advantages of the neumann expansion the proposed algorithm is explained by an example problem

comments on the method of harmonic balance in which jacobi elliptic functions are used abstract a harmonic balance method is presented in which jacobi elliptic functions are used in the trial solution instead of circular functions to obtain approximate periodic solutions of the oscillator x f x dot x 0 conditions for the method to work well are the usual ones of the current method of harmonic balance and that x t must pass through zero the procedure for obtaining a higher order approximation is described and in particular two criteria for chosing the elliptic function parameter m are discussed illustrative examples are presented with f being diverse polynomials of x

a uniqueness theorem for determining conductivity from overspecified boundary data on cherche les coefficients inconnus k u dans l equation elliptique non lineaire k u u f x y x y ou u x y est soumise a des conditions aux limites de dirichlet et de neumann sur la frontiere de

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

a parabolic inverse problem with an unknown boundary condition abstract in this paper we consider a parabolic inverse problem in which an unknown function is involved in the boundary condition and we attempt to recover this function by measuring the value of the temperature at a fixed point on the boundary the motivation for studying this problem arises from some physical models such as a heat conduction system where the heat exchange between the system and its surrounding is unknown we apply the singularity estimates for the fundamental solution of a parabolic equation along with the generalized bellman gronwall inequality to obtain the continuous dependence of the solution upon the known data uniqueness of the solution is established as a direct corollary

the incremental harmonic balance method for nonlinear vibration of axially moving beams in this paper the incremental harmonic balance ihb method is formulated for the nonlinear vibration analysis of axially moving beams the galerkin method is used to discretize the governing equations a high dimensional model that can take nonlinear model coupling into account is derived the forced response of an axially moving strip with internal resonance between the first two transverse modes is studied particular attention is paid to the fundamental superharmonic and subharmonic resonance as the excitation frequency is close to the first second or one third of the first natural frequency of the system numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear vibration of axially moving media

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

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

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

a split frequency harmonic balance method for nonlinear oscillators with multi harmonic forcing abstract a new harmonic balance method hbm is presented for accurately computing the periodic responses of a nonlinear sdof oscillator with multi harmonic forcing and non expansible nonlinearities the presence of multi harmonic forcing requires a large number of solution harmonics with a substantial increase in computational demand for either the conventional or the incremental hbm in this method the oscillator equation error is first defined in terms of two functions originally proposed for obtaining free vibration periods in r e mickens iteration procedure for determining approximate solutions to nonlinear oscillator equations journal of sound and vibration 116 1987 185 187 and more recently r e mickens a generalised iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillations journal of sound and vibration 287 2005 1045 1051 a fourier series solution is assumed in which the total number of harmonics is fixed by the chosen discrete time interval this series is split into two partial sums nominally associated with either low frequency or high frequency harmonics by exploiting a convergence property of the equation error functions the total solution is obtained in a new iterative scheme in which the low frequency components are computed via a conventional hbm using a small number of algebraic equations whereas the high frequency components are obtained in a separate step by updating by gradually increasing the number of harmonics in the low frequency group the equation error can be progressively reduced efficient use is made of fft based algebraic equation generation which allows an important class of non expansible nonlinearities to be handled the proposed method is tested on a duffing type oscillator and an oscillator with a non expansible 7th power stiffness term where in both cases up to 24 component multi harmonic forcing is applied as a comparison a conventional hbm is also used on the duffing model in which the algebraic equations are generated in symbolic form to totally avoid errors from entering the formulation through complicated expansion of the cubic stiffness term as in i senjanovic harmonic analysis of nonlinear oscillations of cubic dynamical systems journal of ship research 38 3 1994 225 238 and in a raghothama s narayanan periodic response and chaos in nonlinear systems with parametric excitation and time delay nonlinear dynamics 27 2002 341 365 the paper shows that in obtaining period 1 solutions the computational accuracy and efficiency of the proposed method is very good