linear damping models for structural vibration abstract linear damping models for structural vibration are examined first the familiar dissipation matrix model then the general linear model in both cases an approximation of small damping is used to obtain simple expressions for damped natural frequencies complex mode shapes and transfer functions results for transfer functions can be expressed in the form of very direct extensions of the familiar expression for the undamped case this allows a detailed discussion of the implications of the various damping models for the interpretation of measured transfer functions especially in the context of experimental modal analysis in the case of a dissipation matrix model it would be possible in principle to determine all the model parameters from measurements in the case of the general model however there is a fundamental ambiguity which prevents full determination of the model from measurements on a single structure

some efficient derivative free methods with memory for solving nonlinear equations abstract based on the family of three point derivative free methods without memory of eighth order convergence proposed by d unic et al j d unic m s petkovic l d petkovic three point methods with and without memory for solving nonlinear equations appl math comput 218 2012 4917 4927 we present three methods with memory by suitable variation of a free parameter in each iterative step this free parameter is calculated using newton s interpolatory polynomial of the third degree in two ways and newton s interpolatory polynomial of the fourth degree consequently the r order of convergence is increased from 8 to 11 137 2 11 352 6 4 2 11 657 and 12 the increase in the convergence order is achieved without any additional function evaluations and therefore the proposed methods possess a very high computational efficiency numerical examples are presented and the performance is compared with the existing three point methods with and without memory of the basic family moreover theoretical order of convergence is verified on the examples

probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation we present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data for which uncertainties are taken into account by using the nonparametric probabilistic approach such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters consequently the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible two formulations are proposed the first one is the mean square method for which a usual differentiable objective function and an unusual non differentiable objective function are proposed the second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics

uncertainty analysis of complex structural systems in this paper methods for uncertainty propagation for complex structural systems are discussed only a limited class of methods is available for this purpose it is shown that only numerical procedures allow the uncertainty assessment and reliability estimation of such systems in particular the perturbation procedure the karhunen loeve reduction scheme the polynomial chaos expansion direct and advanced monte carlo simulation mcs as well as random matrix theory meet these requirements it is shown that advanced mcs procedures proved to be the most versatile approach copyright 2009 john wiley sons ltd

computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties abstract the inherent uncertainties in geometry material properties etc of engineering structures can be represented by stochastic models where the parameters are described by probabilistic laws results from any analysis based on stochastic models inherit probabilistic information as well which can be used e g for reliability analysis particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial a very flexible however computationally intensive way to analyze such systems is direct monte carlo simulation mcs in this paper procedures are shown which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with optimally selected start vectors the procedures are not restricted to any kind of sampling technique

on the geometry of the set of controllability subspaces of a pair a b abstract given a controllable system defined by a pair of matrices a b we investigate the geometry of the set of controllability subspaces this set is a subset of the set of a b invariant subspaces we prove that in fact it is a stratified submanifold and we compute its dimension

nonparametric probabilistic approach of uncertainties for elliptic boundary value problem the paper is devoted to elliptic boundary value problems with uncertainties such a problem has already been analyzed in the context of the parametric probabilistic approach of system parameters uncertainties or for random media model uncertainties are induced by the mathematical physical process which allows the boundary value problem to be constructed from the design system if experiments are not available the bayesian approach cannot be used to take into account model uncertainties recently a nonparametric probabilistic approach of both the model uncertainties and system parameters uncertainties has been proposed by the author to analyze uncertain linear and non linear dynamical systems nevertheless the use of this concept that has to be developed for dynamical systems cannot directly be applied for elliptic boundary value problem for instance for a linear elastostatic problem relative to an elastic bounded domain we then propose an extension of the nonparametric probabilistic approach in order to take into account model uncertainties for strictly elliptic boundary value problems the theory and its validation are presented

a family of second derivative free fourth order continuation method for solving nonlinear equations in this paper we present a parameter based iterative method free from the second derivative for solving nonlinear equations of type f x 0 ezquerro and hernandez 1999 discussed the convergence analysis of a uniparametric family of an iterative method in banach space based on this idea we propose a uniparametric family of second derivative free iterative method in r we discuss the convergence analysis and observe that it possesses a fourth order convergence for solving nonlinear equations in r several numerical examples are worked out with our proposed methods for parameter 1 2 and the existing fourth order iterative method proposed in king 1973 chun 2007 soleymani et al 2012 and maheshwari 2009 finally from the comparison we observe that our method is more efficient than existing methods finally we compare basins of attraction of our methods with the second derivative free fourth order iterative method proposed in king 1973 chun 2007 soleymani et al 2012 and maheshwari 2009 observe that the proposed scheme is more efficient

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

probabilistic equivalence and stochastic model reduction in multiscale analysis this paper presents a probabilistic upscaling of mechanics models a reduced order probabilistic model is constructed as a coarse scale representation of a specified fine scale model whose probabilistic structure can be accurately determined equivalence of the fine and coarse scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained a significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine and coarse scale representations the upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations an illustration in stochastic structural dynamics is provided to demonstrate the proposed framework

modal probabilistic analysis in this paper we first recall a method for estimating the numerical probability density function pdf with parzen rosenblatt estimators of experimental data like the sound absorption coefficient with the kundt s tube then we focus on the model of mikki which is used as a predictive tool we are interested then in inversing the model and in identifying the 3 parameters q of mikki s model but we want to go further and to identify from the estimated pdf directly the pdf of the 3 parameters we explain first when it is possible then we give some examples to illustrate the method this work allows for evaluating the robustness of a model from experimental data

towards a compactification of the set of conditioned invariant subspaces a compactification of the set of conditioned invariant subspaces of fixed dimension for an observable pair c a is proposed it contains the almost conditioned invariant subspaces of the same dimension in certain cases the compactification is shown to be smooth and a complete geometric description is given in the case of a single output system

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

on the convergence of an optimal fourth order family of methods and its dynamics in this paper we present the study of the semilocal and local convergence of an optimal fourth order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior parameter spaces are shown and the study of the stability of all the fixed points is presented

identification of linear non viscous damping with different kernel functions in the time domain abstract the non viscous damping model considers the complete velocity history via a convolution over the kernel functions in this paper a procedure is proposed for identifying a damping model involving different kernel functions in the time domain the method uses responses calculated by the generic time domain integration with the known mass matrix and the stiffness matrix the relaxation parameter in the kernel function is identified through a summation of the error function and the damping coefficient matrix is identified via the linear least squares method two examples show that reasonable estimates could be obtained by the proposed method

calibration of the damping dispersion parameter in a non parametric probabilistic approach abstract response predictions in structural dynamics are in general very sensitive to random uncertainties associated with the underlying predictive mathematical model one way to capture parametric and model uncertainties consists in the construction of a non parametric probabilistic model in which the reduced structural matrices are replaced by random matrices the uncertainties are introduced at a global level and controlled through one dispersion parameter each for the mass damping and stiffness matrix in the present paper the role of the damping dispersion parameter is investigated for the case in which the adopted non parametric model is calibrated with respect to an existing parametric model it is shown that in the non parametric model the influence of the damping dispersion on the scatter in the frequency response is relatively small if a calibration criterion based on matrix norms is used a novel method for calibrating this non parametric model is proposed which enforces the same scatter of the frequency response function frf at the first resonance frequency in the parametric and the non parametric model a case study involving a satellite fe model shows that with this approach the frf scatter of the non parametric model reaches a level similar to that of the parametric model used for its calibration

matrix variate distributions for probabilistic structural dynamics matrix variate distributions are proposed to quantify uncertainty in the mass stiffness and damping matrices arising in linear structural dynamics the proposed approach is based on the so called wishart random matrices it is assumed that the mean of the system matrices are known a new optimal wishart distribution is proposed to model the random system matrices the optimal wishart distribution is such that the mean of the matrix and its inverse produce minimum deviations from their respective deterministic values the method proposed here gives a simple nonparametric approach for uncertainty quantification and propagation for complex aerospace structural systems the new method is illustrated using a numerical example it is shown that wishart random matrices can be used to model uncertainty across a wide range of excitation frequencies

a random matrix approach for quantifying model form uncertainties in turbulence modeling abstract with the ever increasing use of reynolds averaged navier stokes rans simulations in mission critical applications the quantification of model form uncertainty in rans models has attracted attention in the turbulence modeling community recently a physics based nonparametric approach for quantifying model form uncertainty in rans simulations has been proposed where reynolds stresses are projected to physically meaningful dimensions and perturbations are introduced only in the physically realizable limits xiao et al 2016 however a challenge associated with this approach is to assess the amount of information introduced in the prior distribution and to avoid imposing unwarranted constraints in this work we propose a random matrix approach for quantifying model form uncertainties in rans simulations with the realizability of the reynolds stress guaranteed which is achieved by construction from the cholesky factorization of the normalized reynolds stress tensor furthermore the maximum entropy principle is used to identify the probability distribution that satisfies the constraints from available information but without introducing artificial constraints we demonstrate that the proposed approach is able to ensure the realizability of the reynolds stress albeit in a different manner from the physics based approach monte carlo sampling of the obtained probability distribution is achieved by using polynomial chaos expansion to map independent gaussian random fields to the reynolds stress random field with the marginal distributions and correlation structures as specified numerical simulations on a typical flow with separation have shown physically reasonable results which verify the proposed approach therefore the proposed method is a promising alternative to the physics based approach for model form uncertainty quantification of rans simulations the method explored in this work is general and can be extended to other complex physical systems in applied mechanics and engineering

the biflag manifold and the fixed points of a unipotent transformation on the flag manifold abstract we investigate the topology of the submanifold of the partial flag manifold consisting of those flags s 1 s d in f n f r or c that are fixed by a given unipotent a sl n f and for which the restrictions a s i have prescribed jordan type it is shown that this submanifold has a compact nonsingular projective variety as a strong deformation retract this variety the biflag manifold consists of rectangular arrays of subspaces which are nested along each row and column it is proven that the biflag manifold has the homology type mod 2 for f r of a product of grassmannians a partition of the biflag manifold into a finite union of affine spaces is also given in the special case when the biflag manifold is a flag manifold this partition coincides with the bruhat cell decomposition

an interdependency index for the outputs of uncertain systems the study of mechanical systems with uncertain parameters is gaining increasing interest in the field of system analysis to provide an expedient model for the prediction of the system behavior making use of the transformation method the uncertain parameters of the system are modeled by fuzzy numbers in contrast to random numbers used in stochastic approaches as a result of this analysis a quantification of the overall uncertainty of the system outputs including a worst case scenario is obtained the inputs of the resulting fuzzy valued model are a priori uncorrelated but after the uncertainties are propagated through the model interdependency or interaction between the outputs may arise if such interdependency is neglected a misinterpretation of the results may occur for example in the case of applying uncertainty analysis in the early design phase of a product to determine the relevant design parameter space the interdependency between the design variables may reduce significantly the available part of the design space this paper proposes a measure of interdependency between the uncertain system outputs the interdependency index can be derived by a postprocessing of the data gained by the analysis with the transformation method such information can be obtained by a negligible amount of extra computation time

reduced models for the medium frequency dynamics of stochastic systems in this paper a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium frequency range in this frequency range both traditional modal analysis and statistical energy analysis sea procedures well suited for low and high frequency vibration analysis respectively lead to computational and conceptual difficulties the uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical the methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods in particular the karhunen loeve and polynomial chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest

stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion this paper presents the theory and application of the generalized polynomial chaos expansion for the stochastic free vibration of orthotropic plates specifically the stochastic analysis of orthotropic plates under the uncertainties in elasticity moduli is investigated the uncertain moduli eigen frequencies and eigen modes of the plates are represented by truncated polynomial chaos expansions with arbitrary random basis the expansions are substituted in the governing differential equations to calculating the polynomial chaos coefficients of the eigen frequencies and the eigen modes distribution functions of the uncertain moduli are derived from experimental data where the pearson model is used to identify the type of density functions this realization then is employed to construct random orthogonal basis for each uncertain parameter because of available experimental modal analysis data this paper provides a useful practical example on the efficacy of polynomial chaos where the statistical moments and the probability distributions of modal responses are compared with experimental results

a predictive approach to the analysis of designed experiments abstract viewing the analysis of designed experiments as a model selection problem we introduce the use of a predictive bayesian criterion in this context based on the predictive density of a replicate experiment pdre a calibration of the criterion is provided to assist in the model choice the relationships of the proposed criterion to other prevalent criteria such as aic bic and mallows s c p are given an information theoretic criterion based on the pdre s of two competing models is also introduced and compared with the usual f statistic for two nested models examples are given to illustrate the proposed methodology

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

fuzzy structure theory modeling of sound insulation layers in complex vibroacoustic uncertain systems theory and experimental validation the fuzzy structure theory was introduced 20 years ago in order to model the effects of complex subsystems imprecisely known on a master structure this theory was only aimed at structural dynamics in this paper an extension of that theory is proposed in developing an elastoacoustic element useful to model sound insulation layers for computational vibroacoustics of complex systems the simplified model constructed enhances computation time and memory allocation because the number of physical and generalized degrees of freedom in the computational vibroacoustic model is not increased however these simplifications introduce model uncertainties in order to take into account these uncertainties the nonparametric probabilistic approach recently introduced is used a robust simplified model for sound insulation layers is then obtained this model is controlled by a small number of physical and dispersion parameters first the extension of the fuzzy structure theory to elastoacoustic element is presented

the topology of the set of conditioned invariant subspaces we consider the topology of the set of conditioned invariant subspaces of an observable pair c a of a fixed dimension by fixing the observability indices of the restricted system a stratification by finitely many smooth manifolds is obtained termed brunovsky strata it is shown that each brunovsky stratum is homotopy equivalent to a generalized flag manifold from this description an effective formula for the betti numbers of the brunovsky strata can be derived

eigenrelations for nonviscously damped systems multiple degree of freedom linear dynamic systems with nonviscous damping is considered the nonviscous damping is such that the damping forces depend on the past history of the velocities via convolution integrals over some kernel functions for the sake of generality the system matrices are allowed to be asymmetric the mode orthogonality relationships known for undamped or viscously damped systems have been generalized to such nonviscously damped systems some expressions are suggested for the normalization of the eigenvectors a number of useful results relating the system matrices and the eigensolutions have been established numerical examples are provided to illustrate the derived results

random eigenvalues algebraic methods and structural dynamic models a review of the literature of the study of structural vibration problems with random parameters is provided there have been several approaches to this problem these have been encapsulated in this paper for the benefit of those who need to assess such possibilities in particular the algebraic theory of random variables is delineated here with the application in mind being the determination or estimation of the statistics of the eigenvalues of linear dynamical systems various transformation techniques are summarized and discussed with simple examples

derivative free two point methods with and without memory for solving nonlinear equations abstract two families of derivative free two point iterative methods for solving nonlinear equations are constructed these methods use a suitable parametric function and an arbitrary real parameter it is proved that the first family has the convergence order four requiring only three function evaluations per iteration in this way it is demonstrated that the proposed family without memory supports the kung traub hypothesis 1974 on the upper bound 2n of the order of multipoint methods based on n 1 function evaluations further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step this approach leads to a family of two step self accelerating methods with memory whose order of convergence is at least 2 5 4 236 and even 2 6 4 449 in special cases the increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations

on the parametrization of the controllability subspaces of a controllable pair given a controllable linear control system defined by a pair of constant matrices a b the set of controllability subspaces is an stratified submanifold of the set of a b invariant subspaces we parameterize each strata by means of coordinate charts this parametrization has significant dierences to that of a b invariant subspaces showing a more complex geometric structure

generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions a new generalized probabilistic approach of uncertainties is proposed for computational model in structural linear dynamics and can be extended without difficulty to computational linear vibroacoustics and computational non linear structural dynamics this method allows the prior probability model of each type of uncertainties model parameter uncertainties and modeling errors to be separately constructed and identified the modeling errors are not taken into account with the usual output prediction error method but with the nonparametric probabilistic approach of modeling errors recently introduced and based on the use of the random matrix theory the theory an identification procedure and a numerical validation are presented then a chaos decomposition with random coefficients is proposed to represent the prior probabilistic model of random responses the random germ is related to the prior probability model of model parameter uncertainties the random coefficients arc related to the prior probability model of modeling errors and then depends on the random matrices introduced by the nonparametric probabilistic approach of modeling errors a validation is presented finally a future perspective is introduced when experimental data are available the prior probability model of the random coefficients can be improved in constructing a posterior probability model using the bayesian approach

similarity of non everywhere defined linear maps abstract the equivalence between linear maps defined only on a subspace or by duality defined modulo a subspace is studied and applied to block similarity of rectangular matrices in that way we find the complete families of invariants obtained by brunovsky and zaballa and the results of gohberg and others for this particular case also we describe explicit methods for obtaining the reduced matrix and the corresponding bases

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

parametrization of c a invariant subspaces in this paper we classify and parametrize the c a invariant subspaces of a given observable pair c a for that purpose we associate with every c a invariant subspace v a k s module m v of laurent series in s 1 and parametrize these modules via their uniquely determined kronecker hermite bases the discrete and continuous parameters thus obtained are interpreted in slate space terms finally we illustrate the results by presenting a complete overview over the set of all c a invariant subspaces for a given pair c a 6 3 6 in dual brunovsky canonical form

nonparametric stochastic modeling of structures with uncertain boundary conditions coupling between substructures the focus of this investigation is on the formulation and validation of a novel approach for the inclusion of uncertainty in the modeling of the boundary conditions of linear structures and of the coupling between linear substructures this work is particularly relevant to complex structures assembled from simpler substructures as in aerospace applications first a mean structural dynamic model that includes boundary condition coupling flexibility is obtained using classical substructuring concepts the application of the nonparametric stochastic modeling approach to this mean model is next described and thus permits the consideration of both model and parameter uncertainty finally the identification of the two parameters of this stochastic model the mean boundary condition coupling flexibility and the overall level of uncertainty is shown and discussed the methodology is demonstrated on three different structural dynamic models of a rectangular plate and of two different wings

differentiable structure of the set of controllable a b t invariant subspaces abstract given a b t hom c n c n m observable we prove that the set of a b t invariant subspaces having a fixed brunovsky kronecker structure is a connected manifold and we compute its dimension also we include some applications of these results

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

quantification of non viscous damping in discrete linear systems abstract the damping forces in a multiple degree of freedom engineering dynamic system may not be accurately described by the familiar viscous damping model the purpose of this paper is to develop indices to quantify the extent of any departures from this model in other words the amount of non viscosity of damping in discrete linear systems four indices are proposed two of these indices are based on the non viscous damping matrix of the system a third index is based on the residue matrices of the system transfer functions and the fourth is based on the measured complex modes of the system the performance of the proposed indices is examined by considering numerical examples

uncertainty in structural dynamics experimental validation of a wishart random matrix model in the modeling of complex dynamical systems high resolution finite element models are routinely adopted to reduce the discretization error such an approach may fail to enhance the confidence in simulation based predictions when the dynamical systems exhibit significant variability in the parameterization and description of the mathematical model due to parametric data uncertainty and model uncertainty in contrast to parametric uncertainty model uncertainty poses significant challenges as no explicit parameter is available a priori to characterize its behavior in the current investigation a set of sprung mass oscillators randomly attached to a baseline system namely a homogeneous cantilever plate gives rise to model uncertainty contrary to model parametric uncertainty traditionally tackled using stochastic finite element method sfem the model uncertainty arising from randomly attached sprung masses gives rise to a new variant of dynamical system for every sample the feasibility of adopting a wishart random matrix to represent such ensemble of dynamical systems derived from model perturbation is investigated in this paper the experiments conducted on a vibrating plate with randomly attached sprung mass oscillators demonstrate that the wishart random matrix model may provide a reasonable representation of model uncertainty in linear dynamical systems in the medium and high frequency region

anchored analysis of variance petrov galerkin projection schemes for linear stochastic structural dynamics in this paper we propose anchored functional analysis of variance petrovgalerkin aapg projection schemes originally developed in the context of parabolic stochastic partial differential equatio

physical consequences of a nonparametric uncertainty model in structural dynamics abstract one of the main claims of the nonparametric model of random uncertainty introduced by soize 2000 3 is its ability to account for model uncertainty the present paper investigates this claim by examining the statistics of natural frequencies total energy and underlying dispersion equation yielded by the nonparametric approach for two simple systems a thin plate in bending and a one dimensional finite periodic mass spring chain results for the plate show that the average modal density and the underlying dispersion equation of the structure are gradually and systematically altered with increasing uncertainty the findings for the mass spring chain corroborate the findings for the plate and show that the remote coupling of nonadjacent degrees of freedom induced by the approach suppresses the phenomenon of mode localization this remote coupling also leads to an instantaneous response of all points in the chain when one mass is excited in the light of these results it is argued that the nonparametric approach can deal with a certain type of model uncertainty in this case the presence of unknown terms of higher or lower order in the governing differential equation but that certain expectations about the system such as the average modal density may conflict with these results

identification of damping part 1 viscous damping abstract characterization of damping forces in a vibrating structure has long been an active area of research in structural dynamics the most common approach is to use viscous damping where the instantaneous generalized velocities are the only relevant state variables that affect damping forces however viscous damping is by no means the only damping model within the scope of linear analysis any model which makes the energy dissipation functional non negative is a possible candidate for a valid damping model this paper and its companion see pp 63 88 of this issue are devoted to developing methodologies for identification of such general damping models responsible for energy dissipation in a vibrating structure this paper considers identification of viscous damping under circumstances when the actual damping model in the structure is non viscous a method is presented to obtain a full non proportional viscous damping matrix from complex modes and complex natural frequencies it is assumed that the damping is small so that a first order perturbation method is applicable the proposed method and several related issues are discussed by considering numerical examples based on a linear array of damped spring mass oscillators it is shown that the method can predict the spatial location of damping with good accuracy and also give some indication of the correct mechanism of damping

probabilistic approach for damping identification considering uncertainty in experimental modal analysis the system identification technology is essentially an inverse procedure starting from the experimentally measured response to construct mass stiffness and damping matrices of the structure ho

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

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

versal deformations in generalized flag manifolds we find explicit miniversal deformations of flags in the generalized flag manifolds with regard a natural equivalence relation defined by the group action that keeps fixed the reference flag

an optimal steffensen type family for solving nonlinear equations abstract in this paper a general family of steffensen type methods with optimal order of convergence for solving nonlinear equations is constructed by using newton s iteration for the direct newtonian interpolation it satisfies the conjecture proposed by kung and traub h t kung j f traub optimal order of one point and multipoint iteration j assoc comput math 21 1974 634 651 that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2 m 1 its error equations and asymptotic convergence constants are obtained finally it is compared with the related methods for solving nonlinear equations in the numerical examples

on the quantification of damping model uncertainty abstract the characterization of uncertainty in the damping forces in a vibrating structure is of significant interest in structural dynamics the most common approach is to use a viscous damping matrix with uncertain coefficients the viscous damping is not the only damping model within the scope of linear analysis any model which makes the energy dissipation functional non negative is a possible candidate for a valid damping model there are many functional forms which can be used as damping models in this paper two approaches are proposed to quantify such model form uncertainty associated with the use of the viscous damping model not only the model parameters the first approach is based on an ensemble of equivalent damping functions and the second approach is based on the random matrix theory the results obtained from the two methods are compared using numerical examples it was observed that the random matrix theory can be used to quantify damping model uncertainty in some cases

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

high dimensional model representation method for fuzzy structural dynamics abstract uncertainty propagation in multi parameter complex structures possess significant computational challenges this paper investigates the possibility of using the high dimensional model representation hdmr approach when uncertain system parameters are modeled using fuzzy variables in particular the application of hdmr is proposed for fuzzy finite element analysis of linear dynamical systems the hdmr expansion is an efficient formulation for high dimensional mapping in complex systems if the higher order variable correlations are weak thereby permitting the input output relationship behavior to be captured by the terms of low order the computational effort to determine the expansion functions using the cut method scales polynomically with the number of variables rather than exponentially this logic is based on the fundamental assumption underlying the hdmr representation that only low order correlations among the input variables are likely to have significant impacts upon the outputs for most high dimensional complex systems the proposed method is first illustrated for multi parameter nonlinear mathematical test functions with fuzzy variables the method is then integrated with a commercial finite element software adina modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach in the numerical examples triangular membership functions have been used and the results have been validated against direct monte carlo simulations it is shown that using the proposed hdmr approach the number of finite element function calls can be reduced without significantly compromising the accuracy

experimental randomization and the validity of normal theory inference abstract this article examines how randomization helps to validate model based probability statements the asymptotic validity of normal theory tests for means in multivariate linear models is established under general conditions numerical results are presented showing how randomization provides a limited degree of robustness in small experiments the conditional relevance of randomization is also discussed consider the following additive model for a comparative experiment y gu where y is a vector of responses is a vector of treatment effects u is a vector of unit effects and g is a permutation representing the assignment of treatments to units the vector gu is obtained by using g to permute the coordinates of u it is assumed that is fixed and g and u are random and independent the experimenter observes y and g but not or u and chooses the distribution of g that is the randomization strategy in this setup three probabilities may be considered conditional given g conditional g

in sample tests of predictive ability a new approach abstract this paper presents evidence linking in sample tests of predictive content and out of sample forecast accuracy our approach focuses on the negative effect that finite sample estimation error has on forecast accuracy despite the presence of significant population level predictive content we derive in sample tests that assess whether a variable has predictive content and whether this content is estimated precisely enough to improve forecast accuracy our tests are asymptotically non central chi square or non central normal we provide a convenient bootstrap for computing critical values in monte carlo and empirical analysis we examine the effectiveness of our testing procedure

symmetric state space method for a class of nonviscously damped systems multiple degree of freedom linear nonviscously damped systems are considered it is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions the traditional state space approach well known for viscously damped systems is extended to such nonviscously damped systems using a set of internal variables suitable numerical examples are provided to illustrate the proposed approach i introduction v iscousdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems this model e rst introduced by rayleigh 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping viscous damping models are used widely for their simplicity and mathematical convenience even though the behavior of real structural materials is at best poorly mimicked by simple viscous models forthisreason itiswellrecognizedthat ingeneral aphysically realistic model of damping will not be viscous damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models mathematically any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model clearly a wide range of choice is possible either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments here we will use a particular type of damping model that is not viscous and throughout the paper the terminology nonviscous damping will refer to this specie c model only possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions 2 the equations of motion of an n degree of freedom linear system with such damping can be expressed by

an improvement of chebyshev halley methods free from second derivative abstract in this paper a family of modified chebyshev halley s methods free from second derivative is presented per iteration the new methods require three evaluations of the function and one of its first derivatives a detailed convergence analysis of the new methods shows that the new methods are at least fifth order convergent and especially the modified super halley s method is sixth order convergent numerical examples are given to illustrate the efficiency and performance of the new methods

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

a cellular decomposition of the manifold of observable conditioned invariant subspaces abstract given an observable system defined by a pair of matrices c a we obtain a cellular decomposition of the manifold of c a conditioned invariant subspaces with fixed observability indices of the restricted system and we apply this decomposition to compute their homology groups

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

a unifying discussion of correlation analysis for complex random vectors the assessment of multivariate association between two complex random vectors is considered a number of correlation coefficients based on three popular correlation analysis techniques namely canonical correlation analysis multivariate linear regression and partial least squares are reviewed and connected to performance measures in signal processing and communications such as mean squared estimation error mutual information and signal to noise ratio snr for complex data there are three types of correlation coefficients which account for rotational reflectional and total i e rotational and reflectional dependencies between two random vectors these three types are defined and analyzed for different correlation coefficients and a numerical example is given it is often required to compare two complex random vectors in a lower dimensional subspace for the large class of increasing schur convex correlation coefficients it is shown that the low rank approximations of two random vectors maximizing a particular correlation coefficient are determined only by the constraints imposed on the correlation analysis technique in this context the correlation spread is defined as a normalized measure of how much of the overall correlation is contained in a low dimensional subspace

higher order derivative free families of chebyshev halley type methods with or without memory for solving nonlinear equations abstract in this paper we present two new derivative free families of chebyshev halley type methods for solving nonlinear equations numerically both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence furthermore accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step the self accelerating parameter is estimated from the current and previous iteration this self accelerating parameter is calculated using newton s interpolation polynomial of third and fourth degrees consequently the r orders of convergence are increased from 4 to 6 and 8 to 12 respectively without any additional functional evaluation the results require high order derivatives reaching up to the eighth derivative that is why we also present an alternative approach using only the first or at most the fourth derivative we also obtain the radius of convergence and computable error bounds on the distances involved numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency in particular we consider a concrete variety of real life problems coming from different disciplines e g kepler s equation of motion planck s radiation law problem fractional conversion in a chemical reactor the trajectory of an electron in the air gap between two parallel plates van der waal s equation which explains the behavior of a real gas by introducing in the ideal gas equations in order to check the applicability and effectiveness of our proposed methods

numerical solution of constrained non linear algebraic equations on discute la solution numerique de n equations algebriques non lineaires a n inconnues avec des contraintes sur les variables en operant une distinction entre les contraintes physiques et absolues on utilise la methode de prolongement ou des fonctions de penalite dans le premier cas et la methode de newton a pas reduit pour les problemes a contraintes absolues

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

a homeomorphism between observable pairs and conditioned invariant subspaces abstract a bijective correspondence between similarity classes of observable systems c a and n codimensional conditioned invariant subspaces of a pair c a is constructed that leads to a homeomorphism of the spaces this is applied to the parametrization of inner functions of fixed mcmillan degree proofs using state space methods as well as using polynomial models are given

on halley type iterations with free second derivative in this paper we relax the convergence conditions required in ezquerro and hernandez int j pure appl math 6 1 2003 103 for a multipoint third order iteration of halley type where the conditions provided are the known ones for methods of order three

the component mode transformation method a fast implementation of fuzzy arithmetic for uncertainty management in structural dynamics abstract the study of mechanical systems with uncertain parameters is of increasing importance in the analysis and in the design of structures because an adequate description of these uncertainties is necessary to provide a reliable forecasting model in order to update or evaluate an uncertain model different methods have been proposed but many of these such as the monte carlo simulation or the transformation method require multiple sequential evaluations of the model moreover to cover an adequate portion of the uncertain parameter space the number of necessary evaluations of the model grows exponentially with the number of uncertain parameters leading often to a large number of runs the proposed methodology deals with uncertain parameters that are modeled by the use of fuzzy numbers and the transformation method is applied to characterize the dynamic behavior of the structure in order to reduce the order of magnitude of the computational time a reduction of the model dimension obtained through a component mode synthesis is performed advantages and drawbacks of the proposed method are discussed together with an application to the study of the dynamics of a beam

gaussian process emulators for the stochastic finite element method summary this paper explores a method to reduce the computational cost of stochastic finite element codes the method known as gaussian process emulation consists of building a statistical approximation to the output of such codes based on few training runs the incorporation of emulation is explored for two aspects of the stochastic finite element problem first it is applied to approximating realizations of random fields discretized via the karhunen loeve expansion numerical results of emulating realizations of gaussian and lognormal homogeneous two dimensional random fields are presented second it is coupled with the polynomial chaos expansion and the partitioned cholesky decomposition in order to compute the response of the typical sparse linear system that arises due to the discretization of the partial differential equations that govern the response of a stochastic finite element problem the advantages and challenges of adopting the proposed coupling are discussed copyright 2011 john wiley sons ltd

a coordinate atlas of the manifold of observable conditioned invariant subspaces given an observable system c a k mxn x k nxn with k r or c the set of c a invariant subspaces having the restricted system fixed observability indices is a smooth manifold embedded on the corresponding grassmannian we obtain an explicit parametrization of it by means of an atlas of corresponding charts as an application we obtain the set of maximal dimensional outputs making solvable a generic disturbance decoupling problem

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

response of uncertain dynamic systems i this paper and its sequel deal with the characterization of the response of a structure modeled by a single degree of freedom system in which the mass stiffness and damping parameters are only imprecisely known using an informationtheoretic framework closed form expressions are developed for the probability densities of the parameters that control the characteristics of the dynamic response namely the natural frequency of vibration the damped natural frequency and the percentage of critical damping the manner in which these closed form expressions depend on the extent and nature of our knowledge about the uncertain parameters is highlighted some comparisons with perturbation solutions are also presented

basin attractors for various methods abstract there are many methods for the solution of a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several methods of various orders and present the basin of attraction for several examples it can be seen that not all higher order methods were created equal newton s halley s murakami s and neta johnson s methods are consistently better than the others in two of the examples neta s 16th order scheme was also as good

an acceleration of newton s method super halley method from a study of the convexity we give an acceleration for newton s method and obtain a new third order method then we use this method for solving non linear equations in banach spaces establishing conditions on convergence existence and uniqueness of solution as well as error estimates

robust updating of uncertain computational models using experimental modal analysis in this paper a methodology is presented to perform the robust updating of complex uncertain dynamic systems with respect to modal experimental data in the context of structural dynamics because both model uncertainties and parameter uncertainties must be considered in the computational model the uncertain computational model is constructed by using the nonparametric probabilistic approach we present an extension to the probabilistic case of the input error methodology for modal analysis adapted to the deterministic updating problem it is shown that such an extension to the robust updating context induces some conceptual difficulties and is not straightforward the robust updating formulation leads us to solve a mono objective optimization problem in the presence of inequality probabilistic constraints a numerical application is presented to show the efficiency of the proposed method

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

a class of optimal eighth order derivative free methods for solving the danchick gauss problem abstract a derivative free optimal eighth order family of iterative methods for solving nonlinear equations is constructed using weight functions approach with divided first order differences its performance along with several other derivative free methods is studied on the specific problem of danchick s reformulation of gauss method of preliminary orbit determination numerical experiments show that such derivative free high order methods offer significant advantages over both the classical and danchick s newton approach