analytical model updating and model reduction this paper starts with investigating the effect of model reduction on the model updating process and hence demonstrating the practical difficulties associated with reduced order analytical model updating approach the advantages and possibilities of full analytical model updating are then discussed and the performance of some recently developed methods examined

direct updating of damping and stiffness matrices this note has outlined an extension to the available direct methods of model updating to estimate both the damping and stiffness matrices of a structure the method minimizes the change in the damping and stiffness matrices with the constraint that the measured modal data is reproduced

multifractality of nonlinear iterative processes the havlin bunde multifractal hypothesis physica d 38 184 1989 is expanded in the form of the dimension invariance approach to nonlinear iterative recursion processes such as dielectric breakdown phase transitions from periodic attractors to chaos and cascades in turbulence comparison with model and laboratory data of different authors shows that for strong nonlinearity the dimension invariance is broken

optimal matrix pencil approximation problem in structural dynamic model updating abstract the problem of finding the least change adjustment to a given matrix pencil is considered in this paper desired matrix properties including satisfaction of characteristic equation symmetry positive semidefiniteness and sparsity are imposed as side constraints to form the optimal matrix pencil approximation problem such a problem is related to the frequently encountered engineering problem of a structural modification on the dynamic behavior of a structure alternating direction method is applied to this constrained minimization problem numerical results are included to illustrate the performance and application of the proposed method

a symmetric generalized inverse eigenvalue problem in structural dynamics model updating abstract updating an existing but inaccurate structural dynamics model with measured data can be mathematically reduced to the problem of the best approximation to a given matrix pencil in the frobenius norm under a given spectral constraint and a submatrix pencil constraint in this paper a direct method and the associated mathematical theories for solving this problem are proposed it is shown that the best approximation solution is unique and an explicit expression of the solution is derived numerical examples show that the proposed method is quite accurate and efficient

damage detection in framed structures gurdal z and olmedo r in plane response of laminates with spatially varying fiber orientations variable stiffness panel concept aiaa journal vol 31 no 4 1993 pp 751 758 biggers s b and srinivasan s compression buckling response of tailored rectangular composite plates aiaa journal vol 31 no 3 1993 pp 590 596 biggers s b and srinivasan s improved compression buckling for rectangular composite plates by stiffness tailoring enhancing analysis techniques for composite materials nde vol 10 asme new york 1991 pp 187 195 hks hibbitt karlsson and sorensen inc abaqus theory and user s manuals version 4 8 hks pawtucket ri 1989 pp 2 2 7 1 3 4 3 13 4 3 6 dow m b and smith d l properties of two composite materials made of toughened epoxy resin and high strain graphite fiber nasa tp 2826 1988 only dynamic ones in that of the mass matrix the present algorithm is confined for framed structures truss and beam members and does not address the issue of noise as it is based on real test data which are assumed to be accurate

direct update of dynamic mathematical models from modal test data two methods for the direct updating of mathematical models based on modal test data are described a set of minimum required constraints derived from eigendynamic and force equilibrium conditions are presented both methods generate mass and stiffness matrices fitted exactly to the given modal test data one method changes all coefficients of the mass respective to the stiffness matrix and lessens the solution effort by the algebraical elimination of the lagrangian multipliers the other method changes only those selected matrix coefficients requiring additional solutions for the numerical calculation of the lagrangian multipliers examples and comments on the applicability of the methods are given

new array codes for multiple phased burst correction an optimal family of array codes over gf q for correcting multiple phased burst errors and erasures where each phased burst corresponds to an erroneous or erased column in a code array is introduced as for erasures these array codes have an efficient decoding algorithm which avoids multiplications or divisions over extension fields replacing these operations with cyclic shifts of vectors over gf q the erasure decoding algorithm can be adapted easily to handle single column errors as well the codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array thus generalizing a previously known construction for the special case of two erasures algebraically they can be interpreted as reed solomon codes when q is primitive in gf q the resulting codes become conventional reed solomon codes of length p over gf q sup p 1 in which case the new erasure decoding technique can be incorporated into the berlekamp massey algorithm yielding a faster way to compute the values of any prescribed number of errors

structured quadratic inverse eigenvalue problem i serially linked systems quadratic pencils arising from applications are often inherently structured factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters for physical feasibility structural constraints must be respected consequently they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications however the issue of solvability is problem dependent and has to be addressed structure by structure this paper considers one particular structure where the elements of the physical system if modeled as a mass spring system are serially linked the discussion recasts both undamped and damped problems in a framework of inequality systems that can be adapted for numerical computation some open questions are described

x code mds array codes with optimal encoding we present a new class of mds maximum distance separable array codes of size n spl times n n a prime number called x code the x codes are of minimum column distance 3 namely they can correct either one column error or two column erasures the key novelty in x code is that it has a simple geometrical construction which achieves encoding update optimal complexity i e a change of any single information bit affects exactly two parity bits the key idea in our constructions is that all parity symbols are placed in rows rather than columns

eigenstructure assignment approach for structural damage detection in this work a methodology for incorporating measured modal data into an existing refined finite element model is examined with the objective of detecting and locating structural damage the algorithm is based on the partial inverse problem in that only partial spectral information is required the technique utilizes a symmetric eigenstructure assignment algorithm to perform the partial spectral assignment algorithms to enhance mode shape assignability and to preserve sparsity in the damaged fem are developed the sparsity preservation is of particular importance when considering damage detection in trusslike structures several examples are presented to highlight the key points made within the paper

finite element model tuning using automated structural optimization system software a method of adjusting analytical finite element models to measured data is presented the algorithm uses a mathematical optimization strategy to minimize deviations between measured and analytical modal frequencies and partial mode shapes a mode tracking algorithm is used to identify and account for mode switching during the optimization process the algorithm was successfully implemented using the automated structural optimization system software experimental results are presented for tuning a lightly damped 6 m flexible frame structure the results demonstrate excellent agreement between the tuned model and measured data and illustrate the importance of off nominal validation before accepting a model for simulation purposes ccurate prediction and simulation of the dynamic behav ior of large flexible space structures require analytical models that agree with measured data unfortunately uncertainty in a finite element model fem implies less than perfect correlation between analytical and measured data when the disagreement is deemed unacceptable it is necessary for the design engineer to make ad justments to the fem for large problems the number of potential parameters to adjust such as elemental areas elastic moduli inertia moments etc quickly becomes overwhelming thus a systematic method is required to ensure the adjustments produce the desired results to this end a method is introduced that poses a numerical optimization problem namely given a set of measured eigenvalues and partial eigenvectors determine the values of selected physical parameters of the model that minimize the weighted deviations from the analytical eigenvalues and eigenvectors numerous techniques and goals of model tuning also referred to as model refinement or model identification have appeared in the literature the common attribute of these techniques is that they at tempt to minimize the required modification to the analytical mass and stiffness matrices assuming the fem is a reasonable approxi mation to the physical structure sensitivity based approaches have been presented in the literature that adjust the matrices by estab lishing an objective function based on the difference between the experimental and analytical model data 1 5 an advantage of those methods is that the updated models are consistent with the fem formulation and thus the connectivity is preserved an alternative method employing an optimal matrix update has been developed by berman and nogy6 and baruch 7 in this method a perturbation mass damping or stiffness matrix is determined that when added to the analytical matrices produces the measured result the advantage of this method is a tuned analytic model that exactly reproduces the experimental data its shortcoming is that it does not guarantee the closeness to unmeasured modes not used in the tuning process this is a result of potentially unrealistic changes in the stiffness matrix such as the introduction of load paths that physically do not exist to overcome some of the shortcomings of baruch s method of stiff ness matrix adjustments kabe8 and kammer9 introduced objective functions that ensure stiffness terms are corrected in a manner such that the connectivity of the analytical model is preserved similar

correcting finite element models using a symmetric eigenstructure assignment technique improvement of structural models by incorporating measured structural modal parameters is approached from a controls aspect the approach is developed for linear structures that exhibit nonproportional damping residual damping and stiffness matrices are determined such that the improved analytical model eigenstructure matches more closely that obtained experimentally

optimum approximation for residual stiffness in linear system identification a new method of system identification uti 1 i zes projector m atrix theory and the moorepenrose generalized inverse to derive an analytical stiffness matrix which when combined with the analytical mass matrix will more closely p redict modal test results weighting matrices a re used to enforce connectivity and make weighted corrections to the original analytical stiffness matrix a simple and straightforward mathematical formulation is obtained the method is compared and contrasted with other methods found in the literature and a simple numerical example is presented

a symmetric inverse eigenvalue problem in structural dynamic model updating in this paper we first give the representation of the general solution of the following inverse eigenvalue problem iep given email protected r n x p and a diagonal matrix email protected r p x p find real valued symmetric 2r 1 diagonal matrices m and k such that email protected kx we then consider an optimal approximation problem given real valued symmetric 2r 1 diagonal matrices m a k email protected r n x n find email protected email protected s m k such that email protected m email protected email protected email protected k email protected 2 inf m k s m k m m email protected email protected k k email protected 2 where s m k is the solution set of iep we show that the optimal approximation solution email protected email protected is unique and derive an explicit formula for it

stiffness matrix adjustment using incomplete measured modes in the line of the baruch berman and kabe methods this technical note describes a new algorithm for stiffness adjustment

updating finite element dynamic models using an element by element sensitivity methodology a sensitivity based methodology for improving the finite element model of a given structure using test modal data and a few sensors is presented the proposed method searches for both the location and sources of the mass and stiffness errors and does not interfere with the theory behind the finite element model while correcting these errors the updating algorithm is derived from the unconstrained minimization of the squared 2 norms of the modal dynamic residuals via an iterative two step staggered procedure at each iteration the measured mode shapes are first expanded assuming that the model is error free then the model parameters are corrected assuming that the expanded mode shapes are exact the numerical algorithm is implemented in an element by element fashion and is capable of zooming on the detected error locations several simulation examples which demonstrate the potential of the proposed methodology are discussed i introduction e ven though the field of finite element analysis of structural dynamic problems has witnessed tremendous progress in the last two decades greater confidence is still placed in experimental data therefore the finite element model of a given structure is often updated to reflect the results of a particular experiment usually the computed eigenmodes are compared with the measured frequencies and mode shapes if both sets are not in agreement the finite element model is refined via a two step updating procedure which 1 locates the errors and 2 corrects them clearly the first step is the most challenging of the two once the location of the errors is known it is relatively easy to correct them especially if the error sources can be identified however locating and identifying these errors can be a difficult task for the following reasons in general only a few experimental modes are available and these may be contaminated with random and systematic measuring errors moreover only a subset of the degrees of freedom dof in the finite element model can be monitored and as for practical and economical reasons only a few sensors can be utilized

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

a no spill over updating method for undamped structural systems abstract a new numerical method for simultaneously updating mass and stiffness matrices based on incomplete modal measured data is presented by using the kronecker product all the variables that are to be modified can be found out and then can be updated directly the optimal approximation mass and stiffness matrices which satisfy the required eigenvalue equation are found under the frobenius norm sense the large number of unmeasured and unknown eigeninformation and the physical connectivity of the original model are preserved and the updated model will exactly reproduce the modal measured data the method is computationally efficient as neither iteration nor eigenanalysis is required the numerical results show that the method proposed is reliable and attractive

reduced bases for model updating in structural dynamics based on constitutive relation error abstract this paper deals with the application of a model reduction method to the updating of models of industrial structures with many degrees of freedom the updating method is based on the concept of constitutive relation error this is an iterative method in which each iteration consists of a first step in which the most important errors in the model are localized and a second step in which these errors are corrected the reduction method follows a classical approach in which we introduce a truncated modal basis to which the static responses associated with different excitations are added the efficiency of the method is illustrated on one example of a finite element model containing 10 000 degrees of freedom in the first part we update the mass and stiffness properties of the model based on eigenmodes and eigenfrequencies in the second part the damping properties are updated based on the frequency response functions of the structure

new symmetry preserving method for optimal correction of damping and stiffness matrices using measured modes measured and analytical data are unlikely to be equal due to measured noise model inadequacies structural damage etc it is necessary to update the physical parameters of analytical models for proper simulation and design studies starting from simulated measured modal data such as natural frequencies and their corresponding mode shapes a new computationally efficient and symmetry preserving method and associated theories are presented in this paper to update the physical parameters of damping and stiffness matrices simultaneously for analytical modeling a conjecture which is proposed in y x yuan h dai a generalized inverse eigenvalue problem in structural dynamic model updating j comput appl math 226 2009 42 49 is solved two numerical examples are presented to show the efficiency and reliability of the proposed method it is more important that some numerical stability analysis on the model updating problem is given combining with numerical experiments

an iterative method for updating gyroscopic systems based on measured modal data abstract an efficient iterative method for updating the mass gyroscopic and stiffness matrices simultaneously using a few of complex measured modal data is developed by using the proposed iterative method the unique symmetric solution can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrices numerical results show that the presented method can be used to update finite element models to get better agreement between analytical and experimental modal parameters

model updating via weighted reference basis with connectivity constraints abstract the paper considers the problem of updating an analytical model from experimental data using the reference basis approach in the general framework of the reference basis method certain quantities e g natural frequencies or modeshapes are considered to be completely accurate and the others are updated by solving a constrained optimization problem however the underlying structure known as connectivity existing in the model is not preserved and the method is not suited for parametric updating in this paper a method for introducing connectivity constraints into reference basis while maintaining its advantages is presented it brings the reference basis method closer to a broad class of updating methods that use parametric updating the notions of connectivity cost and parameterization cost are defined and used to obtain the best model for a given parameterization and to compare the outcomes of different parameterizations

analytical best approximate hermitian and generalized skew hamiltonian solution of matrix equation axah cych f abstract in this paper we consider the matrix equation a x a h c y c h f where a c and f are given matrices with appropriate sizes and x y is an unknown hermitian and generalized skew hamiltonian matrix pair based on matrix differential calculus and projection theorem in inner product spaces we exploit the best approximate solution x y in the set s to a given matrix pair x y where s signifies the least squares hermitian and generalized skew hamiltonian solution set of the matrix equation a x a h c y c h f the analytical expression of the best approximate solution is presented by applying the canonical correlation decomposition and the generalized singular value decomposition finally a numerical algorithm and an illustrated example are given

bypassing numerical difficulties associated with updating simultaneously mass and stiffness matrices in a model updating problem design parameters or entries of the finite element mass and stiffness matrices are tuned so that the adjusted structural dynamics model matches a set of identified modal parameters as closely as possible numerical difficulties are known to arise during this process because of the inverse nature of the problem in this paper we show that ill conditioning results from the disproportion between the mass derived and stiffness derived equations of the correction system we discuss the effect of the resulting numerical difficulties on a class of sensitivity based updating methods and propose a two step strategy to bypass them in the first step the correction system is nondimensionalized before it is solved in order to prevent large stiffness perturbations from masking mass errors in the second step the implementation of the singular value decomposition factorization is revisited to filter out all nonphysical contributions to the adjustment solution the potential of this two step strategy is demonstrated with the model refinement of a recently published planar frame benchmark problem that exhibits erroneous density parameters results obtained via the sensitivity based element by element updating method are compared with those generated by a commercially available updating software

structural model verification with linear quadratic optimization theory a graphic tool is developed and validated for the verification and updating of large structure finite element models fem for better correlation with modal test data the elaborated linear quadratic optimization lqo theory consists of a linearization and solution of the conventional matrix optimization problem that respect the connectivity of the structure and the symmetry of the stiffness matrix the lqo theory is validated through a numerical and experimental sensitivity analysis the results demonstrate the ability of the lqo algorithm to identify and update the stiffness coefficients associated with the inappropriately modeled load paths in the structure

three dimensional turbulent diffusion versus critical phenomena the critical exponent nu of the correlated or coherent length of a turbulent cluster has been related to the minimal fractal dimension dmin of multifractal isotropic turbulence the relationship has turned out to be nu dmin 3 2 for the homogeneous case dmin 3 and hence nu 1 2 the so called mean field approach of the theory of critical phenomena for dmin approximately 2 36 the well known turbulent value nu approximately 0 63 this result allows one to classify this case as the critical phenomenon of the so called thermal class of universality transition to the percolation class of universality nu approximately 0 9 is determined by the boundary conditions

the inverse eigenvalue problem for hermitian anti reflexive matrices and its approximation in this paper we first consider the inverse eigenvalue problem as follows find a matrix a with specified eigenpairs where a is a hermitian anti reflexive matrix with respect to a given generalized reflection matrix j the sufficient and necessary conditions are obtained and a general representation of such a matrix is presented we denote the set of such matrices by s a then the best approximation problem for the inverse eigenproblem is discussed that is given an arbitrary a find a matrix email protected s a which is nearest to a in the frobenius norm we show that the best approximation is unique and provide an expression for this nearest matrix

stiffness matrix adjustment using mode data a procedure is introduced that uses in addition to mode data structural connectivity information to optimally adjust deficient stiffness matrices the adjustments performed are such that the percentage change to each stiffness coefficient is minimized the physical configuration of the analytical model is preserved and the adjusted model will exactly reproduce the modes used in the identification the theoretical development is presented and the procedure is demonstrated by numerical simulation of a test problem

improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme new approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments specifically the exact eigendata i e the exact natural frequencies and mode shapes of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration this is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions once the updated matrices for the linear structure are found the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices numerical experiments show that for the same number of elements the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model alternatively the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model

frequency response of a combined structure using a modified finite element method a modified finite element method is proposed to analyze the frequency response of an arbitrarily supported linear structure carrying various lumped attachments this is accomplished by updating the finite element mass and stiffness matrices of the linear structure without any lumped attachments using its exact natural frequencies and mode shapes so that the eigensolutions of the updated linear structure coincide with the exact eigendata once the updated matrices for the linear structure are found the finite element assembling technique is then exploited to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices after assembling the global mass and stiffness matrices the frequency response of the combined system can be easily obtained numerical experiments show that using only a few finite elements the proposed scheme returns natural frequencies and frequency responses that are nearly identical to those obtained by using a finite element model with a very fine mesh the new method is easy to apply and efficient to use and it can be extended to determine the frequency response of any combined linear structure over any specified range of excitation frequencies

identification of mechanical properties of elastically restrained laminated composite plates using vibration data abstract a nondestructive evaluation method using vibration data to determine mechanical properties material and spring constants of elastically restrained laminated composite plates is presented the rayleigh ritz method in which a set of legendre s polynomials is adopted to approximate the plate deflection is used to determine the theoretical natural frequencies of the elastically restrained laminated composite plates a number of natural frequencies extracted from the impulse vibration test data of the laminated composite plates supported by elastic restraints at both the edges and centers of the plates are used in the present method to determine the mechanical properties of the plates the sum of the squared differences function which measures the differences between the experimentally and theoretically predicted natural frequencies of the elastically restrained laminated composite plates is established the identification of the plate mechanical properties is then formulated as a constrained minimization problem in which the mechanical properties are determined by making the sum of the squared differences function a global minimum the feasibility and accuracy of the proposed method are studied by means of several numerical examples on the mechanical properties identification of elastically restrained laminated composite plates with different layups made of various composite materials experimental investigation of the mechanical properties identification of several elastically restrained laminated composite plates has been performed to illustrate the applications of the present method it has been shown that the present method can produce good estimates of the mechanical properties of the elastically restrained laminated composite plates in an efficient and effective way

updating structural dynamic models with emphasis on the damping properties the control of complex structural models is a e eld of growing interest the problem studied herein concerns damping updating using experimental frequency response functions although there are several methods improving structural dynamic models only a few of them deal specie cally with damping improvements the method introduced is built on mechanical bases using an error measure on the constitutive relations the tuning strategy uses an iterative process each iteration consisting of two steps the e rst one is the localization of the erroneous regions the second step is the correction of the parameters belonging to these regions examples illustrate the sensitivity of the error to the damping defects and its effectiveness for updating damped structures

spectral basis theory for the identification of structural dynamic systems this paper presents the theoretical formulation of a technique for determining the mathematical model of structural dynamic systems this technique utilizes the fact that the forced responses of a structure can be decomposed into linear combinations of sets of frequency dependent functions the orthonormatized forms of these functions along with the associated transformation matrices are used to obtain the unique pseudoinverse of the matrix of forced responses which satisfies all the required moore penrose conditions variations of this scheme can be used to determine 1 the damping matrix when the mass and stiffness matrices are known 2 the complex stiffness matrix when the mass matrix is known 3 the mass matrix when the stiffness matrix is known and 4 the reduced order mass and stiffness matrices when the larger order matrices are known discussed are problems of redundancy and overtruncation of dynamic models and possible methods of avoiding them

estimation of in plane elastic parameters and stiffener geometry of stiffened plates abstract an investigation on stiffened plates has been conducted for the first time to determine the elastic parameters as well as the cross sectional dimensions of rectangular stiffeners from experimental modal data and finite element predictions using model updating technique the problem is formulated as a global minimization of the error function defined by the difference in modal properties as predicted from the finite element modelling to that obtained experimentally the parameter estimation problem is solved using an iterative gradient based minimization algorithm and can start from different randomly selected set of initial parameters the levenberg marquardt algorithm has been implemented to minimize the error function proper bounds are realistically chosen for the selection of the initial values of the parameters the position physical properties and orientation of stiffeners create considerable variations in the modal properties as compared with the bare plates of similar construction this makes each of the stiffened plate identification problem rather unique the position of stiffeners as well as its orientation makes certain measurement points more suitable for comparing analytical and experimental mode shapes the optimum measurement set of co ordinates which can be used for the purpose of identification is thus case specific the geometrical properties of stiffener cross section also affect the modes inclusion of different sets of modes in noisy environment and the differences in the estimated parameters are studied a few simulated examples are presented to investigate the uniqueness and convergence of results the methodology is found to be very accurate and robust

model correlation using multiple static load and vibration tests in this paper algorithmic approaches to enhance damage detection and model refinement capability are developed for cases in which experimental data from multiple tests are available the tests may include both multiple vibration tests and multiple static load tests the use of multitest measurements are shown to be one method to address a part of the incomplete measurement problem common to system health monitoring and model refinement namely the mismatch in the number of measured vibration modes in comparison to the number of modes included in the analytical finite element model key points made in the development will be highlighted using numerical experiments

finite element model updating with damping identification abstract most of finite element model updating techniques do not employ damping matrices and hence cannot be used for accurate prediction of complex frequency response functions frfs in this paper damped finite element model updating procedure is proposed and tested with the objective that the damped finite element updated model is able to predict the measured frfs accurately the proposed damped updating procedure is a two step procedure in the first step mass and stiffness matrices are updated using frf data and in the second step damping matrix is identified using updated mass and stiffness matrices that are obtained in the previous step the effectiveness of the proposed procedure is demonstrated by numerical examples as well as by actual experimental data firstly a study is performed using a numerical simulation based on fixed fixed beam structure with non proportional viscous damping model the numerical study is followed by case involving actual measured data for the case of f shaped test structure the results have shown that the proposed procedure can be used for accurate prediction of the complex frfs

an efficient iterative updating method for hysteretic damping models abstract finite element model updating techniques are used to update the finite element model of a structure in order to improve its correlation with the experimental dynamic test data this paper presents an efficient iterative method for finite element matrix updating problem in a hysteretic damping model based on a few of complex measured vibration modal data by using the proposed iterative method the unique symmetric solution can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrix triple some theorems are stated and proved numerical results show that the presented method can be used to update finite element models to get better agreement between analytical and experimental modal parameters

the matrix nearness problem for symmetric matrices associated with the matrix equation atxa btxb c d abstract a direct method based on the projection theorem in inner products spaces the generalized singular value decomposition and the canonical correlation decomposition is presented for finding the optimal approximate solution x in the set s e to a given matrix x where s e denotes the least squares symmetric solution set of the matrix equation atxa btxb c d the analytical expression of the optimal approximate solution x is obtained and an algorithm for finding this solution is also suggested

on lowest density mds codes let f sub q denote the finite field gf q and let h be a positive integer mds maximum distance separable codes over the symbol alphabet f sub q sup b are considered that are linear over f sub q and have sparse low density parity check and generator matrices over f sub q that are systematic over f sub q sup b lower bounds are presented on the number of nonzero elements in any systematic parity check or generator matrix of an f sub q linear mds code over f sub q sup b along with upper bounds on the length of any mds code that attains those lower bounds a construction is presented that achieves those bounds for certain redundancy values the building block of the construction is a set of sparse nonsingular matrices over f sub q whose pairwise differences are also nonsingular bounds and constructions are presented also for the case where the systematic condition on the parity check and generator matrices is relaxed to be over f sub q rather than over f sub q sup b

multifractal percolation and growth in intermittent media the havlin bunde multifractal hypothesis for the probability density of a random walker is used to obtain the scaling law of thepth order correlation function of the concentration for percolation and of the height for growing surfaces differences c p r r p in intermittent media it is shown that near the transition to homogeneity p a p in p po wherea andp0 are some constants good agreement with recent experiments and computer simulations of different authors is established

parametric identification of plane frames using static strains in recent years many researchers have begun to investigate the potential of system identification to determine the properties of a structure a lot of studies have adopted the modal data or the dynamic response to identify the global stiffness and mass matrices of a structure in addition to the dynamic approaches sanayei and onipede 1 sanayei and scampoli 2 hajela and soeiro 3 and soeiro and hajela 4 have suggested that static test data be used to compute the stiffness at element level since structural response can be measured precisely in a static test these methods are expected to yield reliable results furthermore it is easy to detect the damage locations from the identification results since the stiffness parameters are attained for each element

minimum rank optimal update of elemental stiffness parameters for structural damage identification a new optimal update method for the correlation of dynamic structural finite element models with modal data is presented the method computes a minimum rank solution for the perturbations of the elemental stiffness parameters while constraining the connectivity of the global stiffness matrix the resulting model contains a more accurate representation of the dynamics of the test structure and the changes between the original model and the updated model can be interpreted as modeling errors or as changes in the structure resulting from damage the motivation for the method is presented in the context of existing optimal matrix update procedures this method is distinct from past minimum rank optimal update procedures because it computes minimum rank updates directly to the elemental stiffness parameters the method is demonstrated numerically on a spring mass system and is also applied to experimental data from the nasa langley research center eight bay truss damage detection experiment the results demonstrate that the proposed procedure may be useful for updating elemental stiffness parameters in the context of damage detection and model refinement

parameter reconstruction of vibration systems from partial eigeninformation quadratic matrix polynomials are fundamental to vibration analysis because of the predetermined interconnectivity among the constituent elements and the mandatory nonnegativity of the physical parameters most given vibration systems will impose some inherent structure on the coefficients of the corresponding quadratic matrix polynomials in the inverse problem of reconstructing a vibration system from its observed or desirable dynamical behavior respecting the intrinsic structure becomes important and challenging both theoretically and practically the issue of whether a structured inverse eigenvalue problem is solvable is problem dependent and has to be addressed structure by structure in an earlier work physical systems that can be modeled under the paradigm of a serially linked mass spring system have been considered via specifically formulated inequality systems in this paper the framework is generalized to arbitrary generally linked systems in particular given any configuration of interconnectivity in a mass spring system this paper presents a mechanism that systematically and automatically generates a corresponding inequality system a numerical approach is proposed to determine whether the inverse problem is solvable and if it is so computes the coefficient matrices while providing an estimate of the residual error the most important feature of this approach is that it is problem independent that is the approach is general and robust for any kind of physical configuration the ideas discussed in this paper have been implemented into a software package by which some numerical experiments are reported

secant method adjustment for structural models on orbit testing will be required for final tuning and validation of any mathematical model of large space structures identification methods using limited response data to produce optimally adjusted property matrices seem ideal for this purpose but difficulties exist in the application of previously published methods to large space truss structures this article presents new stiffness matrix adjustment methods that generalize optimal update secant methods found in quasi newton approaches for nonlinear optimization many aspects of previously published methods of stiffness matrix adjustment may be better understood within this new framework of secant methods one of the new methods preserves realistic structural connectivity with minimal storage requirements and computational effort a method for systematic compensation for errors in measured data is introduced that also preserves structural connectivity two demonstrations are presented to compare the new methods results to those of previously published techniques

the direct updating of damping and gyroscopic matrices in this paper we develop an efficient numerical method for the finite element model updating of damped gyroscopic systems this model updating of damped gyroscopic systems is proposed to incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the damping and gyroscopic matrices that closely match the experimental modal data

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

parameter identification in distributed systems this paper describes a method for the identification of the parameters entering into the equations of motion of distributed systems because the motion of distributed systems is described in terms of partial differential equations these parameters are in general continous functions of the spatial variables for vibrating systems these parameters ordinarily represent the mass stiffness and damping distributions in this paper these distributions are expanded in terms of finite series of known functions of the spatial variables multiplied by undetermined coefficients it is assumed that the nature of the equations of motion is known and that a limited number of eigenvalues and eigenfunctions is identified in advance use is then made of the least squares method in conjunction with the eigenfunctions orthogonality to compute the undetermined coefficients thus identifying the system distributed parameters a method for the identification of the eigensolution is also presented the procedure for the identification of the eigensolution and of the system parameters is demonstrated via a numerical example

an augmented lagrangian dual optimization approach to the h weighted model updating problem abstract model updating for the quadratic eigenvalue problem aims to update the model q 2 m c k by given eigendata in this paper we consider the h weighted model updating problem which can not only preserve the symmetry and definiteness of the original model but also express our confidence in the original model through assigning different confidence weights we propose an augmented lagrangian dual method for the h weighted model updating problem under some mild assumptions our method is shown to converge at least linearly numerical results illustrate the effectiveness of our method in addition we compare our method with the semi definite programming sdp method numerical results illustrate that when the scale of the model becomes large our method still works but the sdp method failed to converge

a new method for the improvement of measured modes through orthogonalization a procedure for the optimal orthogonalization of incomplete measured modes has been described the approach achieves the smallest changes in the measured modes comparaed with other methods a closed form solution as well as an iterative algorithm are derived

model updating using uncorrelated modes abstract fe model updating techniques are used to update dynamic fe models of structures in the light of modal test data iterative methods of model updating that update a set of chosen parameters of the model so as to reduce the difference between the natural frequencies and the mode shapes of the fe model and the corresponding quantities obtained through a modal test on the structure are probably the most widely used methods once experimental modal data has been identified a necessary prior step before updating can be carried out is that of establishing the correspondence between the fe model modes and the experimentally identified modes it is however experienced that many a times a situation is encountered where not all of the modes identified through an experiment can be correlated with certainty with those predicted by the fe model and some experimental modes may be left uncorrelated there could be several reasons for this lack of correlation as identified in the paper but the consequence is that such uncorrelated modes cannot be used in fe model updating using existing iterative methods based on modal data even when they form valid known pieces of information about the structure this is a disadvantage since it reduces the quantity of experimental data available for model updating and hence makes the updating process less effective in yielding an updated model that is a closer representation of the structure this paper identifies this as a limitation of the existing iterative methods of model updating based on modal data and puts forward a notion of fe model updating using uncorrelated modes the paper proposes a solution to overcome this limitation in the form of a new method of fe model updating that accepts both correlated as well as uncorrelated modes for updating this is in contrast to all the current iterative modal data based methods of model updating that are based on the assumption of availability of correlated mode pairs and hence cannot use uncorrelated mode shapes and corresponding natural frequencies in the updating process formulation of the proposed method is described followed by a couple of numerical examples based on a beam structure to validate the method the robustness of the method in the presence of simulated noise is also studied another numerical example of a more complex f shape structure is also presented the method is then validated though an experimental study the proposed method is found to successfully update an fe model yielding correct estimates of the updating parameters in the presence of uncorrelated modes

xi code a family of practical lowest density mds array codes of distance 4 designing the lowest density maximum distance separable mds array codes has gained much attention in recent years due to the optimal redundancy and minimum update penalty lowest density of such codes however the existing lowest density mds array codes of distance 4 have extremely strict constraints on the code length which makes them impractical in particular most of them require the code length to be p or p 1 where p is a prime that satisfies i 3 p 1 and ii 2 is primitive in mathrm gf p in this paper we propose a new family of the lowest density mds array codes of distance 4 called xi code it has the properties of 1 being capable of correcting both triple erasures and a single error combined with one erasure 2 having code length of either p or p 1 with p being an odd prime number 3 achieving optimality in encoding and update and 4 achieving optimality in erasure decoding for certain erasure patterns and near optimality for other erasure patterns it is worth mentioning that xi code is the first discovered family of the lowest density mds array codes of distance 4 that supports the code length of p or p 1 with p being an odd prime

effect of general sparse matrix algorithm on optimization of space structures we investigate the effect of a general sparse matrix approach on the optimization of large structures using the optimality criteria approach and the cholesky lower upper decomposition method for the solution of the resulting linear simultaneous equations

multiple acceptable solutions in structural model improvement this paper examines the question as to whether it is possible to obtain the correct discrete linear analytical model of an actual structure by the use of analysis and test data it is shown that no such correct model exists an illustration of an application is shown where multiple acceptable and very different analytical models are all representative of the structure the implications regarding methods of improving analytical models and detecting damage are discussed

cyclic lowest density mds array codes three new families of lowest density maximum distance separable mds array codes are constructed which are cyclic or quasi cyclic in addition to their optimal redundancy mds and optimal update complexity lowest density the symmetry offered by the new codes can be utilized for simplified implementation in storage applications the proof of the code properties has an indirect structure first mds codes that are not cyclic are constructed and then transformed to cyclic codes by a minimum distance preserving transformation

improvement of a large analytical model using test data a method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure the method directly identifies without iteration a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements an application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure the resulting changes in the model are judged to be small compared to expectations of error in the analysis thus the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics in addition it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure nomenclature analytical matrix matrix of changes identity matrix full improved stiffness and mass matrices n x n full analytical k m matrices n x n partitions of ka ma corresponding to test coordinates partitions of ka ma corresponding to coupling elements partitions of ka ma corresponding to unmea sured coordinates number of measured modes number of degrees of freedom in the model measures of changes eqs 15 17 matrix norm sum of the squares of all the elements rectangular modal matrix normalized n x m th mode th column of measured and unmeasured partitions of diagonal matrix of measured natural frequencies m xm natural frequency of th mode 12 17 sum of the squares of all elements of matrix

extending sensitivity based updating to lightly damped structures the sensitivity based element by element sb ebe method is extended for updating the finite element models of lightly damped structures using experimentally measured complex modes the new updating method proceeds in two steps first the measured mode shapes are expanded and the mass and stiffness matrices are corrected to minimize a set of dynamic residuals next a damping matrix is constructed to absorb the remainder of these dynamic residuals as the mass and stiffness matrices are corrected before the computation of the damping matrix the extension is used for the updating of lightly damped structures the new sb ebe method is illustrated with two examples that highlight its potential for refining the finite element models of lightly damped structures

finite element model update via bayesian estimation and minimization of dynamic residuals an algorithm is presented for updating finite element models based upon a minimization of dynamic residuals the dynamic residual of interest is the force unbalance in the homogeneous form of the equations of motion arising from errors in the model s mass and stiffness when evaluated with the identified modal parameters the present algorithm is a modification and extension of a previously developed sensitivity based element by element sb ebe method for damage detection and finite element model up dating in the present algorithm sb ebe has been generalized to minimize a dynamic displacement residual quantity which is shown to improve test analysis mode correspondence furthermore the algorithm has been modified to include bayesian estimation concepts and the underlying nonlinear optimization problem has been consistently linearized to improve the convergence properties the resulting algorithm is demonstrated via numerical and experimental examples to be an efficient and robust method for both localizing model errors and estimating physical parameters

structural model correlation using large admissible perturbations incognate space a nonlinear perturbation method is developed to solve the problem of correlating a finite element model to a structure for which an incomplete set of natural frequencies and mode shapes and or some static deflections have been measured

improvement of analytical model using uncertain test data the localization and correction of errors in computational models are discussed a parametric optimization technique based on the sensitivity analysis of measured eigensolutions and with regard to stiffness and mass parameters of a e nite element model was used uncertain eigensolutions are considered in thelocalization process due to fuzziness present in the experimental data used to manipulate the uncertainties in the localization process a fuzzy model is introduced and discussed a numerical example is considered to illustrate the computational approach the impact of the uncertain eigensolutions on the localization process is assessed by comparing the present numerical results with those given by the sensitivity based method nomenclature b distance between the structure and its e nite element model 2r m c c 1 1 q b vector of errors fuzzy 2r m c c 1 2 c number of measured degrees of freedom dof m m k m mass and stiffness matrices of the e nite element estimation 2r n n m s k s mass and stiffness matrices of the real structure 2r n n m number of measured eigenmodes mi ki localization indicators for the ith macroelement m quantities related to the initial estimation n number of model dof r number of subdomains introduced for localization s

a model updating method for undamped structural systems in this paper we first give the representation of the general solution of the following least squares problem lsp given a full column rank matrix email protected r n x p a diagonal matrix email protected r p x p and matrices k email protected r r x r m email protected r r x r find nxn matrices k m such that email protected min s t k 1 r k 0 m 1 r m 0 where k 1 r and m 1 r are respectively the rxr leading principal submatrices of k and m we then consider a best approximation problem given nxn matrices k a m a with k a 1 r k 0 m a 1 r m 0 find email protected email protected s e such that k email protected email protected m email protected 2 inf k m s e k email protected email protected m email protected 2 where s e is the solution set of lsp we show that the best approximation solution email protected email protected is unique and derive an explicit formula for it

an iterative updating method for damped structural systems using symmetric eigenstructure assignment an iterative method for updating finite element models with measured modal results using a symmetric eigenstructure assignment technique is developed by the method the updated symmetric damping and stiffness matrices can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrices and the measured data are embedded in the updated model the numerical results show that the proposed method is reliable and attractive

submatrix approach to stiffness matrix correction using modal test data an efficient method six stiffness matrix correction to match modal testing data is presented significant reduction of unknown parameters is achieved by grouping the elements of the same stiffness characteristics and representing them as a multiplication of a scaling factor and a submatrix a concise formulation to identify the scaling factors is found by utilizing a least squares solution the formulation also incorporates a capability of reducing finite element mass and stiffness matrices to test degrees of freedom for a direct analysis experiment correlation

uniqueness aspects of model updating procedures application of model updatin g techniques to obtain improved finite element models for dynamic behavior computations is often associated with the critical observation that model update techniques can yield completely different models in the past this has been a reason for avoiding the use of such procedures in order to obtain more insight into this phenomenon a theoretical study about the uniqueness aspects of model update results is presented a first problem is treated in which the equivalence of stiffness and mass modifications is evaluated in a second step the existence of equivalent finite element models is investigated as a result practical conclusions about the quality of model update results will be drawn nomenclature frequency response function matrix k stiffness matrix ak modification of stiffness matrix m mass matrix am modification of mass matrix mk kth modal mass k k h complex eigenfrequency vk kth damped eigenfrequency i k fcth mode vector uk kih undamped eigenfrequency complex conjugate ters may lose any physical meaning difficulties of this type are related to uniqueness aspects of the model updating approach therefore this study aims at obtaining better insight into this uniqueness quality starting from the definition of identical dynamic behavior two related problems are treated first a study of equivalent mass and stiffness modifications is performed next the related problem of equivalence of finite elemen t models is considered from this study interesting conclusions for practical application of model updating techniques can be drawn related to the necessity of updating mass and or stiffness parameters simultaneously and about the importance of the dimensionality of the vector space of updating parameters as a controlling variable of the updating process i introduction a pplication of finite element models for computation of dynamic behavior is becoming more and more common in industry however practical use of these techniques has made it clear that a thorough verification of the accuracy of these models is still necessary a realistic method is to compare finite element results eigenfrequencies mode shapes generalized masses etc with information obtained from high quality modal survey tests if unacceptable discrepancies occur a model updating step can be considered in which the model parameters are adjusted for optimal improvement of correlation these updated models can then serve as a basis for further analysis of the design under study coupling modification prediction etc several approaches have been suggested to perform the model updating task a classification of these models based on the algorithm applied is suggested by heylen 1 another classification is based on the type of updating parameters selected individual mass and or stiffness matrix elements 2 6 submatrices of the global mass and or stiffness matrices 7 8 or the finite element input data 9 one of the essential obstacles to the increased use of these techniques is the observation that in general it is very difficult to interpret the update of results or that the updated parame

structural damage detection of space truss structures using best achievable eigenvectors a method is presented by which measured modes and frequencies from a modal test can be used to determine the location and magnitude of damage in a space truss structure the damage is located by computing the euclid ean distances between the measured mode shapes and the best achievable eigenvectors the best achievable eigenvectors are the projection of the measured mode shapes onto the subspace defined by the refined analytical model of the structure and the measured frequencies loss of both stiffness and mass properties can be located and quantified to examine the performance of the method when experimentally measured modes are employed various damage detection studies using a laboratory eight bay truss structure were conducted the method per forms well even though the measurement errors inevitably make the damage location more difficult uture structures in space will be orders of magnitude larger and more complex than their predecessors structures such as space station freedom will typically be built around a large flexi ble frame and consist of habitation and experiment modules flexi ble and articulating appendages and numerous utility trays and pay loads along with the need to design the spacecraft to be light weight strong and modular for ease of expansion repair and modification the complexity and size of these structures require that the integrity of the structures be monitored periodically several researchers have studied the possibility of detecting damage or structural anomalies to large space trusses and locating the site of this damage based on changes in the natural frequencies and mode shapes of the structures references 1 and 2 provide a comprehensive overview of many of the proposed methods in this respect and the issues involved with on orbit modal identification and structural verification there are several distinct classes of model improvement damage location methods each of these methods begins with an initial finite element representation of the structure in the case of model improvement the goal is to use test data of the actual structure to modify this initial finite element model so that it may serve as an accurate representation of the un damaged structure in the case of damage location the goal is to use test data from the damaged structure and the updated corre lated finite element model of the undamaged structure to deter mine changes to the stiffness and or mass matrices that would ac count for the observed differences in the measured modes and frequencies the class of update methods that has been explored most widely is known as optimal matrix modification the goal of the optimal matrix modification is to find an updated matrix stiffness and or mass that is as close as possible to the original matrix assumed to be almost correct that accurately produces the measured modes

wheat stem moduli in vivo via reference basis model updating a novel structural analysis method was developed and used to infer the in vivo material properties of an important crop species in particular reference basis model updating was used to reconcile the predictions of an idealized analytical model with the observed forced vibration response of living wheat and thereby determine the effective longitudinal moduli of the tissues comprising the vibrating structure the ensuing study of two varieties of wheat found that the stem tissue moduli decreased nonlinearly with height these findings were in good qualitative agreement with the limited data available in the literature and the updating procedure was highly effective in this unusual nondestructive testing application to our knowledge this is the first time that the reference basis method has been used to infer the spatially varying material properties of a living plant moreover similar approaches could be used to characterize spatial variations in the material properties of other lightweight composite structures of arbitrary geometry

direct identification of non uniform beams using static strains abstract a method for identifying the flexural rigidities of a non uniform beam is developed in this paper the finite element method is used to derive the equilibrium equation of the beam the identification problem is then formulated as an optimization program in which the error norm of the equilibrium equation is minimized the proposed method has the following features 1 it uses longitudinal strains as the primary test data 2 the solution is unique and global minimal 3 the method can be applied to a whole beam as well as a part of a beam 4 the method is very stable the identifiability of the inverse problem is studied in depth two numerical examples are presented to demonstrate the proposed method finally a model test is performed to examine the effectiveness of the method in real applications

probability densities of random walks in random systems abstract we review recent results for the probability distribution of random walkers in random systems where diffusion is anomalous and the mean square displacement scales with time as r 2 t t 2 w d w 2 the random systems are characterized by structural disorder and by random transition rates in general the mean distribution function p r t of the random walkers is a stretched gaussian and scales as log p r t p r 0 r r t u where u d w d w 1 on random fractals the fluctuations of the density distribution p r t for fixed distance r and time t have a broad logarithmic distribution the average moments p q scale in a multifractal way as p q where q q l space the fluctuations of p are narrow and p q p q

location of modeling errors using modal test data the method uses the cross unity check between a flexibility matrix derived from modal test data and the analytical stiffness matrix to locate the errors the method cannot determine the changes needed to correct the errors sensitivity of the method to lack of orthogonality in measured modes is discussed