a nonconfocal generator of involutive systems and levy hierarchy a new nonconfocal generator of involutive systems is obtained and the relation between it and levy equation hierarchy is discussed furthermore the representation of the solution for the levy hierarchy is given

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

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

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 assessment of a nonlinear space antenna structure using quasi linearization methods health monitoring of structural systems for the detection of damage is a critical practice in determining their safety and service life a large amount of research has been done in applying damage detection analysis to linear models of structures although many real structures exhibit nonlinear behavior before or as a result of damage there are currently few tools to apply damage detection analysis to structures that are modeled by nonlinear differential equations the objective of this study is to initiate a facet of damage detection analysis applied to such nonlinear models of structures given a parameterized ordinary differential equation model of a structural system linear or nonlinear parameter estimation can serve as an assessment of possible or known damage to the structure a numerical analysis of estimating parameters in a nonlinear structural model is investigated by applying a quasi linearization method the analysis is successfully performed on a nonlinear eight degree of freedom space antenna model

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

investigation of equivalent system modeling and dynamic characteristics using reduced models animproved approach of updating theinitial e niteelementmodelsincomparison with amodaltestispresented this is accomplished by using the reduced models dynamically equivalent to the initial models the degrees of freedom dof in the reduced models include the generalized coordinates besides the measurement points for a modal test therefore these reduced models can take into account the inertia force at unmeasured dof and preserve the dynamic characteristics of the initial analytical models regardless of the measurement points these reduced models are used for a direct analysis experiment correlation with modal test data numerical simulation resultsarepresentedto showtheeffectivenessofthesereducedmodelsintheparametriccorrectionoftheanalytical e nite element models nomenclature k guyan reduced stiffness matrix ka nominal stiffness matrix of the initial e nite element model kb stiffness matrix for exactly modeled parts in the initial e nite element model k j jth submatrix for the assumed stiffness in the initial e nite element model kk stiffness matrix of the test models of k degreesof freedom dof kk stiffness matrix of the present tam of k dof kn stiffness matrix of the initial e nite element models of n dof kn stiffness matrix of the craig bampton e nite element models of n dof m guyan reduced mass matrix

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

birth delay of a markov process and the stopping distributions for regular processes we consider markov processes with a fixed transition functionp r x t b and with random birth times we show that a process tilde x t tilde p can be obtained from x t p by birth delay if and only if tilde p left tilde x t in b right leqq p left x t in b right for allt andb as an application we give a new version and a new proof of the results of rost r and fitzsimmons f2 on stopping distributions of markov processes the key lemma 1 1 replaces the filling scheme used by the previous authors birth delay was considered from a different prospective in f1

nonlinear perturbation theory for structural dynamic systems a novel nonlinear perturbation theory for structural dynamic systems is developed that can provide an exact relationship between the perturbation of structural parameters and the perturbation of modal parameters a system of governing equations based on the developed theory is further derived which can be utilized for general applications such as structural reanalyses eigendata modification model updating and damage identification suitable for all types of structures including mechanical systems framed structures and continua neither model reduction nor mode shape expansion is required for modal updating and damage identification because information about incomplete measured modal data can be directly employed the developed theory successfully avoids adopting a taylor series expansion procedure and then the derivatives of modal parameters are not required computational procedures based on the derived nonlinear governing equations are presented for eigendata modification model updating and damage identification the jacobi transformation method and the accelerated modal method are introduced to make the proposed techniques particularly suitable for cases with a very large perturbation of structural parameters finally two numerical examples are given to demonstrate the effectiveness of the proposed techniques the results show that the modified modal parameters can be predicted exactly even for cases with a large modification of structural parameters and the analytical model can be adjusted correctly using the information about limited modal data available

bi hamiltonian formulation of the kadomtsev petviashvili and benjamin ono equations it was shown recently that the kadomtsev petviashvili kp equation an integrable equation in 2 1 i e in two spatial and one temporal dimensions admits a bi hamiltonian formulation this was achieved by considering kp as a reduction of a 3 1 dimensional system in the variables x y1 y2 t it is shown here using the kp as a concrete example that equations in 2 1 possess two bi hamiltonian formulations and two recursion operators both hamiltonian operators associated with the x direction are local in contrast only one of the hamiltonian operators associated with the y direction is local furthermore using the benjamin ono equation as a concrete example it is shown that intergrodifferential equations in 1 1 admit an algebraic formulation analogous to that of equations in 2 1

issues in modal identification of flexible structures issues associated with implementation of modal parameter identification procedures that are based upon discretized models are discussed the interest lies in methods where the identification is carried out as a correction process on a postulated model generated by using erroneous system parameters it is shown that the identification approach loses its accuracy because a discretized model has to be used to represent the structure and because extraction of the modal information from the sensors output introduces contamination from the unmodeled modes the level of this inaccuracy is investigated qualitatively and quantitatively

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

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

method for determining minimum order mass and stiffness matrices from modal test data a new method for determining mass and stiffness matrices from modal test data is presented the method builds on the identified modes and the mass normalized mode shapes at the sensor locations and is not limited by either the number of driving points or measurement points and so it is applicable to most general test settings a mixed coordinate basis is defined for the mass and stiffness matrices which is analogous to the craig bampton component mode synthesis method for finite element models the resultant mass and stiffness are of minimal order necessary to span the measured modes and the resulting generalized coordinates provide an objective basis for the test derived matrices to be used as if they are component mode synthesized finite element matrices inclusion of rigid body modes and the relationship of the new method to traditional physical parameter computations based on mobility curves is considered examples of the method as applied to numerical and experimental data are provided

an integral equation for changes in the structural dynamics characteristics of damaged structures this paper presents a relationship between the structural dynamic characteristics of damaged and intact structures such a relationship is derived in the form of an integral equation and is used to develop a method of flaw identification the resulting flaw identification procedure uses the experimental data and relates the flaw to the observed structural dynamic response the algorithm enables one to detect the damage locations and the corresponding damage magnitude simultaneously numerical simulations and experiments are conducted to validate the theory

branching particle systems and superprocesses we start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location then we consider a limit case when the mass of particles and their life times are small and their density is high this way we construct a measure valued process x t which we call a superprocess replacing the underlying markov process xi t by the corresponding historical process xi leq t we construct a measure valued process m t in functional spaces which we call a historical superprocess the moment functions for superprocesses are evaluated linear positive additive functionals are studied they are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path say at the first exit time from a domain a related special markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations the concluding section is devoted to a survey of the literature and the terminology on markov processes used in the paper is explained in the appendix

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

classical liouville completely integrable systems associated with the solutions of boussinesq burgers hierarchy two new finite dimensional completely integrable systems in the liouville sense are obtained the solutions of boussinesq burgers hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems

objective functions for the nonlinear curve fit of frequency response functions iv discussion and conclusions it is concluded that the theoretical model as described herein predicts frequencies and mode shapes that are in good agreement with experiment it has been observed that accurate prediction of higher mode frequencies for plates of the type discussed here often requires inclusion of the effects of rotary inertia of attached masses in the analysis this is not surprising as the masses will undergo harmonic rotation for some of the higher modes extension of the analysis to include these rotary inertia effects is currently underway to the authors knowledge the work reported here represents the first comprehensive analytical and experimental study of this interesting and timely problem the analysis provides the designer with a powerful tool for optimizing the distribution of rigid point supports and attached masses on the plate surface the experimental data will provide other researchers with reference points against which they can compare their theoretical results

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

complex confocal involutive systems associated with the solutions of the akns evolution equations in this paper the complex involutive systems associated with the ablowitz kaup newell and segur akns evolution equations are discussed by means of the complex representations of the standard symplectic form and the standard poisson bracket on the real space r2n using the equivalent relations between the hamiltonian canonical equations it is proved that the complex involutive systems are exactly finite dimensional completely integrable systems in the liouville sense therefore by mapping nonlinearly from the involutive solutions of the compatible hamiltonian canonical equations to the akns evolution equations the representations of the solutions of the akns evolution equations are obtained

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

dual approaches to finite element model updating the problem of finding the least change adjustment to a stiffness matrix modeled by finite element method is considered in this paper desired stiffness matrix properties such as symmetry sparsity positive semidefiniteness and satisfaction of the characteristic equation are imposed as side constraints of the constructed optimal matrix approximation for updating the stiffness matrix which matches measured data better the dual problems of the original constrained minimization are presented and solved by subgradient algorithms with different line search strategies some numerical results are included to illustrate the performance and application of the proposed methods

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

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

a method of non linear modal identification from frequency response tests abstract a theoretical and experimental study of non linear systems governed by duffing s equation is presented in this study the non linear mode concept is applied to analyze the steady state responses of non linear systems driven by a harmonic force in the experimental context the present modal identification method permits one to obtain the non linear modal parameters from frequency response tests by comparison with other methods the proposed methods appear very interesting in regard to computational time formulation and they necessitate fewer computer resources an experimental work on the steady state response of a beam with local non linear stiffness is reported to justify the methods many aspects of theoretical results such as the dependence of resonance frequency on the amplitude of vibration and the downward and upward jump phenomena are observed experimentally the experimental results are in good agreement with theoretical predictions based upon the non linear modal superposition approach

selection of noisy measurement locations for error reduction in static parameter identification an incomplete set of noisy static force and displacement measurements is used for parameter identification of structures at the element level measurement location and the level of accuracy in the measured data can drastically affect the accuracy of the identified parameters a heuristic method is presented to select a limited number of degrees of freedom dof to perform a successful parameter identification and to reduce the impact of measurement errors on the identified parameters this pretest simulation uses an error sensitivity analysis to determine the effect of measurement errors on the parameter estimates the selected dof can be used for nondestructive testing and health monitoring of structures two numerical examples one for a truss and one for a frame are presented to demonstrate that using the measurements at the selected subset of dof can limit the error in the parameter estimates

alternating direction method for structure persevering finite element model updating problem the problem of finding the optimal approximation to the discrete stiffness matrix modeled by the finite element method is considered in this paper desired properties of the updated matrix including symmetry positive semidefiniteness and structure connectivity are imposed as side constraints besides these the optimal approximate matrix should be the least squares solution to the dynamics equation to the best of the author s knowledge the optimal matrix approximation problem containing all these constraints simultaneously has not been proposed in the literature earlier alternating direction method is first applied to this constrained minimization problem numerical examples are performed to illustrate the efficiency of the proposed method

inherently incomplete finite element model and its effects on model updating some of the questions regarding the use of measured data to improve a finite element model of a structure to make it better able to predict the dynamic behavior of the structure are addressed the issues discussed include the following all such models are truly reduced models the parameters of a reduced model of a linear structure vary with frequency a linear model of a linear structure has a limited frequency range of applicability and there are an infinite number of reasonable models that can match measured data these and other related concepts are discussed and demonstrated of course because there is no unique solution questions arise as to how to select the best one and are there different best models for different applications it is concluded that new paths of research in this important field are necessary

structural damage detection based on static and modal analysis the present paper describes an approach for structural damage assessment that has its basis in methods of system identification response of a damaged structure differs from predictions obtained from an analytical model of the original structure where the analytical model is typically a finite element representation the out put error approach of system identification is employed to determine changes in the analytical model necessary to minimize differences between the measured and predicted response structural damage is represented by changes in element stiffness matrices resulting from variations in geometry or material properties of the structure during damage measurements of static deflections and vibration modes are used in the identification procedure the identification methodology is implemented for representative structural systems principal shortcomings in the proposed approach and methods to circumvent these problems are also discussed of obtaining poor results these difficulties are clearly evi denced by the results obtained smith and hendricks8 follow a similar approach using two different identification methods to identify the stiffness matrix based on the minimum deviation approach and using eigenmodes as experimental data similar difficulties are re ported in their work the entries of the stiffness matrix corre sponding to the damaged members do show considerable vari ations however entries corresponding to undamaged members are also affected thereby making the damage detec tion process more uncertain the analysis of changes in the stiffness matrix is typically cumbersome may not always yield correct answers and does not permit the determination of the extent of damage this paper presents an approach that is designed to circum vent the problems just discussed the output error method or structural identification9 is used wherein the analytical model is refined to minimize the difference between the predicted and measured response of the structure iterative nonlinear pro gramming methods are employed to determine a solution to the unconstrained optimization problem damage is repre sented by reduction in the elastic extensional and shear moduli of the element and those are designated as the design varia bles of the problem the use of static structural displacements as the measured response is a departure from the standard practice of using eigenmodes alone for the identification pro blem numerical evidence clearly indicates that when eigen modes alone are used for identification the location and ex tent of damage predicted by the optimization approach is dependent on the number of modes used to match the measured and the predicted response higher modes are diffi cult to determine and measure and the use of static displace ments obtained by a loading that simulates higher modes is proposed as a solution to this problem the paper also presents an implementation of the proposed damage assessment strategies with special focus on problems of practical significance in this context the use of incomplete modal or static displacement information in the identification problem is discussed further the approach of treating the modulus of each structural element as an independent design variable results in a large dimensionality problem this results in significant computational costs when using a gradient based nonlinear programming algorithm for function minimization the use of a reduced set of dominant design variables and the construction of equivalent reduced order models for damage assessment are explored with some success

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

a generalized inverse eigenvalue problem in structural dynamic model updating this paper is concerned with the problem of the best approximation for a given matrix pencil under a given spectral constraint and a submatrix pencil constraint such a problem arises in structural dynamic model updating by using the moore penrose generalized inverse and the singular value decomposition svd matrices the solvability condition and the expression for the solution of the problem are presented a numerical algorithm for solving the problem is developed

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

method for structural model update using dynamically measured static flexibility matrices a new method is presented for parametric correction of full order analytical stiffness matrices from reduced order dynamically measured static flexibility matrices the measured static flexibility matrix is formed using modal data in conjunction with a residual flexibility estimate the algorithm corrects model parameters by minimizing a static flexibility matrix error residual constructed for measurement degrees of freedom only by utilizing a flexibility matrix residual instead of one derived using stiffness matrices numerical problems associated with inverting a rank deficient measured flexibility matrix are avoided posing the problem in terms of flexibility matrices also avoids the problems of modal correspondence mode selection and modal truncation in addition incorporating static reduction relationships into the error residual makes prior eigenvector expansion or model reduction unnecessary two formulations for the error residual are presented an explicit inverse formulation and one derived from a pseudoinverse relationship the second leads to an exact linear update problem when all of the model degrees of freedom are measured numerical simulation results demonstrate that both formulations are capable of localizing and quantifying local stiffness errors in a full order finite element model from reduced order measurements experimental results for a cantilever beam structure are also presented

a flexibility based continuum damage identification approach the present work presents a flexibility based continuum damage identification approach here it is considered that the current integrity state of a structure is described by means of a continuum damage model this model enables one to parameterize the stiffness matrix of the finite element fe model of the system such that in an updating process properties such as symmetry and sparsity of the original fe model are naturally preserved the damage identification is accomplished by minimizing the frobenius norm of the difference between the experimental and analytical flexibility matrix only related to the measured degrees of freedom dof of the structure by utilizing an error only related to the measured dof one avoids the need for model expansion or model reduction difficulties commonly present in the model updating process the modal information required for the experimental flexibility matrix calculation are obtained by means of eigensystem realization algorithm era and common based structure identification cbsi the assessment of the present approach has been performed by means of simulations on a beam like structure considering typical shortcomings occurring in real applications such as small number of sensors different levels of signal to noise ratio and limited spectral information it has also been analyzed the use of a regularization term in the error function to be minimized in order to improve the performance of the proposed method

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

finite band solutions of the classical boussinesq burgers equations the classical boussinesq burgers hierarchy is decomposed into two systems of solvable ordinary differential equations with the help of the lax representations of stationary evolution equations from which the finite band solutions of the higher order classical boussinesq burgers equations are obtained

reductions of 2 1 dimensional integrable systems via mixed potential eigenfunction constraints new types of reductions of 2 1 dimensional integrable systems which are associated with mixed potential eigenfunction constraints are discussed necessary and sufficient conditions for the admissibility of such constraints are presented classification results for integrable equations will be used for applying these conditions several typical examples are considered including the kp mkp 2 1 dimensional savada kotera kaup kupershmidt harry dym and nizhnik veselov novikov equations

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

a constrained matrix least squares problem in structural dynamics model updating a constrained matrix least squares problem in structural dynamics model updating is considered in this paper desired matrix properties including the satisfaction of the linear equation symmetric positive semidefiniteness and sparsity are imposed as side constraints alternating projection method is applied to solve the constrained minimization problem the results of the numerical examples show that the proposed method works well all necessary constraints are considered in our model alternating projection method is first used to update the finite element model numerical results are compared with the recent results

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

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

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

enhanced structural damage detection using alternating projection methods alternating projection algorithms are examined for the solution of damage detection problems in structures the damage detection problem is formulated as a feasibility problem to find a damaged stiffness matrix that is close to the refined stiffness matrix of the undamaged structure and that satisfies the necessary symmetry sparsity positive definiteness eigenequation and damage localization constraints alternating projection methods are proposed to utilize the orthogonal projections onto these constraint sets in an iterative fashion to find a solution that best satisfies these constraints in addition directional alternating projections that exploit the geometry of the damage detection feasibility problem are introduced to improve the computational efficiency of the approach the techniques are applied to detect damage in a simulated cantilever beam model and in the nasa eight bay truss damage detection experimental test bed

two finite dimensional completely integrable hamiltonian systems associated with the solutions of the mkdv hierarchy in this paper the equations of the mkdv hierarchy and their lax pairs are obtained from zero curvature equations using trace formulas of the spectral problem and the lax pairs two finite dimensional completely integrable systems are found the solving equations of the mkdv hierarchy are transformed into the solving hamiltonian canonical equations

the complete characterization of a general class of superprocesses in an important paper 7 dynkin kuznetsov and skorohod showed that under mild conditions the log laplace functional of every branching measure valued process is the solution of an evolution equation determined by three parameters q and l this paper essentially deals with the converse of this result we consider a general class of bmv subject to some mild conditions first we derive from 7 that every process x is a k superprocesses where the triples k are subject to some conditions ans1 ans3 conversely we show that for each of these triples satisfying ans1 ans3 there exists a version x of the k superprocess which belongs to consequently ans1 ans3 fully characterizes and subject to mild conditions bmv and superprocesses are equivalent concepts this requires to prove a general existence theorem for superprocesses the existence of a regular version of these processes and that for processes in branching characteristics q and l are continuous

comparing model update error residuals and effects on model predictive accuracy the effect of error residual choice on the predictive capability of an updated structural model is examined first analytical expressions are developed that relate errors in dynamically measured static flexibility matrices stiffness matrices and modal matrices the analysis shows that flexibility stiffness and modal error residuals correspond to unique weightings on the error in the measured modal data next the flexibility weighting indicator and the stiffness weighting indicator are defined to provide a simple means of ranking individual modes in terms of their potential contribution to errors in a measured static flexibility or stiffness matrix finally expressions are developed for bounding model prediction errors when static flexibility stiffness or modal error residuals are used the results show that the amount of uncertainty in predicted static displacements static loads and mode shapes and frequencies is directly related to the choice of error residual and is different in each case these results are also illustrated using numerical simulations for a modified version of kabe s problem which includes model form errors

submatrix constrained least squares inverse problem for symmetric matrices from the design of vibrating structures abstract given a full column rank matrix x r n m a matrix b r m m and a symmetric matrix a 0 r p p in structural dynamic model updating yuan and dai 2007 considered the matrix equation x t a x b with a leading principal submatrix a 0 constraint for a given matrix a r n n they updated the mass and stiffness matrix in the frobenius norm sense such that the corrected matrices satisfy the generalized eigenvalue equation and orthogonality conditions but due to measurement errors the measured mass and stiffness matrices will not always satisfy these requirements since they still contain some useful information we would like to retrieve their least squares approximations to correct these matrices then we obtain the least squares symmetric solutions of the equation x t a x b with a trailing principal submatrix a 0 constraint by using the matrix differential calculus and canonical correlation decomposition furthermore by applying the generalized singular value decomposition and projection theorem we get the best frobenius norm approximate symmetric solution of this equation according to a given matrix a r n n with a 0 as its trailing principal submatrix finally a numerical algorithm for computing the best approximate solution is established some illustrated numerical examples are also presented

direct least squares formulation of a stiffness adjustment method a direct least squares formulation of the stiffness matrix adjustment method known as the kma method kabe a m stiffness matrix adjustment using mode data aiaa journal vol 23 no 9 1985 pp 1431 1436 is presented the kma method belongs to a class of procedures that refine dynamic models using test measured modes and structural connectivity viewed as a constrained minimization problem most of these methods have applied lagrange multiplier techniques in their development it is shown that these two approaches result in linear systems of equations that are algebraically equivalent and therefore have identical solutions by virtue of this equivalence the direct least squares formulation of the kma method is shown to be equivalent to its original lagrange multiplier formulation the direct least squares versions of other model refinement methods that preserve structural connectivity are also discussed in this respect this approach provides a rigorous framework for unifying these optimal update procedures and indicates how their solutions should compare numerical results are presented that illustrate the equivalence between the two formulations and also the conditions under which the various methods yield identical solutions the results also indicate that because of improved numerical conditioning the direct least squares approach yields more accurate computational solutions and generally requires less computer storage than methods that were developed using lagrange multipliers

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

on regularity of superprocesses three theorems on regularity of measure valued processesx with branching property are established which improve earlier results of fitzsimmons f1 and the author d5 the main difference is that we treatx as a family of random measures associated with finely open setsq in time space heuristically x describes an evolution of a cloud of infinitesimal particles to everyq there corresponds a random measurex which arises if each particle is observed at its first exit time fromq the statex t at a fixed timet is a particular case we consider a monotone increasing familyq t of finely open sets and we establish regularity properties of bar x t x tau t as a function oft the results are used in d6 d7 and d10 for investigating the relations between superprocesses and non linear partial differential equations basic definitions on markov processes and superprocesses are introduced in sect 1 the next three sections are devoted to proving the regularity theorems they are applied in sect 5 to study parts of superprocess the relation to the previous work is discussed in more detail in the concluding section it may be helpful to look briefly through this section before reading sects 2 5

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

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

damage detection in elastic structures using vibratory residual forces and weighted sensitivity a methodology is presented for detecting structural damage in elastic structures by nondestructiv e means measured modal test data along with a correlated analytical structural model are used to locate potentially damaged regions using residual modal force vectors and to conduct a weighted sensitivity analysis to assess the extent of mass and or stiffness variations where damage is characterized as a stiffness reduction the current approach is unique among other approaches in that it accounts for 1 variations in system mass system stiffness and mass center locations 2 perturbations of both the natural frequencies and modal vectors and 3 statistical confidence factors for the structural parameters and potential experimental instrumentation error moreover this procedure can be used with either full or reduced models a wide variety of numerical examples are presented that show that the current method provides a precise indication of both the location and the extent of structural damage lexible space structures launch vehicles and satellites are susceptible to structural damage over their operating lives from impact operating loads and fatigue undetected and uncorrected damage can lead to structural deterioration of their members and consequently jeopardize the flight safety of the structure thus numerous damage inspection methods and monitoring procedures have been developed examples include x ray ultrasonic testing magnetic resonance coin tap dye penetration and visual inspection these methods can be time consuming and are local assessments often requir ing the exposure of structural elements to the inspector and equipment for detecting damage as a monitoring system for detecting damage of spacecraft in orbit i e space station freedom none of these methods are appropriate an alterna tive approach which formed the basis for our current study is to recognize the fact that modal vibration test data struc tural natural frequencies and mode shapes characterize the state of the structure 1 therefore postflight and in flight e g monitoring data can be used to distinguish whether changes damage have occurred to the structure by comparing these data with a set of baseline data modal testing as a means of inspection has several advantages direct exposure of struc tural elements is not required and at the same time more of the complete structure can be inspected in one modal test by having appropriately placed sensors the consequences of this are a reduction in schedule and cost the use of vibration test data to locate structural damage has been attempted to evaluate the integrity of offshore oil and gas platforms 2 3 composite laminates 4 continuum struc tures 5 8 and recently large space structures 9 11 the early ana lytical and experimental approaches2 3 had limited success as a result of inadequate instrumentation and addressing only fre quency shifts associated with stiffness changes while ignoring changes in the modal vectors the continuum based meth ods5 8 provide valuable insight into the magnitude of fre quency shifts but their usefulness is severely limited to simple structures bars beams plates with unrealistic boundary con

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

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

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