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

a framework for iterative analysis of non classically damped dynamical systems abstract in this paper we propose a general iterative framework to solve the dynamic problem for linear systems with non classical viscous damping a systematic approach is used to derive families of stationary iterative schemes that as an instance of particular interest decouple the equations of motion for numerical study of the system response for such schemes we present a detailed convergence analysis and propose several solution strategies suitable for a broad class of systems these techniques are based on spectral analysis of particular iteration matrices arising in the derivation and aim at optimizing the convergence performance of the method we demonstrate that the proposed systematic framework based on a novel application of the homotopy analysis method generalizes iterative schemes previously reported in the literature and importantly provides a unified perspective for the study of iterative solutions of dynamic problems further we establish a connection between our results and the theory of iterative schemes for algebraic linear systems thus providing insights on convergence results and applicability of the method numerical examples illustrate the effectiveness of the approach and indicate future research directions

eigenderivative analysis of asymmetric non conservative systems in general the damping matrix of a dynamic system or structure is such that it can not be simultaneously diagonalized with the mass and stiffness matrices by any linear transformation for this reason the eigenvalues and eigenvectors and consequently their derivatives become complex expressions for the first and second order derivatives of the eigenvalues and eigenvectors of these linear non conservative systems are given traditional restrictions of symmetry and positive definiteness have not been imposed on the mass damping and stiffness matrices the results are derived in terms of the eigenvalues and left and right eigenvectors of the second order system so that the undesirable use of the first order representation of the equations of motion can be avoided the usefulness of the derived expressions is demonstrated by considering a non proportionally damped two degree of freedom symmetric system and a damped rigid rotor on flexible supports copyright 2001 john wiley sons ltd

numerical presentation of eigenvalue curve veering and mode localization in coupled pendulums to be 0 01 0 01 0 01 0 0 and 0 01 0 the e rst variations and the eigenvalues obtained by the classic method and the improved calculation are given in table 1 the relative errors of the variations are shown in fig 2 the result of the improved calculation is in the very good agreementwith the exact one apartfromthe third mode the e rst variations given by the improved calculation are even better than those with the second variations by the classic method a large modie cation of the truss is considered by multiplying a and b with a factor from 1 through 10 the relative errors of the e rst variations are shown in fig 3 the errors in the e rst mode increase from 0 0827 to 0 8410 whereas those of the classic method change from 0 4159 to 4 1834 the fourth mode shows the biggest difference as the largest error of the improved calculation is 0 0023 but 9 9982 for the classic method conclusions an improved calculation of eigenvalue variations has been presented the eigenvalue approximation from the e rst variation is a rayleigh quotient the variations are different from those in the classic method when there is a variation of b which is the inertiarelated matrix in a structural dynamics although there are more terms in the variations of eigenvalues no extra matrix operation is required thegivenexamplecone rmsthattheaccuracyoftheeigenvalue variations is much better than that obtained with the classic method

approximate eigensolutions for arbitrarily damped nearly proportional systems two efficient methods for determining the approximate eigenvalues and eigenvectors for arbitrarily damped nearly proportional systems are developed both approaches are formulated by means of a first order perturbation technique whereby the real modes of vibration of the undamped system are used to derive approximate expressions for the complex eigenvalues and eigenvectors of a nearly proportionally damped system using either approach the unperturbed configuration corresponds to a damped one whose damping matrix can be diagonalized by the same transformation that uncouples the undamped system and the perturbation consists of the deviation of this diagonalizable damping matrix from the actual damping matrix the proposed approaches are easy to code implement and solve and do not require forming state equations numerical examples are presented to validate the effectiveness of the current methods

on the asymptotic behaviour of spitzer s model for evolution of one dimensional point systems a nearest neighbor gradient dynamics of one dimensional infinite particle systems is considered the model admits a two parameter family of stationary configurations some domains of attraction of stationary configurations are described and the continuum hydrodynamical limit of the system is investigated it is shown that the mean density of points satisfies a nonlinear diffusion equation in the hydrodynamical limit

particularities of newton s method in space frame force determination utilising eigenpair functions abstract redundant space frame dynamics are modelled through the structural eigenproblem which is formulated to account for the geometric stiffness due to self equilibrating membrane forces the resulting mathematical eigenvalues and eigenvectors respectively indicative of natural frequencies and modes of oscillation are then assumed to be continuous functions of a scalar factor upon a normalised set of forces in equilibrium newton s method or sensitivity analysis provides a means for minimising the difference between mathematical and physically observed eigenvalues whereupon the axial forces are inferred from the model issues of convergence and root uniqueness are anticipated eigenvalue coalescence in which eigenvalues transitorily coalesce to permute their arbitrarily numerical ordering is seen to define non smooth eigenvalue functions and hence encumber newton s method mode tracing strategies to overcome this are discussed the existence of various degrees of eigenvalue degeneracy owing to frame cyclic symmetry or otherwise is anticipated in terms of the influence upon newton s method if the degree of frame static redundancy is greater than one and there exist a number of linearly independent force distributions then the required force determination is a multidimensional newton method problems encountered in these manifold dimensions including those arising from frame spatial periodicity and the adversity of newton s method are overcome emphasis is placed upon minimising the dimensionality of newton s method through enforcement of equilibrium constraints illustrative numerical simulations are given and some applications of the method are proposed

second order sensitivity analysis in vibration and buckling problems a perturbation approach for the first and second order sensitivity analysis of eigenvalue problems as they arise in buckling and vibration of structural systems is presented a particular feature introduced in the formulation is the dependence of all matrices that model the problem not only on the design parameter but also on a pre critical state thus the sensitivity of the response of the fundamental path in buckling problems or a pre stressed state in vibration problems are introduced in the analysis two techniques are considered in this work the direct and the adjoint methods first and second order sensitivity equations for eigenvalues and eigenvectors are obtained and it is shown that second order sensitivity of an eigenvalue can be accomplished by using only one adjoint problem

co ordinate transformations for second order systems part i general transformations abstract when the dynamics of any general second order system are cast in a state space format the initial choice of the state vector usually comprises one partition representing system displacements and another representing system velocities co ordinate transformations can be defined which result in more general definitions of the state vector this paper discusses the general case of co ordinate transformations of state space representations for second order systems it identifies one extremely important subset of such coordinate transformations namely the set of structure preserving transformations for second order systems and it highlights the importance of these it shows that one particular structure preserving transformation results in a new system characterized by real diagonal matrices and presents a forceful case that this structure preserving transformation should be considered to be the fundamental definition for the characteristic behaviour of general second order systems in preference to the eigenvalue eigenvector solutions conventionally accepted

efficient modal analysis of systems with local stiffness uncertainties the characterization of the uncertainty in modal quantities of an uncertain linear structural system is essential to the rapid determination of its response to arbitrary loadings although the size of many computational structural models used is extremely large i e thousands of equations the uncertainty to be analyzed is oftentimes localized to very small regions of the model this paper addresses the development of an efficient computational methodology for the modal analysis of linear structural systems with local stiffness uncertainties the newly developed methodology utilizes an enriched basis that consists of the sub spectrum of a nominal structural system augmented with additional basis vectors generated from a knowledge of the structure of the stiffness uncertainty in addition methods for determining bounds on the approximate modal frequencies and mode shapes are discussed numerical results demonstrate that the algorithm produces highly accurate results with greatly reduced computational effort copyright 2009 john wiley sons ltd

iterative algorithms for non linear eigenvalue problems application to vibrations of viscoelastic shells in this paper two numerical iterative algorithms are developed for the vibrations of damped sandwich structures these methods associate homotopy asymptotic numerical techniques and pade approximants the first one is a sort of high order newton method and the second one uses a more or less arbitrary matrix so one can determine the natural frequencies and the loss factors of viscoelastically damped sandwich structures to assess their efficiency a few sandwich beams and plates have been considered the techniques can be applied to large scale structures to large damping and to strongly non linear viscoelastic modulus

differential equation based method for accurate approximations in optimization the key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations approximations were developed for vibration frequencies mode shapes and static displacements

unstable asymptotics for nonstationary queues we relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes we apply these results to develop approximations for queues that are unstable in a nonstationary manner we obtain unstable nonstationary analogs of the queueing relation l w and associated central limit theorem versions for modeling and to obtain the first limits we can construct nonstationary point processes as random time transformations of familiar point processes such as renewal processes and stationary point processes we deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation

the asymptotic efficiency of simulation estimators a decision theoretic framework is proposed for evaluating the efficiency of simulation estimators the framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate the cost of obtaining the estimate and the estimate itself are represented as realizations of jointly distributed stochastic processes in this context the efficiency of a simulation estimator based on a given computational budget is defined as the reciprocal of the risk the overall expected cost this framework is appealing philosophically but it is often difficult to apply in practice e g to compare the efficiency of two different estimators because only rarely can the efficiency associated with a given computational budget be calculated however a useful practical framework emerges in a large sample context when we consider the limiting behavior as the computational budget increases a limit theorem established for this model supports and extends a fairly well known efficiency principle

indirect estimation via l w for a large class of queueing systems little s law l w helps provide a variety of statistical estimators for the long run time average queue length l and the long run customer average waiting time w we apply central limit theorem versions of little s law to investigate the asymptotic efficiency of these estimators we show that an indirect estimator for l using the natural estimator for w plus the known arrival rate is more efficient than a direct estimator for l provided that the interarrival and waiting times are negatively correlated thus extending a variance reduction principle for the gi g s model due to a m law and j s carson we also introduce a general framework for indirect estimation which can be applied to other problems besides l w we show that the issue of indirect versus direct estimation is related to estimation using nonlinear control variables we also show under mild regularity conditions that any nonlinear control variable scheme is equivalent to a linear control va

improved first order approximation of eigenvalues and eigenvectors a method based on reduced basis approximation concepts is presented for improved first order approximation of eigenvalues and eigenvectors of modified structural dynamic systems the terms of a local approximation based on taylor or matrix power series are used as basis vectors for approximating the perturbed eigenparameters for each eigenmode a reduced eigensystem is generated by using the baseline eigenvector and the first order approximation term as ritz vectors the solution of the reduced eigensystem leads to two possible estimates of each perturbed eigenvalue and eigenvector criteria for selection of the best approximation are presented the zero and first order rayleigh quotient approximation can be directly recovered as special cases of the present method results are presented for approximate dynamic reanalysis of a 25 bar planar truss structure it is shown that high quality approximation of the perturbed eigenparameters can be obtained for moderate perturbations in the stiffness and mass matrices for very large local perturbations of the structure including deletion of some structural members it is shown that the present method yields reasonable quality approximations

eigensolutions of non proportionally damped systems based on continuous damping sensitivity abstract the viscous damping model has been widely used to represent dissipative forces in structures under mechanical vibrations in multiple degree of freedom systems such behavior is mathematically modeled by a damping matrix which in general presents non proportionality that is it does not become diagonal in the modal space of the undamped problem eigensolutions of non proportional systems are usually estimated assuming that the modal damping matrix is diagonally dominant neglecting the off diagonal terms or in the general case using the state space approach in this paper a new closed form expression for the complex eigenvalues of non proportionally damped system is proposed the approach is derived assuming small damping and involves not only the diagonal terms of the modal damping matrix but also the off diagonal terms which appear under higher order the validity of the proposed approach is illustrated through a numerical example

comparison of several methods for calculating vibration mode shape derivatives four methods for the calculation of derivatives of vibration mode shapes eigenvectors with respect to design parameters are described these are finite difference method modal method a modified modal method and nelson s method the methods are implemented in a general purpose commercial finite element program and applied to the following test problems a cantilever beam and a stiffened cylinder with a cutout design variables are a beam tip mass a beam root height and specific dimensions of the cylinder model the methods are compared on the basis of central processor cp seconds required to obtain the derivatives and two of the methods are also evaluated for the rapidity of convergence data is presented showing the amount of cp time used to compute the first four eigenvector derivatives for each example problem a scalar measure of the error in the mode shape derivative is defined and numerical results illustrating the rapidity of convergence of the approximate derivative to the exact derivative are presented results indicate an advantage in using nelson s method because this method is exact and requires less cp time especially when derivatives with respect to several design variables are computed

high quality approximation of eigenvalues in structural optimization a new function for approximating natural frequency constraints during structural optimization is presented the rayleigh quotient approximation rqa presented here icreases the quality of the approximate frequency constraint by approximating the modal strain and kinetic energies instead of the frequency eigenvalue itself

eigenvalue reanalysis by improved perturbation a new efficient method of eigenvalue reanalysis has been developed based on the perturbation method such that first the differential equations which describe the change of the eigenvalues and eigenvectors of a system being modified are derived from the perturbation method and then the solutions of these equations are obtained by following them up with the solutions of more simple equations the special features of this method are that it can give a clear correspondence between the eigenpairs before and after system modification and that the computer run time spent by this method is far shorter than that of revised analysis by fem when only a few selected modes are used in the form of an approximate computation in addition this approximate computation can give more accurate results than the perturbation method the efficiency of the proposed method has been verified by applying it to a simple spring mass model as well as a 600 degrees of freedom model which receive structural modification

diagonal dominance of damping and the decoupling approximation in linear vibratory systems a common approximation in the analysis of non classically damped systems is to ignore the off diagonal elements of the modal damping matrix this procedure is termed the decoupling approximation it is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant in addition the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant it is shown numerically in this paper that over a finite range errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off diagonal elements decreasing continuously in magnitude an explanation for these unexpected drifts of decoupling errors is provided with the use of complex coupling coefficients small off diagonal elements in the modal damping matrix are not sufficient to ensure small errors due to the decoupling approximation any error criterion based solely upon the diagonal dominance of the modal damping matrix would not be accurate

passive vibration suppression of flexible space structures via optimal geometric redesign acomputationalframework is presented for the design of large flexible space structures with nonperiodicgeometries to achieve passive vibration suppression the present system combines an approximationmodel management framework ammf developed for evolutionary optimization algorithms eas with reduced basis approximate dynamic reanalysis techniques a coevolutionary genetic search strategy is employed to ensure that design changes during the optimization iterations lead to low rank perturbations of the structural system matrices for which thereduced basis methods give high quality approximations the k means algorithm is employed for cluster analysis of the population of designs to determine design points at which exact analysis should be carried out the fitness of the designs in an ea generation is then approximated using reduced basis models constructed around the points where exact analysis is carried out results are presented for the optimal design of a two dimensional cantilevered space structure to achieve passive vibration suppression it is shown that significant vibration isolation of the order of 50 db over a 100 hz bandwidth can be achieved further it is demonstrated that the ammf can potentially arrive at a better design compared to conventional approaches when a constraint is imposed on the computational budget available for optimization

improving service by informing customers about anticipated delays this paper investigates the effect upon performance in a service system such as a telephone call center of giving waiting customers state information in particular the paper studies two m m s r queueing models with balking and reneging for simplicity it is assumed that each customer is willing to wait a fixed time before beginning service however customers differ so the delay tolerances for successive customers are random in particular it is assumed that the delay tolerance of each customer is zero with probability and is exponentially distributed with mean 1 conditional on the delay tolerance being positive let n be the number of customers found by an arrival in model 1 no state information is provided so that if n s the customer balks with probability if the customer enters the system he reneges after an exponentially distributed time with mean 1 if he has not begun service by that time in model 2 if n s k s then the customer is told the system state k and the remaining service times of all customers in the system so that he balks with probability 1 1 qk where qk p t sk t is exponentially distributed with mean 1 sk is the sum of k 1 independent exponential random variables each with mean s 1 and 1 is the mean service time in model 2 all reneging is replaced by balking the number of customers in the system for model 1 is shown to be larger than that for model 2 in the likelihood ratio stochastic ordering thus customers are more likely to be blocked in model 1 and are more likely to be served without waiting in model 2 algorithms are also developed for computing important performance measures in these and more general birth and death models

an lil version of l λw this paper establishes a law of the iterated logarithm lil version of the fundamental queueing formula l w under regularity conditions the continuous time arrival counting process and queue length process jointly obey an lil when the discrete time sequence of interarrival times and waiting times jointly obey an lil and the limit sets are related the standard relation l w appears as a corollary lils for inverse processes and random sums are also established which are of general probabilistic interest because the usual independence identical distribution and moment assumptions are not made moreover an lil for regenerative processes is established which can be used to obtain the other lils

mode tracking issues in structural optimization within the context of optimization of the structural dynamics properties of finite element models methodology is developed for the tracking of eigenpairs through changes in the structural eigenvalue problem the goal is to eliminate difficulties caused by mode switching i e frequency crossing out of several candidate methods two methods for mode tracking are successful the first method the higher order eigenpair perturbation algorithm is based on a perturbation expansion of the eigenproblem it iteratively computes changes in the eigenpairs due to parameter perturbations with the important feature of maintaining the correspondence between the baseline and perturbed eigenpairs the second method is a cross orthogonality check method which uses mass orthogonality to reestablish correspondence after a standard reanalysis modified eigenpair extraction routines lanczos subspace iteration inverse power were unsuccessful in tracking modes applications of mode tracking technology that are presented are frequency constrained optimization and optimization with mode shape constraints each application procedure is outlined and examples are given recommendations are made based on method efficiency and robustness in the example problems

scaling limit for a mechanical system of interacting particles a system of a large number of classical particles moving on a onedimensional segment with virtually reflecting boundaries is studied the particles interact with one another through repulsive pair potential forces and are subjected to resistance proportional to their velocities because of the latter it is only the number of particles that is conserved under the evolution of the system it is proved that in the hydrodynamic limit of diffusion type scaling the normalized counting measure of particle locations converges and its limiting density is governed by a non linear diffusion equation which in typical cases is of porous media equation type

local stability and hydrodynamical limit of spitzer s one dimensional lattice model an infinite system of ordinary differential equations is considered the right hand side is just the negative gradient of potential energy of a one dimensional system of unbounded spins interacting by a symmetric and convex pair potential constant configurations are stationary points and the mean spin is conserved it is shown that each of these stationary points has its own domain of attraction the initial distribution need not be translation invariant as a consequence we obtain that the mean spin satisfies the heat equation in the hydrodynamical limit

rates of change of eigenvalues and eigenvectors in damped dynamic system rates of change of eigenvalues and eigenvectors of a damped linear discrete dynamic system with respect to the system parameters are presented a nonproportional viscous damping model is assumed because of the nonproportional nature of the damping the mode shapes and natural frequencies become complex and as a consequence the sensitivities of eigenvalues and elgenvectors are also complex the results are presented in terms of the complex modes and frequencies of the second order system and the use of rather undesirable state space representation is avoided the usefulness of the derived expressions is demonstrated by considering an example of a nonproportionally damped two degree of freedom system

convergence to stationary states for infinite harmonic systems we study the evolution of the states for one dimensional infinite harmonic systems interacting through a translation invariant force of rapid decrease we prove that for a large class of initial states convergence to a gaussian limiting state as time goes to infinity is equivalent to convergence of the covariance the main assumption on the initial states is a kind of weak dependence between distant regions mixing condition we prove also convergence of the covariance under some general assumptions we show furthermore that there are two countable families of intensive constants of the motion which are inherited from the corresponding finite systems the translation invariant limiting states are in one to one correspondence with the admissible values of these constants of the motion moreover under some additional regularity assumption such states are shown to be gibbs states obtained by a boltzmann gibbs prescription

higher order eigenpair perturbations a matrix method is presented for the exact calculation of large eigenpair perturbations arising from design variable changes this method of forward analysis is presented that takes into account all order of the perturbation expansion

one dimensional hard rod caricature of hydrodynamics we give here a rigorous deduction of the hydrodynamic equation which holds in the hydrodynamic limit for a model system of one dimensional identical hard rods interacting through elastic collisions the equation should be considered as the analog of the euler equation of real hydrodynamics owing to the degeneracy of the model it is written in terms of a functiong q v t expressing the density of particles with velocityv at the pointq at timet for this equation we prove an existence and uniqueness theorem in some natural class of functions our main result is the proof that if 0 is a class of initial states which are homogeneous on a scale much less than e 1 and if the corresponding particle densities tend ase 0 in the proper scale to the initial hydrodynamic densitygo q v then under some general assumptions on the states and ong0 the particle densities of the evolved states at timee 1t tend ase 0 to the unique solution of the hydrodynamic equation with initial conditiong0 the proof is completed by exhibiting a large class of initial families 0 which possess the required properties

on eigenproblem solution of damped vibrations associated with gyroscopic moments a new efficient approach is presented for solving the quadratic eigenvalue problem of weakly nonproportionally damped vibration systems in the analysis of these systems gyroscopic moments and external damping are both considered traditional restriction of symmetry of inertia damping and stiffness matrices is slightly relaxed a second order perturbation theory is developed such that the perturbed solution is based on the eigensolution of an unperturbed subproblem this subproblem considers the unperturbed system in two different forms i a conservative gyroscopic part of an original problem or ii a nonconservative gyroscopic part of an original problem that is proportionally damped to cope with asymmetry of the system matrices a duncan s like state formulation is used to bring these matrices into a suitable form for perturbations two numerical examples are introduced for explaining the detailed implementation of the presented approach additionally a practical problem of rotor supported by two tilting pad bearings is investigated the eigensolutions obtained by the current approach match to a great extent other solutions obtained by time consuming exact methods the investigation procedure given here gives a framework to handle vibration problems of weakly nonproportional damping and or weakly asymmetric inertia damping and stiffness matrices

an approximate solution scheme for the algebraic random eigenvalue problem a reduced basis formulation is presented for the efficient solution of large scale algebraic random eigenvalue problems this formulation aims to improve the accuracy of the first order perturbation method and also allow the efficient computation of higher order statistical moments of the eigenparameters in the present method the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for ritz analysis of the governing random eigenvalue problem this leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest since only two basis vectors are used to represent each eigenvector explicit expressions for the random eigenvalues and eigenvectors can readily be derived this enables the statistics of the random eigenparameters and the forced response to be efficiently computed numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system it is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method

a method for modal reanalysis of topological modifications of structures a method for structural modal reanalysis for three cases of topological modifications the number of degrees of freedom dofs is unchanged decreased and increased is presented in this method the newly added dofs are linked to the original dofs of the modified structure by means of the dynamic reduction so as to obtain the condensed equation the methods for forming the stiffness and mass increments k and m resulting from the three cases of topological modifications of structures are discussed the extended kirsch method is used to improve the accuracy of the starting solutions of the initial structure and then the eigenvectors of newly added dofs resulting from topological modification can be recovered at last the rayleigh ritz analysis is used to evaluate the eigenvalues and eigenvectors for the modified structure three numerical examples are given to illustrate the applications of the present approach the results show that the proposed method is effective for structural modal reanalysis of three cases of the topological modifications and it is easy to implement on a computer by comparing with previous method it can be seen that the present method can give good approximate eigenvalues and eigenvectors even if the topological modifications are very large copyright 2005 john wiley sons ltd

eigenvector derivatives with repeated eigenvalues in this paper an algorithm is derived for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues where the matrices are functions of real parameters such as mass density or moment of inertia the algorithm is an extension of recent work by i u ojalvo the key step in this extended derivation is to differentiate the eigenvalue equation twice the algorithm preserves the symmetry and band structure of the matrices allowing efficient computer storage and solution techniques applications include sensitivity analysis and optimization of the normal modes of finite element modeled structures such as large space structures a cantilever beam finite element example is included

an improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem a technique is developed and presented which improves the accuracy of the predictions for the eigenvalues of the usual first order perturbation scheme for the generalized symmetric linear algebraic eigenvalue problem the improved method very simply replaces the usual b norms appearing in the denominator of the expressions for the first order changes in the eigenvalues by bp norms where bp is the perturbed b matrix with increasing sizes for the perturbations the improved technique produces results whose accuracy approaches those of the usual second order predictions numerical studies are presented to support the analysis

ordinary clt and wlln versions of l λw the familiar queueing principle expressed by the formula l w little s law can be interpreted as a relation among strong laws of large numbers sllns here we prove central limit theorem clt and weak law of large numbers wlln versions of l w for example if the sequence of ordered pairs of interarrival times and waiting times is strictly stationary and satisfies a joint clt then the queue length process also obeys a clt with a related limiting distribution in a previous paper we proved a functional central limit theorem version of l w without stationarity by very different arguments the two papers highlight the differences between establishing ordinary limit theorems and their functional limit theorem counterparts

perturbation method for the eigenvalue problem of lightly damped systems abstract an efficient method for the determination of the eigenvalues and eigenvectors of lightly damped systems is developed by means of a perturbation technique the second order matrix differential equation containing mass stiffness and damping matrices is normally transformed into a first order state equation to deal with the general damping matrix however the method described in this paper enables us to predict the complex eigenvalues and eigenvectors with relative ease without forming the state equation the light damping implies that the eigensolution of the damped system differs slightly from the eigensolution of the undamped system which also implies that we can express the eigensolution of the lightly damped system in terms of a power series expanded from the eigensolution of the undamped system once the eigenvalue problem of the undamped system is solved the higher order terms which reflect the effect of damping can be obtained from the matrix equations which reduce to simple algebraic equations a numerical example illustrates the effectiveness of the new method

new mode tracking methods in aeroelastic analysis within the context of analysis of finite element models a methodology is developed for the tracking of complex eigenvalues and eigenvectors through changes in aeroelastic eigenvalue problems the goal is to eliminate difficulties caused by mode switching i e frequency crossing two new methods for mode tracking are presented and compared with an existing method the first new method the complex higher order eigenpair perturbation algorithm is based on a perturbation expansion of the eigenproblem it iteratively computes changes in the eigenpairs due to parameter perturbations with the important feature of maintaining the correspondence between the baseline and perturbed eigenpairs the second new method is the complex cross orthogonality check method which uses mass biorthogonality of the left and right eigenvectors to re establish correspondence after a standard reanalysis applications of mode tracking technology are presented in v g and fc subsonic aeroelastic analyses each application procedure is outlined and examples are given recommendations are made based on method ease of use and robustness in the example problems

extended kirsch combined method for eigenvalue reanalysis based on the kirsch combined method and the rayleigh quotient a new eigenvalue reanalysis method for modified structures is developed using this method the accuracy of the eigenvalues reanalysis of large modifications of structural parameters is improved as an application of the method a numerical example is given to illustrate the accuracy by comparing the second order perturbation method with the proposed method if the large modifications of structural parameters are made the calculated results show that the proposed method is effective

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

hybrid perturbation polynomial chaos approaches to the random algebraic eigenvalue problem the analysis of structures is affected by uncertainty in the structure s material properties geometric parameters boundary conditions and applied loads these uncertainties can be modelled by random variables and random fields amongst the various problems affected by uncertainty the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure the methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on monte carlo simulation in the past few years methods based on polynomial chaos pc have been developed for this problem where each eigenvalue and eigenvector are represented by a pc expansion in this paper four variants of a method hybridizing perturbation and pc expansion approaches are proposed and compared the methods use rayleigh quotient the power method the inverse power method and the eigenvalue equation pc expansions of eigenvalues and eigenvectors are obtained with the proposed methods the new methods are applied to the problem of an euler bernoulli beam and a thin plate with stochastic properties

reduced basis approximations of structural displacements for optimaldesign a new solution procedure for improved approximate reanalyses of structures using results of a single precise analysis is presented the proposed procedure is based on combining the computed terms of a series expansion used as high quality basis vectors with coefficients of a reduced basis expression the latter coefficients can readily be determined for each trial design by various criteria the results are measured by the errors in satisfying the analysis equations and are compared in terms of the quality of the approximations the proposed approach is suitable for various types of design variables and can be used with a general finite element model a general reanalysis procedure is introduced and the physical significance of first order approximations is demonstrated several numerical examples illustrate the effectiveness of the solution process it is shown that high quality approximations of displacements can be obtained with a small computational effort for very large changes in the design variables

co ordinate transformations for second order systems part ii elementary structure preserving transformations abstract it has been shown in a previous paper that there is a real valued transformation from the generaln degree of freedom second order system to a second order system characterized by diagonal matrices an immediate extension of this fact is that for any second order system there is a set of real valued transformations thestructure preserving transformations which transform this system to a different second order system having identical characteristic behaviour there are several possible reasons why it may be very useful to achieve a particular structure in the transformed system it is obvious that a diagonal structure is extremely useful and a method has been devised for determining the diagonalizing transformation from the solution of the usual complex eigenvalue eigenvector problem this paper begins by outlining the usefulness of some other structures then it defines a class of elementary structure preserving co ordinate transformations that transform from one n degree of freedom second order system to another the termelementary is applied because any one of these transformations is the minimum rank modification of the identity transformation the changes occurring in the system matrices as a result of the application of one such elementary transformation transpire to be very simple in form they are low rank and they can be computed very efficiently this paper provides the fundamental tools to enable the design of structure preserving co ordinate transformations which transform a second order system originally characterized by three general matrices in stages into a mathematically similar second order system characterized by three diagonal matrices the procedure by which the individual elementary transformations are obtained is still under development and it is not discussed in this paper however an illustration is given of a five degree of freedom self adjoint system being transformed into tridiagonal form

mode traces in degenerate eigensystems and augmented assurance eigenpairs contextually indicative of the frequencies and oscillatory modes of structural systems are considered as functions of a single parameter undesired permutation of modal designations is noted to arise following events of transitory eigenvalue coalescence with respect to the parameter resulting in the definition of nonsmooth functions the tracing of eigenpairs across such events is outlined for the permanently degenerate general eigenproblem as the general case the cases of transitorily degenerate and distinct eigenvalues are thus accounted for the foundation for mode tracing is the assumption of eigenvector consistency across parameter intervals used as a means of eigenpair reconciliation the resulting traced modes are smooth and their variations physically pertinent which is necessary in iterative schemes in which convergence may otherwise be jeopardized the proposal of an augmented modal assurance routine that potentially augments the assurance of tracing and therefore the maximum permissible parameter perturbation is given a threefold increase in the insight to consistency entails the notion of the routine is to forward and backward cast eigenvectors utilizing their derivatives in affine modeling a numerical example of the modes of a cyclic frame as functions of a cyclic distribution of membrane forces demonstrates the concepts and utility of the proposal

numerical method for sensitivity analysis of eigensystems with non repeated and repeated eigenvalues abstract a numerical method is presented for computation of eigenvector derivatives used an iterative procedure with guaranteed convergence an approach for treating the singularity in calculating the eigenvector derivatives is presented in which a shift in each eigenvalue is introduced to avoid the singularity if the shift is selected properly the proposed method can give very satisfactory results after only one iteration a criterion for choosing an adequate shift dependent on computer hardware is suggested it is directly dependent on the eigenvalue magnitudes and the number of bits per numeral of the computer another merit of this method is that eigenvector derivatives with repeated eigenvalues can be easily obtained if the new eigenvectors are calculated these new eigenvectors lie adjacent to them number of repeated eigenvalues distinct eigenvectors which appear when the design parameter varies as an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues a cantilever plate is considered the results are compared with those of nelson s method which can find the exact eigenvector derivatives for the case of repeated eigenvalues a cantilever beam is considered the results are compared with those of dailey s method which also can find the exact eigenvector derivatives the design parameter of the cantilever plate is its thickness and that of the cantilever beam its height

a perturbation method for reduction of eigenvalue analysis of structures with large stiffnesses and small masses a perturbation method to reduce the computing time for the eigenvalue analysis of structural dynamics is introduced in this paper a simplified structural model in which the degrees of freedom associated with the large stiffnesses are eliminated from the eigenvalue equation and the small masses are not considered is set up and analysed the eigenvalues and eigenvectors are served as the zeroth order solution and a perturbation procedure is performed to restore the eigensolution for the model of full degrees of freedom the method is successfully applied to the plane frame the numerical results show that the computing time can be greatly reduced while the high accuracy for the eigensolution is achieved the good perturbed result of both eigenvalues and eigenvectors is obtained even from the first order perturbation and the accurate normality of eigenvectors is achieved in the higher order perturbation for the perturbation with small masses a step perturbation is suggested to obtain a better result 1997 by john wiley sons ltd

accelerated subspace iteration for eigenvector derivatives an accelerated subspace iteration method for calculating eigenvector derivatives has been developed factors affecting the effectiveness and the reliability of the subspace iteration are identified and effective strategies concerning these factors are presented the method has been implemented and the results of a demonstration problem are presented

improved approximations via rayleigh s quotient two methods for obtaining excellent results based on one term approximations via rayleigh s quotient are described along with their applications a combination of these methods can be used to obtain accurate resutls for a wide class of problems as well after a few iterations depending on the starting function the stodola vianello method sometimes leads to overly complicated formulae so that symbolic manipulation software may become bogged down

a perturbation technique for gyroscopic systems with small internal and external damping abstract the response of linear damped gyroscopic systems can be obtained by means of techniques of linear systems theory which involves the computation of the transition matrix the response is in terms of complex quantities which is likely to cause computational difficulties as the order of the system increases in the absence of damping it is possible to derive the response of a linear gyroscopic system with relative ease by working with real quantities alone when damping is small one can use a perturbation approach to produce the response by regarding the undamped gyroscopic system as the unperturbed system in a previous paper a perturbation analysis was used to derive the response of a gyroscopic system with small internal damping this paper extends the approach to the case of external damping which is characterized not only by symmetric coefficients multiplying velocities but also by skew symmetric coefficients multiplying displacements where the latter terms are known as circulatory a numerical example is presented

planning queueing simulations simple heuristic formulas are developed to estimate the simulation run lengths required to achieve desired statistical precision in queueing simulations the formulas are intended to help in the early planning stages before any data have been collected the queueing simulations considered are single replications one long run conducted to estimate steady state characteristics such as expected equilibrium queue lengths the formulas can be applied to design simulation experiments to develop and evaluate queueing approximations in fact this work was motivated by efforts to develop approximations for packet communication networks with multiple classes of traffic having different service characteristics and bursty arrival processes in addition to indicating the approximate simulation run length required in each case of a designed experiment the formulas can help determine what cases to consider what statistical precision to aim for and even whether to conduct the experiment at all the formulas are bas

estimating customer and time averages in this paper we establish a joint central limit theorem for customer and time averages by applying a martingale central limit theorem in a markov framework the limiting values of the two averages appear in the translation terms this central limit theorem helps to construct confidence intervals for estimators and perform statistical tests it thus helps determine which finite average is a more asymptotically efficient estimator of its limit as a basis for testing for pasta poisson arrivals see time averages we determine the variance constant associated with the central limit theorem for the difference between the two averages when pasta holds

eigensolution reanalysis of modified structures using epsilon algorithm based on the neumann series expansion and epsilon algorithm a new eigensolution reanalysis method is developed in the solution process the basis vectors can be obtained using the matrix perturbation or the neumann series expansion to construct the vector sequence and then using the epsilon algorithm table to obtain the approximate eigenvectors the approximate eigenvalues are computed from the rayleigh quotients the solution steps are straightforward and it is easy to implement with the general finite element analysis system two numerical examples a 40 storey frame and a chassis structure are given to demonstrate the application of the present method by comparing with the exact solutions and the kirsch method solutions it is shown that the excellent results are obtained for very large changes in the design and that the accuracy of the epsilon algorithm is higher than that of the kirsch method and the computation time is less than that of the kirsch method

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

time evolution of a one dimensional point system a note on fritz s paper in a companion paper ref 5 fritz studies the time evolution of a one dimensional point system which was introduced by spitzer as a model of traffic in the present paper we improve some of the results of ref 5 by using a different approach our results are obtained in a very simple and straightforward way but the techniques employed require conditions somewhat stronger than those assumed in ref 5

improvement of normalization methods for eigenvector derivatives the subject algorithms are all forward analysis i e the eigenvector changes are being driven by design parameter perturbations the new method emphasizes proper mass normalization and is most needed when iteratively computing eigenvector perturbations within a reduced dimension space

discrete flow networks bottleneck analysis and fluid approximations we conduct bottleneck analysis of a deterministic dynamic discrete flow network the analysis presupposes only the existence of long run averages and is based on a continuous fluid approximation to the network in terms of these averages the results provide functional strong laws of large numbers for stochastic jackson queueing networks since they apply to their sample paths with probability one

rapid reanalysis of the generalized eigenvalue problem of locally modified linear dynamical systems abstract a computationally efficient method for the reanalysis of the damped generalized eigenvalue problems associated with linear dynamical systems with local damping and stiffness modifications is developed no restrictions are imposed on the damping of the system and the modifications in the damping and the stiffness can be independent of each other the application of the developed method to the undamped generalized eigenvalue problem with mass and stiffness modifications is also discussed the proposed method achieves up to three orders of magnitude gains in computational efficiency compared to the conventional solution procedures

accelerated iterative procedure for calculating eigenvector derivatives an iterative method for the computation of eigenvector derivatives of real valued symmetric systems has been proposed the process is shown to converge to the exact solution with any initial values and it is numerically stable however since the convergence rate is determined by the ratios of the eigenvalues the convergence rate will be prohibitively slow when the ratio is close to 1 two ef cient accelerated algorithms are presented numerical examples show that when the eigenvalue ratio is greater than 0 70 the computational effort in the iterative process is drastically reduced the procedure incorporates the method of frequency shift to deal with a singular stiffness matrix that also provides an additional improvement for the convergence when the eigenvalue ratios are extremely close to 1 the second generalized stiffness matrix inverse is suggested to reestablish a high convergence rate the procedure can be used as an exact as well as an approximate method it requires no more eigenvectors than those whose derivatives are to be calculated and can be applied to systems with repeated eigenvalues

a lanczos based method for structural dynamic reanalysis problems in this paper a new solution method for the modified eigenvalue problem with specific application to structural dynamic reanalysis is presented the method which is based on the block lanczos algorithm is developed for multiple low rank modifications to a system and calculates a few selected eigenpairs given the solution to the original system ax x procedures are developed for the modified standard eigenvalue problem a a x x where 1 a jbsjbt where sj s p p p n and b n p is constant for all the perturbations sj 2 a i jbisjbit where bi n p may vary with the pertubations sj the procedures are then extended for the reciprocal and generalized eigenvalue problems so that they are directly applicable to the structural dynamic reanalysis problem numerical examples are given to demonstrate the applications of the method