on the solution of s ω x 0 by a newtonian procedure the method 1 developed for the solution of e a x 0 where e and a are real symmetric matrices and a is positive definite is extended to deal with s x 0 where s is a real symmetric structural dynamic stiffness matrix whose elements are trancendental functions of the radian natural frequency the method enables all natural frequencies latent roots 2 and mode shapes latent vectors x across a prescribed frequency range to be determined infallibly and in ascending frequency order at a speed approaching twice that associated with bisection and sign counting methods 3 it is ideally suited for use on micro and minicomputers where single precision working is the norm and on the commodore 4032 computer can cope with matrices s of order n and bandwidth p provided that np 4200 much larger problems of course may be dealt with on mainframe machines

maximizing the minimum load for selfish agents we consider the problem of maximizing the minimum load completion time for machines that are controlled by selfish agents who are only interested in maximizing their own profit unlike the classical load balancing problem this problem has not been considered for selfish agents until now the goal is to design a truthful mechanism i e one in which all users have an incentive to tell the truth about the speeds of their machines this then allows us to find good job assignments it is known that this requires monotone approximation algorithms in which the amount of work assigned to an agent does not increase if its bid claimed cost per unit work increases for a constant number of machines m we show a monotone polynomial time approximation scheme ptas with running time that is linear in the number of jobs it uses a new technique for reducing the number of jobs while remaining close to the optimal solution we use an fptas for the classical problem i e where no selfish agents are involved to give a monotone fptas additionally we give a monotone approximation algorithm with approximation ratio min m email protected s 1 s m where e 0 can be chosen arbitrarily small and s i is the real speed of machine i finally we give improved results for two machines

a newtonian procedure for the solutions of ex λax abstract newtonian iteration on a scalar function which is obtained by condensation of the pencil e a is used in conjunction with sturm sequence properties of the pencil to provide an infallible quadratically convergent algorithm for all eignevalues of a 1 e within any stated range the algorithm has been programmed for use on a desk top machine and large sparse matrix eigenvalue problems have been solved a brief outline of the essential steps in the program is given along with some particular solutions

spectral super elements for wave propagation in structures with local non uniformities the spectral element method for the analysis of wave propagation in structures is extended by the introduction of a super element concept to handle local regions of non uniformity the region of interest is meshed with conventional finite elements and its stiffness and mass matrices dynamically condensed leaving only the connection degrees of freedom these are further condensed to just waveguide connectivities thus making the element suitable for assembly with other spectral elements the formulation is verified by comparison with 2 d finite element solutions examples of its use in waveguides with cracks holes and extended joints are demonstrated

procedure to reduce the effects of modal truncation in eigensolutionreanalysis the influence of structural perturbations on the eigensolutions of a mechanical system is calculated through the use of a discrete model composed of the first normal modes of the base system it consists of modifying the structural perturbations before the application of the classical sensitivity relationships

an exact model reduction procedure for mechanical systems abstract this paper gives details of an exact model reduction procedure for the dynamic analysis of general mechanical systems in this procedure the system is first transformed into an equivalent one dimensional id lumped mass spring damping system by coordinate transformations the 1d lumped system is further reduced into a lower order model with arbitrarily selected degrees of freedom three numerical examples are given for a distributed lumped mass spring system a truss frame structure and a cantilever plate

formulation and solution of the non linear damped eigenvalue problem for skeletal systems this paper presents and discusses an arnoldi based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures the new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices this modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems the eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non linear eigenvalue problem at given values of the parameter numerical examples illustrate that this method used in conjunction with a secant interpolation accurately evaluates the complex natural frequencies and modes of the quadratic non linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models

the resource dependent assignment problem with a convex agent cost function abstract we study the resource dependent assignment problem rdap for which the cost of assigning agent j to task i is a multiplication of task i s cost parameter by a cost function of agent j which is a convex function of the amount of resource allocated for the agent to execute his task the quality of a solution is measured by two criteria the first is the total assignment cost and the second is the total weighted resource consumption we consider four different variations of the rdap and prove that three are np hard while the last is polynomially solvable under some acceptable assumptions in our np hardness proof we use a very general instance which makes the proof applicable to a large set of special cases of the rdap including several important scheduling problems whose complexity was unresolved heretofore in addition we design a novel approximation algorithm for one of the np hard variations of the rdap with a very tight approximation ratio for any practical problem found in the literature

computational procedure for sturm liouville problems the approaches of datzeff c r acad bulg sci 12 1959 113 285 and wittrick and williams quart j mech appl math xxiv 1971 263 int j mech sci 16 1974 209 have been incorporated to develop a procedure for the automatic computation of the eigen values and the eigen functions of one dimensional linear sturm liouville boundary value eigenproblems both singular and nonsingular the continuous coefficients of a regular sturm liouville problem have been approximated by a finite number of step functions in each step the resulting boundary value problem has been integrated exactly and the solutions have then been matched to construct the continuously differentiable solution of the original problem and the corresponding eigencondition step sizes have been chosen automatically so that the local error has been held in a predetermined interval representative test examples have been computed to illustrate the accuracy reliability and efficiency of the algorithm proposed

dynamic analysis of space frames the method of reverberation ray matrix and the orthogonality of normal modes the formulation of reverberation ray matrix analysis has been proposed to study wave propagation in planar frames it is applied here to modal analysis of complex three dimensional framed structures optionally with lumped masses and or elastic supports furthermore by means of betti s reciprocity theorem orthogonal conditions are established for different natural modes and hence transient response analysis based on mode superposition is developed both the reverberation ray matrix analysis for free vibration and the mode superposition method for transient response are illustrated by numerical examples

characterization of time series via renyi complexity entropy curves abstract one of the most useful tools for distinguishing between chaotic and stochastic time series is the so called complexity entropy causality plane this diagram involves two complexity measures the shannon entropy and the statistical complexity recently this idea has been generalized by considering the tsallis monoparametric generalization of the shannon entropy yielding complexity entropy curves these curves have proven to enhance the discrimination among different time series related to stochastic and chaotic processes of numerical and experimental nature here we further explore these complexity entropy curves in the context of the renyi entropy which is another monoparametric generalization of the shannon entropy by combining the renyi entropy with the proper generalization of the statistical complexity we associate a parametric curve the renyi complexity entropy curve with a given time series we explore this approach in a series of numerical and experimental applications demonstrating the usefulness of this new technique for time series analysis we show that the renyi complexity entropy curves enable the differentiation among time series of chaotic stochastic and periodic nature in particular time series of stochastic nature are associated with curves displaying positive curvature in a neighborhood of their initial points whereas curves related to chaotic phenomena have a negative curvature finally periodic time series are represented by vertical straight lines

characterizing time series via complexity entropy curves the search for patterns in time series is a very common task when dealing with complex systems this is usually accomplished by employing a complexity measure such as entropies and fractal dimensions however such measures usually only capture a single aspect of the system dynamics here we propose a family of complexity measures for time series based on a generalization of the complexity entropy causality plane by replacing the shannon entropy by a monoparametric entropy tsallis q entropy and after considering the proper generalization of the statistical complexity q complexity we build up a parametric curve the q complexity entropy curve that is used for characterizing and classifying time series based on simple exact results and numerical simulations of stochastic processes we show that these curves can distinguish among different long range short range and oscillating correlated behaviors also we verify that simulated chaotic and stochastic time series can be distinguished based on whether these curves are open or closed we further test this technique in experimental scenarios related to chaotic laser intensity stock price sunspot and geomagnetic dynamics confirming its usefulness finally we prove that these curves enhance the automatic classification of time series with long range correlations and interbeat intervals of healthy subjects and patients with heart disease

extrapolation locking and its sanitization in koiter s asymptotic analysis abstract this paper shows that the fem implementation of koiter s asymptotic method w t koiter on the stability of elastic equilibrium 1970 ph d thesis delft 1945 english transl nasa tt f10 883 1967 affdl tr70 25 outlined by casciaro et al finite element asymptotic analysis of slender elastic structures a simple approach int j num meth eng 35 1992 1397 1426 provides accurate and reliable results in the critical and post critical analysis of non linear elastic structures care however does have to be taken in implementing apparently minor details to avoid locking effects which adversely affect accuracy and which can destroy the method reliability as the effects related to the finite element interpolation have been discussed before this paper focuses on the non linear locking due to the use implicit in the method of finite distance extrapolations within this scope it is shown that perturbation algorithms based on compatible formulations can imply a strong critical and post critical locking when analysing structures characterized by high stiffness ratios in the presence of moderate pre critical rotations on the contrary perturbation algorithms based on independent extrapolations of displacements and stresses furnish reliable results in excellent agreement with those provided by step by step analysis at a small fraction of its computational cost

the member stiffness determinant and its uses for the transcendental eigenproblems of structural engineering and other disciplines transcendental stiffness matrices are well established in vibration and buckling analysis having been derived from exact analytical solutions of the differential equations for many structural members without recourse to finite element discretization their assembly into the overall structural stiffness matrix gives a transcendental eigenproblem solvable with certainty by the wittrick williams ww algorithm instead of the usual linear algebraic eigenproblem this paper establishes a normalized member stiffness determinant being the value of the stiffness matrix determinant if the member were modelled by infinitely many finite elements and its ends were clamped it is derived for beams with uncoupled axial and bernoulli euler flexural behaviour by methods applicable to any member possessing transcendental stiffnesses multiplying the product of a structure s member stiffness determinants by the determinant of its transcendental overall stiffness matrix gives a determinant which unlike the determinant of the transcendental overall stiffness matrix itself has no poles when plotted against the eigenparameter leading to a more secure eigenvalue location numerical results demonstrate the resulting advantages in addition the member stiffness determinant can give exact determinants for members with various standard end conditions and also provides limits for finite element solutions as the number of elements approaches infinity analogous advantages occur in all other disciplines where the ww algorithm is used e g fluid vibrating in pipes heat and mass diffusion and sturm liouville problems

substructure mode synthesis with overlapping connecting elements abstract an approach is presented to synthesize the modal properties of two substructures to obtain the modal properties of the combined structure the substructures are assumed to be fixed at the interfacing boundary and have overlapping elements the eigenproperties of the overlapping elements are also used in the synthesis process the approach can be used with any finite element discretization scheme the approach avoids the solution of a second eigenvalue problem by conventional means instead the eigenvalues are obtained as the solution of characteristic equations and eigenvectors are defined in closed form if the complete set of eigenproperties are used the approach provides the exact eigenproperties

a lanczos based technique for exact vibration analysis of skeletal structures this paper presents and discusses a lanczos based eigensolution technique for evaluating the natural frequencies and modes from frequency dependent eigenproblems in structural dynamics the new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices the method is well suited to the solution of large scale problems the new solution methodology presented here is based on the ability to evaluate a specific set of parameterized non linear eigenvalue curves at given values of the parameter through an implicitly restarted lanczos technique numerical examples illustrate that the implicitly restarted lanczos method with secant interpolation accurately evaluates the exact natural frequencies and modes of the non linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models in addition the eigenvalue technique presented here is shown to be far more computationally efficient on large scale problems than the determinant search techniques traditionally employed for solving exact vibration problems

dynamic substructure response abstract dynamic substructure methods enable one to analyze a large scale dynamic system by considering its constituents individually and the combined action afterwards steady state harmonic oscillation has been studied almost exclusively the methods are extended to general non steady state excitations the orthonormality condition of the modal vectors in terms of the masters is introduced a mixed mass matrix for initial conditions is derived the dynamic flexibility is expressed in spectral form for response calculation modal participation factors for both distributed and concentrated loads are given explicitly

sunspot cycle a driven nonlinear oscillator a property of nonlinear oscillators mutual dependence between their instantaneous amplitude and frequency is tested in the yearly and monthly records of the sunspot numbers using the histogramadjusted isospectral surrogate data and the barnes model as the autoregressive moving average surrogates the instantaneous amplitudes and frequencies are obtained by means of the analytic signal approach using the discrete hilbert transform in several tests the amplitude frequency correlation has been found significant on levels ranging from p 0 03 to p 0 07 which supports the hypothesis of a driven nonlinear oscillator as a mechanism underlying the sunspot cycle

a highly stable and accurate computational method for eigensolutions in structural dynamics a new computational method for the linear eigensolution of structural dynamics is proposed the eigenvalue problem is theoretically transformed into a specific initial value problem of an ordinary differential equation based on the physical meaning of the sign count of the dynamic stiffness matrix a stability control device is designed and combined with the fourth order runge kutta method the resulting method finds the eigenvalues and eigenvectors at the same time with high accuracy and high stability numerical examples show that the proposed method still gives high accuracy solutions when there is a great difference in magnitude among the eigenvalues and also when some eigenvalues are very close to each other

exact substructuring in recursive newton s method for solving transcendental eigenproblems the well established wittrick williams w w algorithm guarantees accurate convergence on natural frequencies or critical buckling loads of structures in the transcendental eigenproblems arising from the use of exact member stiffnesses i e dynamic member stiffnesses for vibration the associated mode calculations had no such guarantee until they were recently greatly improved by solving the transcendental eigenproblem exactly by reducing it to a generalised linear eigenproblem which is solved by a guided recursive newton method involving inverse iteration the present paper demonstrates the benefits of using frequency squared rather than frequency as the eigenparameter next exact substructuring is introduced into the recursive newton method with accuracy retained because the inverse iteration includes the substructure nodes if member fixed end eigenvalues lie close to the sought eigenvalue they can cause inaccuracy or even wrong results and so they are removed efficiently by inserting interior nodes to create simple substructures numerical results for a moderately large structure show that exact substructuring reduces the transcendental eigensolution time without reducing accuracy simpler examples designed to be numerically ill conditioned in the absence of interior nodes show that such ill conditioning is almost removed by inserting interior nodes within substructures exact substructuring is also applicable to other inverse iteration and w w algorithm based methods

second order mode finding method in dynamic stiffness matrix methods abstract this paper addresses the transcendental eigenvalue problem which arises for those structures e g frames for which dynamic member stiffnesses are available and are exact in the sense that the appropriate differential equations have been solved thus it does not relate to traditional finite element vibration or buckling methods which are approximate and give a linear eigenvalue problem which can be solved by many excellent methods the well established wittrick williams algorithm is a powerful reliable and efficient means of obtaining the desired number of natural frequencies for transcendental eigenvalue problems but except for one very recent paper involving explicit derivatives and a recursive method the corresponding vibration mode computations suffer from various difficulties and far from match the elegance of the frequency computations this paper presents a newly developed mathematically elegant and computationally efficient algorithm for accurate and reliable computation of vibration modes it uses standard inverse iteration with approximate natural frequencies of only first order accuracy to produce the corresponding vibration modes to second order accuracy extrapolation can then be used to improve the original natural frequencies to second order accuracy moreover the algorithm automatically provides a helpful so called check which enables the acceptability of the mode accuracy to be judged numerical examples presented include some demanding ones e g with coincident natural frequencies these show the excellent performance of the method two advantages over the recent recursive paper are that explicit derivatives are replaced by differencing and recursion is avoided in combination these two advantages make the new method much more suitable for retrofitting into existing computer programs

a newtonian procedure for the solution of the kron characteristic value problem abstract in this paper a quadratically convergent newtonian procedure is presented to determine the latent roots of the kron matrix the roots are found in ascending order with the certainty that none has been missed since the procedure is allied to a sturm sequence algorithm the procedure has been programmed and it performs at least 50 faster than the bisection scanning method available for the kron matrix examples of successful use of the procedure to solve for the natural frequencies of frames with large numbers of degrees of freedom are presented

tight results for next fit and worst fit with resource augmentation it is well known that the two simple algorithms for the classic bin packing problem nf and wf both have an approximation ratio of 2 however wf seems to be a more reasonable algorithm since it never opens a new bin if an existing bin can still be used using resource augmented analysis where the output of an approximation algorithm which can use bins of size b 1 is compared to an optimal packing into bins of size 1 we give a complete analysis of the asymptotic approximation ratio of wf and of nf and use it to show that wf is strictly better than nf for any 1 2 and for b 1

damped dynamic substructures the dynamic substructure method is extended to lightly or heavily damped systems both internal and external dampings are considered the damped dynamic flexibility associated with the slave co ordinates is first expanded in terms of the damped fixed interface natural modes and the condensed dynamic stiffness associated with the master co ordinates is formed subsequently the convergence of the condensed dynamic stiffness with respect to the damped natural modes can be improved by means of the static matrices since the dynamic stiffness method is equivalent to the modal synthesis method the component mode method and kron s method the theory presented here is readily applicable to these methods are restricted to symmetric damping matrices

safeguarded use of the implicit restarted lanczos technique for solving non linear structural eigensystems this paper presents a new algorithm for evaluating the eigenvalues and their corresponding eigenvectors for large scale non linear eigensystems in structural dynamics the algorithm is based on solving a sequence of algebraic eigenproblems and updating the parameter the implicitly restarted lanczos method had been determined to be well suited for solving the linear eigenproblems that arise in this context a zero finder approach that uses rational interpolation to approximate the generalized eigenvalues has been developed to update the methodology of the new algorithm developed here is designed to evaluate a subset of the parametrized non linear eigencurves at specific values of numerical experiments show that the new eigensolution technique is superior to existing approaches for large scale problems and competitive for small size problems the main emphasis of this contribution is the derivation and analysis of this scheme for non linear structural eigensystems

development of a block lanczos algorithm for free vibration analysis of spinning structures this paper is concerned with the development of an efficient eigenproblem solution algorithm and an associated computer program for the economical solution of the free vibration problem of complex practical spinning structural systems thus a detailed description of a newly developed block lanczos procedure is presented in this paper that employs only real numbers in all relevant computations and also fully exploits sparsity of associated matrices the procedure is capable of computing multiple roots and proves to be most efficient compared to other existing similar techniques

an accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies abstract a new computational method for the transcendental eigensolution of structural dynamics is proposed it is based on a new criterion for detecting the eigenfrequency instead of trying to find the roots of the determinant of the dynamic stiffness matrix this method seeks the frequency that makes the last energy norm vanish so matching the physical definition of eigenfrequency except for those eigenfrequencies for which the last degree of freedom is nodal in the mode for which the method is suitably modified by using the property of derivatives of energy norms the eigenproblem is transformed safely into a specific initial value problem of an ordinary differential equation among many available methods to solve the resulting ordinary differential equation the one step runge kutta method is proved to be a simple and efficient way to obtain eigensolutions as confirmed by a numerical example

realization of mechanical systems from second order models congruent coordinate transformations are used to convert second order models to a form in which the mass damping and stiffness matrices can be interpreted as a passive mechanical system for those systems which can be constructed from interconnected mass stiffness and damping elements it is shown that the input output preserving transformations can be parametrized by an orthogonal matrix whose dimension corresponds to the number of internal masses those masses at which an input is not applied nor an output measured only a subset of these transformations results in mechanically realizable models for models with a small number of internal masses complete discrete mapping of the transformation space is possible permitting enumeration of all mechanically realizable models sharing the original model s input output behavior when the number of internal masses is large a nonlinear search of transformation space can be employed to identify mechanically realizable models applications include scale model

time variant random interval natural frequency analysis of structures abstract this paper presents a new robust method namely unified interval chebyshev based random perturbation method to tackle hybrid random interval structural natural frequency problem in the proposed approach random perturbation method is implemented to furnish the statistical features i e mean and standard deviation and chebyshev surrogate model strategy is incorporated to formulate the statistical information of natural frequency with regards to the interval inputs the comprehensive analysis framework combines the superiority of both methods in a way that computational cost is dramatically reduced this presented method is thus capable of investigating the day to day based time variant natural frequency of structures accurately and efficiently under concrete intrinsic creep effect with probabilistic and interval uncertain variables the extreme bounds of the mean and standard deviation of natural frequency are captured through the embedded optimization strategy within the analysis procedure three particularly motivated numerical examples with progressive relationship in perspective of both structure type and uncertainty variables are demonstrated to justify the computational applicability accuracy and efficiency of the proposed method

quadratic representation of a nonlinear dynamic stiffness matrix and related eigenvalue problems the approximate representation of an exact dynamic stiffness matrix k rho by a quadratic matrix formulation a rho b rho 2 c is studied theoretically in this paper the matrix formulation is formed by expressing the elements of k rho as parabolic functions based on choosing three fixed values of the eigenparameter rho the general bounding properties of the approximate eigenvalues provided by the quadratic matrix formulation are shown to exist provided that the three fixed values are below the lowest pole of the nonlinear stiffness matrix and that the three coefficient matrices a b and c are positive definite it is shown theoretically in this paper that the approximate eigenvalues are either upper or lower bounds of the corresponding exact ones of the exact dynamic stiffness matrix

on roots of functional lambda matrices the problem of determining the roots or eigenvalues of functional lambda matrices and their sensitivities are considered an accurate and efficient computational technique is proposed it is shown that sensitivities are computed virtually for free as a byproduct of the algorithm examples demonstrate applications

a solution of the eigenproblem for undamped gyroscopic systems with the lanczos algorithm this paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems such problems are known to reduce to a generalized linear eigenproblem defined by two real non singular matrices one symmetric and one skew symmetric for this class of problems the general lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems numerical examples are presented which demonstrate the efficiency and accuracy of the solution procedure

fluid vibration in piping systems a structural mechanics approach i theory abstract a general computational technique is developed for the study of uniaxial linear vibration of compressible fluids in piping systems an exact finite element mobility method is applied the elements included are finite and semi infinite fluid columns intermediate and boundary mass spring damper systems cavity springs and modal bodies the method is analogous to the stiffness method in structural mechanics branches and loops in the piping system do not complicate the solution pressures and volume velocities accelerations can be prescribed forced harmonic responses at a fixed frequency are solved from the global matrix equation established eigenfrequencies are found through the wittrick williams algorithm transient responses are calculated by use of modal superposition numerical examples are given in a companion paper

reliable use of determinants to solve non linear structural eigenvalue problems efficiently a multiple determinant parabolic interpolation method is described and evaluated principally by using a plane frame test bed program it is intended primarily for solving the transcendental eigenvalue problems arising when the exact member equations obtained by solving the governing differential equations of members are used to find the eigenvalues i e critical buckling loads or undamped natural frequencies of structures the method has five stages which together ensure successful convergence on the required eigenvalues in all circumstances thus whenever checks indicate its suitability parabolic interpolation is used to obtain eigenvalues more rapidly than would the popular bisection alternative the checks also ensure a wise choice of the determinant used by the interpolation the determinants available are all usually zero at eigenvalues and comprise the determinant of the overall stiffness matrix kn and the determinants which result with negligible extra computation from effectively considering all except the last m m 1 2 n 1 freedoms to which kn corresponds as internal substructure freedoms tests showed time savings compared to bisection of 31 per cent when finding non coincident eigenvalues to relative accuracy 10 4 increasing to 62 per cent when 10 8 the tests also showed time savings of about 10 per cent compared with an earlier newtonian approach the method requires no derivatives and its use in the widely available space frame program bunvis rg has demonstrated how easily it can replace bisection which was used in the earlier program bunvis

an exact model reduction technique for beam structures combination of transfer and dynamic stiffness matrices an exact and direct modelling technique is presented in this paper for modelling beam structures this technique is based on the combination of transfer and dynamic stiffness matrices in this technique the whole structure is first divided into substructures based on the required master degrees of freedom for each substructure the global dynamic stiffness matrix dsm is obtained directly by rearranging the corresponding global transfer matrix in this way the internal degrees of freedom of each substructure are not used in the model therefore the order of the model is greatly reduced the efficiency and simplicity of the method is demonstrated by two numerical examples the results are compared with those obtained by other model reduction methods such as static and dynamic reduction methods and with experimental results it is shown that the method presented in this paper can predict accurately all the eigensolutions in a large frequency range in terms of a low dimension dynamic stiffness matrix

bicriterion single machine scheduling with resource dependent processing times a bicriterion problem of scheduling jobs on a single machine is studied the processing time of each job is a linear decreasing function of the amount of a common discrete resource allocated to the job a solution is specified by a sequence of the jobs and a resource allocation the quality of a solution is measured by two criteria f1 and f2 the first criterion is the maximal or total weighted resource consumption and the second criterion is a regular scheduling criterion depending on the job completion times both criteria have to be minimized general schemes for the construction of the pareto set and the pareto set epsilon approximation are presented computational complexities of problems to minimize f1 subject to f 2 le k and to minimize f2 subject to f 1 le k where k is any number are studied for various functions f1 and f2 algorithms for solving these problems and for the construction of the pareto set and the pareto set epsilon approximation for the corresponding bicriterion problems are presented

on development of a finite dynamic element and solution of associated eigenproblem by a block lanczos procedure the paper first presents the details of the development of a new six noded plane triangular finite dynamic element a block lanczos algorithm is developed next for the accurate and efficient solution of the quadratic matrix eigenvalue problem associated with the finite dynamic element formulation the resulting computer program fully exploits matrix sparsity inherent in such a discretization and proves to be most efficient for the extraction of the usually required first few roots and vectors including repeated ones most importantly the present eigenproblem solution effort is shown to be comparable to that of the corresponding finite element analysis thereby rendering the associated dynamic element method rather attractive owing to superior convergence characteristics of such elements presented herein

chebyshev filtering lanczos process in the subspace iteration method bathe s basic algorithm of subspace iteration for the solution of the symmetric eigenvalue problem is improved by including a chebyshev filtering mechanism to obtain satisfactory convergence for the largest eigenvalues a shifting strategy is adopted the shift factor is approximately computed by the lanczos process

exact solutions for sturm liouville problems on trees via novel substitute systems and the wittrick williams algorithm since 1970 the wittrick williams algorithm has been applied with increasing sophistication in structural mechanics to guarantee that eigenvalues cannot be missed and are calculated accurately the underlying theorem enables its application to any discipline requiring eigenvalues of self adjoint systems of differential equations its value in mathematics was recently illustrated by studying sturm liouville equations on large homogeneous trees with dirichlet boundary conditions and n 43 levels recursive subsysteming was applied n 1 times to assemble the tree progressively from sub trees hence numerical results confirmed the recent theoretical bounds of sobolev solomyak for n in addition a structural mechanics analogy yielded a proof that many eigenvalues had high multiplicities determined by n and the branching number b inspired by the structural mechanics analogy we now prove that all eigenvalues of the tree are obtainable from n substitute chains r 1 2 n which involve only r consecutively linked differential equations and which have only singlefold eigenvalues equations are also derived for the multiplicities these eigenvalues have for the tree hence double precision calculations on a pc readily gave eigenvalues for n 10 6 and b 10 i e 10 999 999 linked sturm liouville equations moreover a simple equation is derived which gives all the eigenvalues of uniform trees with dirichlet conditions at both ends and band gap spectra are numerically demonstrated and theoretically justified for trees with the dirichlet conditions at either end replaced by neumann ones additionally even if each multiple eigenvalue would be counted as if it were singlefold the proportion of eigenvalues that are multiple is proved to approach unity as n

tsallis non extensive statistics intermittent turbulence soc and chaos in the solar plasma part one sunspot dynamics abstract in this study the non linear analysis of the sunspot index is embedded in the non extensive statistical theory of tsallis 1988 2004 2009 7 9 10 the q triplet of tsallis as well as the correlation dimension and the lyapunov exponent spectrum were estimated for the svd components of the sunspot index timeseries also the multifractal scaling exponent spectrum f a the generalized renyi dimension spectrum d q and the spectrum j p of the structure function exponents were estimated experimentally and theoretically by using the q entropy principle included in tsallis non extensive statistical theory following arimitsu and arimitsu 2001 2000 76 77 our analysis showed clearly the following a a phase transition process in the solar dynamics from high dimensional non gaussian soc state to a low dimensional non gaussian chaotic state b strong intermittent solar turbulence and anomalous multifractal diffusion solar process which is strengthened as the solar dynamics makes a phase transition to low dimensional chaos in accordance to ruzmaikin zeleny and milovanov s studies zelenyi and milovanov 1991 21 milovanov and zelenyi 1993 22 ruzmakin et al 1996 26 c faithful agreement of tsallis non equilibrium statistical theory with the experimental estimations of i non gaussian probability distribution function p x ii multifractal scaling exponent spectrum f a and generalized renyi dimension spectrum d q iii exponent spectrum j p of the structure functions estimated for the sunspot index and its underlying non equilibrium solar dynamics

exact bernoulli euler dynamic stiffness matrix for a range of tapered beams bernoulli euler theory and bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial torsional and flexural vibrations of any beam which is tapered such that a varies as yn and gj and i both vary as y n 2 where a gj and i have their usual meanings y cx l 1 c is a constant such that c 1 l is the length of the beam and x is the distance from one end of the beam numerical checks give better than seven figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements a procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped this enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members

introducing the use of positive and negative inertial functions in asymptotic modelling the work presented here shows that the natural frequencies of constrained systems may be obtained from asymptotic models of corresponding systems where the constraints are replaced by artificial mass or moment of inertia of very large positive and negative values this offers a convenient alternative to the current practice of using artificial elastic restraints of large stiffness a concept introduced by courant in 1943 to remove a limitation on the choice of admissible functions recent publications show that in order to control the error caused by approximating constraints with restraints of large stiffness it is necessary to use both positive and negative stiffness values however the negative stiffness introduces instability near the lower modes of vibration and the magnitude of negative stiffness parameter used must be greater than the highest critical stiffness to ensure bounded results are obtained the use of positive and negative artificial inertial parameters overcomes this problem as they do not introduce instability near the lower modes allowing the natural frequencies of constrained systems to be delimited to any desired accuracy

the use of the reciprocals of positive and negative inertial functions in asymptotic modelling this paper shows how the recently introduced concept of positive and negative inertial functions in asymptotic modelling may be used effectively noting an almost linear variation of the natural frequency over a wide range of values of the reciprocal of the inertial parameter the results presented show that both the natural frequencies of constrained systems can be calculated accurately using only moderately large artificial inertial parameters e g which are only about 10 times that of the system mass and also that values about a million times more than this do not precipitate ill conditioning when using 16 figure computer accuracy this helps to minimize the chances of numerical problems occurring due to the limitation of machine precision

a method to solve the nonlinear eigenvalue problem of timoshenko plane frames with rigid offsets and end releases abstract the study reported here makes use of the dynamic stiffness matrix dsm for timoshenko beams hence considering rotatory inertia and shear deflection to obtain the exact solution of the associated free vibration problem the natural frequencies and vibration modes in the presence of end releases and rigid offsets are all exactly obtained for different combinations of boundary conditions and in plane arrangements of members this is possible by solving the resulted nonlinear and transcendental eigenvalue problem using two general methods one based on the wittrick williams algorithm adapted to handle end release and rigid offset and the other on the power deflation secant method the latter offered here as a novelty is proven to be an efficient technique judging from the examples given using both algorithms and the finite element method it is concluded that the power deflation method is a powerful tool to solve the nonlinear eigenvalue problem leading to exact solutions of the free vibration of complex skeleton like structures

application of the wittrick williams algorithm to the sturm liouville problem on homogeneous trees a structural mechanics analogy the wittrickwilliams ww algorithm was developed over 30 years ago and has been applied with increasing sophistication to problems in structural mechanics ever since much wider applications to a

wittrick williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters the well established wittrick williams algorithm is used to derive novel and general proofs that show that the eigenvalues of systems with constraints can be bracketed by replacing the constraints by positive and negative pairs of either ordinary or inertial penalty parameters it is also shown that convergence occurs from both above and below when the numerical values of these parameters are increased towards infinity the proofs are applicable in many contexts but are derived in that of structural systems for which the eigenvalues are either buckling load factors or the squares of natural frequencies of vibration ordinary penalty parameters are stiffnesses of translational and rotational springs and inertial penalty parameters are either masses or rotary inertias the penalty parameters can be used to constrain a system or to impose constraints between systems it is shown that the use of inertial penalty parameters has several advantages compared with using ordinary ones then the pth eigenvalue of a system with n constraints is bounded closely from above by the p n th eigenvalue of the system with very large positive inertial penalty parameters and from below by the pth eigenvalue when large negative values are used instead this work is expected to enhance the versatility of numerical eigenproblem methods e g the rayleigh ritz method copyright 2007 john wiley sons ltd

beam bending torsion dynamic stiffness method for calculation of exact vibration modes abstract the exact dynamic stiffness matrix is derived for a straight and uniform beam element whose elastic and inertial axes are not coincident elementary bending torsion beam theory is used and bending translation is restricted to one direction the element matrix can be used in the dynamic stiffness method for calculation of exact natural frequencies mode shapes and generalized masses for planar assemblages of connected bending torsion beams the dynamic stiffness method is outlined and details pertinent to the bending torsion beam element are given an example of a three beam assemblage representing an airplane wing is posed and exact numerical solutions are tabulated for natural frequencies and generalized masses solutions calculated by the finite element method are also tabulated for comparison

recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices when exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures a challenging transcendental eigenvalue problem results the present paper presents a newly developed mathematically elegant and computationally efficient method for accurate and reliable computation of both natural frequencies and vibration modes the method can also be applied to buckling problems the transcendental eigenvalue problem is first reduced to a generalized linear eigenvalue problem by using newton s method in the vicinity of an exact natural frequency identified by the wittrick williams algorithm then the generalized linear eigenvalue problem is effectively solved by using a standard inverse iteration or subspace iteration method the recursive use of the newton method employing the wittrick williams algorithm to guide and guard each newton correction gives secure second order convergence on both natural frequencies and mode vectors the second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods which typically give modes of much lower accuracy than that of the natural frequencies the excellent performance of the method is demonstrated by numerical examples including some demanding problems e g with coincident natural frequencies with rigid body motions and large scale structures copyright 2003 john wiley sons ltd

insight into the dynamic condensation technique of non classically damped models the dynamic condensation technique fully depends on the definition and computational scheme of the dynamic condensation matrix four definitions for the dynamic condensation matrix in the single mode m mode response dependent dynamic condensation and modal reduction of non classically damped models are presented they are respectively defined as the relations of the single eigenvector m eigenvectors p eigenvectors and responses between the master and the slave degrees of freedom using the complex mode superposition technique the response dependent dynamic condensation matrix may be interpreted as any mode including whole mode m mode p mode and single mode dependent condensation matrix computational equations for the dynamic condensation matrix are derived for each of definitions after the proper introduction of the assumptions for the single mode and the response dependent dynamic condensation the same computational equation is obtained from the former three definitions in the modal reduction the dynamic condensation matrix is directly computed from the eigenvector matrix of full model because the eigenvector matrix of the non classically damped models is generally complex the complex numerical operations are required in the commonly used expression an alternative expression is derived in which only the real numerical operations are necessary furthermore it is proven that the dynamic condensation matrix and the reduced system matrices resulted from the modal reduction all have real values

a successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations an alternative algorithm for finding exact natural frequencies or buckling loads from the transcendental e g dynamic stiffness matrix method is presented in this paper and evaluated by using the plate assembly testbed program viconopt the method is based on the bounding properties of the eigenvalues provided by either linear or quadratic matrix pencils on the exact solutions of the transcendental eigenvalue problem the procedure presented has five stages including two accuracy checking stages which prevent unnecessary calculations numerical tests on buckling of general anisotropic plate assemblies show that significant time savings can be achieved compared with an earlier multiple determinant parabolic interpolation method

natural vibration and buckling of general periodic lattice structures a method is presented for vibration and buckling analysis of arbitrary lattice structures having repetitive geometry in any combination of coordinate directions the approach is based on exact member theory representing the stiffness of an individual member subject to axial load and in the case of vibration undergoing harmonic oscillation the method is an extension of previous work that was limited to specific geometries the resulting eigenvalue problem is of the size associated with the repeating element of the structure a computer program has been developed incorporating the theory and results are given for vibration of rectangular platforms and a large antenna structure having rotational symmetry buckling and vibration results for cablestiffened rings are also given

dynamic buckling of columns by biaxial moments and uniform end torque a new concept of uniform torque is proposed for the dynamic torsional buckling analysis a dynamic biaxial moments and torque buckling theory is presented for analysis in structural dynamics second order effects of the axial force biaxial moments and torque are considered the consistent natural boundary moments and forces are derived to ensure the symmetry of the dynamic stiffness matrix in fulfilling the requirement of the reciprocal theorem and conservation of energy the exact dynamic stiffness matrix is obtained using power series expansion the derivatives of the analytical dynamic stiffness matrix with respect to different loading and geometric parameters are derived explicitly for sensitivity and continuation analyses generally distributed axial force can be analyzed without difficulty it is pointed out that non uniform sections may not be handled by power series due to the convergent problem global pictures for all kinds of linear dynamic buckling are given for the first time the methodology is based on finite element formulation and therefore it can easily be extended to analyze structural frames

the use of asymptotic modelling in vibration and stability analysis of structures abstract the use of springs with very large stiffness to model constraints in vibratory systems has been a popular approach to overcome the limitations on the choice of admissible functions in the rayleigh ritz method the maximum possible error resulting from this asymptotic modelling can be determined by using positive and negative stiffness values or in general terms using positive and negative penalty functions this paper illustrates how this method could be used to determine the critical loads of structures

the lanczos algorithm applied to kron s method in this paper kron s primitive composite system matrix is shown to be reducible to a symmetric indefinite matrix this matrix although never formed explicitly can then be transformed to a tridiagonal matrix of reduced order by the lanczos algorithm eigenvalue solutions of this partially tridiagonalized matrix give very good approximations to the eigenvalues of the composite system the method has multiplication counts which are over 90 per cent lower than when the condensed kron matrix is solved by the newton raphson iteration applied to the kron scalar equation numerical examples illustrating the kron lanczos method when solving for the natural frequencies of frames with large numbers of degrees of freedom are presented

dynamic condensation with synthesis of substructure eigenproperties abstract for structures divided into substructures which are defined by their eigenproperties a combination of the node synthesis and condensation techniques is presented to obtain a first few important eigenproperties of the combined system accurately the accuracy of the calculated modal properties can be increased by iteratively solving an improved and condensed eigenvalue problem if desired the higher modal properties can also be obtained by solving a complementry eigenvalue problem associated with the higher modes the availability of the substructure modes in this approach also provides a very convenient and efficient means of selecting the most appropriate kept degrees of freedom in the condensed eigenvalue problem to expediate the convergence to the correct eigenproperties the methods for coupling of the substructure eigenproperties obtained with different boundary conditions at the interface are described several examples are presented to demonstrate the application of the approach

convex resource allocation problems on directed acyclic graphs duality complexity special cases and extensions consider the following resource allocation problem on a directed acyclic graph the precedence graph each vertex has a known work load and a fixed amount of total resource is available the time required to process a vertex is inversely proportional to the amount of the resource allocated to it the time to complete all of the work is the length of time to complete a longest chain in the graph the problem of finding an allocation which minimizes the time required to complete all of the work subject to the limited resource availability can be formulated as a separable convex programming problem we use results from lagrangian duality for convex programs the length width inequality and dilworth s theorem for directed acyclic graphs to obtain a strong relationship between optimal solutions of this problem and its dual this allows us to obtain closed form solutions for certain special classes of graphs and leads to a generalization of the lym property for partially ordered sets the computational complexity of the general problem is an open question however the ellipsoid method yields a fully polynomial approximation scheme and some light can be shed on the associated decision problem the results of this paper are shown to extend to resource allocation problems on perfect graphs

exact determinant for infinite order fem representation of a timoshenko beam column via improved transcendental member stiffness matrices transcendental stiffness matrices for vibration or buckling have been derived from exact analytical solutions of the governing differential equations for many structural members without recourse to the discretization of conventional finite element methods fem their assembly into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem whose eigenvalues natural frequencies or critical load factors can be found with certainty using the wittrick williams algorithm a very recently discovered analytical property is the member stiffness determinant which equals the fem stiffness matrix determinant of a clamped ended member modelled by infinitely many elements normalized by dividing by its value at zero frequency or load factor curve following convergence methods for transcendental eigenproblems are greatly simplified by multiplying the transcendental overall stiffness matrix determinant by all the member stiffness determinants to remove its poles in this paper the transcendental stiffness matrix for a vibrating axially loaded timoshenko member is expressed in a new convenient notation enabling the first ever derivation of its member stiffness determinant to be obtained further expressions are derived also for the first time for unloaded and for static loaded timoshenko members these new expressions have the advantage that they readily reduce to corresponding expressions for bernoulli euler members when shear rigidity and rotatory inertia are neglected additionally the total equivalence of the normalized transcendental determinant with that of an infinite order fem formulation aids understanding and evaluation of conventional fem results examples are presented to illustrate the use of the member stiffness determinant

a technique for the systematic choice of admissible functions in the rayleigh ritz method abstract a simple and systematic choice of admissible functions which are the eigenfunctions of the closest simple problem extracted from the one considered is proposed the extracted problem must be less constrained than the original one in other words it must be a problem where some constraints or other complications e g added masses are eliminated elastic constraints replace the eliminated rigid ones the convergence is also analyzed this approach has practical applications when it is possible to extract a problem with eigenfunctions expressed in closed form it also allows a very simple calculation of the potential energy of the system solutions for several cases involving beams are given in order to show the power of the method application of the method to circular plates and shells is also addressed

a contribution to the exact modal solution of in plane beam structures abstract the exact vibration modes and natural frequencies of planar structures and mechanisms comprised euler bernoulli beams are obtained by solving a transcendental non linear eigenvalue problem stated by the dynamic stiffness matrix dsm to solve this kind of problem the most employed technique is the wittrick williams algorithm developed in the early seventies by formulating a new type of eigenvalue problem which preserves the internal degrees of freedom for all members in the model the present study offers an alternative to the use of this algorithm the new proposed eigenvalue problem presents no poles so the roots of the problem can be found by any suitable iterative numerical method by avoiding a standard formulation for the dsm the local mode shapes are directly calculated and any extension to the beam theory can be easily incorporated it is shown that the method here adopted leads to exact solutions as confirmed by various examples extensions of the formulation are also given where rotary inertia end release skewed edges and rigid offsets are all included

exact dynamic stiffness method for non uniform timoshenko beam vibrations and bernoulli euler column buckling abstract the exact dynamic stiffness method is further extended using a recently developed approach for vibration of bernoulli euler members to flexural free vibration of non uniform timoshenko beams with gradual or stepwise non uniformity of geometric and or material properties and to euler buckling of similarly non uniform columns two key strategies are emphasized i formulation of the governing ordinary differential equations ode for dynamic stiffnesses and their derivatives and the solution of the ode problem by standard ode solvers and ii establishment of mesh generation rules for the two problems extension of the method to three dimensional frames with non uniform members poses no major theoretical hurdles numerical examples including challenging problems are given to show the effectiveness efficiency accuracy and reliability of the proposed method which unlike the finite element method is exact and so can be iterated until any preset accuracy is achieved

bounding properties for eigenvalues of a transcendental dynamic stiffness matrix by using a quadratic matrix pencil an approximate representation of a transcendental dynamic stiffness matrix k rho by a simple quadratic matrix pencil a rho b rho 2 c is studied in this paper the matrix pencil is formed by expressing the elements of k as parabolic functions based on choosing three fixed values of the eigenparameter rho general bounds on the exact eigenvalues of the transcendental eigenvalue problem provided by the quadratic matrix pencil are shown to exist provided that the three fixed values are below the lowest pole of the transcendental stiffness matrix considered and that the three coefficient matrices are positive definite numerical examples illustrate and confirm these bounding properties furthermore the bounding properties are extended to constrained dynamic stiffness matrices e g matrices formed by using lagrangian multipliers to couple individual stiffness matrices of several different responses when a single response does not satisfy the desired boundary conditions