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

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

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

the detection of the source of acoustical noise in two dimensions we consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin mathematically this field satisfies the helmholtz equation in this paper we consider the model two dimensional case we show that any regular solution of this equation admits a unique representation by a single layer potential so that the problem is reduced to the solution of a linear integral equation of the first kind we prove uniqueness of reconstruction and obtain a sharp stability estimate finally for two geometries and sources of noise simulating the cabin of the aircraft and two engines we give results of the numerical solution of this integral equation comparing regularization by the truncated singular value decomposition and the conjugate gradient method

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

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

dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition abstract the dual formulation of the multiple reciprocity method mrm is employed to solve the acoustic mode of a cavity with a thin partition in order to avoid the spurious eigenvalues and non uniqueness of the solution in the case of zero thickness for the partition the hypersingular equation for mrm is considered to determine the invariant quantity of influence coefficients more efficiently an objectivity concept i e a frame of indifference is used by means of co ordinate transformation a generalized eigenvalue problem is derived and transformed into a standard eigenvalue problem through the state space formulation the spurious roots in mrm are examined and filtered out by using the hypersingular formulation three examples including finite thickness of the partition zero thickness and no partition are shown to check the validity of the proposed method also the analytical solution if available the fem results obtained by petyt and by abaqus and experimental measurements are compared with those of the proposed method and it is found that the agreement between them is very good

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

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

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

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

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

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

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

modified tikhonov regularization method for the cauchy problem of the helmholtz equation in this paper the cauchy problem for the helmholtz equation is investigated by green s formulation the problem can be transformed into a moment problem then we propose a modified tikhonov regularization algorithm for obtaining an approximate solution to the neumann data on the unspecified boundary error estimation and convergence analysis have been given finally we present numerical results for several examples and show the effectiveness of the proposed method

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

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

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

iterative tikhonov regularization for the cauchy problem for the helmholtz equation abstract the cauchy problem for the helmholtz equation appears in various applications the problem is severely ill posed and regularization is needed to obtain accurate solutions we start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in h 1 2 spaces by introducing an artificial boundary in the interior of the domain we obtain an inner product for this hilbert space in terms of a quadratic form associated with the helmholtz equation perturbed by an integral over the artificial boundary the perturbation guarantees positivity property of the quadratic form this inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well posed boundary value problem for the helmholtz equation with transmission boundary conditions on the artificial boundary in an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the cauchy problem in this work we instead use the conjugate gradient method for minimizing a tikhonov functional the added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data numerical tests show that the proposed algorithm works well

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

a potential function method for the cauchy problem of elliptic operators abstract in this paper we deal with the cauchy problem of elliptic operators through the use of a single layer potential function we devise a numerical method for approximating the solution of the cauchy problem of elliptic operators which are well known to be highly ill posed in nature the method is based on the denseness of single layer potential functions convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method it has been observed that the use of more cauchy data will greatly improve the accuracy of the approximate solutions

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