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

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

efficient non linear solid fluid interaction analysis by an iterative bem fem coupling an iterative coupling of finite element and boundary element methods for the time domain modelling of coupled fluid solid systems is presented while finite elements are used to model the solid the adjacent fluid is represented by boundary elements in order to perform the coupling of the two numerical methods a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved in the case of local non linearities within the finite element subdomain it is straightforward to perform the iterative coupling together with the iterations needed to solve the non linear system in particular a more efficient and a more stable performance of the new coupling procedure is achieved by a special formulation that allows to use different time steps in each subdomain copyright 2005 john wiley sons ltd

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

on a hybrid boundary element method in this paper we study a symmetric boundary element method based on a hybrid discretization of the steklov poincare operator well suited for a symmetric coupling of finite and boundary elements the representation used involves only single and double layer potentials and does not require the discretization of the hypersingular integral operator as in the symmetric formulation the stability of the hybrid galerkin discretization is based on a bbl like stability condition for the trial spaces numerical examples confirm the theoretical results

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

closed form eigensolutions of nonviscously nonproportionally damped systems based on continuous damping sensitivity abstract in this paper nonviscous nonproportional vibrating structures are considered nonviscously damped systems are characterized by dissipative mechanisms which depend on the history of the response velocities via hereditary kernel functions solutions of the free motion equation lead to a nonlinear eigenvalue problem involving mass stiffness and damping matrices viscoelasticity leads to a frequency dependence of this latter in this work a novel closed form expression to estimate complex eigenvalues is derived the key point is to consider the damping model as perturbed by a continuous fictitious parameter assuming then the eigensolutions as function of this parameter the computation of the eigenvalues sensitivity leads to an ordinary differential equation from whose solution arises the proposed analytical formula the resulting expression explicitly depends on the viscoelasticity frequency derivatives of the damping function the nonproportionality influence of the modal damping matrix off diagonal terms eigenvectors are obtained using existing methods requiring only the corresponding eigenvalue the method is validated using a numerical example which compares proposed with exact ones and with those determined from the linear first order approximation in terms of the damping matrix frequency response functions are also plotted showing that the proposed approach is valid even for moderately or highly damped systems

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

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

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

solitons and discrete eigenfunctions of the recursion operator of non linear evolution equations i the caudrey dodd gibbon sawada kotera equation the solution of the third order isospectral equation of the caudrey dodd gibbon sawada kotera equation cdgske for soliton potential is obtained recursively from the riccati equation derived by iterating once the auto backlund transformation it is then shown that the discrete eigenfunctions of the sixth order recursion operator for this equation can be written in terms of the solutions of the isospectral equation the behaviour of the 1 soliton solution which has certain novel features is studied a sine gordon like equation resembling the double sine gordon equation is derived from the cdgske

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

symbolic computation and new families of exact soliton like solutions to the integrable broer kaup bk equations in 2 1 dimensional spaces in this paper new families of soliton like solutions are obtained for 2 1 dimensional integrable broer kaup equations by using the symbolic computation method developed by gao and tian sample solutions obtained from these methods are presented solitary wave solutions are merely a special case in one family the method can also be extended to other types of nonlinear evolution equations in mathematical physics

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

time domain bem analysis for the 2d scalar wave equation initial conditions contributions to space and time derivatives the complete expressions corresponding to space and time derivatives of volterra s integral representation of the time dependent scalar wave equation are presented in this paper the concept of finite part of an integral is used in order to compute boundary kernels contributions time integration of boundary kernels is performed analytically for linear and constant time approximations for the potential and its normal derivative respectively the new integral representations presented in this text are quite general as contributions due to the initial conditions have also been included two examples are presented in order to assess the accuracy of the formulation

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

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

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

a variationally coupled fe be method for transient problems a variationally coupled finite element boundary element method is developed for transient problems a single variational statement is obtained for the entire domain and the unknown tractions which may be discontinuous on the interface and are often a source of difficulties are eliminated moreover no interface conditions need be taken into consideration at the level of the discretized equation the discrete equations for the coupled system can be obtained directly without any intermediate steps the method generalizes a coupling method previously developed by the authors for statics numerical examples show that the solutions obtained by the present method agree very well with those obtained by analytical solutions

on the combinatorics of the hirota d operators a generic formula is presented which relates the hirota d operators to simple combinatorics particular classes of partition polynomials bell polynomials and generalizations are found to play an important role in the characterization of bilinearizable equations as a consequence it is shown that bilinear backlund transformations for single field bilinearizable equations linearize systematically into corresponding laxpairs

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

n soliton solutions for the combined kdv cdg equation and the kdv lax equation abstract the hirota ito approach presented in r hirota m ito resonance of solitons in one dimension j phys soc jpn 52 3 1983 744 748 for extending fifth order integrable equations with a nonvanishing boundary conditions to combined equations is used in this work the generalized fifth order caudrey dodd gibbon cdg and lax equations are extended to combined integrable equations the hirota s bilinear method is used to derive multiple soliton solutions for the extended kdv cdg and kdv lax equations

seismic soil structure interaction analysis by direct boundary element methods abstract upon establishing the mathematical framework necessary for a proper understanding of the analytical theory a regularized form of the conventional direct boundary integral equation formulation for three dimensional elastodynamics is presented for a general anisotropic medium founded on the basis of a full decomposition of the greens functions into regular and singular parts the alternative boundary integral equation format is compact and without demand ingmathematical and numerical complexities such as cauchy principal values extended to deal with general seismic soil structure interaction problems in semi infinite media the formulation is implemented computationally together with a rigorous treatment of singular dynamic multi layered viscoelastic half space greens functions and interfacial boundary tractions arising in typical soil structure foundation configurations a set of new benchmark numerical results are included

delta dressing and exact solutions for the 2 1 dimensional harry dym equation exact implicit solutions with functional parameters of the 2 1 dimensional harry dym hd equation are constructed by the delta dressing method the interrelation between the hd and modified kadomtsev petviashvili mkp equations is established within the framework of the delta dressing method knowledge of the mkp eigenfunctions allows the solutions of the 2 1 dimensional hd equation to be represented in a simple parametric form

hourglass control in linear and nonlinear problems abstract mesh stabilization techniques for controlling the hourglass modes in under integrated hexahedral and quadrilateral elements are described it is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly it is also shown that this leads to an hourglass control that satisfies the patch test the nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed finally effective means of implementing these hourglass procedure in computer codes are described applications to both the laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered

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

backlund transformation lax pair and solutions for the caudrey dodd gibbon equation by using bell polynomials and symbolic computation we investigate the caudrey dodd gibbon equation analytically through a generalization of bells polynomials its bilinear form is derived based on which the periodic wave solution and soliton solutions are presented and the soliton solutions with graphic analysis are also given furthermore backlund transformation and lax pair are derived via the bells exponential polynomials finally the ablowitz kaup newell segur system is constructed

advanced development of time domain bem for two dimensional scalar wave propagation an improved formulation for solving 2d transient scalar wave propagation problems by the boundary element method bem is presented the kernels presented are simpler and better behaved than those that have appeared in the published literature an appropriate set of temporal shape functions for linear variation is used for spatial variations isoparametric quadratic elements are used all of these represent significant improvements over the present level of sophistication in the analysis of 2d transient scalar problems the algorithm is implemented in a general purpose boundary element code known as gpbest

a fem bem coupling procedure to model the propagation of interacting acoustic acoustic acoustic elastic waves through axisymmetric media abstract in the present work the propagation of waves through axisymmetric bodies is analysed taking into account acoustic acoustic acoustic elastic interacting media and coupled numerical methodologies the boundary element method bem is here employed to model acoustic media its time convolution integrals are evaluated analytically and the theta method is adopted to advance the solution on time the finite element method fem on the other hand models wave propagation through elastic and acoustic media the newmark method being employed for time stepping the fem bem coupling is accomplished through an iterative process in which variables at common interfaces are updated until convergence is achieved thus each sub domain is analysed independently so that the numerical algorithm can take full advantage of the favourable features of the different methods involved relaxation parameters which are introduced in order to speed up and or ensure convergence and the possibility to consider different time steps within each sub domain are two factors that improve substantially the efficiency stability accuracy and flexibility of the coupled analysis the potentialities of the proposed procedures are illustrated through numerical examples presented at the end of the paper

coupling of the mesh free finite cloud method with the boundary element method a collocation approach abstract meshless and mesh based methods are among the tools frequently applied in the numerical treatment of partial differential equations pdes this paper presents a coupling of the meshless finite cloud method fcm and the standard mesh based boundary element method bem which is motivated by the complementary properties of both methods while the bem is appropriate for solving linear pdes with constant coefficients in bounded or unbounded domains the fcm is appropriate for either homogeneous inhomogeneous or even nonlinear problems in bounded domains both techniques fcm and bem use a collocation procedure in the numerical approximation no mesh is required in the fcm subdomain and its nodal point distribution is completely independent of the bem subdomain the coupling approach is demonstrated for linear elasticity problems because both fcm and bem employ traction displacement relations no transformations between forces and tractions typical of bem and finite element coupling are needed several numerical examples are given to demonstrate the proposed approach

eigensolutions of nonviscously damped systems based on the fixed point iteration abstract in this paper nonviscous nonproportional symmetric vibrating structures are considered nonviscously damped systems present dissipative forces depending on the time history of the response via kernel hereditary functions solutions of the free motion equation leads to a nonlinear eigenvalue problem involving mass stiffness and damping matrices this latter as dependent on frequency viscous damping can be considered as a particular case involving damping forces as function of the instantaneous velocity of the degrees of freedom in this work a new numerical procedure to compute eigensolutions is proposed the method is based on the construction of certain recursive functions which under a iterative scheme allow to reach eigenvalues and eigenvectors simultaneously and avoiding computation of eigensensitivities eigenvalues can be read then as fixed points of those functions a deep analysis of the convergence is carried out focusing specially on relating the convergence conditions and error decay rate to the damping model features such as the nonproportionality and the viscoelasticity the method is validated using two 6 degrees of freedom numerical examples involving both nonviscous and viscous damping and a continuous system with a local nonviscous damper the convergence and the sequences behavior are in agreement with the results foreseen by the theory

the patch test a condition for assessing fem convergence this paper discusses various forms of the irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency further it is shown that the test can be applied for verification of stability requirements thus providing also a sufficient condition of convergence the patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

dynamic analysis of fluid soil structure interaction problems by the boundary element method the present paper describes an iterative procedure for bem bem coupling the paper presents suitable interface conditions and algorithms for iteratively coupling sub domains modeled by three different boundary element time domain formulations namely acoustic and elastodynamic bem formulations based on time dependent green s functions and non linear time domain approach which employs elastostatic green s functions and therefore requires domain discretization two examples are analyzed and at the end of the paper conclusions of the study are presented

computing eigenvalues eigenvectors and frequency responses of structures with non proportional damping abstract this paper presents algorithms to compute eigenvalues eigenvectors and frequency response matrix of a linear structure with non proportional damping the method is based on derivatives of repeated or unrepeated complex eigenvalues and complex eigenvectors since eigenvectors are not unique for a repeated eigenvalue a special algorithm is developed to select the correct eigenvector for a given non proportional damping the advantage of the proposed method is that it only requires the knowledge of natural frequencies and mode shapes of the undamped structure the validity of this method is established by comparing numerical results from this algorithm to exact answers

fem and bem coupling in elastostatics using localized lagrange multipliers the equations produced by the finite and boundary element methods in structural mechanics are expressed in different variables and cannot be linked without modifications conventional coupling methods of these numerical techniques have traditionally been based on the use of classical lagrange multipliers making a direct connection of the two solids through their common interfaces using matching meshes and altering the formulation of one of the methods to make it compatible with the other in this work a discrete surface called frame is interposed between the connected subdomains to approximate their common interface displacements it is treated using a finite element discretization and connected to each substructure using localized lagrange multipliers collocated at the interface nodes this methodology facilitates the connection of non matching finite and boundary element meshes avoiding modifications to the numerical methods used and providing a partitioned formulation which preserves software modularity