partial eigenstructure assignment for the quadratic pencil abstract it is shown in this paper that by the appropriate choice of gain and input influence matrices certain eigenpairs of a vibrating system may be assigned while the other eigenpairs remain unchanged the system under consideration is modelled by a set of second order differential equations and the assignment is carried by multi input state feedback control the solution may be of particular interest in the stabilization and control of flexible structures using smart materials where only a small part of the eigenstructure is to be reassigned and the rest is required to remain unchanged the method presented is illustrated with a numerical example

a receptance based method for predicting latent roots and critical points in friction induced vibration problems of asymmetric systems this paper studies the latent roots and critical points of friction induced vibration problems in which the stiffness matrix is asymmetric the asymmetric terms are represented by a parameter or parameters related to the friction coefficient as the parameter value increases some latent roots of the asymmetric system change and even become complex with positive real parts at a critical point indicating flutter instability a method is put forward for computing the latent roots and predicting the critical value of this parameter at the flutter instability boundary of the asymmetric system based on the receptance of the symmetric system when measured receptances of the symmetric system at those locations where friction forces would be acting in the corresponding asymmetric system are available the simulated numerical example shows that this method is efficient

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

efficient computation of modal sensitivities for systems with repeated frequencies the physical origin and ambiguity of repeated frequency modes is discussed followed by a mathematical derivation of the governing equations for the partial derivatives of repeated eigenvalues with respect to changes in the system parameters this is followed by a derivation for the associated eigenvector derivative equations physical interpretations are included to help explain the analytical results an efficient computation procedure which preserves the bandwidth for very large systems is also proposed for systems with repeated or closely spaced eigenvalues two small examples to illustrate the solution procedure are presented 8 references

on the sensitivity of multiple eigenvalues of nonsymmetric matrix pencils this paper considers the sensitivity of semisimple multiple eigenvalues and corresponding generalized eigenvector matrices of a nonsymmetric matrix pencil analytically dependent on several parameters the directional derivatives of the multiple eigenvalues are obtained and the average of eigenvalues and corresponding generalized eigenvector matrices are proved to be analytic the results can be used to define the sensitivity of the semisimple multiple eigenvalues and corresponding generalized eigenvector matrices these are useful for investigating structural optimal design model updating and structural damage detection

free vibrations and sensitivity analysis of a defective two degree of freedom system free oscillations of a two degree of freedom system with honproportional damping are analyzed by a suitable choice of parameters a family of defective systems having a noncomplete set of eigenvectors is selected free motions of underdamped and overdamped defective systems are studied in the four dimensional state space and their main characteristics are discussed in particular the rate at which the trajectories are attracted by the eigenvectors is determined small perturbations of order e of the parameters are then considered and asymptotic expressions for the modified system eigensolutions are obtained these allow qualitative discussion of the effects of modifications on the mechanical behavior of nearly defective systems marked sensitivities of order 2 or e are found these depend strongly on the damping magnitude an extensive numerical analysis is performed n structural dynamics it is by no means rare to encounter systems having multiple eigenvalues this does not pose any substantial difficulties when the system is conservative but it involves specific problems when the system is nonconservative very often indeed the geometric multiplicity of the eigenvalue is less than the algebraic multiplicity and so the system has an incomplete set of eigenvectors insufficient to form a base for the state space systems of this type are called defective the free evolution of defective systems is well known to persons working in the automatic control or system theory field moreover basic notions can be found in any good book on linear algebra however in the author s opinion the problem is not sufficiently known to the structural analyst indeed it is common practice to assume damping of proportional type so that the eigenvectors coincide with those of the corresponding undamped system always forming a complete set and defective systems cannot occur

structural modification part 2 assignment of natural frequencies and antiresonances by an added beam in this paper the inverse structural modification problem is solved in order to determine the dimensions of the cross section of a beam that when added to an original structure will assign natural frequencies or antiresonances as specified in order for this to be accomplished rotational receptances must be measured as presented in the companion paper when the modification is cast as an additional forcing term on the original structure a system of multivariate polynomials in the parameters of the beam cross section are revealed the solution of this system yields the beam parameters that assign the specified vibration behaviour

structured quadratic inverse eigenvalue problem i serially linked systems quadratic pencils arising from applications are often inherently structured factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters for physical feasibility structural constraints must be respected consequently they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications however the issue of solvability is problem dependent and has to be addressed structure by structure this paper considers one particular structure where the elements of the physical system if modeled as a mass spring system are serially linked the discussion recasts both undamped and damped problems in a framework of inequality systems that can be adapted for numerical computation some open questions are described

lyapunov coefficients for an invisible fold fold singularity in planar piecewise hamiltonian systems abstract we study the stability of an invisible fold fold singularity of planar piecewise smooth hamiltonian vector fields by computing some kind of lyapunov coefficients we obtain the general expressions for the first five lyapunov coefficients as a consequence the bifurcation diagrams illustrating the number types and positions of the bifurcating small amplitude crossing limit cycles for these vector fields are determined

fast sensitivity analysis of defective system abstract sensitivity analysis of linear vibration system is of wide interest in this paper sensitivity analysis based on non defective system and defective system is summarized in all cases specially for the defective systems a fast method for the perturbation problem of state vectors is constructed in terms of the theories of generalized eigenvectors and adjoint matrices by this method the state vector derivatives can be expressed by a linear combination of generalized eigenvectors the expansion coefficients can be obtained without solving large scale equations based on eigensolutions of original system numerical results demonstrate the effectiveness and the stability of the method

quadratic model updating with no spill over and incomplete measured data existence and computation of solution abstract quadratic finite element model updating problem qfemup to be studied in this paper is concerned with updating a symmetric nonsingular quadratic pencil in such a way that a small set of measured eigenvalues and eigenvectors is reproduced by the updated model if in addition the updated model preserves the large number of unupdated eigenpairs of the original model the model is said to be updated with no spill over qfemup is in general a difficult and computationally challenging problem due to the practical constraint that only a very small number of eigenvalues and eigenvectors of the associated quadratic eigenvalue problem are available from computation or measurement additionally for practical effectiveness engineering concerns such as nonorthogonality and incompleteness of the measured eigenvectors must be considered most of the existing methods including those used in industrial settings deal with updating a linear model only ignoring damping only in the last few years a small number of papers been published on the quadratic model updating several of the above issues have been dealt with both from theoretical and computational point of views however mathematical criterion for existence of solution has not been fully developed in this paper we first i prove a set of necessary and sufficient conditions for the existence of a solution of the no spill over qfemup then ii present a parametric representation of the solution assuming a solution exists and finally iii propose an algorithm for qfemup with no spill over and incomplete measured eigenvectors interestingly it is shown that the parametric representation can be constructed with the knowledge of only the few eigenvalues and eigenvectors that are to be updated and the corresponding measured eigenvalues and eigenvectors complete knowledge of eigenvalues and eigenvectors of the original pencil is not needed which makes the solution readily applicable to real life structures

pole zero cancellation in structures repeated roots abstract the conditions for the creation of nodes of normal modes of vibration from the cancellation of poles and zeros are established when either the poles or the zeros or both appear as repeated eigenvalues the analysis is illustrated by numerical examples including the case of a pole zero cancellation at every co ordinate at the same frequency which is shown to occur whenever there are repeated poles if there are repeated poles and repeated zeros at the same frequency then the number of poles must be either one more one less or equal to the number of zeros

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

low rank damping modifications and defective systems the structural modification of dynamical systems is an important issue in a wide range of applications for example in vibration suppression or in active and passive control it is well known that for a proportionally or classically damped system there always exists a real matrix of eigenvectors which simultaneously diagonalizes the three system matrices of inertia damping and stiffness even if the system possesses repeated eigenvalues for general viscously damped systems the eigenvalue analysis must be performed in state space and for systems with distinct eigenvalues the corresponding eigenvectors diagonalize the state space matrices however with general viscous damping systems with repeated complex eigenvalues may have insufficient linearly independent complex eigenvectors these systems are termed defective in contrast to non defective or simple systems for defective systems only a jordan decomposition exists in this paper conditions on rank 1 rank 2 and higher rank modifications are derived which ensure that the modified system is simple if none of the eigenvalues of the unmodified system is an eigenvalue of the modified system then every rank 1 modification that produces a pair of repeated complex eigenvalues leads to a defective system under the same assumptions there exist higher rank modifications which lead to simple systems either these modifications produce a proportionally damped system or the restrictions on these modifications are rather strict which suggests that in practical cases every rank 2 or higher modification that produces repeated pairs of complex eigenvalues will lead to a defective system the findings are demonstrated by simulated examples

regularization and singular perturbation techniques for non smooth systems abstract this paper is concerned with some aspects of the qualitative geometric theory of non smooth systems we present a survey of the state of the art on the connection between the regularization process of non smooth vector fields and the singular perturbation problems as a matter of fact the main results in our setting fill a gap between these areas exploiting such results applications to constrained differential equations are discussed

on the inverse eigenvalue problem for t alternating and t palindromic matrix polynomials abstract the inverse eigenvalue problem for t alternating matrix polynomials over arbitrary algebraically closed fields of characteristic different from two is considered the main result shows that the necessary conditions obtained in 9 for a matrix polynomial to be the smith form of a t alternating matrix polynomial are under mild conditions also sufficient to be the smith form of a t alternating matrix polynomial with invertible leading coefficient which is additionally in anti triangular form in particular this result implies that any t alternating matrix polynomial with invertible leading coefficient is equivalent to a t alternating matrix polynomial in anti triangular form that has the same finite and infinite elementary divisors as the original matrix polynomial finally the inverse eigenvalue problem for t palindromic matrix polynomials is considered excluding the case that both 1 and 1 are eigenvalues

vibration nodes and the cancellation of poles and zeros by unit rank modifications to structures abstract the main focus of this paper is on pole zero cancellation in structures by unit rank modifications the modifications may be achieved by passive stiffnesses or by active control techniques a review of the classical theory on unit rank modification is included and new results are obtained to explain the existence of coincident zeros in point and cross receptances the necessary and sufficient conditions are established for the production of a vibration node by pole zero cancellation the classical theory is used to bring about the cancellation of poles and zeros and numerical examples are used to illustrate the application of the technique

multiple eigenvalue sensitivity analysis abstract this paper surveys the development of the sensitivity analysis of multiple eigenvalues and the associated invariant subspaces of two kinds of matrix eigenvalue problems in which the matrices are analytically dependent on several parameters

the relationship between the real and imaginary parts of complex modes abstract it is shown that a simple relationship exists between the real and imaginary parts of complex modes of all systems which can be represented by real and symmetric mass stiffness and damping matrices the relationship is most simply expressed in those cases where all roots are complex and where the real parts of all roots have the same sign in these cases the relationship can be expressed in a form where the imaginary part of the modal matrix is equal to the real part of the modal matrix post multiplied by a matrix which involves an arbitrary real orthogonal matrix and some diagonal matrices which are determined directly from the complex roots in other cases there remains a relationship between the real and imaginary parts but this must be expressed in a different way

regularization around a generic codimension one fold fold singularity abstract this paper is devoted to study the generic fold fold singularity of filippov systems on the plane its unfoldings and its sotomayor teixeira regularization we work with general filippov systems and provide the bifurcation diagrams of the fold fold singularity and their unfoldings proving that under some generic conditions is a codimension one embedded submanifold of the set of all filippov systems the regularization of this singularity is studied and its bifurcation diagram is shown in the visible invisible case the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding specially the appearance and disappearance of periodic orbits that are not present in the filippov vector field in the case of a linear regularization we prove that the regularized system is equivalent to a general slow fast system studied by krupa and szmolyan 10

derivatives of complex eigenvectors using nelson s method 6shyy w an adaptive grid method for navier stokes flow computation applied mathematics and computation vol 21 no 3 1987 pp 201 219 7jeng y n andliou s c modi edmultipleone dimensional adaptive grid method numerical heat transfer part b vol 15 no 2 1989 pp 241 247 8jeng y n and lee z s revisit to the modi ed muliple onedimensional adaptive grid method numerical heat transfer part b vol 29 no 3 1996 pp 305 323 9bockelie m j and smith r e an adaptive grid method for computing the high speed 3d viscous flow about a re entry vehicle aiaa paper 92 2685 june 1992 10yee h c and harten a implicit tvd schemes for hyperbolic conservation laws in curvilinear coordinates aiaa journal vol 25 no 2 1987 pp 266 274 11jeng y n and liou y c adaptive grid generation by elliptic equations with grid control at all of the boundaries numerical heat transfer part b vol 23 no 2 1993 pp 135 151

natural frequency and mode shape sensitivities of damped systems part ii multiple natural frequencies abstract an efficient algorithm with proven numerical stability is derived for computation of eigenvalue and eigenvector derivatives of damped vibratory systems with multiple eigenvalues in the proposed method adjacent eigenvectors and orthonormal condition are used to compose an algebraic equation whose order is n m n m where n is the number of co ordinates and m the number of multiplicity of a multiple natural frequency the mode shape derivatives of the damped systems can be obtained by solving the algebraic equation the method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters as an example of a structural system to demonstrate the theory of the proposed methods and its possibilities in the case of multiple eigenvalues the finite element model of the cantilever beam is considered and also a 5 dof mechanical system in the case of a non proportionally damped system the design parameter of the cantilever beam is its height and that of the 5 dof mechanical system is a spring

inverse vibration problem for un damped 3 dimensional multi story shear building models various researchers have contributed to the identification of the mass and stiffness matrices of two dimensional 2 d shear building structural models for a given set of vibratory frequencies the suggested methods are based on the specific characteristics of the jacobi matrices i e symmetric tri diagonal and semi positive definite matrices however in case of three dimensional 3 d structural models those methods are no longer applicable since their stiffness matrices are not tri diagonal in this paper the inverse problem for a special class of vibratory structural systems i e 3 d shear building models is investigated a practical algorithm is proposed for solving the inverse eigenvalue problem for un damped 3 d shear buildings the problem is addressed in two steps first using the target frequencies a so called normalized eigenvector matrix which is a banded matrix containing the information related to the frequencies and mode shapes of the target structural system is determined in this regard similar to the solution of inverse problem for 2 d shear building structural models in which an auxiliary structure is constructed by adding constraints or springs to the original system three auxiliary structures are proposed to solve the problem for 3 d cases in the second step the normalized eigenvector matrix is utilized to obtain the normalized stiffness matrix in turn this matrix is decomposed into the stiffness and mass matrices of the system finally a numerical example is presented to demonstrate the efficiency of the proposed algorithm in determining the mass and stiffness matrices of a 3 d structural model for a given set of target vibrational frequencies

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

degenerate grazing sliding bifurcations in planar filippov systems abstract in this paper we contribute to a class of degenerate grazing sliding bifurcations in planar filippov systems assuming that the unperturbed system has a generalized homoclinic loop to a fold fold point depending on the type of this fold fold point we exhibit some codimension 2 bifurcation diagrams as well as the asymptotics of all bifurcation curves our work not only reports several complete unfoldings in the investigation of degenerate grazing sliding bifurcations but also answers the open problem about the maximum number of limit cycles bifurcating from the generalized homoclinic loop for the case of fold fold presented in liang et al 2013 30

isospectral vibrating systems part 1 the spectral method as tudy is made of inverse problems forn n systems of the form l m 2 d k t his paper concerns the determination of systems in an equivalence class def ined by afi xed 2n 2n admissible jordan matrix i e a class of isospectral systems constructive methods are obtained for complex or real systems with no symmetry constraints it is also shown how isospectral families of complex hermitian matrices can be formed the case of real symmetric matrices is more difficult some partial solutions are obtained but in this case the theory remains incomplete examples are given 2 005 elsevier inc all rights reserved

calculation of eigensolution derivatives for nonviscously damped systems using nelson s method a method to calculate the derivatives of the eigenvalues and eigenvectors of multiple degree of freedom damped linear dynamic systems with respect to arbitrary design parameters is presented in contrast to the traditional viscous damping model a more general nonviscous damping model is considered the nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over given kernel functions because of the general nature of the damping eigensolutions are generally complex valued and eigenvectors do not satisfy the classical orthogonality relationship the proposed method to calculate the eigenvector derivative depends only on the eigenvector concerned numerical examples are provided to illustrate the derived results nomenclature c j constant associated with the derivative of u j d s dynamic stiffness matrix d j constant associated with the derivative of v j g s laplace transform of g t g t nonviscous damping matrix k stiffness matrix l laplace transform of m mass matrix m total number of eigenvectors n number of degrees of freedom of the system n number of relaxation parameters p design parameter rank rank of a matrix s laplace domain parameter u j jth eigenvector t displacement vector v j jth adjoint left eigenvector x j vector associated with the derivative of u j yj vector associated with the derivative of v j j normalization constant for the jth eigenvector j jth eigenvalue k relaxation parameters subscripts t matrix transpose derivative with respect to s

solutions to an inverse monic quadratic eigenvalue problem abstract given n 1 pairs of complex numbers and vectors closed under complex conjugation the inverse quadratic eigenvalue problem is to construct real symmetric or anti symmetric matrix c and real symmetric matrix k of size n n so that the quadratic pencil q 2 i n c k has the given n 1 pairs as eigenpairs necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are obtained numerical algorithms for solving the problem are developed numerical examples illustrating these solutions are presented

peres horodecki separability criterion for continuous variable systems the peres horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states the partial transpose operation admits in the continuous case a geometric interpretation as mirror reflection in phase space this recognition leads to uncertainty principles stronger than the traditional ones to be obeyed by all separable states for all bipartite gaussian states the peres horodecki criterion turns out to be a necessary and sufficient condition for separability

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

bound entangled gaussian states we discuss the entanglement properties of bipartite states with gaussian wigner functions for the separability and the positivity of the partial transpose we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state it is shown that for systems composed of a single oscillator for alice and an arbitrary number for bob positivity of the partial transpose implies separability however this implication fails with two oscillators on each side as we show by constructing a five parameter family of bound entangled gaussian states

computation of eigensolution derivatives for nonviscously damped systems using the algebraic method t he computation of the eigensolution derivatives plays a significant role in dynamic model updating structural design optimization structural dynamic modification damage detection andmany other applications themethods to calculate eigensolution derivatives are well established for undamped and viscous damped systems these common methods can be divided into the modal method nelson s method and the algebraic method fox and kapoor 1 first proposed the modal method for symmetric undamped systems by approximating the derivative of each eigenvector as a linear combination of all undamped eigenvectors later adhikari and friswell 2 and adhikari 3 extended the modal method to the more general asymmetric systems with viscous and nonviscous damping respectively to simplify the computation of eigensolution derivatives nelson 4 proposed a method which requires only the eigenvector of interest by expressing the derivative of each eigenvector as a particular solution and a homogeneous solution for symmetric undamped systems later friswell and adhikari 5 extended nelson s method to symmetric and asymmetric systemswith viscous damping recently adhikari and friswell 6 extended nelson s method to symmetric and asymmetric nonviscously damped systems however nelson s method is lengthy and clumsy for programming lee et al 7 derived an efficient algebraic method which has a compact form to compute the eigensolution derivatives by solving a nonsingular linear system of algebraic equations for symmetric systems with viscous damping later guedria et al 8 extended the algebraic method to general asymmetric systems with viscous damping recently chouchane et al 9 wrote an excellent review of the algebraic method for symmetric and asymmetric systems with viscous damping and extended their method to the second order and high order derivatives of eigensolutions in this note the algebraic method will be extended to symmetric and asymmetric systems with nonviscous damping the equations of motion describing free vibration of ann degreeof freedom dof linear system with nonviscous viscoelastic damping can be expressed by 3 6

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

quantification of gaussian quantum steering einstein podolsky rosen steering incarnates a useful nonclassical correlation which sits between entanglement and bell nonlocality while a number of qualitative steering criteria exist very little has been achieved for what concerns quantifying steerability we introduce a computable measure of steering for arbitrary bipartite gaussian states of continuous variable systems for two mode gaussian states the measure reduces to a form of coherent information which is proven never to exceed entanglement and to reduce to it on pure states we provide an operational connection between our measure and the key rate in one sided device independent quantum key distribution we further prove that peres conjecture holds in its stronger form within the fully gaussian regime namely steering bound entangled gaussian states by gaussian measurements is impossible

spectral decomposition of real symmetric quadratic λ matrices and its applications spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions the canonical form often facilitates discussions which otherwise would be complicated and involved spectral decomposition is of fundamental importance in many applications the well known glr theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees but its development is based on complex numbers this paper modifies the glr theory for the special application to real symmetric quadratic matrix polynomials q a mx2 cx k m nonsingular subject to the specific restriction that all matrices in the representation be realvalued it is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which in turn can be constructed from the spectral data applications to a variety of challenging inverse problems are discussed

an assessment of damping identification methods a study is carried out into the philosophy and performance of different approaches for the determination of linear viscous damping in elasto mechanical systems the methods studied include a closed form solution identification methods based on inverting the matrix of receptances energy expressions developed from single frequency excitation and responses as well as first order perturbation methods the work is concentrated particularly upon modal truncation and how this affects the distribution of matrix terms and the ability of the identified damping together with known mass and stiffness terms to reproduce the complex eigenvalues and eigenvectors of the full order system a simulated example is used to illustrate various points covered in the theoretical discussion of the methods considered

parameter reconstruction of vibration systems from partial eigeninformation quadratic matrix polynomials are fundamental to vibration analysis because of the predetermined interconnectivity among the constituent elements and the mandatory nonnegativity of the physical parameters most given vibration systems will impose some inherent structure on the coefficients of the corresponding quadratic matrix polynomials in the inverse problem of reconstructing a vibration system from its observed or desirable dynamical behavior respecting the intrinsic structure becomes important and challenging both theoretically and practically the issue of whether a structured inverse eigenvalue problem is solvable is problem dependent and has to be addressed structure by structure in an earlier work physical systems that can be modeled under the paradigm of a serially linked mass spring system have been considered via specifically formulated inequality systems in this paper the framework is generalized to arbitrary generally linked systems in particular given any configuration of interconnectivity in a mass spring system this paper presents a mechanism that systematically and automatically generates a corresponding inequality system a numerical approach is proposed to determine whether the inverse problem is solvable and if it is so computes the coefficient matrices while providing an estimate of the residual error the most important feature of this approach is that it is problem independent that is the approach is general and robust for any kind of physical configuration the ideas discussed in this paper have been implemented into a software package by which some numerical experiments are reported

calculation of eigenvalue and eigenvector derivatives of a defective matrix based on puiseux expansions of perturbation parameter for the solution of the perturbed problem a modal expansion method for the eigensensitivity analysis of a defective matrix is developed in which any of eigenvector derivatives is expressed as a linear combination of all the eigenvectors and principal vectors of the matrix first an eigenvalue problem related to the differentiable eigenvectors and the first order eigenvalue derivatives associated with the same order jordan blocks corresponding to a concerned eigenvalue of the matrix is formulated then under the condition that all of the eigenvalues of the derived eigenvalue problem are simple the formulas for calculating the differentiable eigenvectors the first to third order eigenvalue derivatives and the first to second order eigenvector derivatives are derived two numerical examples show the validity of the method and how to program based on the mathematical derivations

inertia characteristics of self adjoint matrix polynomials abstract the main concern of this work is a description of inertia characteristics applicable to both a self adjoint matrix polynomial and an associated perturbed polynomial the description is given in terms of the inertia of a bezout matrix and a hankel matrix applications of the results include the calculation of lower bounds for the number of real eigenvalues skew perturbations and a problem concerning controllability and linear vibrations

eigensolutions sensitivity for quadratic eigenproblems abstract when a dynamic system to be analyzed is subjected to velocity dependent force s a quadratic eigenproblem arises this paper presents an efficient algorithm for the calculation of the eigensolutions sensitivity in quadratic eigenproblems the attention being focused on how to extract eigenvalue and eigenvector derivatives of a general non defective eigenpair directly in the configuration space the method preserves the banded nature of the structural matrices and requires knowledge of only the eigenvectors to be differentiated which makes it powerful in dealing with large scale systems

the decoupling of defective linear dynamical systems in free motion abstract it was demonstrated in two earlier papers that there exists a real linear time varying transformation that decouples any non defective linear dynamical system in free vibration in the configuration space as an extension of this work the present paper represents the first systematic effort to decouple defective systems it is shown that the decoupling of defective systems is a rather delicate procedure that depends on the multiplicities of the system eigenvalues while any defective system can be decoupled with the eigenvalues kept invariant the geometric multiplicities of these eigenvalues may not be preserved several numerical examples are provided to illustrate the theoretical developments

a development of modal expansion method for eigensensitivity analysis of a defective matrix abstract in zhang and zhang 2006 z y zhang h s zhang calculation of eigenvalue and eigenvector derivatives of a defective matrix applied mathematics and computation 176 2006 7 26 a modal expansion method for eigensensitivity analysis of a defective matrix was developed where all of the eigenvalues of the derived eigenvalue problem for the first order eigenvalues derivatives are simple in this paper the work is extended to the case where the derived eigenvalue problem has repeated eigenvalues the formulas for calculating the differentiable eigenvectors the first to third order eigenvalue derivatives and the first order eigenvector derivatives are derived a numerical example shows the validity of the method

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

second order eigensensitivity analysis of asymmetric damped systems using nelson s method first order eigensensitivity analysis using nelson s method is first reviewed then nelson s approach is extended to the computation of the second order derivatives of the eigenvalues and eigenvectors of symmetric and asymmetric damped systems the computation of second order derivatives may be required for large variation of design parameters and for some optimization algorithms nelson s method has the advantage of requiring only the knowledge of the eigenvector to be differentiated only systems with distinct eigenvalues are considered in this paper two numerical examples have been selected to illustrate the application and the utility of the derived expressions of the first and second order eigensensitivities a four degree of freedom symmetric damped system is chosen for the first example whereas a finite element asymmetric 20 dof damped rotor model is considered in the second example

solutions to a quadratic inverse eigenvalue problem abstract in this paper we consider the quadratic inverse eigenvalue problem qiep of constructing real symmetric matrices m c and k of size n n with m c k 0 so that the quadratic matrix polynomial q 2 m c k has m n m 2 n prescribed eigenpairs it is shown that for almost all prescribed eigenpairs the qiep has a solution with m nonsingular if m m and has only solutions with q 0 otherwise where m n 1 1 8 n 2 we also derive the expression of the general solution of the qiep for both cases furthermore we develop an algorithm for finding a particular solution to the qiep with m positive definite if it exists

new adaptive technique of kirsch method for structural reanalysis the combined approximations method proposed by kirsch is an efficient reanalysis method providing high quality results to integrate the kirsch method into a structural optimization process this paper presents a new adaptive kirsch method to determine the minimum number of basis vectors using the k condition number consequently a closer relationship between the kirsch method and structural optimization is bridged moreover the effectiveness of the kirsch method is improved for application in optimal design problems the presented numerical results prove that the adaptive kirsch method is significantly more efficient than the kirsch method because of the advantage of adaptively determining the minimum number of basis vectors to satisfy the predefined reanalysis error

inverse problems for damped vibrating systems abstract linear damped vibrating systems are defined by three real definite matrices m 0 d 0 and k 0 the mass damping and stiffness matrices respectively it is assumed that all eigenvalues of the system are simple and nonreal so that the eigenvectors columns of a matrix x c c n n are also complex it is shown that when properly defined the eigenvectors have a special structure consistent with x c x r i i where x r r n n x r is nonsingular and is orthogonal by taking advantage of this structure solutions of the inverse problem are obtained i e given complete information on the eigenvalues and eigenvectors it is shown how m d and k can be found three points of view are developed and compared namely using spectral theory structure preserving similarities and factorisation theory

derivative of eigensolutions of nonviscously damped linear systems derivatives of eigenvalues and eigenvectors of multiple degree of freedom damped linear dynamic systems with respect to arbitrary design parameters are presented in contrast to the traditional viscous damping model a more general nonviscous damping model is considered the nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions because of the general nature of the damping eigensolutions are generally complex valued and eigenvectors do not satisfy any orthogonality relationship it is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems numerical examples are provided to illustrate the derived results

calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues summary an algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues in the proposed method the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution by introducing an additional normalization condition we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution the method is numerically stable and the homogeneous solutions are computed by the second order derivatives of the eigenequations two numerical examples are used to illustrate the validity of the proposed method copyright 2015 john wiley sons ltd

generic bifurcations of low codimension of planar filippov systems abstract in this article some qualitative and geometric aspects of non smooth dynamical systems theory are discussed the main aim of this article is to develop a systematic method for studying local and global bifurcations in non smooth dynamical systems our results deal with the classification and characterization of generic codimension 2 singularities of planar filippov systems as well as the presentation of the bifurcation diagrams and some dynamical consequences

model updating for self adjoint quadratic eigenvalue problems this paper concerns quadratic matrix functions of the form l m 2 d k where m d k are hermitian n n matrices with m 0 it is shown how new systems of the same type can be generated with some eigenvalues and or eigenvectors updated and this is accomplished without spill over i e other spectral data remain undisturbed furthermore symmetry is preserved the methods also apply for hermitian matrix polynomials of higher degree

inverse eigenvalue problems for damped vibrating systems first let us describe what is understood here by a damped vibrating system consider a system that is determined by n generalized coordinates ql r q t and associtated kinetic energy potential energy and dis sipation functions these are supposed to be homogeneous quadratic forms in the generalized velocities coordinates and velocities respectively and independent of the time motions of such a system are then governed by differential equations of the form aij j cq o 0 1 where q is the column vector of coordinates the dots denote derivatives with respect to time and a b c are n x n matrices associated with the respective quadratic forms in particular they are generally real and sym metric with a positive definite and b c nonnegative or positive definite it is well known that solutions of such a system are intimately connected with the algebraic properties of the matrix valued function l i a3 2 ba c 0 2 in the terminology of 1 21 l i is known as a selfadjoint matrix polynomial a number l and a nonzero vector x are called an eigenvalue and associated right eigenvector of l 3 if det l o and l x o generalized eigenvectors or jordan chains which will be defined shortly 238

an iterative method for finite dimensional structural optimization problems with repeated eigenvalues a numerical method for solving finite dimensional structural optimization problems with constraints on natural frequency and on buckling is formulated the method treats nondifferentiable repeated eigenvalues that have been shown to systematically arise in structural optimization recent results on differentiability of eigenvalues are used to develop a generalized gradient projection method for structural optimization the algorithm is shown to overcome technical difficulties associated with nondifferentiability of repeated eigenvalues the method is used to solve buckling and vibration optimization problems in which repeated eigenvalues occur

eigensolutions of perturbed nearly defective matrices abstract perturbation methods for approximating eigenvalues and eigenvectors of perturbed nearly defective mistuned systems have been discussed it is shown that the mistuning causes taylor expansions to be not uniformly valid even in small intervals of the perturbation parameter subsequently rendering them useless for practical purposes the problem is overcome by starting the perturbation expansion from an exactly defective tuned system close to the mistuned one asymptotic expansions are then obtained in terms of fractional powers of two parameters the modification and the mistuning parameter however as the tuned system is unknown an inverse problem has to be solved in order to determine it an algorithm valid for the particular but frequent case of several couples of nearly coincident eigenvalues has been detailed for this problem an outline of a more general case is also given illustrative examples are presented

a singular approach to discontinuous vector fields on the plane in this paper we deal with discontinuous vector fields on r 2 and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation the regularization process developed by sotomayor and teixeira is crucial for the development of this work

quadratic inverse eigenvalue problem for damped gyroscopic systems in this paper we discuss the quadratic inverse eigenvalue problem for damped gyroscopic systems that is we are to construct a damped gyroscopic second order system having given 2n eigenvalues and associated eigenvectors as its full eigenstructure under some mild assumptions the coefficient matrices are constructed in a recursive process we also discuss how to obtain a damped gyroscopic system with the leading coefficient matrix m being positive definite

logarithmic negativity a full entanglement monotone that is not convex it is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation a set of operations that incorporate local operations and classical communication as a subset and in the process a further proof is provided that the negativity does not increase on average under the same set of operations given that the logarithmic negativity is not a convex function this result is surprising as it is generally considered that convexity describes the local physical process of losing information the role of convexity and in particular its relation or lack thereof to physical processes is discussed and importance of continuity in this context is stressed

multimode uncertainty relations and separability of continuous variable states a multimode uncertainty relation generalizing the robertson schrodinger relation is derived as a necessary constraint on the second moments of n pairs of canonical operators in turn necessary conditions for the separability of multimode continuous variable states under m n mode bipartitions are derived from the uncertainty relation these conditions are proven to be necessary and sufficient for 1 n mode gaussian states and for m n mode bisymmetric gaussian states

strong monogamy of bipartite and genuine multipartite entanglement the gaussian case we demonstrate the existence of general constraints on distributed quantum correlations which impose a trade off on bipartite and multipartite entanglement at once for all n mode gaussian states under permutation invariance we establish exactly a monogamy inequality stronger than the traditional one that by recursion defines a proper measure of genuine n partite entanglement strong monogamy holds as well for subsystems of arbitrary size and the emerging multipartite entanglement measure is found to be scale invariant we unveil its operational connection with the optimal fidelity of continuous variable teleportation networks

strong subadditivity for log determinant of covariance matrices and its applications we prove that the log determinant of the covariance matrix obeys the strong subadditivity inequality for arbitrary tripartite states of multimode continuous variable quantum systems this establishes general limitations on the distribution of information encoded in the second moments of canonically conjugate operators the inequality is shown to be stronger than the conventional strong subadditivity inequality for von neumann entropy in a class of pure tripartite gaussian states we finally show that such an inequality implies a strict monogamy type constraint for joint einstein podolsky rosen steerability of single modes by gaussian measurements performed on multiple groups of modes

eigensensitivity analysis for asymmetric nonviscous systems t he eigensensitivities of mechanical systems with respect to structural design parameters have become an integral part of many engineering design methodologies including optimization structural health monitoring structural reliability model updating dynamic modification reanalysis techniques and many other applications fox and kapoor 1 computed the derivative of each eigenvector as a linear combination of all of the undamped eigenvectors later adhikari and friswell 2 and adhikari 3 extended the modal method to the more general asymmetric systems with viscous and nonviscous damping respectively nelson 4 presented a method which requires only the eigenvector of interest by expressing the derivative of each undamped eigenvector as a particular solution and a homogeneous solution later friswell and adhikari 5 extended nelson s method to symmetric and asymmetric systems with viscous damping recently adhikari and friswell 6 extendednelson s method to symmetric and asymmetric nonviscously damped systems fox and kapoor 1 also suggested a direct algebraic method to calculate the eigensensitivity for symmetric undamped systems by solving a nonsingular linear system of algebraic equations lee et al 7 derived an efficient algebraic method which has a compact linear system with a symmetric coefficient matrix for symmetric systems with viscous damping later guedria et al 8 extended the algebraic method to general asymmetric viscous damped systems chouchane et al 9 reviewed the algebraic method and extended their method to the second order and high order derivatives of eigensolutions li et al 10 extended the algebraic method to symmetric and asymmetric nonviscously damped systems xu andwu 11 proposed a new normalization and presented a method for the computation of eigensolution derivatives of asymmetric systemswith viscously damping recently mirzaeifar et al 12 proposed a new method based on a combination of algebraic and modal methods for generally asymmetric viscously damped systems more recently li et al 13 proposed a method of design sensitivity analysis of asymmetric viscously damped systems with distinct and repeated eigenvalues which can compute the left and right eigenvector derivatives separately and independently all of the methods mentioned previously compute the eigensensitivities of asymmetric damped systems by using the left eigenvector however these methods have disadvantages in computational cost and storage capacity for the left eigenvector should be calculated to avoid using the left eigenvector an algebraic method is presented 14 which does not require the left eigenvector for asymmetric damped systems but this method is restricted to the case of viscous damping it should be noted that the coefficientmatrices of the algebraicmethodmay be ill conditioned due to the components of the additional constraints and system matrices in the coefficient matrices are not all of the same order of magnitude in addition 15 the normalization adapted in 14 and 12 will fail in some cases because it can equal zero or a very small number this note will present a method which is well conditioned and can calculate the eigensensitivity of asymmetric nonviscous damped systems without using the left eigenvector considering an n degree of freedom linear system with nonviscous viscoelastic damping 3 6 10

a canonical form of the equation of motion of linear dynamical systems the equation of motion of a discrete linear system has the form of a second order ordinary differential equation with three real and square coefficient matrices it is shown that for almost all linear systems such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement this canonical form of the equation of motion is unique up to an equivalence class for non defective systems as an important by product a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system

hermitian quadratic matrix polynomials solvents and inverse problems a monic quadratic hermitian matrix polynomial l can be factorized into a product of two linear matrix polynomials say l i s i a for the inverse problem of finding a quadratic matrix polynomial with prescribed spectral data eigenvalues and eigenvectors it is natural to prescribe a right solvent a and then determine compatible left solvents s this problem is explored in the present paper the splitting of the spectrum between real eigenvalues and nonreal conjugate pairs plays an important role special attention is paid to the case of real symmetric quadratic polynomials and the allocation of the canonical sign characteristics as well as the eigenvalues themselves

an inverse problem for undamped gyroscopic systems linear undamped gyroscopic systems are defined by three real matrices m 0 k 0 and g g t g the mass stiffness and gyroscopic matrices respectively in this paper an inverse problem is considered given complete information about eigenvalues and eigenvectors l diag l 1 l 2 l 2 n 1 l 2 n c 2 n x 2 n and x x 1 x 2 x 2 n 1 x 2 n c n x 2 n where the diagonal elements of l are all purely imaginary x is of full row rank n and both l and x are closed under complex conjugation in the sense that l 2 email protected 2 j email protected c x 2 email protected 2 j email protected c n for j 1 n find m k and g such that email protected email protected kx 0 the solvability condition for the inverse problem and a solution to the problem are presented and the results of the inverse problem are applied to develop a method for model updating

an efficient algebraic method for computing eigensolution sensitivity of asymmetric damped systems in this paper an efficient algebraic method for the computation of eigensolution derivatives of asymmetric damped systems with distinct eigenvalues is presented by introducing an additional and new normalization condition we construct two extended systems of linear equations with nonsingular coefficient matrices which are transpose to each other we can compute the derivatives of the eigenvalues and their associated right and left eigenvectors by solving the two systems respectively in this way the cpu computation time and the storage space are considerably reduced finally a numerical example is included to demonstrate the validity of the proposed method