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

calculating derivatives of repeated and nonrepeated eigenvalues without explicit use of eigenvectors the analysis of inverse problems in parametric model updating often require the sensitivities of eigenvalues the calculation of these sensitivities is mathematically related to the derivatives of the eigenvalues with respect to the model parameters a common method to calculate these derivatives is the nelson method which requires the eigenvectors the method introduced in this paper is derived from the characteristic equation of the underlying general eigenvalue problem and allows the derivatives of eigenvalues with respect to the model parameters to be calculated without explicit use of the eigenvectors the method is extended for the case of repeated eigenvalues which leads to restrictions on the parameterization for repeated eigenvalues of multiplicity two these restrictions are formulated expicitly applications and limitations of the method are demonstrated by examples

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

on the relationship between continuous and discrete time quantum walk quantum walk is one of the main tools for quantum algorithms defined by analogy to classical random walk a quantum walk is a time homogeneous quantum process on a graph both random and quantum walks can be defined either in continuous or discrete time but whereas a continuous time random walk can be obtained as the limit of a sequence of discrete time random walks the two types of quantum walk appear fundamentally different owing to the need for extra degrees of freedom in the discrete time case

preserving poisson structure and orthogonality in numerical integration of differential equations we consider the numerical integration of two types of systems of differential equations we first consider hamiltonian systems of differential equations with a poisson structure we show that symplectic runge kutta methods preserve this structure when the poisson tensor is constant using nonlinear changes of coordinates this structure can also be preserved for nonconstant poisson tensors as exemplified on the euler equations for the free rigid body we also consider orthogonal flows and the closely related class of isospectral flows to numerically preserve the orthogonality property we take the approach of formulating an equivalent system of differential algebraic equations daes and of integrating the system with a special combination of a particular class of runge kutta methods this approach requires only matrix matrix products and can preserve geometric properties of the flow such as reversibility

a quantum interior point predictor corrector algorithm for linear programming we introduce a new quantum optimization algorithm for dense linear programming problems which can be seen as the quantization of the interior point predictor corrector algorithm 1 using a quantum linear system algorithm 2 the worst case work complexity of our method is up to polylogarithmic factors o l root n n m parallel to m parallel to over bar f kappa over bar over bar epsilon 2 for n the number of variables in the cost function mthe number of constraints epsilon 1 the target precision lthe bit length of the input data parallel to m parallel to over bar f over bar f and kappa over bar an upper bound to the condition number kappa of those systems of equations this represents a quantum speed up in the number n of variables in the cost function with respect to the comparable classical interior point algorithms when the initial matrix of the problem a is dense if we substitute the quantum part of the algorithm by classical algorithms such as conjugate gradient descent that would mean the whole algorithm has complexity o l root n n m 2 kappa over bar log epsilon 1 or with exact methods at least o l root n n m 2 373 also in contrast with any quantum linear system algorithm the algorithm described in this article outputs a classical description of the solution vector and the value of the optimal solution

variational quantum simulation of general processes variational quantum algorithms have been proposed to solve static and dynamic problems of closed many body quantum systems here we investigate variational quantum simulation of three general types of tasks generalized time evolution with a non hermitian hamiltonian linear algebra problems and open quantum system dynamics the algorithm for generalized time evolution provides a unified framework for variational quantum simulation in particular we show its application in solving linear systems of equations and matrix vector multiplications by converting these algebraic problems into generalized time evolution meanwhile assuming a tensor product structure of the matrices we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution finally we introduce variational quantum simulation for open system dynamics we variationally implement the stochastic schrodinger equation which consists of dissipative evolution and stochastic jump processes we numerically test the algorithm with a 6 qubit 2d transverse field ising model under dissipation

highly accurate modal method for calculating eigenvector derivatives in viscous damping systems a highly accurate modal superposition method for computing complex eigenvector derivatives in viscous damp ing systems has been developed when higher modes are truncated the conventional modal superposition method cannot give an accurate solution and the errors may become significant in this paper calculating the derivatives is regarded as calculating the structural response to harmonic exciting using multiple modal accelerations and shifted poles highly accurate results would be obtained when only few modes are used all of the available modal superposition methods would be directly obtained from the proposed method numerical examples show that it achieves better calculation efficiency than all of the available modal methods and nelson s method when more than one eigenvector derivative is of interest moreover the presented method can be used to improve response calculations and substructure syntheses igenvector derivatives with respect to system parameters are widely used in structural design structural stability anal ysis finite element model updating and structural control since the earlier work of fox and kapoor 1 many methods have been developed 2 3 all of the available methods could be categorized into three groups modal method 1 3 4 direct method 1 4 and iter ation method 5 6 most of the iteration methods suffer from slow convergence rates and are less efficient recently ting6 improved the effectiveness of the method because the direct methods give out the exact solutions they are widely used however when the derivatives of a large number of eigenvectors are demanded the cal culation is very time consuming here the modal methods are more suitable the modal methods employ a modal superposition idea there fore the accuracy is dependent on the number of modes used in calculation when the complete set of eigenvectors is used the ex act solution could be obtained to guarantee the accuracy the clas sical modal method1 needs higher eigenvectors recently wang4 presented a modified modal method in which a modal acceleration type approach was used to approximate the contribution of trun cated higher modes therefore the modified modal method con verges faster than the classical method ma and hagiwara7 im proved the modified method with an appropriate shift value better convergence than that of the modified method is achieved and sys tems having zero eigenvalues can be directly analyzed to further improve the convergence the improved modified modal method8 10 and accurate modal superposition method9 have been proposed by several researchers the basic original idea of the aforementioned improved modal methods was found in the hu s book 11 the idea was used to improve the convergency of dynamic flexibility in series form for no damping systems all of the aforementioned methods were used for real eigen vectors for the cases of complex eigenvectors in viscous damp ing systems the three groups of methods were presented 12 13 the classical modal method for the complex eigenvectors which was similar to the method presented in ref 1 for real eigenvectors suffers from heavy modal truncation errors recently akgun14 presented a new family of modal methods for the calculation of the derivatives in non self adjo int systems the modal accelera tion approach was used to improve the convergence in this paper calculating the derivatives in viscous damping systems is regarded as calculating a structural response to a harmonic exciting using multiple modal accelerations and shifted poles a highly accurate modal method for calculating the derivatives has been developed there are clear mathematical and physical meanings in conduct ing the proposed method from which all of the available meth ods would be directly obtained furthermore fewer eigenvectors even only the specific eigenvector that is being considered are re quired for the predetermined accuracy as a result the method is more efficient in computation than akgun s method and nelson s method when more than one eigenvector derivative is of inter est numerical examples show that the proposed method is cor rect and drastically improves solution accuracy the method can directly be applied in systems with zero eigenvalues although the method is developed to calculate eigenvectors derivatives it can be used to improve response calculations and substructure syntheses

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

computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed point iteration abstract linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions the free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency dependent damping matrix in the present paper a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non proportional viscoelastic structures is developed the central idea is the construction of two complex valued functions of a complex variable whose fixed points are precisely the eigenvalues this important property allows the use of these functions in a fixed point iterative scheme with help of some results in fixed point theory necessary conditions for global and local convergence are provided it is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping in addition under certain conditions the recursive method can also be used for the calculation of non viscous real eigenvalues in order to validate the mathematical results two numerical examples are analyzed one for single degree of freedom systems and another for multiple ones

modal expansion of structural systems with time delays conventional modal analyses are not applicable to structural systems with time delays because time delays destroy any possible orthogonality among system eigenvectors a new series solution method is proposed for transient response analysis of a class of delayed linear structural systems in this method inverse laplace transform is combined with a root locus sensitivity analysis leading to a relation between the transfer function residues and eigensolutions of the delayed system with this relation closed form time domain response can be accurately and efficiently estimated in series of eigenvectors without involving transfer function singularities and s domain integration which are usually encountered in inverse laplace transform the proposed method is illustrated on a gyroscopic dynamic system under delayed feedback control

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

approximate method for eigensensitivity analysis of a defective matrix based on the exact modal expansion method an arbitrary high order approximate method is developed for calculating the second order eigenvalue derivatives and the first order eigenvector derivatives of a defective matrix the numerical example shows the validity of the method if the different eigenvalues m 1 m q of the matrix are arranged so that m 1 m q and satisfy the condition that m q 1 m q 1 1 for some q 1

numerical integrators that preserve symmetries and reversing symmetries we consider properties of flows the relationships between them and whether numerical integrators can be made to preserve these properties this is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields this new framework unifies several known constructions we also use the concept of covariance of a numerical method with respect to a group of coordinate transformations the main application is to explore the relationship between spatial symmetries reversing symmetries and time symmetry of flows and numerical integrators

eigenrelations for nonviscously damped systems multiple degree of freedom linear dynamic systems with nonviscous damping is considered the nonviscous damping is such that the damping forces depend on the past history of the velocities via convolution integrals over some kernel functions for the sake of generality the system matrices are allowed to be asymmetric the mode orthogonality relationships known for undamped or viscously damped systems have been generalized to such nonviscously damped systems some expressions are suggested for the normalization of the eigenvectors a number of useful results relating the system matrices and the eigensolutions have been established numerical examples are provided to illustrate the derived results

poisson integrators for lie poisson structures on mathbb r 3 this paper is concerned with the study of poisson integrators we are interested in lie poisson systems on first we focus on poisson integrators for constant poisson systems and the transformations used for transforming lie poisson structures to constant poisson structures then we construct local poisson integrators for lie poisson systems on finally we present the results of numerical experiments for two lie poisson systems and compare our poisson integrators with other known methods

poisson schemes for hamiltonian systems on poisson manifolds abstract in this paper we construct poisson difference schemes of any order accuracy based on pade approximation for linear hamiltonian systems on poisson manifolds with constant coefficients for nonlinear hamiltonian systems on poisson manifolds we point out that symplectic diagonal implicit runge kutta methods are also poisson schemes the preservation of distinguished functions and quadratic first integrals of the original hamiltonian systems of these schemes are also discussed

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

quantum linear system algorithm for dense matrices solving linear systems of equations is a frequently encountered problem in machine learning and optimization given a matrix a and a vector mathbf b the task is to find the vector mathbf x such that a mathbf x mathbf b we describe a quantum algorithm that achieves a sparsity independent runtime scaling of mathcal o mathbf ensuremath kappa 2 sqrt n text polylog n ensuremath epsilon mathbf for an n ifmmode times else texttimes fi n dimensional a with bounded spectral norm where ensuremath kappa denotes the condition number of a and ensuremath epsilon is the desired precision parameter this amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices and poses a new state of the art for solving dense linear systems on a quantum computer furthermore an exponential improvement is achievable if the rank of a is polylogarithmic in the matrix dimension our algorithm is built upon a singular value estimation subroutine which makes use of a memory architecture that allows for efficient preparation of quantum states that correspond to the rows of a and the vector of euclidean norms of the rows of a

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

derivation and significance of second order modal design sensitivities if the system is considered to be unsatisfactory in some way e g a natural frequency to the system coinciding with a constant shaft speed then it is of interest to find the design change which will give maximum benefit with minimum disturbance to the system this may be quantified in terms of design sensitivities of the structure fox and kapoor derived an expression for the sensitivity of an eigenvalue for self adjoint systems i e where the left and right eigenvectors are simple transposes typical of conservative systems and duncan forms garg extended this result to apply to non self adjoint systems e g where dissipative and gyroscopic effects are present the analysis for eigenvectors was presented by plaut and huseyin for the general case most recently nelson has provided an alternative approach to eigenvector sensitivity the sensitivities derived by all of the above methods are expressed in the form of first derivatives and hence are properties local to the original solution of the system in a recent paper by vanhonacker there is a clear implication that these first derivatives may be extrapolated for gross system changes clearly this assumption can be valid only if the value of the second derivative is negligible

computational methods for complex eigenproblems in finite element analysis of structural systems with viscoelastic damping treatments in this paper efficient numerical methods to approximate the complex eigenvalues and eigenvectors in non proportional and non viscous systems are presented these methods are specially conceived for practical engineering applications making use of the finite element analysis to determinate the effect that potential damping treatments have on the natural frequencies and mode shapes of structural systems considering the solution of the undamped problem the complex eigenpair is estimated by finite increments using the eigenvector derivatives for non proportional viscous systems with low and medium damping a simple single step technique is presented whose rapidity and accuracy is verified by means of numerical applications for higher damped systems an incremental approach is proposed that keeps the accuracy without significantly increasing the computational time for non viscously damped systems a fast iterative modality is suggested which allows to approximate in an efficient and simple way the complex eigenpair as numerical applications the study of a metallic beam with free layer damping treatment is completed using finite element procedures where the damping material is modelized by an exponential model whose parameters are obtained from curve fitting to experimental data

transient response of one dimensional distributed systems a closed form eigenfunction expansion realization abstract the exact and closed form transient response of general one dimensional distributed dynamic systems subject to arbitrary external initial and boundary disturbances is determined non self adjoint operators characterizing damping gyroscopic and circulatory effects and eigenvalue dependent boundary conditions are considered through introduction of augmented operators a closed form modal expansion of the displacement and internal forces of the distributed system is derived the eigenfunction expansion is realized in a spatial state formulation which systematically yields exact eigensolutions eigenfunction normalization coefficients and modal co ordinates the proposed method is illustrated on a cantilever beam with end mass viscous damper and spring

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

stability and optimal feedback controls for time delayed linear periodic systems an effective numerical method for stability analysis of feedback controls with time delay and for identifying optimal gains of the control is presented the method develops a mapping of the system response in a finite dimensional state space minimization of the largest absolute value of eigenvalues of the mapping leads to optimal control gains numerical examples of both time invariant and periodic linear systems are presented to demonstrate the method we have found that the proposed method provides accurate stability boundaries and performance contours in the parametric space of control gains

analytical solutions in the time domain for vibration problems of discrete viscoelastic systems in this study isotropic homogeneous hereditary materials are considered the material is supposed to be in an isothermal state

integral formulas for non self adjoint distributed dynamic systems integral formulas for analytical prediction of the dynamic response of a class of non self adjo int distributed systems are studied combined non self adjoint effects of damping gyroscopic and circulatory forces are examined the response of the distributed system subject to arbitrary external boundary and initial disturbances is obtained in a closed form green s function integral the green s function is expanded in an eigenfunction series without assuming completeness of system eigenfunctions in addition a generalized reciprocal theorem is derived istributed dynamic systems whose parameters are dependent on the spatial domain are common in many branches of science and technology the mathematical description of these systems usually leads to boundary init ial value problems associated with partial differential equations the development of modern technologies including optimal design and active control of flexible structures requires accurate prediction of the dynamic response of distributed systems this work investigates integral formulas for analytical prediction of the dynamic response of a class of linear distributed systems that have combined non self adjoint effects of damping gyroscopic and circulatory forces and are under arbitrary external boundary and initial disturbances mainly three issues are addressed 1 green s function formulation for transient response prediction 2 modal expansion of system green s function and 3 a generalized reciprocal principle these issues are important to structural dynamics and structural control and lay a foundation for the development of analytical and numerical solution methods the concept of green s functions is by no means new it was first introduced by g green as early as 1828 and since has been applied to various problems in mathematical physics several excellent monographs on the subject have been published 1 4 instead of its many advantages the green s function method has been for formal and

eigenvalue curve veering in stressed structures an experimental study there have been extensive research works on the veering phenomenon in dynamic systems as eigenvalues change under parametric variation converging loci are commonly seen to veer away suddenly in a small region of the graph such that the modes swap trajectories all of the modal properties are swapped in the process leading to some curious behaviour in the transition zone theoretical studies of this behaviour have been reported for half a century but despite this heritage explicit references to experimental results are scarce in this paper detailed experimental and numerical investigations on veering are reported the parameter varied is the internal pre load of a redundant frame an fe model is presented and stress stiffening approximations are employed to obtain a tangent stiffness for a nonlinear static solution which is then used in a linear dynamic analysis experimental results are given and the behaviour is found to correspond well with the analytical results in particular the mode shape variation is found to be consistent and the implications of this finding with regard to modal correlation and model validation are noted analyses of the mode shapes in terms of eigenvector rotations are presented and are found to form a valuable tool for the interpretation of experimental results

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

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

quantum algorithm for linear systems of equations solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems given a matrix a and a vector b find a vector x such that ax b we consider the case where one does not need to know the solution x itself but rather an approximation of the expectation value of some operator associated with x e g x dagger mx for some matrix m in this case when a is sparse n x n and has condition number kappa the fastest known classical algorithms can find x and estimate x dagger mx in time scaling roughly as n square root kappa here we exhibit a quantum algorithm for estimating x dagger mx whose runtime is a polynomial of log n and kappa indeed for small values of kappa i e poly log n we prove using some common complexity theoretic assumptions that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm

analysis of errors and approximations in the use of modal co ordinates abstract a basic operation in several updating procedures and other modal analysis applications concerns the transformation of the complex modes identified from experimental tests into the real ones typical of theoretical models in this transformation several approximations are normally adopted this paper presents a discussion of two of these approximations the first refers to a series expansion of the complex modes into the subspace of real modes commonly used for normalization of complex modes the second used in advanced modal analysis applications relates the physical damped co ordinates to the modal ones through the real modal matrix an explanation of the limitations of these assumptions is given through analytical developments and numerical results obtained from a discrete model

second order design sensitivities to assess the applicability of sensitivity analysis a quality criterion based on the cauchy ratio test for convergence of infinite series is suggested to examine the liability of systems to perform poorly under extrapolation

derivatives of eigenvalues and eigenvectors of a general complex matrix a survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non hermitian matrices is presented in addition a modification of one method based on a better normalizing condition is proposed methods are classified as direct or adjoint and are evaluated for efficiency operation counts are presented in terms of matrix size number of design variables and number of eigenvalues and eigenvectors of interest the effect of the sparsity of the matrix and its derivatives is also considered and typical solution times are given general guidelines are established for the selection of the most efficient method

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