an inverse strategy for relocation of eigenfrequencies in structural design part ii second order approximate solutions this paper proposes an inverse strategy for relocation of structural natural frequencies using first order formulation and solution algorithm in the proposed method a sensitivity analysis of the systems eigenvalues with respect to material or geometrical parameters of the structure is conducted initially the required parameter variation to achieve a desired frequency shift for the structure is then computed the proposed technique incorporates the design constraints or objective functions in the system equations in such a way that a square system of equations is always preserved the formulations are general and applicable to all finite element structures because the sensitivity analysis is based on the stiffness and mass matrices regardless of the type of elements used an algorithm suitable for implementation of the technique for practical purposes is developed the accuracies of the proposed methods are tested conducting several case studies and the results are validated against exact solutions

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

inverse sensitivity method for asymmetric multivariable nondefective systems an available parametric correction method for asymmetric multivariable nondefective systems is investigated it will be proved that the eigensolutions associated with the repeated eigenvalues are not differentiable although their partial derivatives exist a calculable generalized jacobian matrix is presented based on the first order taylor expansion of the repeated root eigenpairs as the case of symmetric systems which ensures the feasibility of the inverse sensitivity method for parametric correction the first order directional derivatives of the repeated root eigenpairs are also found two numerical asymmetric examples are given to verify the validity of the method by use of simulated data

computation of eigenvector derivatives with repeated eigenvalues using a complete modal space this paper presents a complete modal space cms method that is an exact modal expansion method there is no modal truncation error existing in the derivation

characteristics of sensitivity analysis of repeated frequencies 5huang x z and zan s wing and fin buffet on the standard dynamic model rto rto tr 26 ac 323 avt tp 19 neuilly sur seine france oct 2000 6goura g s l time marching analysis of flutter using computational fluid dynamics ph d dissertation dept of aerospace engineering univ of glasgow glasgow scotland u k sept 2001 7goura g s l badcock k j woodgate m a and richards b e a data exchange method for fluid stucture interaction problems aeronautical journal vol 105 no 1046 2001 pp 215 221 8goura g s l badcock k j woodgate m a and richards b e implicit method for the time marching analysis of flutter aeronautical journal vol 105 no 1046 2001 pp 199 214 9melville r nonlinear simulation of aeroelastic instability aiaa paper 2001 0570 jan 2001 e livne associate editor

eigenvalue and eigenvector approximate analysis for repeated eigenvalue problems the method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problems in terms of a single positive valued parameter the resulting equations yield first order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem

optimization of support positions to maximize the fundamental frequency of structures in this paper the position optimization of simple supports is implemented to maximize the fundamental frequency of a beam or plate structure both elastic and rigid supports are taken into account first the frequency sensitivity with respect to the movement of a simple support is derived using the discrete method by means of the shape functions of the finite element method closed form sensitivity formulations are developed straightforwardly then a heuristic approach called evolutionary shift method is presented for optimizing support positions with a fixed grid mesh scheme based on the design sensitivity analysis the support with the highest efficiency is shifted in priority along the elementary edges with the interval step of the elementary size to facilitate the convergence of the process the interpolation technique is employed to evaluate the solution more accurately finally three numerical examples are presented to demonstrate the validity of the sensitivity analysis and the effectiveness of the optimization method copyright 2004 john wiley sons ltd

truss optimization on shape and sizing with frequency constraints an optimality criteria algorithm is presented for three dimentional truss structure optimization with multiple constraints on its natural frequencies both nodal coordinates and element cross sectional areas which are quite different in their natures are treated simultaneously in a unified design space for structural weight minimization first the optimality criterion is developed for a single constraint based on differentiation of the lagrangian function it states that at the optimum all of the variables should have equal efficiencies then the global sensitivity numbers are introduced to solve multiple constraints of frequencies avoiding computation of the lagrange multipliers finally upon the sensitivity analysis the most efficient variables are identified and modified in priority the optimal solution is achieved gradually from the initial design with a minimum weight increment four typical trusses are used to demonstrate the feasibility and validity of the proposed method

eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues a perturbation algorithm is developed for evaluating eigenvalue and eigenvector directional derivatives of nonsymmetric defective matrices some properties are recalled in particular chains of generalized right and left eigenvectors and their orthogonal properties are defined small perturbations of the matrix are then considered an asymptotic expansion of the eigensolutions of the perturbed problem is obtained in terms of noninteger powers of the perturbation parameter particular attention is devoted to the eigenvectors of the perturbed system and to the strong coupling that occurs between the chains

macdonald denominators for affine root systems orthogonal theta functions and elliptic determinantal point processes rosengren and schlosser introduced notions of rn theta functions for the seven types of irreducible reduced affine root systems rn an 1 bn bn cn cn bcn dn n n and gave the macdonald denominator formulas we prove that if the variables of the rn theta functions are properly scaled with n they construct seven sets of biorthogonal functions each of which has a continuous parameter t 0 t with given 0 t following the standard method in random matrix theory we introduce seven types of one parameter t 0 t families of determinantal point processes in one dimension in which the correlation kernels are expressed by the biorthogonal theta functions we demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions in the scaling limits associated with n we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t 0 t we give new expressions for the macdonald denominators using the karlin mcgregor lindstrom gessel viennot determinants for noncolliding brownian paths and show the realization of the associated elliptic determinantal point processes as noncolliding brownian brides with a time duration t which are specified by the pinned configurations at time t 0 and t t rosengren and schlosser introduced notions of rn theta functions for the seven types of irreducible reduced affine root systems rn an 1 bn bn cn cn bcn dn n n and gave the macdonald denominator formulas we prove that if the variables of the rn theta functions are properly scaled with n they construct seven sets of biorthogonal functions each of which has a continuous parameter t 0 t with given 0 t following the standard method in random matrix theory we introduce seven types of one parameter t 0 t families of determinantal point processes in one dimension in which the correlation kernels are expressed by the biorthogonal theta functions we demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions in the scaling limits associated with n we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t

mode tracking issues in structural optimization within the context of optimization of the structural dynamics properties of finite element models methodology is developed for the tracking of eigenpairs through changes in the structural eigenvalue problem the goal is to eliminate difficulties caused by mode switching i e frequency crossing out of several candidate methods two methods for mode tracking are successful the first method the higher order eigenpair perturbation algorithm is based on a perturbation expansion of the eigenproblem it iteratively computes changes in the eigenpairs due to parameter perturbations with the important feature of maintaining the correspondence between the baseline and perturbed eigenpairs the second method is a cross orthogonality check method which uses mass orthogonality to reestablish correspondence after a standard reanalysis modified eigenpair extraction routines lanczos subspace iteration inverse power were unsuccessful in tracking modes applications of mode tracking technology that are presented are frequency constrained optimization and optimization with mode shape constraints each application procedure is outlined and examples are given recommendations are made based on method efficiency and robustness in the example problems

design sensitivity analysis in large deformation elasto plastic and elasto viscoplastic problems implicit time integration algorithm derived by simo for his large deformation elasto plastic constitutive model is generalized for the case of isotropy and associative flow rule towards viscoplastic material behaviour and consistently differentiated with respect to its input parameters combining it with the general formulation of design sensitivity analysis dsa for non linear finite element transient equilibrium problem we come at a numerically efficient closed form finite element formulation of dsa for large deformation elasto plastic and elasto viscoplastic problems with various types of design variables material constants shape parameters the paper handles several specific issues like the use of a non algorithmic coefficient matrix or sensitivity discontinuities at points of instantaneous structural stiffness change computational examples demonstrate abilities of the formulation and quality of results copyright 2005 john wiley sons ltd

combined method for calculating eigenvector derivatives with repeated eigenvalues a combined method for calculating particular solutions of eigenvector derivative governing equations in the generalized eigenproblem with groups of repeated eigenvalues is presented which finds the first set of the particular solutions by a direct method and the other particular solutions by a family of modal methods the direct method is based on gauss elimination by which the constraint generalized inverse of the frequency shift stiffness matrix can be obtained as well as the particular solution and the method is applicable to all nondefective systems a formula for finding a constraint generalized inverse from another is derived by which the total calculation can be reduced a simple calculating example is included

modal expansion method for eigensensitivity with repeated roots extensions are made to the modal expansion eigensensitivity procedure for self adjoint systems originally formulated by fox and kapoor these allow the computation of sensitivity information for mode shapes associated with repeated roots and repeated root derivatives the extensions follow a similar line to additions to nelson s direct method proposed first by ojalvo and later generalized independently by mills curran and dailey the formulation based on modal expansion is simpler and more computationally efficient than the direct method although some accuracy is sacrificed by modal truncation the method generally yields acceptable results for the primary application in optimization

eigensensitivity analysis of a defective matrix the formulas for calculating the first to third order perturbation coefficients of the eigenvalues and the first order perturbation coefficients of eigenvectors of a defective matrix are derived by use of the direct perturbation method for the case where the eigenvalue problem for the first order perturbation coefficients of the eigenvalues of the defective matrix has repeated eigenvalues associated with th order jordan blocks of the matrix the perturbation equations are derived by expanding the perturbed eigenvalues and eigenvectors in power series of 1 where is the small parameter of the problem substituting the series into the perturbed eigenvalue problem and comparing the powers of the kth order perturbation equations for k 1 1 can be solved formally the solutions contain some undetermined quantities by use of the solvability conditions of the kth order equations for k v 1 the undetermined quantities in the solutions can be calculated in turns when we run all orders of the jordan blocks the problem is solved the numerical example shows the validity of the method

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

hybrid perturbation polynomial chaos approaches to the random algebraic eigenvalue problem the analysis of structures is affected by uncertainty in the structure s material properties geometric parameters boundary conditions and applied loads these uncertainties can be modelled by random variables and random fields amongst the various problems affected by uncertainty the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure the methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on monte carlo simulation in the past few years methods based on polynomial chaos pc have been developed for this problem where each eigenvalue and eigenvector are represented by a pc expansion in this paper four variants of a method hybridizing perturbation and pc expansion approaches are proposed and compared the methods use rayleigh quotient the power method the inverse power method and the eigenvalue equation pc expansions of eigenvalues and eigenvectors are obtained with the proposed methods the new methods are applied to the problem of an euler bernoulli beam and a thin plate with stochastic properties

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

optimum design of structures with both natural frequency and frequency response constraints an efficient method is developed to compute the sensitivities of eigenvalues eigenvectors and frequency responses in a reduced approximate rnodel the reanalyses of a modified structure for those dynamic responses are also derived in the approximate model an optimization process utilizing the efficiently calculated sensitivities and reanalyses of the modified structure is formed the difficulties of finding eigenvector sensitivities for repeated modes have also been addressed and averted in the approximate model two numerical examples show satisfactory outcomes by using this proposed approach

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

frequency optimization with respect to lumped mass position the sensitivity analysis of a natural frequency with respect to the change of a lumped mass position has been a research topic forsometime however relatively littlework hasbeen doneon optimizing thelumped massposition the frequency sensitivity with respect to both the position and magnitude of a lumped mass is investigated by the discrete method furthermore an evolutionary algorithm is presented for frequency optimization of beam and plate structures to positions of the lumped mass attachments which can move in the structural domain based on the design sensitivity analysis the most efe cient lumped mass is shifted to optimize the frequencies asymptotically three model examples are provided to demonstrate the validity of the sensitivity analysis and the effectiveness of the optimization method nomenclature a x coordinate of a lumped mass b y coordinate of a lumped mass d design region e young s modulus k m global stiffness and mass matrices respectively k number of the movable lumped masses l length of the beam m attached lumped mass m l equivalent mass matrix of a lumped mass n shape function of an element n number of the constrained natural frequencies t kinetic energy

efficient computation of mode shape derivatives for large dynamic systems this paper is concerned with the simplified numerical computation of eigenvector derivatives for very large problems with symmetric banded matrices such as those ordinarily encountered in structural dynamic systems starting with a brief review of existing alternate computation schemes the paper focuses on nelson s work which is at present generally accepted as the most computationally efficient method two competitive and more traditional alternatives to nelson s approach one direct and one iterative are offered no claim is made that either of these methods are superior to nelson s but they do offer viable and more conventional alternatives also discussed is the common case of repeated or closely spaced eigenvalues which was not treated by nelson a procedure for extending nelson s ideas for such cases is proposed

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

eigenmode computation of cavities with perturbed geometry using matrix perturbation methods applied on generalized eigenvalue problems abstract generalized eigenvalue problems are standard problems in computational sciences they may arise in electromagnetic fields from the discretization of the helmholtz equation by for example the finite element method fem geometrical perturbations of the structure under concern lead to a new generalized eigenvalue problems with different system matrices geometrical perturbations may arise by manufacturing tolerances harsh operating conditions or during shape optimization directly solving the eigenvalue problem for each perturbation is computationally costly the perturbed eigenpairs can be approximated using eigenpair derivatives two common approaches for the calculation of eigenpair derivatives namely modal superposition method and direct algebraic methods are discussed in this paper based on the direct algebraic methods an iterative algorithm is developed for efficiently calculating the eigenvalues and eigenvectors of the perturbed geometry from the eigenvalues and eigenvectors of the unperturbed geometry

derivatives of complex eigenvectors with distinct and repeated eigenvalues in this paper we investigate the computation of the first order derivatives of complex eigenvectors for general non defective eigensystems a new normalization condition is proposed with which we can compute unique first order derivatives of arbitrary differentiable eigenvectors of systems with distinct and repeated eigenvalues we also present an efficient algorithm to compute the particular solutions to the governing equations of the first order derivatives of eigenvectors finally numerical examples are included to demonstrate the validity of the proposed method copyright 2007 john wiley sons ltd

structural eigenderivative analysis using practical and simplified dynamic flexibility method a practical and simplified technique for computing a large number of eigenvector derivatives of a complex structural system has been developed this technique uses the dynamic flexibility method and is based on the practical complete modal space approach the practical complete modal space technique is an exact modal method there is no modal truncation error existing in the derivation the dynamic flexibility method requires only the solving of the system governing equation once regardless of the number of eigenvector derivatives needed to be computed using this method the computer time required to obtain more than one eigenvector derivative can be dramatically reduced this method gives the mathematical expression of the solution for eigenvector derivatives and is easier for engineers to use to perform theoretical formulation this method can be applied to systems with and without repeated eigenvalues this method gives better numerical precision and may be a very good tool for engineers to compute many eigenvector derivatives

a new method for calculating derivatives of eigenvalues and eigenvectors for discrete structural systems abstract a new method is presented for calculating the derivatives of eigenvalues and eigenvectors for discrete structural systems in the cases of distinctive eigenvalues proportional damping is assumed and the exact eigen pair derivatives of arbitrary order are calculated for a repeated eigenvalue its first order derivatives are calculated by solving a reduced eigenvalue problem of order equal to the eigenvalue s geometric multiplicity the new method is computationally efficient and numerically robust examples are presented to illustrate the use and effectiveness of the new method

damage detection in elastic structures using vibratory residual forces and weighted sensitivity a methodology is presented for detecting structural damage in elastic structures by nondestructiv e means measured modal test data along with a correlated analytical structural model are used to locate potentially damaged regions using residual modal force vectors and to conduct a weighted sensitivity analysis to assess the extent of mass and or stiffness variations where damage is characterized as a stiffness reduction the current approach is unique among other approaches in that it accounts for 1 variations in system mass system stiffness and mass center locations 2 perturbations of both the natural frequencies and modal vectors and 3 statistical confidence factors for the structural parameters and potential experimental instrumentation error moreover this procedure can be used with either full or reduced models a wide variety of numerical examples are presented that show that the current method provides a precise indication of both the location and the extent of structural damage lexible space structures launch vehicles and satellites are susceptible to structural damage over their operating lives from impact operating loads and fatigue undetected and uncorrected damage can lead to structural deterioration of their members and consequently jeopardize the flight safety of the structure thus numerous damage inspection methods and monitoring procedures have been developed examples include x ray ultrasonic testing magnetic resonance coin tap dye penetration and visual inspection these methods can be time consuming and are local assessments often requir ing the exposure of structural elements to the inspector and equipment for detecting damage as a monitoring system for detecting damage of spacecraft in orbit i e space station freedom none of these methods are appropriate an alterna tive approach which formed the basis for our current study is to recognize the fact that modal vibration test data struc tural natural frequencies and mode shapes characterize the state of the structure 1 therefore postflight and in flight e g monitoring data can be used to distinguish whether changes damage have occurred to the structure by comparing these data with a set of baseline data modal testing as a means of inspection has several advantages direct exposure of struc tural elements is not required and at the same time more of the complete structure can be inspected in one modal test by having appropriately placed sensors the consequences of this are a reduction in schedule and cost the use of vibration test data to locate structural damage has been attempted to evaluate the integrity of offshore oil and gas platforms 2 3 composite laminates 4 continuum struc tures 5 8 and recently large space structures 9 11 the early ana lytical and experimental approaches2 3 had limited success as a result of inadequate instrumentation and addressing only fre quency shifts associated with stiffness changes while ignoring changes in the modal vectors the continuum based meth ods5 8 provide valuable insight into the magnitude of fre quency shifts but their usefulness is severely limited to simple structures bars beams plates with unrealistic boundary con

new method for eigenvector sensitivity analysis with repeated eigenvalues and eigenvalue derivatives this paper proposes a new method of eigenvector sensitivity analysis for real symmetric systems with repeated eigenvalues and eigenvalue derivatives the derivation is completed by using information from the second and third derivatives of the eigenproblem and is applicable to the case of repeated eigenvalue derivatives the extended systems with a nonsingular coefficient matrix are constructed to calculate the particular solutions an efficient approach is developed to compute the homogeneous solutions associated with the repeated eigenvalue derivatives different constraints for calculating the eigenvector sensitivities are derived and discussed four numerical examples are presented to demonstrate the validity of the proposed method

computation of derivatives of repeated eigenvalues and corresponding eigenvectors by simultaneous iteration an earlier simultaneous iteration method for computing derivatives of eigenvalues and the corresponding eigenvectors of nonsymmetric matrices depending smoothly on some real parameters is extended to the case of repeated eigenvalues

improved nelson s method for computing eigenvector derivatives with distinct and repeated eigenvalues t he computation of derivatives of eigenvalues and eigenvectors with respect to model parameters is crucial for many applications 1 nelson 2 proposed an approach for general real eigensystemswith distinct eigenvalueswhere only the eigenvector of interest was required ojalvo 3 mills curran 4 dailey 5 and shaw and jayasuriya 6 developed nelson s method for real symmetric eigensystems with repeated eigenvalues for computation of a particular solution themethods described in 3 5 may fail in some circumstances other methods have been suggested by millscurran 4 and song et al 7 but they are difficult to implement when a system of linear equations is solved one hopes that the calculated solution is a close representation of the true solution the computed solution for a system with a large condition number may be inexact due to extreme sensitivity of the solution to small changes in its coefficient matrix and right column vectors 8 because some diagonal elements in the coefficient matrix for particular solutions are set to units its coefficients are not all of the same order of magnitude so the coefficient matrix has a large condition number in this note nelson s method is first extended to the case of repeated eigenvalues for the symmetric real eigensystems and its improvements are then presented the condition number of the coefficient matrix of the improved method is considerably reduced