Pdf inverse eigenvalue problems in structural dynamics. Two iterative algorithms are devised, as well as a restriction method for simplifying the system behavior away from the desired flutter points. Updating an existing but inaccurate structural dynamics model with measured data can be mathematically reduced to the problem of the best approximation to a given matrix pencil in the frobenius norm under a given spectral constraint and a submatrix pencil constraint. In lumped mass analysis m can have in general zero elements on the diagonal.
Solution methods for the generalized eigenvalue problem these slides are based on the recommended textbook. Eigenvalue sensitivity analysis in structural dynamics. Natural frequency2 so, solution for u ut is where a depends on the initial conditions 2 cos. Unesco eolss sample chapters experimental mechanics structural dynamics and modal analysis d. Pdf inverse eigenvalue problem in structural dynamics design. It is common to use the finite element method fem to perform this analysis because, like other calculations using the fem, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable. Solution techniques for large eigenvalue problems in structural dynamics by i. Random eigenvalue problems the random eigenvalue problem of undamped or proportionally. Multigrid solution procedures for structural dynamics. Nr 064183 reproduction in while or in part is permitted. An approach to some nonclassical eigenvalue problems of structural dynamics.
The problem is formulated as an optimization problem. The diagonal blocks can be ordered in ascending order of their eigenfrequencies 8. In the dynamic response analysis of an assemblage of structural elements using conventional mode super position the generalized eigenvalue problem. Natural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems. Structural dynamics design is to design a structure subject to the dynamic characteristics requirement, i. Eigenvalue problem also arises in the context of stability analysis of structures. Random eigenvalue problems in structural dynamics citeseerx. A symmetric generalized inverse eigenvalue problem in. Free vibration frequencies and load magnitudes in stability analysis are computed by solving large and sparse generalized symmetric eigenvalue problems. Structural dynamics final year structural engineering.
Considering that the system parameters are known only probabilistically, we obtain the moments and the probability density. In this case it is necessary to use first static condensation on the massless degrees of freedom. Inverse eigenvalue problem in structural dynamics design. Steffen, jr encyclopedia of life support systems eolss eigenvalues. Friswell university of bristol, bristol, united kingdom dynamic characteristics of linear structural systems are governed by the natural frequencies and the modeshapes.
In general the system matrices for real structures are not gue or goe. Pdf we consider the numerical solution of inverse eigenvalue problems iep. Eigenvalue problems existence, uniqueness, and conditioning computing eigenvalues and eigenvectors eigenvalue problems eigenvalues and eigenvectors geometric interpretation eigenvalues and eigenvectors standard eigenvalue problem. In this study it is shown that structural dynamic modification is important in structural reanalysis. Using the interstory drifts story distortions as the coordinates, say 1 and 2, the equations of motion for free vibration can be written as remember the freebody diagrams and force balances we did in class. Inverse eigenvalue problems in structural dynamics article pdf available in pamm 61. This interval finiteelementbased method is capable of obtaining the bounds on dynamic response of a structure with interval uncertainty. Matlab programming eigenvalue problems and mechanical. This problem often arises in engineering connected with vibration. Three multigrid methods are described for solving the generalized symmetric eigenvalue problem encountered in structural dynamics.
Ec efficient solution of the fuzzy eigenvalue problem in. In general,they could be transformed into nonlinear equations to solve. The rapid computation of random eigenvalue problems of uncertain structures is the key point in structural dynamics, and it is prerequisite to the efficient dynamic analysis and optimal design of. Eigenvalue problems in structural mechanics 215 it is computationally efficient to use as s the cholesky factor em of my i. If the frequencies are close, the operation of the fan may lead to structural damage or failure. The resulting eigenvalues of this problem are complex. Solution techniques for large eigenvalue problems in structural dynamics. The methods are applied to a certain mechanical system. On eigenvalue problem of bar structures with stochastic spatial stiffness variations structural engineering and mechanics, vol. This problem could either be a differential eigenvalue problem or a matrix eigenvalue problem, depending on whether a continuous model or a discrete model is used to describe the given vibrating system.
Efficient solution of the fuzzy eigenvalue problem in structural dynamics. Structural dynamic modification implies the incorporation, into an existing model, of new information gained either from experimental testing or some other source, which questions or improves the accuracy of the model. Pdf purpose many analysis and design problems in engineering and science involve uncertainty to varying degrees. The properties of this problem are analyzed, and the. In this work, we devise solution algorithms for nonlinear multiparameter eigenvalue problems arising in the analysis of aeroelastic flutter. Siam journal on numerical analysis siam society for. The structural dynamics problems,such as structural design,parameter identification and model correction,are considered as a kind of the inverse generalized eigenvalue problems mathematically. We consider the numerical solution of inverse eigenvalue problems iep. Random eigenvalue problems in structural dynamics s. Solving the interval problem as a generalized interval eigenvalue problem in interval mathematics will produce conservative bounds on the eigenvalues. Efficient solution of the fuzzy eigenvalue problem in. This paper is concerned with the structural vibration problem.
If h has wishart distribution then the exact joint pdf of the eigenvalues can be obtained from muirhead 30, theorem 3. Random matrix eigenvalue problems in probabilistic. A survey of probably the most efficient solution methods currently in use for the problems k. This is in contrast to the classical linear eigenvalue problem which results when frequencyindependent matrices are used. The eigenvalue problem for damped gyroscopic systems. Pdf efficient solution of the fuzzy eigenvalue problem in structural.
An approach to some nonclassical eigenvalue problems of. The book by parlett 148 is an excellent treatise of the problem. Eigenvalues and eigenvectors projections have d 0 and 1. A symmetric generalized inverse eigenvalue problem in structural dynamics model updating a symmetric generalized inverse eigenvalue problem in structural dynamics model updating jiang, jiashang. In this equation k is the stiffness matrix and m is the mass matrix of the element assemblage, both are of order n. Probability density function like a continuous histogram of response. Friswell college of engineering, swansea university, swansea, uk abstract purpose many analysis and design problems in engineering and science involve uncertainty to varying degrees. This study treats the determination of eigenvalues and eigenvectors of large algebraic systems.
Pdf random matrix eigenvalue problems in structural. Solution methods for eigenvalue problems in structural. A kind of inverse eigenvalue problem in structural dynamics design is considered. Random eigenvalue problems in structural analysis aiaa. The purpose of this paper is to investigate strategies to efficiently solve the fuzzy eigenvalue problem. Random matrix eigenvalue problems in structural dynamics. Algorithms for the nonlinear eigenvalue problem siam. The model updating problem can be regarded as a special case of the inverse eigenvalue problem which occurs in the design and modification of massspring systems and dynamic structures. Due to the special structure of the mass and stiffness matrix we benefit from a. Caprani importance as they are relatively easily analysed mathematically, are easy to understand intuitively, and structures usually dealt with by structural engineers can be modelled approximately using an sdof. Description of reallife engineering structural systems is in. Dynamic analysis of structures with interval uncertainty abstract by mehdi modarreszadeh a new method for dynamic response spectrum analysis of a structural system with interval uncertainty is developed.
K is the stiffness matrix, v is the matrix containing all the eigenvectors, m is the mass matrix, and d is a diagonal matrix containing the eigenvalues v,deigk,m cite as. Announcements sept 01 welcome to cee511 structural dynamics nov 25 final exam. All problems in structural dynamics can be formulated based on the above equation of motion 1. The symmetric inverse eigenvalue problem and generalized inverse eigenvalue problem with submatrix constraint in structural dynamic model updating have been. Distribution of potential and kinetic energy in every finite element is used for analysis. Structural dynamics of rocket engines andy brown, ph. Matrix k can be either positivedefinite or positive semidefinite, according to the boundary conditions kinematic constraints of the system. The trans formation is then a stable process provided m is wellconditioned with respect to inversion. Eigenvalue solvers for structural dynamics physics forums. Several books dealing with numerical methods for solving eigenvalue problems involving symmetric or hermitian matrices have been written and there are a few software packages both public and commercial available. This paper is concerned with the structural vibration problem involving uncertain. The sensitivity approach is based on the prior selection of updating parameters design variables in the initial fe model.
In the eigenvalue problems the stiffness matrices k and k g and the mass matrix m can be full or banded. Efficient solution of the fuzzy eigenvalue problem in structural dynamics efficient solution of the fuzzy eigenvalue problem in structural dynamics yuying xia. For example maximizing the eigenvalue representing the load magnitude subject to a constraint on structural weight. The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration. Eigenvalues in optimum structural design springerlink. If the inline pdf is not rendering correctly, you can download the pdf file here. Solution techniques for large eigenvalue problems in. Eig can also operate on the eigenvalue equation in this form where. Robinson a technical report of research sponsored by the office of naval research department of the navy contract no. An inverse eigenvalue problem of hermitehamilton matrices. Two implicit algorithms are discussed that use a multigrid method to solve the linear matrix equations encountered in each iteration of the standard subspace and block lanczos methods. The use of frequencydependent matrices in structural dynamics leads to eigenvalue problems that are nonlinear in the eigenvalueh.
910 1507 1674 762 620 1415 253 1412 70 148 102 816 1552 1426 1128 320 140 1251 1203 41 1012 33 1167 1299 918 776 638 733 1091 509 590