Natural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems. Three multigrid methods are described for solving the generalized symmetric eigenvalue problem encountered in structural dynamics. Random eigenvalue problems the random eigenvalue problem of undamped or proportionally. We consider the numerical solution of inverse eigenvalue problems iep. 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. Efficient solution of the fuzzy eigenvalue problem in structural dynamics. Structural dynamics of rocket engines andy brown, ph. In lumped mass analysis m can have in general zero elements on the diagonal. For example maximizing the eigenvalue representing the load magnitude subject to a constraint on structural weight.
Steffen, jr encyclopedia of life support systems eolss eigenvalues. Nr 064183 reproduction in while or in part is permitted. Matlab programming eigenvalue problems and mechanical. 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. 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. If the inline pdf is not rendering correctly, you can download the pdf file here. The trans formation is then a stable process provided m is wellconditioned with respect to inversion. An approach to some nonclassical eigenvalue problems of structural dynamics.
In this study it is shown that structural dynamic modification is important in structural reanalysis. Structural dynamics final year structural engineering. Description of reallife engineering structural systems is in. Pdf inverse eigenvalue problems in structural dynamics. 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. The methods are applied to a certain mechanical system. Natural frequency2 so, solution for u ut is where a depends on the initial conditions 2 cos.
Eigenvalue problems in structural mechanics 215 it is computationally efficient to use as s the cholesky factor em of my i. Solving the interval problem as a generalized interval eigenvalue problem in interval mathematics will produce conservative bounds on the eigenvalues. This study treats the determination of eigenvalues and eigenvectors of large algebraic systems. In general the system matrices for real structures are not gue or goe. An inverse eigenvalue problem of hermitehamilton matrices. A symmetric generalized inverse eigenvalue problem in. Inverse eigenvalue problem in structural dynamics design. Random matrix eigenvalue problems in structural dynamics. This is in contrast to the classical linear eigenvalue problem which results when frequencyindependent matrices are used. Structural dynamics design is to design a structure subject to the dynamic characteristics requirement, i. The eigenvalue problem for damped gyroscopic systems. Solution techniques for large eigenvalue problems in.
Unesco eolss sample chapters experimental mechanics structural dynamics and modal analysis d. 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. Solution methods for the generalized eigenvalue problem these slides are based on the recommended textbook. Ec efficient solution of the fuzzy eigenvalue problem in. Announcements sept 01 welcome to cee511 structural dynamics nov 25 final exam. This paper is concerned with the structural vibration problem involving uncertain. 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. Probability density function like a continuous histogram of response. A symmetric generalized inverse eigenvalue problem in structural dynamics model updating a symmetric generalized inverse eigenvalue problem in structural dynamics model updating jiang, jiashang. Friswell university of bristol, bristol, united kingdom dynamic characteristics of linear structural systems are governed by the natural frequencies and the modeshapes. Distribution of potential and kinetic energy in every finite element is used for analysis. Robinson a technical report of research sponsored by the office of naval research department of the navy contract no. 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.
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. Efficient solution of the fuzzy eigenvalue problem in structural dynamics efficient solution of the fuzzy eigenvalue problem in structural dynamics yuying xia. The problem is formulated as an optimization problem. An approach to some nonclassical eigenvalue problems of. Random matrix eigenvalue problems in probabilistic. Pdf efficient solution of the fuzzy eigenvalue problem in structural. A survey of probably the most efficient solution methods currently in use for the problems k. Siam journal on numerical analysis siam society for. Random eigenvalue problems in structural analysis aiaa. In this equation k is the stiffness matrix and m is the mass matrix of the element assemblage, both are of order n. Eigenvalues in optimum structural design springerlink. The sensitivity approach is based on the prior selection of updating parameters design variables in the initial fe model.
The properties of this problem are analyzed, and the. The book by parlett 148 is an excellent treatise of the problem. If h has wishart distribution then the exact joint pdf of the eigenvalues can be obtained from muirhead 30, theorem 3. 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. 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. Considering that the system parameters are known only probabilistically, we obtain the moments and the probability density. In the eigenvalue problems the stiffness matrices k and k g and the mass matrix m can be full or banded. The resulting eigenvalues of this problem are complex. The symmetric inverse eigenvalue problem and generalized inverse eigenvalue problem with submatrix constraint in structural dynamic model updating have been. The purpose of this paper is to investigate strategies to efficiently solve the fuzzy eigenvalue problem. Random eigenvalue problems in structural dynamics citeseerx. On eigenvalue problem of bar structures with stochastic spatial stiffness variations structural engineering and mechanics, vol. Eig can also operate on the eigenvalue equation in this form where. Free vibration frequencies and load magnitudes in stability analysis are computed by solving large and sparse generalized symmetric eigenvalue problems.
The use of frequencydependent matrices in structural dynamics leads to eigenvalue problems that are nonlinear in the eigenvalueh. Pdf we consider the numerical solution of inverse eigenvalue problems iep. Pdf random matrix eigenvalue problems in structural. 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 case it is necessary to use first static condensation on the massless degrees of freedom. In the dynamic response analysis of an assemblage of structural elements using conventional mode super position the generalized eigenvalue problem. 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. Inverse eigenvalue problems in structural dynamics article pdf available in pamm 61. Matrix k can be either positivedefinite or positive semidefinite, according to the boundary conditions kinematic constraints of the system.
Eigenvalue solvers for structural dynamics physics forums. The diagonal blocks can be ordered in ascending order of their eigenfrequencies 8. Pdf purpose many analysis and design problems in engineering and science involve uncertainty to varying degrees. A kind of inverse eigenvalue problem in structural dynamics design is considered. Eigenvalue sensitivity analysis in structural dynamics. Eigenvalues and eigenvectors projections have d 0 and 1.
Algorithms for the nonlinear eigenvalue problem siam. This paper is concerned with the structural vibration problem. All problems in structural dynamics can be formulated based on the above equation of motion 1. Each real part of an eigenvalue physically represents the damping coefficient multiplied with the. Two iterative algorithms are devised, as well as a restriction method for simplifying the system behavior away from the desired flutter points. Eigenvalue problem also arises in the context of stability analysis of structures. Pdf inverse eigenvalue problem in structural dynamics design. In this work, we devise solution algorithms for nonlinear multiparameter eigenvalue problems arising in the analysis of aeroelastic flutter. Random eigenvalue problems in structural dynamics s. Solution techniques for large eigenvalue problems in structural dynamics. 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. This problem often arises in engineering connected with vibration. If the frequencies are close, the operation of the fan may lead to structural damage or failure. Multigrid solution procedures for structural dynamics.
The majority of structural failures occur because physical phenomena are overlooked, or greatly underestimated, rather than as a result of compu tational errors e. In general,they could be transformed into nonlinear equations to solve. 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. 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. Solution methods for eigenvalue problems in structural. Solution techniques for large eigenvalue problems in structural dynamics by i. This interval finiteelementbased method is capable of obtaining the bounds on dynamic response of a structure with interval uncertainty. Friswell college of engineering, swansea university, swansea, uk abstract purpose many analysis and design problems in engineering and science involve uncertainty to varying degrees. Due to the special structure of the mass and stiffness matrix we benefit from a.
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