Derivatives of eigenvalues and eigenvectors
01 May 1970-AIAA Journal (American Institute of Aeronautics and Astronautics (AIAA))-Vol. 8, Iss: 5, pp 943-944
About: This article is published in AIAA Journal.The article was published on 1970-05-01. It has received 189 citations till now. The article focuses on the topics: Matrix differential equation & Defective matrix.
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TL;DR: A simplified procedure is presented for the determination of the derivatives of eigenvectors of nth order algebraic eigensystems, applicable to symmetric or nonsymmetric systems, and requires knowledge of only one eigenvalue and its associated right and left eigenavectors.
Abstract: A simplified procedure is presented for the determination of the derivatives of eigenvectors of nth order algebraic eigensystems. The method is applicable to symmetric or nonsymmetric systems, and requires knowledge of only one eigenvalue and its associated right and left eigenvectors. In the procedure, the matrix of the original eigensystem of rank (/?-!) is modified to convert it to a matrix of rank /?, which then is solved directly for a vector which, together with the eigenvector, gives the eigenvector derivative to within an arbitrary constant. The norm of the eigenvector is used to determine this constant and complete the calculation. The method is simple, since the modified n rank matrix is formed without matrix multiplication or extensive manipulation. Since the matrix has the same bandedness as the original eigensystems, it can be treated efficiently using the same banded equation solution algorithms that are used to find the eigenvectors.
878 citations
TL;DR: In this paper, a survey of sensitivity derivatives for discrete structural systems is presented, primarily focusing on publications developed in nonstructural fields such as electronics, control, and physical chemistry which are directly applicable to structural problems.
Abstract: Methods for calculating sensitivity derivatives for discrete structural systems are surveyed, primarily covering literature published during the past two decades. Methods are described for calculating derivatives of static displacements and stresses, eigenvalues and eigenvectors, transient structural response, and derivatives of optimum structural designs with respect to problem parameters. The survey is focused on publications developed in nonstructural fields such as electronics, controls, and physical chemistry which are directly applicable to structural problems. Most notable among the nonstructural-based methods are the adjoint variable technique from control theory, and the Green's function and FAST methods from physical chemistry.
489 citations
TL;DR: In this article, a review article provides an overview of the problems pertaining to structural dynamics with bolted joints, including energy dissipation, dynamic properties of the joints, parameter uncertainties and relaxation, and active control of the joint preload.
397 citations
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TL;DR: In this paper, the authors present a review of different approaches to sensitivity analysis in structural problems, including global finite differences, continuum derivatives, discrete derivatives, and computational or automated differentiation.
296 citations
TL;DR: The algorithm preserves the symmetry and band structure of the matrices, allowing efficient computer storage and solution techniques, and applications include sensitivity analysis and optimization of the normal modes of finite-element modeled structures, such as large space structures.
Abstract: In this paper an algorithm is derived for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues, where the matrices are functions of real parameters such as mass density or moment of inertia. The algorithm is an extension of recent work by I. U. Ojalvo; the key step in this extended derivation is to differentiate the eigenvalue equation twice. The algorithm preserves the symmetry and band structure of the matrices, allowing efficient computer storage and solution techniques. Applications include sensitivity analysis and optimization of the normal modes of finite-element modeled structures, such as large space structures. A cantilever beam finite-element example is included.
190 citations
References
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TL;DR: Exact expressions for rates of change of eigenvalues and eigenvector to facilitate computerized design of complex structures are presented.
Abstract: Exact expressions for rates of change of eigenvalues and eigenvector to facilitate computerized design of complex structures
1,094 citations
TL;DR: In this paper, the effects of atmospheric dissociation on the airflow, and hence on the drag, stability and aerodynamic heating of such projectiles, are investigated. But this paper is intended mainly as a source of ideas for later, more comprehensive investigations.
Abstract: Atmospheric dissociation will be appreciable in the neighbourhood of projectiles travelling at speeds greater than 2 km/sec. This introductory paper on possible effects of dissociation on the airflow, and hence on the drag, stability and aerodynamic heating of such projectiles, is intended mainly as a source of ideas for later, more comprehensive investigations.
381 citations
265 citations
TL;DR: In this paper, the boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number.
Abstract: Prandtl's boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number. The technique of inner and outer expansions is used to treat this singular-perturbation problem. Only analytic semi-infinite bodies free of separation are considered. The second approximation is analysed in detail for steady laminar flow past plane or axisymmetric solid bodies. Attention is restricted to low speeds and small temperature changes, so that the velocity field is that for an incompressible fluid, the temperature field being calculated subsequently. The additive effects are distinguished of longitudinal curvature, transverse curvature, external vorticity, external stagnation enthalpy gradient, and displacement speed. The effect of changing co-ordinates is examined, and the behaviour of the boundary-layer solution far downstream discussed. Application to specific problems will be made in subsequent papers.
164 citations
TL;DR: In this article, the relationship between the eigenvalue and one or more of the basic parameters of the physical system has been investigated in the fields of buckling and vibration. But the problem of finding the relation between the two types of load at buckling has not been addressed.
Abstract: There are many problems in practice, particularly in the fields of buckling and vibration, where the required end result is a curve showing the relation between the eigenvalue and one or more of the basic parameters of the physical system. For example, the curve relating the buckling stress and side ratio of a rectangular plate, with some specified set of edge conditions, under uni-axial compression may be required. Or the problem may be the buckling of a rectangular plate of given proportions, under combined compression and shear, and the determination of the relation between the two types of load at buckling. Alternatively, the effect on the frequency of vibration of varying a concentrated mass, or the stiffness of an elastic constraint at a given point in an elastic system may be required.
111 citations