Eigenvalue problems. The Rayleigh-Ritz method

1965 ◽  
pp. 112-119
Author(s):  
Gaetano Fichera
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method

scholarly journals A note on the Ritz method with an application to overtone stellar pulsation theory

1968 ◽  
Vol 8 (2) ◽  
pp. 275-286 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Numerical Solution ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Linear Operators ◽  
Matrix Equations ◽  
Infinite Dimensional ◽  
Stellar Pulsation ◽  
Finite Matrix ◽  
The Matrix ◽  
Matrix Eigenvalue Problems

The Ritz method reduces eigenvalue problems involving linear operators on infinite dimensional spaces to finite matrix eigenvalue problems. This paper shows that for a certain class of linear operators it is possible to choose the coordinate functions so that numerical solution of the matrix equations is considerably simplified, especially when the matrices are large. The method is applied to the problem of overtone pulsations of stars.

A Separation Principle for Gyroscopic Conservative Systems

1997 ◽  
Vol 119 (1) ◽  
pp. 110-119 ◽  
Author(s):  
L. Meirovitch
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Separation Theorem ◽  
Separation Principle ◽  
Numerical Illustration ◽  
Closed Form Solutions ◽  
Conservative Systems ◽  
Algebraic Eigenvalue Problem

Closed-form solutions to differential eigenvalue problems associated with natural conservative systems, albeit self-adjoint, can be obtained in only a limited number of cases. Approximate solutions generally require spatial discretization, which amounts to approximating the differential eigenvalue problem by an algebraic eigenvalue problem. If the discretization process is carried out by the Rayleigh-Ritz method in conjunction with the variational approach, then the approximate eigenvalues can be characterized by means of the Courant and Fischer maximin theorem and the separation theorem. The latter theorem can be used to demonstrate the convergence of the approximate eigenvalues thus derived to the actual eigenvalues. This paper develops a maximin theorem and a separation theorem for discretized gyroscopic conservative systems, and provides a numerical illustration.

scholarly journals Application of the Ritz method to non-standard eigenvalue problems

1969 ◽  
Vol 10 (3-4) ◽  
pp. 367-384 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method ◽  
Positive Definite ◽  
Linear Operators ◽  
Extensive Literature ◽  
Maximum Eigenvalue ◽  
Special Cases ◽  
Image Position ◽  
Linear Manner

There is an extensive literature on application of the Ritz method to eigenvalue problems of the type where L1, L2 are positive definite linear operators in a Hilbert space (see for example [1]). The classical theory concerns the case in which there exists a minimum (or maximum) eigenvalue, and subsequent eigenvalues can be located by a well-known mini-max principle [2; p. 405]. This paper considers the possibility of application of the Ritz method to eigenvalue problems of the type (1) where the linear operators L1L2 are not necessarily positive definite and a minimum (or maximum) eigenvalue may not exist. The special cases considered may be written with the eigenvalue occurring in a non-linear manner.

scholarly journals A Short Theory of the Rayleigh–Ritz Method

2013 ◽  
Vol 13 (4) ◽  
pp. 495-502 ◽  
Author(s):  
Harry Yserentant
Keyword(s):  
Finite Element ◽  
Variational Method ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Approximation Error ◽  
Eigenvalues And Eigenfunctions ◽  
Element Space ◽  
The Best Approximation ◽  
The Common ◽  
The Given

Abstract. We present some new error estimates for the eigenvalues and eigenfunctions obtained by the Rayleigh–Ritz method, the common variational method to solve eigenproblems. The errors are bounded in terms of the error of the best approximation of the eigenfunction under consideration by functions in the ansatz space. In contrast to the classical theory, the approximation error of eigenfunctions other than the given one does not enter into these estimates. The estimates are based on a bound for the norm of a certain projection operator, e.g., in finite element methods for second order eigenvalue problems, the H1-norm of the L2-projection onto the finite element space.

The Rayleigh-Ritz Method With Quasi-Comparison Functions in Nonself-Adjoint Problems

1993 ◽  
Vol 115 (3) ◽  
pp. 280-284 ◽  
Author(s):  
P. Hagedorn
Keyword(s):  
Multibody Systems ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Continuous Linear ◽  
Linear Elastic ◽  
Comparison Functions ◽  
Elastic Systems ◽  
Nonlinear Motions ◽  
Admissible Functions

In the determination of the first eigenmodes of continuous linear elastic systems the Rayleigh-Ritz method is often used. It is also very useful in the discretization of the elastic members of multibody systems undergoing large nonlinear motions. Recently the concept of quasi-comparison functions has been introduced for the Rayleigh-Ritz discretization in self-adjoint eigenvalue problems, where it may lead to a considerable improvement of the convergence when compared with other classes of admissible functions. In this paper it is shown with a simple example that a similar phenomenon also holds for nonself-adjoint problems. Since the exact solutions are known, precise information on the errors can be given.

scholarly journals Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
D. V. Bambill ◽  
C. A. Rossit
Keyword(s):  
Transverse Vibration ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Mechanical Performance ◽  
Ritz Method ◽  
Normal Modes ◽  
Thin Plates ◽  
Transverse Vibrations ◽  
Edge Conditions ◽  
Distributed Forces

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

scholarly journals On Rayleigh-Ritz Method in Three-Parameter Eigenvalue Problems

2014 ◽  
Vol 86 (3) ◽  
pp. 38-42
Author(s):  
Surashmi Bhattacharyya ◽  
Arun Kumar Baruah
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method

scholarly journals Buckling and Free Vibrations of Nanoplates – Comparison of Nonlocal Strain and Stress Approaches

2019 ◽  
Vol 9 (7) ◽  
pp. 1409 ◽  
Author(s):  
Małgorzata Chwał ◽  
Aleksander Muc
Keyword(s):  
Eigenvalue Problems ◽  
Scale Effects ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Small Scale ◽  
Governing Equations ◽  
First Order ◽  
Fundamental Equations

The buckling and free vibrations of rectangular nanoplates are considered in this paper. The refined continuum transverse shear deformation theory (third and first order) is introduced to formulate the fundamental equations of the nanoplate. In addition, the analysis involves the nonlocal strain and stress theories of elasticity to take into account the small-scale effects encountered in nanostructures/nanocomposites. Hamilton’s principle is used to establish the governing equations of the nanoplate. The Rayleigh-Ritz method is proposed to solve eigenvalue problems dealing with the buckling and free vibration analysis of the nanoplates considered. Some examples are presented to investigate and illustrate the effects of various formulations.

Sign in / Sign up

Export Citation Format

Share Document