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.

Convergence of the Ritz Method

1995 ◽  
Vol 48 (11S) ◽  
pp. S90-S95 ◽  
Author(s):  
Arthur W. Leissa ◽  
Samir M. Shihada
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method ◽  
Equation Of Motion ◽  
Upper Bounds ◽  
Free Vibrations ◽  
Bending Moments ◽  
Shear Forces ◽  
Buckling Loads ◽  
Derivatives Of ◽  
Admissible Functions

The Ritz method is widely used for the solution of problems in structural mechanics, especially eigenvalue problems where the free vibration frequencies or buckling loads are sought. It is well-known that the method yields upper bounds for these eigenvalues, and that convergence to exact eigenvalues will occur if proper admissible functions are used to represent the displacements (eigenfunctions). However, little is known about the convergence of the derivatives of the eigenfunctions. In this paper the method is studied for the problem of free vibrations of a cantilever beam. Convergences of the eigenfunctions and their second and third derivatives (i.e., bending moments and shear forces) are examined, as well as convergence to satisfy the differential equation of motion (which involves fourth derivatives). Another study examines the effects of deliberately omitting one term from the set of admissible displacement functions which would otherwise be complete. It is found that in such cases, when orthogonal polynomials are used to represent the displacement, one eigenvalue (such as the lowest frequency) may be completely missed, and that the others will converge incorrectly. When ordinary polynomials are used, correct convergence is obtained even with a missing term.

Control of Elastic Systems via Passivity-Based Methods

2004 ◽  
Vol 10 (11) ◽  
pp. 1699-1735 ◽  
Author(s):  
A. G. Kelkar ◽  
S. M. Joshi
Keyword(s):  
Flexible Structures ◽  
Robotic Manipulators ◽  
Mathematical Concept ◽  
Controller Synthesis ◽  
Control Laws ◽  
Linear Elastic ◽  
Elastic Systems ◽  
Different Types ◽  
Linear Controllers ◽  
Synthesis Approach

In this paper we present a controller synthesis approach for elastic systems based on the mathematical concept of passivity. For nonlinear and linear elastic systems that are inherently passive, robust control laws are presented that guarantee stability. Examples of such systems include flexible structures with col-located and compatible actuators and sensors, and multibody space-based robotic manipulators. For linear elastic systems that are not inherently passive, methods are presented for rendering them passive by compensation. The “passified” systems can then be robustly controlled by a class of passive linear controllers that guarantee stability despite uncertainties and inaccuracies in the mathematical models. The controller synthesis approach is demonstrated by application to five different types of elastic systems.

A Systematic Topology Optimization Approach for Optimal Stiffener Design

1998 ◽  
Author(s):  
Jian Hui Luo ◽  
Hae Chang Gea
Keyword(s):  
Topology Optimization ◽  
Location Problem ◽  
Eigenvalue Problems ◽  
Three Dimensional ◽  
Orthotropic Materials ◽  
Optimization Approach ◽  
Design Domain ◽  
Plate Structures ◽  
Shell Plate

Abstract A systematic topology optimization approach is developed to design the optimal stiffener of three dimensional shell/plate structures in static and eigenvalue problems. Optimal stiffener design involves the determination of the best location and orientation. In this paper, the stiffener location problem is solved by a microstructure-based design domain method and the orientation probelm is modeled as an optimal orientation problem of equivalent orthotropic materials, which is solved by a newly developed energy based method. Examples are presented to demonstrate the application of the proposed approach.

scholarly journals Automatic discontinuity of intertwining operators

1982 ◽  
Vol 25 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Spaces ◽  
Linear Transformation ◽  
Linear Operators ◽  
Intertwining Operators ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Converse Problem ◽  
Simple Modification ◽  
Continuous Linear Operators

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

scholarly journals Independent coordinate coupling method for vibration analysis of a functionally graded carbon nanotube–reinforced plate with central hole

2019 ◽  
Vol 11 (8) ◽  
pp. 168781401987292 ◽  
Author(s):  
Yan Guo ◽  
Yanan Jiang ◽  
Bin Huang
Keyword(s):  
Carbon Nanotube ◽  
Volume Fraction ◽  
Ritz Method ◽  
Matching Condition ◽  
Functionally Graded ◽  
Central Hole ◽  
Lagrange’S Equation ◽  
Reinforced Plate ◽  
Distribution Type ◽  
Admissible Functions

In this article, the free vibration of a functionally graded carbon nanotube–reinforced plate with central hole is investigated by means of the independent coordinates-based Rayleigh–Ritz method. For the proposed method, the kinematic and potential energies are substituted into Lagrange’s equation in order to obtain the equation of motion. However, the total energies are computed by the difference of energies between the hole domain and the plate domain. By applying the displacement matching condition at the hole domain, two coordinate systems are coupled. For the Rayleigh–Ritz method, the mode shape functions of uniform beams are assumed as admissible functions. By this method, convergent results can be obtained with certain number of terms of admissible functions. The present results clearly reflect the effects of the carbon nanotube distribution type, carbon nanotube volume fraction, hole size, and boundary condition on the nondimensional natural frequencies. The provided results show that the present method is efficient in studying the vibration problems of functionally graded carbon nanotube–reinforced plate with central hole.

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.

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