scholarly journals Curve Veering: Inherent Behaviour of Some Vibrating Systems

2000 ◽  
Vol 7 (4) ◽  
pp. 241-249 ◽  
Author(s):  
B.B. Bhat
Keyword(s):  
System Parameter ◽  
Ritz Method ◽  
Approximate Solutions ◽  
System Model ◽  
Exact Form ◽  
Actual System ◽  
Vibrating Structures ◽  
Curve Veering ◽  
The Mathematical Model ◽  
Vibrating Systems

The plots of variation of eigenvalues of vibrating structures with a system parameter often cross each other or abruptly veer away avoiding the crossing. The phenomenon is termed as curve veering and has been observed both in approximate solutions as well as in exact solutions associated with vibration of some vibrating systems. An explanation to such behavior is provided and illustrated by solving a simple example. The curve veering behavior is induced into a membrane by introducing a parameter that can change the mathematical model from an exact form to an approximate one. Approximate deflection functions such as those used in Galerkin's method or the Rayleigh Ritz method invariably create an approximate or a ficticious system model in lieu of the actual system. The ficticious system may exhibit curve veering while the corresponding real system has no such behaviour. When the ficticious nature of the system is minimized by using large number of terms in the approximate techniques or by discretisation of the domain as in finite difference or by assuming spline type deflection functions, the curve veering behavior subsides and in some cases almost vanishes.

Disordered Dynamic Systems Resulting From Approximate Modeling and Analysis

2000 ◽  
Author(s):  
R. B. Bhat ◽  
I. Stiharu
Keyword(s):  
Dynamic Systems ◽  
Disordered Systems ◽  
Ritz Method ◽  
Actual System ◽  
Approximate Methods ◽  
Modeling And Analysis ◽  
Mode Localization ◽  
Exact Analysis ◽  
Curve Veering ◽  
Curve Crossing

Abstract Some dynamic systems exhibit curve veering behavior or avoided crossings when their natural frequencies are plotted against a system parameter, while some other show a curve crossing behavior or frequency coalescence. The curve veering behavior is also observed in disordered systems where the symmetry of the system is slightly perturbed and a mode localization takes place. In some systems while the exact analysis shows a curve crossing trend, approximate analyses show a curve veering behavior. Earlier studies have shown that there is a common pattern in curve veering systems and disordered systems. In the present study the exact analysis is recognized as representing the actual system while the approximate analysis of the same system renders it a disordered system by perturbing the eigenvalues and eigenfunctions from their true values. Since the responses of disordered systems can sometimes show violent changes for small perturbations in the system parameters, the response of a simply supported plate has been obtained both exactly and approximately using the Rayleigh Ritz method, and compared. The conclusions have far reaching implications from the point of the accuracy of the response quantities obtained by approximate methods such as finite element method, the Rayleigh Ritz and Galerkin methods.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Approximate Solutions in Linear, Coupled Thermoelasticity

1968 ◽  
Vol 35 (2) ◽  
pp. 255-266 ◽  
Author(s):  
R. E. Nickell ◽  
J. L. Sackman
Keyword(s):  
Half Space ◽  
Rapid Heating ◽  
Ritz Method ◽  
Gaussian Quadrature ◽  
Boundary Surface ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Coupled Thermoelasticity ◽  
Initial Boundary Value ◽  
Thermally Conducting

A method for obtaining approximate solutions to initial-boundary-value problems in the linear theory of coupled thermoelasticity is developed. This procedure is a direct variational method representing an extension of the Ritz method. As an illustration of the procedure, it is applied to a class of one-dimensional, transient problems involving weak thermal shocks. The problems considered are: (a) Rapid heating of a half space through a thermally conducting boundary layer, and (b) gradual heating of the boundary surface of a half space. The solutions generated by the extended Ritz method are compared, for accuracy, to solutions obtained from a numerical inversion scheme for the Laplace transform based on Gaussian quadrature. These comparisons indicate that the variational procedure developed here can yield accurate results.

The Finite-Element Method—Part I

1989 ◽  
Author(s):  
Shiro Kobayashi ◽  
Soo-Ik Oh ◽  
Taylan Altan
Keyword(s):  
Heat Transfer ◽  
Finite Element Method ◽  
Finite Element ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Plane Elasticity ◽  
Heat Transfer Analysis ◽  
Deformation Analysis ◽  
The Finite Element Method ◽  
Element Method

The concept of the finite-element procedure may be dated back to 1943 when Courant approximated the warping function linearly in each of an assemblage of triangular elements to the St. Venant torsion problem and proceeded to formulate the problem using the principle of minimum potential energy. Similar ideas were used later by several investigators to obtain the approximate solutions to certain boundary-value problems. It was Clough who first introduced the term “finite elements” in the study of plane elasticity problems. The equivalence of this method with the well-known Ritz method was established at a later date, which made it possible to extend the applications to a broad spectrum of problems for which a variational formulation is possible. Since then numerous studies have been reported on the theory and applications of the finite-element method. In this and next chapters the finite-element formulations necessary for the deformation analysis of metal-forming processes are presented. For hot forming processes, heat transfer analysis should also be carried out as well as deformation analysis. Discretization for temperature calculations and coupling of heat transfer and deformation are discussed in Chap. 12. More detailed descriptions of the method in general and the solution techniques can be found in References [3-5], in addition to the books on the finite-element method listed in Chap. 1. The path to the solution of a problem formulated in finite-element form is described in Chap. 1 (Section 1.2). Discretization of a problem consists of the following steps: (1) describing the element, (2) setting up the element equation, and (3) assembling the element equations. Numerical analysis techniques are then applied for obtaining the solution of the global equations. The basis of the element equations and the assembling into global equations is derived in Chap. 5. The solution satisfying eq. (5.20) is obtained from the admissible velocity fields that are constructed by introducing the shape function in such a way that a continuous velocity field over each element can be denned uniquely in terms of velocities of associated nodal points.

Optimization of a Snap Through Spring for a Hydraulic Valve With Hysteresis Response Behavior

2019 ◽  
Author(s):  
Matthias Scherrer ◽  
Rudolf Scheidl ◽  
Bernhard Manhartsgruber
Keyword(s):  
Ritz Method ◽  
High Strength ◽  
Axial Direction ◽  
Approximate Solutions ◽  
Element Model ◽  
Lateral Force ◽  
Spring Steel ◽  
Nonlinear Buckling ◽  
Buckling Beam ◽  
Compact Design

Abstract The hydraulic binary counter requires switching valves with a hysteretic response. In this paper an elastic snap through element is studied as means for that. The concept is based on a buckling beam which is elastically supported in axial direction in order to adjust its buckling properties with moderate manufacturing precision and to assure a well defined snap through behavior. The elastic support is provided by a cantilever beam. A rigorous optimization is performed heading for a most compact and fatigue durable design which exhibits the required lateral force displacement characteristics. A genetic algorithm is used to find the global design optimum. The stress/displacement properties of each design variant are computed by a compact model of the snap through system. It is derived by a Ritz method to obtain approximate solutions of the nonlinear buckling beam behavior. Its validity is checked by a Finite Element model. A compact design is possible if high strength spring steel is used for the elastic elements.

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.

On the Use of Beam Functions for Problems of Plates Involving Free Edges

1975 ◽  
Vol 42 (4) ◽  
pp. 858-864 ◽  
Author(s):  
S. F. Bassily ◽  
S. M. Dickinson
Keyword(s):  
Free Vibration ◽  
Vibration Mode ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Static Deflection ◽  
Numerical Examples ◽  
Accurate Treatment ◽  
Beam Mode ◽  
Free Edges

The inadequacy of beam vibration mode shapes when used in the Ritz method to obtain approximate solutions for flexural problems concerning plates involving free edges is demonstrated. A new set of functions, related to beam mode shapes, is postulated which allows considerably more accurate treatment of such plates. Several numerical examples concerning static deflection and free vibration of plates involving free edges are examined and serve to illustrate the applicability and accuracy of the new functions and to further demonstrate the inadequacy of the ordinary beam functions.

Stability of Velocity-Loop-Based Hydraulic Control Loading System

2012 ◽  
Vol 619 ◽  
pp. 472-475
Author(s):  
Pan Guo Qi ◽  
Li Wei Zhao ◽  
Pei Chao Cong ◽  
Hui Wang
Keyword(s):  
Mathematical Model ◽  
Block Diagram ◽  
System Model ◽  
Flight Simulator ◽  
Stability Conditions ◽  
Hydraulic Control ◽  
Function Block ◽  
Loading System ◽  
The Stability ◽  
The Mathematical Model

A hydraulic Control Loading System (CLS) based on velocity-loop was developed and installed on our flight simulator earlier days, but the CLS cannot keep stable in some conditions. This problem is discussed in this paper. The mathematical model of velocity-loop-based CLS is firstly developed with the method of transfer function block diagram. Then, system’s stability conditions are put forward using Roth criterion based on the system model developed. At last, the experiments proves the stability

scholarly journals Study of electrical R-L circuits composed of resistors and inductors and driven by a voltage of the current source: Simulation with implementing an accurate method

2020 ◽  
Author(s):  
Mohamed Adel ◽  
Hari M. Srivastava ◽  
Mohamed Khader
Keyword(s):  
Current Source ◽  
Numerical Procedure ◽  
Classical Case ◽  
Approximate Solutions ◽  
Accurate Method ◽  
Order Model ◽  
The Third ◽  
Fractional Order Model ◽  
Proposed Model ◽  
The Mathematical Model

In this study, we propose to derive an accurate numerical procedure to solve the mathematical model which describes the electrical R-L circuit composed of resistors and inductors driven by a voltage of current source, which is the fractional-order model for the electrical RL-circuit. Our study depends on the spectral collocation method via the useful properties of the Chebyshev polynomials of the third-kind. Some theorems about the convergence analysis are given. The study concludes by comparing the resulting approximate solutions of the proposed model with the exact solution in the classical case. Illustrative graphical and numerical analysis of the derived results are also included in this study.

scholarly journals Use of Pareto Principle in Power System Mode Analysis

2010 ◽  
Vol 26 (1) ◽  
pp. 54-57
Author(s):  
Anatoly Mahnitko ◽  
Alexander Gavrilov
Keyword(s):  
Power System ◽  
System Model ◽  
Mode Analysis ◽  
Pareto Principle ◽  
Optimal Regime ◽  
Optimal Power ◽  
United Power System ◽  
System Mode ◽  
The Mathematical Model ◽  
The Given

Use of Pareto Principle in Power System Mode AnalysisThe optimal power dispatch problem in the power system is looked out in the given work. The mathematical model of power system optimal regime searching approach in the market conditions in accordance with Pareto principle is described. The theoretical layout is illustrated on a real power system model of the united power system, which consists of 17 nodes and 21 lines. The procedure is realized using the GAMS software.

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