The secular variations in the restricted problem's sixth-order theory and stability: An explicit numerical approach

1990 ◽  
Vol 167 (1) ◽  
pp. 139-158 ◽  
Author(s):  
Catherine J. Flogaitis ◽  
V. S. Geroyannis ◽  
G. A. Antonacopoulos
Keyword(s):  
Numerical Approach ◽  
Order Theory ◽  
Sixth Order ◽  
Secular Variations

The secular variations in the restricted problem: An explicit numerical approach

1989 ◽  
Vol 155 (1) ◽  
pp. 85-103 ◽  
Author(s):  
V. S. Geroyannis ◽  
Catherine J. Flogaitis
Keyword(s):  
Restricted Problem ◽  
Numerical Approach ◽  
Secular Variations

scholarly journals Numerical description of mode coupled waves in inhomogeneous magnetized plasmas

2019 ◽  
Vol 203 ◽  
pp. 01011
Author(s):  
Kota Yanagihara ◽  
Shin Kubo ◽  
Toru Tsujimura
Keyword(s):  
Ray Tracing ◽  
Electron Cyclotron Resonance ◽  
Numerical Approach ◽  
Geometrical Optics ◽  
Order Theory ◽  
Complex Amplitude ◽  
First Order ◽  
First Order Theory ◽  
Full Wave Analysis ◽  
Order Part

Geometrical optics (GO) ray tracing has been widely used for a description of electron cyclotron resonance waves in inhomogeneous magnetized fusion plasmas. However, this reduced approach is not correct in sufficient low density plasmas with a sheared magnetic field, where mode coupling between two electromagnetic-like cold plasma modes can occur. Here, we extend a ray tracing method based on the first-order theory of extended geometrical optics (XGO), which captures mode coupled complex amplitude between O and X mode along the ray trajectory. In our approach, reference ray is calculated with ray equation to satisfy the lowest-order part of XGO theory and an evolution of complex amplitude profile along the reference ray is calculated with partial differential equation derived from first-order terms. Calculation results performed by extended ray tracing are in good agreement with 1D full wave analysis. By introducing second-order terms into our numerical approach, diffraction will be treated.

A Combined Analytical and Numerical Approach for the Solution of an Edge Loaded Semi-Infinite Elastic Sheet

1996 ◽  
Author(s):  
Xin Sun ◽  
Dale G. Karr ◽  
Chunhui Han
Keyword(s):  
Fourier Transforms ◽  
Thin Sheet ◽  
Three Dimensional ◽  
Numerical Approach ◽  
Order Theory ◽  
Thick Sheet ◽  
Approximate Theory ◽  
Loading Conditions ◽  
Elastic Sheet ◽  
Limit Solutions

Abstract Edge loading of an semi-infinite elastic sheet is of interest to many engineering applications. In this paper the penetration problem involving a rigid indentor and an elastic semi-infinite sheet of uniform thickness is addressed using a combined analytical and numerical approach. The tenth-order approximate theory of stretching of an isotropic sheet is applied to formulate the governing differential equations. Solutions are then obtained using Fourier transforms for various loading conditions and numerical schemes are employed to calculate the three dimensional state of stress throughout the sheet. The influence of the aspect ratio on the resulting stress state is studied. Limit solutions for thin sheet and thick sheet are presented. Finite element analyses of the same loading conditions are also performed. Results are compared with those of the tenth-order theory and the stress distribution assumptions of the tenth-order theory are examined.

A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Guangyu Shi ◽  
George Z. Voyiadjis
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Cross Sections ◽  
Beam Theory ◽  
Order Theory ◽  
Sixth Order ◽  
Equilibrium Equations ◽  
Flexible Beams ◽  
Generalized Displacements ◽  
Shear Deformable

This paper presents the derivation of a new beam theory with the sixth-order differential equilibrium equations for the analysis of shear deformable beams. A sixth-order beam theory is desirable since the displacement constraints of some typical shear flexible beams clearly indicate that the boundary conditions corresponding to these constraints can be properly satisfied only by the boundary conditions associated with the sixth-order differential equilibrium equations as opposed to the fourth-order equilibrium equations in Timoshenko beam theory. The present beam theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the authors, a system of sixth-order differential equilibrium equations in terms of two generalized displacements w and ϕx of beam cross sections, and three boundary conditions at each end of shear deformable beams. A technique for the analytical solution of the new beam theory is also presented. To demonstrate the advantages and accuracy of the new sixth-order beam theory for the analysis of shear flexible beams, the proposed beam theory is applied to solve analytically three classical beam bending problems to which the fourth-order beam theory of Timoshenko has created some questions on the boundary conditions. The present solutions of these examples agree well with the elasticity solutions, and in particular they also show that the present sixth-order beam theory is capable of characterizing some boundary layer behavior near the beam ends or loading points.

Reflections on the Theory of Elastic Plates

1985 ◽  
Vol 38 (11) ◽  
pp. 1453-1464 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Shear Deformation ◽  
Fourth Order ◽  
Transverse Shear ◽  
Three Dimensional ◽  
Order Theory ◽  
Sixth Order ◽  
Interior Solution ◽  
Transverse Shear Deformation ◽  
Two Dimensional ◽  
Elastic Plates

We depart from a three-dimensional statement of the problem of small bending of elastic plates, for a survey of approximate two-dimensional theories, beginning with Kirchhoff’s fourth-order formulation. After discussing various variational statements of the three-dimensional problem, we describe the development of two-dimensional sixth-order theories by Bolle´, Hencky, Mindlin, and Reissner which take account of the effect of transverse shear deformation. Additionally, we report on an early analysis by Le´vy, on a direct two-dimensional formulation of sixth-order theory, on constitutive coupling of bending and stretching of laminated plates, on higher than sixth-order theories, and on an asymptotic analysis of sixth-order theory which leads to a fourth-order interior solution contribution with first-order transverse shear deformation effects included, as well as to a sequentially determined second-order edge zone solution contribution.

A Sixth-Order Plate Theory—Derivation and Error Estimates

1987 ◽  
Vol 54 (2) ◽  
pp. 275-279 ◽  
Author(s):  
Z. Rychter
Keyword(s):  
Plate Theory ◽  
Three Dimensional ◽  
Composite Plates ◽  
Order Theory ◽  
Sixth Order ◽  
Plate Bending ◽  
Mean Square ◽  
Reissner’S Theory ◽  
Elasticity Solutions ◽  
Admissible Solutions

Two consistent variants of a sixth-order theory for elastic plate bending are developed together with corresponding three-dimensional statically and kinematically admissible solutions. The relative mean square error of these solutions as compared with exact elasticity solutions is found to be proportional to the thickness squared in analogy with previous estimates for Reissner’s theory, but the contribution to the error governed by transverse shear deformability is reduced and shown to be of the order of the thickness cubed, this contribution being decisive in composite plates.

An Analytical Model for the Vibration of Laminated Beams Including the Effects of Both Shear and Thickness Deformation in the Adhesive Layer

1986 ◽  
Vol 108 (1) ◽  
pp. 56-64 ◽  
Author(s):  
R. N. Miles ◽  
P. G. Reinhall
Keyword(s):  
Shear Deformation ◽  
Analytical Model ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Adhesive Layer ◽  
Order Theory ◽  
Sixth Order ◽  
Viscoelastic Damping ◽  
Laminated Beams ◽  
Metal Layers

The equations of motion governing the vibration of a beam consisting of two metal layers bonded together with a soft viscoelastic damping adhesive are derived and solved. The adhesive is assumed to undergo both shear and thickness deformations during the vibration of the beam. In previous investigations the thickness deformation has been assumed to have negligible effect on the total damping. However, if the adhesive is very soft, and if at least one of the metal layers is stiff in bending, the thickness deformation in the adhesive can become the dominant damping mechanism. The analysis presented here comprises an extension of the well-known sixth order theory of DiTaranto, Mead, and Markus to include thickness deformation. The equations of motion are derived using Hamilton’s Principle and solutions are obtained by the Ritz method. It is shown that the use of a lightweight constraining layer which is stiff in bending will result in a design which is considerably more damped than a conventional configuration in which the adhesive is undergoing predominant shear deformation.

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