Analysis of Cylindrically Curved Panels Based on a Two Field Variable Variational Principle

1993 ◽  
Vol 46 (11S) ◽  
pp. S71-S78 ◽  
Author(s):  
Maher N. Bismarck-Nasr
Keyword(s):  
Variational Principle ◽  
Variational Formulation ◽  
Free Vibrations ◽  
Field Variable ◽  
Numerical Calculations ◽  
Transverse Displacement ◽  
Panel Flutter ◽  
Lagrange Equations ◽  
Experimental Findings ◽  
Curved Panels

A variational formulation of thin cylindrically curved panels having transverse displacement and Airy stress function as field variable is presented. Euler-Lagrange equations and boundary conditions are obtained. A finite element based on this variational principle preserving C(1) continuity is formulated. Applications for free vibrations, buckling, and supersonic panel flutter analyses are given. Several numerical calculations are presented. The results obtained are discussed and are compared with previous analytical solutions, numerical calculations, and experimental findings.

On the equations governing the motion of an anisotropic poroelastic material

2006 ◽  
Vol 462 (2072) ◽  
pp. 2373-2396 ◽  
Author(s):  
Gülay Altay ◽  
M. Cengİz Dökmecİ
Keyword(s):  
Variational Principle ◽  
Heterogeneous Media ◽  
Variational Principles ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Anisotropic Fluid ◽  
Governing Equations ◽  
Lagrange Equations ◽  
Poroelastic Material

We address Biot's equations governing the motion of an anisotropic fluid-saturated poroelastic material with certain properties. First, we investigate the uniqueness in solutions of the three-dimensional governing equations for the regular region of the poroelastic material and enumerate the conditions sufficient for the uniqueness. Next, by applying Hamilton's principle to the motion of the region, we obtain a variational principle that generates only the Biot–Newton equations and the associated natural boundary conditions. Then, by extending the variational principle for the region with an internal fixed surface of discontinuity through Legendre's transformation, we derive a six-field variational principle that operates on all the poroelastic field variables. The variational principle leads, as its Euler–Lagrange equations, to all the governing equations, including the jump conditions but the initial conditions, as a generalized version of the Hellinger–Reissner variational principle. Moreover, we consider the free vibrations of the region, and we discuss some basic properties of eigenvalues and present a variational formulation by Rayleigh's quotient. This work provides a standard tool with the features of variational principles when numerically solving the governing equations in heterogeneous media with finite element methods, treating the free vibrations and consistently deriving some one-dimensional/two-dimensional equations of the poroelastic region.

SYMMETRY DECOUPLING IN LINEAR ALGEBRAIC VARIATIONAL SCATTERING PROBLEMS

1995 ◽  
Vol 06 (01) ◽  
pp. 105-121
Author(s):  
MEISHAN ZHAO
Keyword(s):  
Angular Momentum ◽  
Variational Principle ◽  
Total Angular Momentum ◽  
Numerical Calculations ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Scattering Problems ◽  
Identical Particles ◽  
Generalized Newton

This paper discusses the symmetry decoupling in quantum mechanical algebraic variational scattering calculations by the generalized Newton variational principle. Symmetry decoupling for collisions involving identical particles is briefly discussed. Detailed discussion is given to decoupling from evaluation of matrix elements with nonzero total angular momentum. Example numerical calculations are presented for BrH2 and DH2 systems to illustrate accuracy and efficiency.

scholarly journals A VARIATIONAL FORMULATION OF SYMPLECTIC NONCOMMUTATIVE MECHANICS

2007 ◽  
Vol 04 (05) ◽  
pp. 789-805 ◽  
Author(s):  
IGNACIO CORTESE ◽  
J. ANTONIO GARCÍA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principle ◽  
Configuration Space ◽  
Variational Formulation ◽  
Noncommutative Space ◽  
First Order ◽  
Dynamical Properties ◽  
Definition Of ◽  
Space Noncommutativity

The standard lore in noncommutative physics is the use of first order variational description of a dynamical system to probe the space noncommutativity and its consequences in the dynamics in phase space. As the ultimate goal is to understand the inherent space noncommutativity, we propose a variational principle for noncommutative dynamical systems in configuration space, based on results of our previous work [18]. We hope that this variational formulation in configuration space can be of help to elucidate the definition of some global and dynamical properties of classical and quantum noncommutative space.

New Solitary Wave Solution for the Boussinesq Wave Equation Using the Semi-Inverse Method and the Exp-Function Method

2009 ◽  
Vol 64 (11) ◽  
pp. 709-712 ◽  
Author(s):  
Wenjun Liu
Keyword(s):  
Wave Equation ◽  
Variational Principle ◽  
Solitary Wave ◽  
Wave Solution ◽  
Variational Formulation ◽  
Function Method ◽  
Inverse Method ◽  
Solitary Wave Solution ◽  
Solitary Solutions

Using the semi-inverse method, a variational formulation is established for the Boussinesq wave equation. Based on the obtained variational principle, solitary solutions in the sech-function and expfunction forms are obtained

scholarly journals Peridynamic Higher-Order Beam Formulation

2020 ◽  
Author(s):  
Zhenghao Yang ◽  
Erkan Oterkus ◽  
Selda Oterkus
Keyword(s):  
Finite Element Method ◽  
Cantilever Beam ◽  
Higher Order ◽  
Point Load ◽  
Transverse Displacement ◽  
Lagrange Equations ◽  
Concentrated Load ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Good Agreement

Abstract In this study, a novel higher-order peridynamic beam formulation is presented. The formulation is obtained by using Euler-Lagrange equations and Taylor’s expansion. To demonstrate the capability of the presented approach, several different beam configurations are considered including simply supported beam subjected to distributed loading, simply supported beam with concentrated load, clamped-clamped beam subjected to distributed loading, cantilever beam subjected to a point load at its free end and cantilever beam subjected to a moment at its free end. Transverse displacement results along the beam obtained from peridynamics and finite element method are compared with each other and very good agreement is obtained between the two approaches.

scholarly journals Covariant brackets for particles and fields

2017 ◽  
Vol 32 (19) ◽  
pp. 1750100 ◽  
Author(s):  
M. Asorey ◽  
M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort
Keyword(s):  
Variational Principle ◽  
Geometrical Approach ◽  
Covariant Formulation ◽  
Lagrange Equations ◽  
Relativistic Systems

A geometrical approach to the covariant formulation of the dynamics of relativistic systems is introduced. A realization of Peierls brackets by means of a bivector field over the space of solutions of the Euler–Lagrange equations of a variational principle is presented. The method is illustrated with some relevant examples.

On the Nonuniqueness of Solutions Obtained With Simplified Variational Principles

1996 ◽  
Vol 63 (3) ◽  
pp. 820-827 ◽  
Author(s):  
H. Mang ◽  
P. Helnwein ◽  
R. H. Gallagher
Keyword(s):  
Variational Principle ◽  
Lagrange Multipliers ◽  
Variational Principles ◽  
Boundary Value ◽  
Geometrically Nonlinear ◽  
Lagrange Equations ◽  
Geometrically Nonlinear Theory ◽  
Nonuniqueness Of Solutions ◽  
Uniqueness Of The Solution ◽  
Bending Of Beams

The attempt to satisfy subsidiary conditions in boundary value problems without additional independent unknowns in the form of Lagrange multipliers has led to the development of so-called “simplified variational principles.” They are based on using the Euler-Lagrange equations for the Lagrange multipliers to express the multipliers in terms of the original variables. It is shown that the conversion of a variational principle with subsidiary conditions to such a simplified variational principle may lead to the loss of uniqueness of the solution of a boundary value problem. A particularly simple form of the geometrically nonlinear theory of bending of beams is used as the vehicle for this proof. The development given in this paper is entirely analytical. Hence, the deficiencies of the investigated simplified variational principle are fundamental.

Hamiltonian Formulation for Continuous Third-order Systems Using Fractional Derivatives

2021 ◽  
Vol 14 (1) ◽  
pp. 35-47
Keyword(s):  
Variational Principle ◽  
Fractional Derivatives ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Lagrange Equations ◽  
Third Order ◽  
Derivative Operator ◽  
Hamilton Equations ◽  
Liouville Fractional Derivative ◽  
Continuous Field

Abstract: We constructed the Hamiltonian formulation of continuous field systems with third order. A combined Riemann–Liouville fractional derivative operator is defined and a fractional variational principle under this definition is established. The fractional Euler equations and the fractional Hamilton equations are obtained from the fractional variational principle. Besides, it is shown that the Hamilton equations of motion are in agreement with the Euler-Lagrange equations for these systems. We have examined one example to illustrate the formalism. Keywords: Fractional derivatives, Lagrangian formulation, Hamiltonian formulation, Euler-lagrange equations, Third-order lagrangian.

Exact Solutions for the Free Vibrations and Buckling of Rectangular Plates With Linearly Varying In-Plane Loading

2001 ◽  
Author(s):  
Arthur W. Leissa ◽  
Jae-Hoon Kang
Keyword(s):  
Solution Procedure ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Buckling Loads ◽  
Simply Supported ◽  
Edge Conditions ◽  
Elastically Supported

Abstract An exact solution procedure is formulated for the free vibration and buckling analysis of rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying normal stresses. The other two edges may be clamped, simply supported or free, or they may be elastically supported. The transverse displacement (w) is assumed as sinusoidal in the direction of loading (x), and a power series is assumed in the lateral (y) direction (i.e., the method of Frobenius). Applying the boundary conditions yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to obtain adequate convergence for accurate vibration frequencies and buckling loads, as is demonstrated by two convergence tables. Some interesting and useful results for vibration frequencies and buckling loads, and their mode shapes, are presented for a variety of edge conditions and in-plane loadings, especially pure in-plane moments.

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