Strongest Columns and Isoperimetric Inequalities for Eigenvalues

1962 ◽  
Vol 29 (1) ◽  
pp. 159-164 ◽  
Author(s):  
I. Tadjbakhsh ◽  
J. B. Keller
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Cross Sections ◽  
Isoperimetric Inequalities ◽  
Second Order ◽  
Buckling Load ◽  
The Other ◽  
Critical Buckling Load ◽  
Various Boundary Conditions

We consider the problem of determining what shape column has the largest critical buckling load of all columns of given length and volume. This problem was previously solved for a column hinged (pinned) at both ends. We solve it for columns clamped at one end and clamped, hinged, or free at the other end, assuming that all cross sections of the column are similar and similarly oriented. We also prove that the column previously obtained in the hinged-hinged case is actually strongest and not merely stationary. Graphs of the areas of the strongest columns as functions of distance along the columns are given for the various cases. The results are also expressed as isoperimetric inequalities for eigenvalues of second-order ordinary differential equations with various boundary conditions. Certain additional inequalities of this type are also obtained.

The interlacing of eigenvalues for periodic multi-parameter problems

1978 ◽  
Vol 80 (3-4) ◽  
pp. 357-362 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Image Position ◽  
Periodic Equation

SynopsisThis paper studies a linked system of second order ordinary differential equationswhere xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

Stability regions for multi-parameter systems of periodic second order ordinary differential equations

1979 ◽  
Vol 84 (3-4) ◽  
pp. 249-257 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Parameter System ◽  
Stability Regions ◽  
The Stability ◽  
Image Position

SynopsisThis paper studies the stability regions associated with the multi-parameter systemwhere the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.

Nonlinear eigenvalue problems for a class of ordinary differential equations

2002 ◽  
Vol 132 (6) ◽  
pp. 1333-1359 ◽  
Author(s):  
Uri Elias ◽  
Allan Pinkus
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Various Boundary Conditions ◽  
Image Position ◽  
Nonlinear Eigenvalue

We consider the class of nonlinear eigenvalue problems where yp* = |y|p sgn y, pi > 0 and p0p1 … pn−1 = r, with various boundary conditions. We prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions.

FIRST INTEGRALS OF FIN EQUATIONS FOR STRAIGHT FINS

2009 ◽  
Vol 23 (30) ◽  
pp. 3659-3666 ◽  
Author(s):  
E. MOMONIAT ◽  
C. HARLEY ◽  
T. HAYAT
Keyword(s):  
Temperature Distribution ◽  
Temperature Gradient ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations

First integrals admitted by second-order nonlinear ordinary differential equations modeling the temperature distribution in a straight fin are obtained. After imposing the boundary conditions these first integrals give a relationship between temperature at the fin tip and the temperature gradient at the base of the fin in terms of the fin parameters. These first integrals are plotted and analyzed. The results obtained show how the temperature at the fin tip can be controlled by the temperature gradient at the base for fixed fin parameters.

Torsional-Flexural Critical Force for Various Boundary Conditions

2016 ◽  
Vol 710 ◽  
pp. 303-308 ◽  
Author(s):  
Ivan Balaz ◽  
Michal Kovac ◽  
Tomáš Živner ◽  
Yvona Kolekova
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Approximate Method ◽  
Cross Sections ◽  
Critical Force ◽  
Various Boundary Conditions ◽  
Element Method

The system of governing differential equations of stability of members with the rigid open cross-sections was developed by Vlasov [1] in 1940. Goľdenvejzer [2] published in 1941 solution of this system by an approximate method. He proposed formula for torsional-flexural critical force Ncr.TF calculation which is modified and used in EN 1999-1-1 [3] (I.19). By introducing factor αzw he take into account any combination of boundary conditions (BCs).The purpose of this paper is to verify this formula and explore the possibility to improve the factor αzw. In the large parametrical study the authors investigated a lot of different shape of cross-sections, all 100 theoretical possible combinations of BCs and various member lengths. All results are evaluated regarding the reference results by finite element method (FEM).

FORCES, MOMENTS, ROTATIONS, AND DISPLACEMENTS OF POLYNOMIAL-SHAPED CURVED BEAMS

2010 ◽  
Vol 10 (01) ◽  
pp. 77-89 ◽  
Author(s):  
LAZARO GIMENA ◽  
PEDRO GONZAGA ◽  
FAUSTINO GIMENA
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Mechanical Behavior ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
The Other ◽  
Curved Beams ◽  
Fourth Degree ◽  
Linear Ordinary Differential Equations ◽  
Parabolic Shape

This paper deals with curved beams with polynomial free geometry. The problem is approached analytically and the differential equations that govern the mechanical behavior of curved beams are presented. A system of twelve linear ordinary differential equations is solved using either an analytical or a customized numerical method with boundary conditions. Results of the different components of forces, moments, rotations, and displacements are given and plotted in the examples for different polynomial-shaped beams of the fourth degree. It is concluded from the present analyses that the parabolic shape has better response to distributed loads than the other polynomial-shaped beams considered.

A HYBRID DIFFERENCE SCHEME FOR SINGULARLY PERTURBED SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS CONVECTION COEFFICIENT AND MIXED TYPE BOUNDARY CONDITIONS

2008 ◽  
Vol 05 (04) ◽  
pp. 575-593 ◽  
Author(s):  
R. MYTHILI PRIYADHARSHINI ◽  
N. RAMANUJAM
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Ordinary Differential Equations ◽  
Mixed Type ◽  
Second Order ◽  
Singularly Perturbed ◽  
Convection Coefficient ◽  
Type Boundary ◽  
Numerical Derivative

This paper presents, a hybrid difference scheme for singularly perturbed second order ordinary differential equations with a small parameter multiplying the highest derivative with a discontinuous convection coefficient subject to mixed type boundary conditions. Error bounds for the numerical solution and numerical derivative are established. Numerical results are provided to illustrate the theoretical results.

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