On the Optimal Shape of a Rotating Rod

2001 ◽  
Vol 68 (6) ◽  
pp. 860-864 ◽  
Author(s):  
T. M. Atanackovic
Keyword(s):  
Maximum Principle ◽  
Boundary Value Problem ◽  
Numerical Integration ◽  
Nonlinear Boundary ◽  
Variational Principles ◽  
Boundary Value ◽  
First Integral ◽  
Optimal Shape ◽  
Sectional Area ◽  
Cross Sectional

By using Pontryagin’s maximum principle we determine the shape of the lightest rotating rod, stable against buckling. It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. Three variational principles for this boundary value problem are formulated and a first integral is constructed. The optimal shape of a rod is determined by numerical integration.

Optimal Shape of a Rotating Rod With Unsymmetrical Boundary Conditions

2007 ◽  
Vol 74 (6) ◽  
pp. 1234-1238 ◽  
Author(s):  
Teodor M. Atanackovic
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Optimal Shape ◽  
Elastic Rods ◽  
Sectional Area ◽  
Governing Equations ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Multiplicity Of An Eigenvalue

Governing equations of a compressed rotating rod with clamped–elastically clamped (hinged with a torsional spring) boundary conditions is derived. It is shown that the multiplicity of an eigenvalue of this system can be at most equal to two. The optimality conditions, via Pontryagin’s maximum principle, are derived in the case of bimodal optimization. When these conditions are used the problem of determining the optimal cross-sectional area function is reduced to the solution of a nonlinear boundary value problem. The problem treated here generalizes our earlier results presented in Atanackovic, 1997, Stability Theory of Elastic Rods, World Scientific, River Edge, NJ. The optimal shape of a rod is determined by numerical integration for several values of parameters.

scholarly journals MAXIMUM PRINCIPLE AND THE FOURTH ORDER BOUNDARY VALUE PROBLEM

2011 ◽  
Vol 16 (1) ◽  
pp. 143-152
Author(s):  
Inara Yermachenko
Keyword(s):  
Maximum Principle ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problem ◽  
Nonlinear Boundary Value ◽  
Lower And Upper Functions

This paper is devoted to the study the nonlinear boundary value problem . We state a maximum principle related with the operator and apply it to prove the existence and approximation of solutions to the problem (i), (ii), in presence of properly ordered lower and upper functions.

APPLICATION OF PONTRYAGIN'S PRINCIPLE TO BIMODAL OPTIMIZATION OF NANO RODS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250012 ◽  
Author(s):  
TEODOR M. ATANACKOVIC ◽  
BRANISLAVA N. NOVAKOVIC ◽  
ZORA VRCELJ
Keyword(s):  
Maximum Principle ◽  
Characteristic Length ◽  
Optimization Procedure ◽  
Optimal Shape ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Elastic Rod ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Nano Rods

By using the Pontryagin's maximum principle, we determine optimal shape of a nonlocal elastic rod clamped at both ends. In the optimization procedure, we imposed restriction on the minimal value of the cross-sectional area. We showed that the optimization may be both unimodal and bimodal depending on the value of the restrictions and the value of characteristic length. Several concrete examples are treated in detail and the increase in buckling capacity is determined.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

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