Existence and Uniqueness of Equilibrium States of a Rotating Elastic Rod

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Abstract

A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two-point boundary value problem which depends on two parameters A and a representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α = 0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the “trivial” solution is lost. The computations of Ac required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α ≠ 0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each λc provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.

Original languageEnglish
Pages (from-to)315-322
Number of pages8
JournalInternational Journal of Mathematics and Mathematical Sciences
Volume17
Issue number2
DOIs
Publication statusPublished - 1994
Externally publishedYes

Fingerprint

Elastic Rods
Equilibrium State
Rotating
Existence and Uniqueness
Existence and Uniqueness of Solutions
Critical value
Bifurcation
Nonlinear Boundary Value Problems
Two-point Boundary Value Problem
Unique Solution
Rigidity
Fourth Order
Two Parameters
Trivial
Uniqueness
Angle

Keywords

  • Existence and uniqueness of equilibrium states
  • fourth order two-point nonlinear boundary value problems
  • nonlinear eigenvalue problems
  • perturbation solution
  • perturbed bifurcation problems
  • rotating rods
  • Schauder fixed point theorem

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

Cite this

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title = "Existence and Uniqueness of Equilibrium States of a Rotating Elastic Rod",
abstract = "A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two-point boundary value problem which depends on two parameters A and a representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α = 0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the “trivial” solution is lost. The computations of Ac required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α ≠ 0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each λc provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.",
keywords = "Existence and uniqueness of equilibrium states, fourth order two-point nonlinear boundary value problems, nonlinear eigenvalue problems, perturbation solution, perturbed bifurcation problems, rotating rods, Schauder fixed point theorem",
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PY - 1994

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N2 - A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two-point boundary value problem which depends on two parameters A and a representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α = 0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the “trivial” solution is lost. The computations of Ac required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α ≠ 0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each λc provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.

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