### 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 λ_{c} beyond which the uniqueness of the “trivial” solution is lost. The computations of A_{c} 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 λ_{c} such that the equilibrium equation has a unique solution for each λ_{c} provided λ < λ_{c}. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.

Original language | English |
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Pages (from-to) | 315-322 |

Number of pages | 8 |

Journal | International Journal of Mathematics and Mathematical Sciences |

Volume | 17 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1994 |

Externally published | Yes |

### Fingerprint

### 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

**Existence and Uniqueness of Equilibrium States of a Rotating Elastic Rod.** / Elgindi, Mohamed.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Existence and Uniqueness of Equilibrium States of a Rotating Elastic Rod

AU - Elgindi, Mohamed

PY - 1994

Y1 - 1994

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.

AB - 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.

KW - Existence and uniqueness of equilibrium states

KW - fourth order two-point nonlinear boundary value problems

KW - nonlinear eigenvalue problems

KW - perturbation solution

KW - perturbed bifurcation problems

KW - rotating rods

KW - Schauder fixed point theorem

UR - http://www.scopus.com/inward/record.url?scp=84958301787&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958301787&partnerID=8YFLogxK

U2 - 10.1155/S0161171294000451

DO - 10.1155/S0161171294000451

M3 - Article

AN - SCOPUS:84958301787

VL - 17

SP - 315

EP - 322

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

SN - 0161-1712

IS - 2

ER -