### 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 |

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### 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)