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 λ and α 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 λc 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 α ≠ 0 provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.