Bimodal optimization of a compressed rotating rod

Abstract

By using Pontryagin’s maximum principle we determine the shape of the lightest compressed rotating rod, stable against buckling. It is assumed that boundary conditions correspond to clamped ends. This boundary condition leads to a bimodal optimization problem. The necessary conditions of optimality, given by Eq. (17), are derived. In the special case when the rod is not rotating, this optimality conditions reduce to the previously obtained condition for the compressed rod only. It is shown that the cross-sectional area as a function of arc-length is determined from the solution of a nonlinear boundary value problem. A first integral for the system of equations is constructed. The optimal cross-sectional area and the post-buckling shape are determined by numerical integration. The stability of the bifurcation branches is investigated by use of the energy method.