This paper develops a variational formulation for determining the optimal shape of a rotating rod that maximizes stability against buckling. The problem is expressed as a fifth-order nonlinear boundary value problem, solved through a specialized numerical–analytical integration method. The study establishes a linear relation between rod volume and squared critical rotating speed and shows that the critical speed can be determined from the lowest eigenvalue, validating the optimization approach.
Why it matters: Structural stability under rotation is a key consideration in engineering design. This work provides both theoretical and computational insights into optimizing rotating rods, contributing to the broader understanding of stability in nonlinear mechanical systems.