Vibration and buckling of rotating flexible rods at transitional parameter values

Summary We consider the connection between vibration and buckling problems for a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The rod is assumed off-clamped, i.e. the axis of rotation does not pass through the rod's clamped end. The resulting fourth-order boundary value problem with a turning point for the free vibrations is solved using uniform approximations in a transitional parameter range where high rotation rates balance small off-clampings. Second approximations to the vibration eigenvalues are used to determine critical buckling rotation rates for the slightly off-clamped rods. Buckling is unexpected in this situation as the rod is wholly under tension.