Abstract: This work contains a generalization of Kapitsa’s classical problem. The stability of the vertical position of a flexible rod with a lower support point under gravity and vibrations is considered. It has been shown that an unstable position can become stable in the presence of vertical harmonic vibrations of the base. Both rigid and hinge fixing of the lower rod end are considered. In the linear approximation, the problem is reduced to transverse oscillations of the rod under the action of periodic axial compression. The solution is obtained in two formulations—taking into account the propagation of longitudinal waves in the rod and without regard for it. It turns out that longitudinal waves significantly reduce the base vibration level necessary for the stability.