Summary: Chelomei's pendulum has a sliding disc on its rod, and is mounted on a support that vibrates vertically with small amplitude and high frequency. In 1982, V. N. Chelomei demonstrated experimentally that a configuration can be stable where the pendulum points against gravity, and the disc 'floats' on the rod. This phenomenon has never been satisfactorily explained. The present work considers why and where the disc floats. It suggests that the phenomenon is caused by resonant flexural rod vibrations, which are excited through small symmetry-breaking imperfections, such as a small deviation from perfectly vertical excitation. This hypothesis is supported by laboratory experiments, and by perturbation analysis and numerical analysis of a new mathematical three-degree-of-freedom model of the system. Simple analytical expressions for the prediction of stable states of the system are set up, providing frequency responses that agree closely with numerical simulation, and agree qualitatively with experimental observations.