We study mechanical systems each of which admits a family of periodic motions
when the systems are not coupled. It is proved that a necessary condition for the existence of
a cycle in a coupled system is the nondegeneracy of periodic motions in all possibly but one
subsystem. The structure and specific type of the coupling control are found. The problems of
existence, stability, and natural stabilization of oscillations are solved. It is shown that the cycle
synchronizes the oscillations of mechanical systems in frequency and phase. The paper develops
the idea of stabilizing the oscillations of a coupled system by selecting a suitable coupling control
between subsystems.