Conservative mechanical systems are considered. Each system is supposed to admit a family of non-degenerate symmetric periodic motions. For the entire set of systems, they form a family of conditionally periodic oscillations, which is assumed to contain a symmetrical periodic motion. Universal couplings for mechanical systems are proposed such that they play the role of controls that ensure the existence, stability, and stabilization of oscillations at the same time for the whole system. Those couplings provide the natural, i.e. without involving any additional controls, stabilization of the oscillation. In addition, mechanical systems perform the synchronization in frequency and phase.