Certain　classes　of　essentially　nonlinear　nonstationary　systems　are　studied.　With　the　aid　of　the　Lyapunov　direct　method　and　the　averaging　technique,　conditions　are　obtained　under　which　the　zero　solutions　of　the　considered　systems　are　stable　with　respect　to　all　variables　and　asymptotically　stable　with　respect　to　a　part　of　variables.　It　is　shown　that　the　proposed　approach　can　be　used　for　the　stability　analysis　of　equilibrium　positions　of　nonlinear　mechanical　systems.　Some　examples　are　presented　to　demonstrate　the　effectiveness　of　our　results.