Using the example of Mlodzievsky's pendulum os-cillations of a heavy rigid body with one fixed point, we show how nonlinear dissipation is introduced in the vicinity of the oscillation of a mechanical system in the case when the motion belongs to a two-dimensional integral manifold. In this case, a new characteristic function of the oscillation family is used, an explicit form of dissipation is found and the problem of stabilizing the oscillation of the “adjusted” body is solved.