The problem of quenching small vibrations of a mechanical system near its equilibrium position is considered. A method for finding switching points of piecewise constant control for simultaneous damping of several vibrations of the same frequency is proposed. As an optimized functional "the Expenditure"is considered. The method leads to explicit formulas. All results of research are formulated as the theorem. As an example the problem of stabilizing the spacecraft's rotation axis is considered. The mathematical model of this problem coincides with the problem of a solid body rotational motion in a uniform gravity field (the "sleeping top"problem). Presented results can be applied not only in mechanical controlled systems, but also in any problem that can be described by the system of ordinary differential equations with control.