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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 m...