*Problem statement.* The development of energy-efficient and high-performance vibratory lapping machines demands the improvement of their design diagrams and calculation techniques. *Purpose.* The main objectives of this research consist in detailed analysis of existent design diagrams and mathematical models of vibratory lapping machines; designing the three-mass hanger-type structures of such machines providing circular oscillations of laps; derivation of differential equations describing the motion of their oscillatory systems. *Methodology.* The technique of the research is based on fundamental concepts of engineering mechanics, strength of materials and theory of mechanical vibrations. *Findings (results)*. The improved design diagrams of vibratory lapping machines with circular oscillations of laps are proposed and the corresponding differential equations describing the motion of the working elements are derived. *Originality (novelty)*. The mathematical model of circular oscillations of the three-mass mechanical oscillatory system of vibratory lapping machine is developed. The possibilities of performing the double-side lapping of cylindrical and prismatic parts are considered. *Practical value.* The results of the performed investigations can be used during creating new designs and improving existent structures of vibratory finishing machines for lapping flat surfaces of cylindrical and prismatic parts. *Scopes of further investigations*. In further investigations, it is necessary to perform the numerical modelling of the system’s motion using the derived differential equations, and to compare the obtained theoretical results with the results of experimental investigations.

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