Problem statement. Designing and manufacturing of efficient resonant vibratory lapping machines with linear oscillations of laps demand an accurate and detailed calculation of parameters of their elastic systems and electromagnetic drives. Purpose. The main objective of this research consists in derivation of analytical dependencies for calculating the stiffness and excitation parameters of mechanical oscillatory system of vibratory finishing machine in order to ensure its resonance operation mode. 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 design diagram of mechanical oscillatory system of vibratory finishing machine with linear oscillations of laps is considered and corresponding equations of motion are presented. Analytical dependencies for calculating stiffness and excitation parameters of the system are deduced. The example of parameters calculation is given and time dependencies of the system's motion are constructed. Originality (novelty). The mathematical model of linear oscillations of the three-mass mechanical system of vibratory finishing machine was developed. The possibilities of performing the laps dressing using "lap over lap" method were substantiated. Practical value. The results of the performed investigations can be used during designing new designs and improving existing structures of vibratory finishing machines for lapping flat surfaces of cylindric and prismatic parts. Scopes of further investigations. In further investigations, it is necessary to analyse the influence of the viscous damping on the system's motion. In order to substantiate (justify) the obtained theoretical results, the experimental investigations should be carried out.

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