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.
 R. Mediratta, K. Ahluwalia, and S. H. Yeo, “State-of-the-art on vibratory finishing in the aviation industry: an industrial and academic perspective,” Int. J. Adv. Manuf. Technol., vol. 85, no. 1–4, pp. 415–429, Jul. 2016. https://doi.org/10.1007/s00170-015-7942-0
 D. Huo, and F. Wardle, “A holistic integrated dynamic design and modelling approach applied to the development of ultraprecision micro-milling machines,” Int. J. Mach. Tools Manuf., vol. 50, no. 4, pp. 335–343, Apr. 2010. https://doi.org/10.1016/j.ijmachtools.2009.10.009
 V. Tsiakoumis, and A. Batako, “Vibration-Assisted Grinding of Mild and Hardened Steel: A Novel Design Vibrating Jig and Process Performance,” in Volume 4: Advanced Manufacturing Processes; Biomedical Engineering; Multiscale Mechanics of Biological Tissues; Sciences, Engineering and Education; Multiphysics; Emerging Technologies for Inspection, 2012, pp. 141–146. https://doi.org/10.1115/ESDA2012-82702
 J. Yuan, W. Yao, P. Zhao, B. Lyu, Z. Chen, and M. Zhong, “Kinematics and trajectory of both-sides cylindrical lapping process in planetary motion type,” Int. J. Mach. Tools Manuf., vol. 92, pp. 60–71, May 2015. https://doi.org/10.1016/j.ijmachtools.2015.02.004
 D. Wen, H. Qi, L. Ma, C. Lu, and G. Li, “Kinematics and trajectory analysis of the fixed abrasive lapping process in machining of interdigitated micro-channels on bipolar plates,” Precis. Eng., vol. 44, pp. 192–202, Apr. 2016. https://doi.org/10.1016/j.precisioneng.2015.12.005
 F. Hashimoto, and D. B. DeBra, “Modelling and Optimization of Vibratory Finishing Process,” CIRP Ann., vol. 45, no. 1, pp. 303–306, Jan. 1996. https://doi.org/10.1016/S0007-8506(07)63068-6
 F. Hashimoto, and S. P. Johnson, “Modeling of vibratory finishing machines,” CIRP Ann., vol. 64, no. 1, pp. 345–348, Jan. 2015. https://doi.org/10.1016/j.cirp.2015.04.004
 Y. B. Tian, Z. W. Zhong, and S .J. Tan, “Kinematic analysis and experimental investigation on vibratory finishing,” Int. J. Adv. Manuf. Technol., vol. 86, no. 9-12, pp. 3113-3121, October 2016. https://doi.org/10.1007/s00170-016-8378-x
 P. P. Kumar, and S. Sathyan, “Simulation of 1D Abrasive Vibratory Finishing Process,” Adv. Mater. Res., vol. 565, pp. 290–295, Sep. 2012. https://doi.org/10.4028/www.scientific.net/AMR.565.290
 O. V. Havrylchenko, V. M. Korendiy, and V. M. Zakharov, “Formuvannia matematychnykh modelei try- i chotyrymasovykh kolyvnykh system vibrovykinchuvalnykh verstativ” [“Formation mathematical model three and four mass oscillatory systems of vibrating finishing machines”], Visnyk Natsionalnoho universytetu "Lvivska politekhnika" [Bulletin of Lviv Polytechnic National University], no. 866, p. 3-12, 2017. [in Ukrainian].
 V. Korendiy, and V. Zakharov, “Substantiation of Parameters and Analysis of Operational Characteristics of Oscillating Systems of Vibratory Finishing Machines”, Ukrainian Journal of Mechanical Engineering and Materials Science, vol. 3, no. 2, pp. 67-78, 2017. https://doi.org/10.23939/ujmems2017.02.067
 I. V. Kuzio, V. M. Zakharov, and V. M. Korendiy, “Modelling the process of dressing the laps of vibratory finishing machine”, Avtomatizacìâ virobničih procesìv u mašinobuduvannì ta priladobuduvannì [Industrial Process Automation in Engineering and Instrumentation], no. 52, p. 81-93, 2018.
 O. Lanets, Osnovy rozrakhunku ta konstruiuvannia vibratsiinykh mashyn [Fundamentals of Analysis and Design of Vibratory Machines], Lviv, Ukraine: Lviv Polytechnic Publishing House, 2018. [in Ukrainian].
 V. M. Gursky, Bahatokryterialnyi analiz i syntez neliniinykh rezonansnykh vibratsiinykh mashyn [Multi-Criteria Analysis and Synthesis of the Nonlinear Resonant Vibratory Machines], Lviv, Ukraine: Lviv Polytechnic Publishing House, 2017. [in Ukrainian].