621.787.4

technology, cylindrical surface, quality parameters, vibration processing, torsional vibrations, mathematical model.

The article presents the results of analysis of modern literature sources in search of mathematical models describing the dynamics of the process of forming regular microrelief on the inner cylindrical surfaces of parts operating in difficult conditions, in order to increase their life cycle. The absence of mathematical models describing this process and the peculiarities of its implementation with the point action of the deforming element on the surface of the body part are established. The movements of the tool during the process of forming a regular microrelief on the inner cylindrical surface of the body of the part are considered and the driving forces that follow this process are analyzed. Based on the results of the analysis, a mathematical model of the dynamic process of forming regular microrelief on the inner cylindrical surface of the body of the part was developed. The peculiarity of this process is that microrelief is formed by concentrated force, the point of application of which is constantly changing in the radial and axial directions relative to the part. Therefore, the mathematical model that describes this process will have a discrete right-hand side. It is proposed to model such an action using Dirac delta functions with linear and temporal variables, using the method of regularization of these features, in particular, existing methods of integrating the corresponding nonlinear mathematical models of torsional vibrations of a part. Analytical relations describing these vibrations in the process of forming a regular microrelief are obtained. Using Maple software 3D changes in torsion angle depending on different values of the source data are constructed. The conducted research will allow to consider torsional oscillations that is crucial for long cylindrical details, such as sleeves of hydraulic cylinders, parts of drilling mechanisms and others.

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1. The device and operation of gas turbine plants: a tutorial. Edited by Yu. D. Zemenkov. Tyumen: TyumSOGU, 2015. 434 p. [In Russian].
2. E. A. Petrovsky, Kirill Bashmur, Yu. N. Shadchina, V. V. Bukhtoyarov, V. S. Tynchenko, (2019). Study of microrelief forming technology on sliding bearings for oil and gas centrifugal units. Journal of Physics: Conference Series. 1399. 055032. 10.1088/1742-6596/1399/5/055032.
3. Hassan A. M. The effects of ball and roller burnishing on the surface roughness and hardness of some non-ferrous metals. J. Mater. Process Technol. 1997, 72, 385–391. [CrossRef]. DOI: https://doi.org/10.1016/ S0924-0136(97)00199-4
4. Andrzej Dzierwa, Angelos P. Markopoulos. Influence of ball-burnishing process on surface topography parameters and tribological properties of hardened steel. Machines 2019. 7. 11. DOI: https://doi.org/ 10.3390/machines7010011.
5. Hamdi Amine (2020). Effect of cutting variables on bearing area curve parameters (BAC-P) during hard turning process. Archive of Mechanical Engineering. 67. Р. 73–95. 10.24425/ame.2020.131684.
6. Kubatova D. & Melichar M. Roughness Evaluation Using Abbott-Firestone Curve Parameters, Proceedings of the 30th DAAAM International Symposium. 2019. P. 0467–0475. Published by DAAAM International. ISBN 978-3-902734-22-8, ISSN 1726-9679, Vienna, Austria DOI: 10.2507/30th.daaam. proceedings.063.
7. Sheider Yu. G.. Service properties of parts with regular microrelief, 2nd ed., Revised and augmented, Leningrad: Mashinostroenie, 1982, 247 p. [In Russian].
8. GOST 24773-81 Surfaces with regular microshape. Classification, parameters and characteristics, Moscow: Izd. Stand., 1988, 14 p. [In Russian].
9. Aftanaziv I. S., Kyrychok P. O., Melnychuk, P. P. et al. Improving the reliability of machine parts by surface plastic deformation. Zhytomyr: ZhTI Publishing, 2001. 516 p. [in Ukrainian].
10. Slavov S., Dimitrov D., Iliev I. “Variability of Regular Relief Cells Formed on Complex Functional Surfaces by Simultaneous Five-Axis Ball Burnishing,” UPB Scientific Bulletin, Series D: Mechanical Engineering 82. No. 3. August 2020. P. 195–206.
11. Slavov S. D., Dimitrov D. M. A study for determining the most significant parameters of the ball-burnishing process over some roughness parameters of planar surfaces carried out on CNC milling machine, MATEC Web of Conferences 2018 178, 02005. DOI: https://doi.org/10.1051/matecconf/201817802005.
12. Dzyura V. O. Modeling of partially regular microreliefs formed on the end faces of rotation bodies by a vibration method, UJMEMS. 2020, 6 (1), 30–38. DOI: https://doi.org/10.23939/ujmems2020.01.030.
13. Lacalle Luis. (2012). Ball burnishing application for finishing sculptured surfaces in multi-axis machines. International Journal of Mechatronics and Manufacturing Systems. P. 997–1003.
14. Aftanaziv I. S., Lytvynyak Ya. M., Kusyy Ya. M. Doslidzhennya dynamichnykh kharakterystyk vibratsiyno-vidtsentrovoho zmitsnennya dovho vymirnykh tsylindrychnykh detaley. Visnyk Natsional'noho universytetu “L'vivs'ka politekhnika”. 2004. No. 515: Optymizatsiya vyrobnychykh protsesiv i tekhnichnyy kontrol' u mashynobuduvanni ta pryladobuduvanni. P. 55–64.
15. Tsizh B. R., Sokil B. I., Sokil M. B. Teoretychna mekhanika: pidruchnyk. L'viv: Spolom, 2008. P. 458.
16. Pavlovs'kyy M. A. Teoretychna mekhanika. Kyiv: Tekhnika, 2002. 512 p.
17. Dzyura V. Dynamics of regular microrelief formation on internal cylindric surfaces. Scientific Journal of TNTU. Ternopil: TNTU, 2021. Vol. 101. No. 1. P. 115–128.
18. Markovych B. M. Equations of Mathematical Physics: A Textbook. L'viv: “L'vivs'ka politekhnika” Publishing, 2010. 384 p. [In Ukrainian].
19. Delta-funktsyya. “Matematyka”. URL: https//math world.wolfram.com/ DeltaFunction.html.
20. Cveticanin L. Period of vibration of axially vibrating truly nonlinear rod. Journal of Sound and Vibration. 2016. 374. Р. 199–210. DOI: https://doi.org/10.1016/j.jsv.2016.03.027.
21. Cveticanin L., PoganyT. Oscillator with a sum of non-integer order non-linearities. Journal of Applied Mathematics. 2012. Article ID 649050. 20 p.