Shevchenko H., Tytov O., Samodryha O. Determination of energy absorption level in the process of contact interaction of grinding bodies
- Details
- Parent Category: Geo-Technical Mechanics, 2024
- Category: Geo-Technical Mechanics, 2024, Issue 171
Geoteh. meh. 2024, 171, 167-177
https://doi.org/10.15407/geotm2024.171.167
DETERMINATION OF ENERGY ABSORPTION LEVEL IN THE PROCESS OF CONTACT INTERACTION OF GRINDING BODIES
Shevchenko H. 
Tytov O.
Samodryha O.
M.S. Poliakov Institute of Geotechnical Mechanics of the National Academy of Sciences of Ukraine
UDC 621.926: 622.7
Language: English
Abstract. This paper addresses the problem of quantitatively describing energy absorption during the contact interaction of grinding bodies in mills of various types. The aim of the study is to develop an analytical framework for describing the motion of grinding bodies, taking into account their dissipative properties, which are defined through dimensionless parameters, particularly the coefficient of restitution. A new approach is proposed that does not rely on the traditional Rayleigh energy dissipation formula or the classical equation of damped harmonic oscillations, but instead employs nonlinear differential equations of motion with dissipative forces proportional to the gradient of the potential energy of elastic deformation. Numerical methods, specifically the fourth-order Runge–Kutta method, are used to solve these equations. The study investigates interaction models for the most typical scenarios: collisions between two balls, a ball and a massive plate, a rod and a plate, etc. Equations of motion are derived for each of these scenarios, accounting for body geometry, type of contact, and loss coefficient. For ball collisions, the elastic contact stiffness considers the nonlinear deformation behavior described by Hertzian theory. For rod impacts, a simplified one-sided elastic connection model is applied. It is shown that as the energy loss coefficient increases, the restitution coefficient decreases, which corresponds to the physical nature of the process. The results of numerical modeling are illustrated by displacement and velocity graphs, from which the relationships between the restitution coefficient and the energy loss coefficient are also derived. It is demonstrated that in many cases this relationship approaches linearity for small loss coefficients but becomes significantly nonlinear at higher values. The obtained results have important practical significance, as they enable more accurate estimation of energy losses in mills, improve dynamic modeling of mill structures, and justify parameters for effective vibro-impact operating modes. It is noted that the majority of energy is absorbed during the contact interactions between grinding bodies themselves, even in the absence of processed material in the contact zone, which partly explains the low efficiency of milling equipment. Therefore, the findings of this study can be effectively used both in designing energy-efficient grinding equipment and in optimizing technological processes within “smart manufacturing” systems, where precise control of energy consumption and particle dynamics is essential.
Keywords: mill, grinding body, energy absorption, restitution coefficient, nonlinear differential equation, numerical methods.
REFERENCES
1. Babii, K., Chetveryk, M., Perehudov, V., Kovalov, K., Kiriia, R. and Pshenychnyi, V. (2022), “Features of using equipment for in-pit crushing and conveying technology on the open pit walls with complex structure”, Mining of Mineral Deposits, vol. 16(4), 96–102. https://doi.org/10.33271/mining16.04.096
2. Babii, K., Ikol, O. and Malieiev, Y. (2019), “Substantiating a technique to process hard rock involving extraction of valuable components and use of wastes to form mesorelief”, E3S Web of Conferences “International Conference Essays of Mining Science and Practice”, vol. 109, 00003–00003. https://doi.org/10.1051/e3sconf/201910900003
3. Biletskii, V.S., Oliinyk, T.A., Smyrnov, V.O. and Skliar, L.V. (2019), Tekhnikatatekhnologiiazbahachenniakorysnykhkopalyn. Chastyna I. Pidhotovchi protsesy [Equipment and technology of mineral dressing. Part I. Preparatory processes], FOP Cherniavskiy D.O., Kryvyi Rig, Ukraine.
4. Bozhyk, D.P., Sokur, M.I, Biletskii, V.S., Yehurnov, O.I., Vorobyov, O.M. and Smyrnov, V.O. (2017), Pidhotovka korysnykh kopalyn do zbahachennia: Monohrafiia [Preparation of minerals for dressing: Monograph], PP Shcherbatykh O.V., Kremenchuk, Ukraine.
5. Shevchenko, H. and Shevchenko, V. (2024), “Analytical researches of a vibrating mill with vibroimpact excitation of a crushing chamber”, IOP Conf. Series: Earth and Environmental Science, V International Conference "Essays of mining science and practice", Online, vol. 1348,012050, pp. 1–10. https://doi.org/10.1088/1755-1315/1348/1/012050
6. Shevchenko H., Tytov O., Haddad J., Samodryha O. and Krasnokutskyi, O. (2024), “Determination of stiffness of three-dimensional composition of elastic balls under condition of uniaxial compression”, Geotechnical Mechanics, vol. 169, pp. 160-170. https://doi.org/10.15407/geotm2024.169.160
7. Timoshenko, S. and Goodier, J. N. (2020),Theory of elasticity, 3rd ed., McGraw-Hill, New York, USA.
8. Johnson, K. L. (2021), Contact mechanics, 2nd ed., Cambridge University Press, Cambridge, UK.
9. Brach, R. M. (2006), Mechanical Impact Dynamics: Rigid Body Collisions, 2nd ed., Wiley-Interscience, New York, USA.
10. Kovacic, I. and Brennan, M. J. (2011), The Duffing Equation: Nonlinear Oscillators and their Behaviour, Wiley, Hoboken, New Jersey, USA. https://doi.org/10.1002/9780470977859
11. Zhang, J., Fan, L.-S., Zhu, C., Pfeffer, R. and Qi, D. W. (1999), “Dynamic behavior of collision of elastic spheres in viscous fluids”, Powder Technology, vol. 106(1–2), pp. 98–109. https://doi.org/10.1016/S0032-5910(99)00053-4
12. Zhang, Y., Li, M., Huang, X., Wang, J. and Liu, C. (2023), "Experimental investigation of energy dissipation in steel ball collisions: Effects of impact velocity and material properties", Journal of Sound and Vibration, vol. 543, 117382.
13. Liu, H., Zhou, G., Zhang, Q. and Wang, P.(2021), "Experimental study on the coefficient of restitution for elastic-plastic collisions", International Journal of Impact Engineering, vol. 156, 103951. https://doi.org/10.1016/j.ijimpeng.2021.103951
14. Wang, L. and Chen, X. (2022). "Energy absorption characteristics of granular materials under dynamic impact", Powder Technology, vol. 398, 117102.
15. Meirovitch, L. (1986), Elements of Vibration Analysis, 2nd ed., McGraw-Hill, New York, USA.
16. Puglisi, G. and Truskinovsky, L. (2022), "Nonlinear dissipation in Hertzian contacts: Theory and experiments", Journal of the Mechanics and Physics of Solids, vol. 159, 104735.
17. Antonyuk, S., Müller, M., Börner, T., Wittmann, J. and Schubert, R. (2020), "Modeling of energy dissipation in particle collisions using nonlinear damping models", Chemical Engineering Science, vol. 227, 115925. https://doi.org/10.1016/j.ces.2020.115925
18. Brilliantov, N. V. and Pöschel, T. (2023), "Nonlinear viscoelastic contacts and their role in energy dissipation", Granular Matter, vol. 25(1), 12.
19. Maugis, D. (2021), Contact, adhesion, and rupture of elastic solids, 2nd. ed., Springer, Berlin, Germany.
20. Stronge, W. J. (2021), Impact mechanics with applications to energy dissipation, 2nd. ed., Cambridge University Press, Cambridge, UK.
21. Bilbao, S., Martínez, J., García, P. and Fernández, L. (2022), "Numerical simulation of impact dynamics using high-order Runge-Kutta methods", Journal of Computational Physics, vol. 434, 110216.
About the authors:
Shevchenko Heorhii, Doctor of Technical Sciences (D.Sc.), Head of Department of Mechanics of Mineral Processing Machines and Processes, M.S. Poliakov Institute of Geotechnical Mechanics of the National Academy of Sciences of Ukraine (IGTM of the NAS of Ukraine), Dnipro, Ukraine, This email address is being protected from spambots. You need JavaScript enabled to view it. , ORCID0000-0002-8047-7014
Tytov Oleksandr, Candidate of Technical Sciences (Ph.D), Researcher in Department of Mechanics of Mineral Processing Machines and Processes, M.S. Poliakov Institute for Geotechnical Mechanics of the National Academy of Sciences of Ukraine (IGTM of the NAS of Ukraine), Dnipro, Ukraine, This email address is being protected from spambots. You need JavaScript enabled to view it. (Corresponding author) ORCID0000-0002-5562-7543
Samodryha Oleh, Postgraduate Student, M.S. Poliakov Institute of Geotechnical Mechanics of the National Academy of Sciences of Ukraine (IGTM of the NAS of Ukraine), Dnipro, Ukraine, This email address is being protected from spambots. You need JavaScript enabled to view it. , ORCID0009-0007-6095-0031