# Науковий вісник НГУ

## The tumbling mill rotation stability

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Authors:

K.Yu. Deineka,CandidateofTechnicalSciences,Technical College of the National University of Water and Environmental Engineering,Lecturer of the Highest Category, Rivne, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it., orcid.org/0000-0001-7376-6734

Yu.V. Naumenko, Doctor of TechnicalSciences, Associate Professor, NationalUniversityofWaterandEnvironmentalEngineering, Professor of the Department of Construction, Road, Reclamation, Agricultural Machines and Equipment, Rivne, Ukraine, orcid.org/0000-0003-3658-3087

Abstract:

Purpose. Creation of a mathematical model of conditions and factors of stability of the established motion of a machine unit whose working machine is a drum, permanently rotating around the horizontal axis, with a fluid fill of the chamber.

Methodology. A filled drum is considered to be a system with permanent composition with variable inertial parameters whose variability is due to the redistribution at relative motion of masses of fluid fill of the chamber on the body of the drum. Based on the application of the principle of hardening of a mechanical system, the drum rotation equations take into account the whole mass of the fill, regardless of the nature of its interaction with the surface of the chamber. We employed methods of the mechanics of relative motion for the mathematical description of the undisturbed motion of the system. The principle of establishing a hierarchy of variables was applied. A hypothesis is accepted about proximity of the system's motion to the rotation with a slowly changing angular velocity. It is believed that dynamic and inertial parameters of the filled drum are determined by the quasi-static dependences on the rotation velocity. We used a second-order Lagrange equation for a system with variable inertial parameters in order to determine the transitional motion. A direct Lyapunov method is applied to search for conditions of the motion stability.

Findings. Dynamic and inertial parameters of the filled drum are formalized using differential equations of undisturbed steady and transitional rotation. We obtained a condition of asymptotic stability of the established motion of a tumbling mill machine unit. Instability factors of the system's motion are determined.

Originality. Regularities of steady rotation of a drum mill are identified. It was established that factors of instability in the established, rotational around the horizontal axis, motion of the drum filled with a fluid medium, are the variations of variable inertial parameters – the axial moment of inertia and the moment of resistance of the fill to rotation. It is shown that the failure to comply with conditions of the motion stability could be caused by achieving the extreme negative values of derivatives from the inertial parameters of the in-chamber fill by the angular velocity of drum rotation.

Practical value. The developed mathematical model allows us to qualitatively determine conditions of steady rotational motion of the drum with a fluid fill of the chamber. Conditions for the occurrence of unstable motion have considerable practical significance because they cause self-excitation of auto-oscillations of the filled drum and determine improvement of effectiveness of grinding process in tumbling mills with traditional design solutions.

References.

1. Vynohradov, B. V., 2016. Statics and dynamics of drum mill drive. Dnipropetrovsk: DVNZ UDKhTU.

2. Dube, O., Alizadeh, E., Chaouki, J. and Bertrand, F., 2013. Dynamics of non-spherical particles in a rotating drum. Chemical Engineering Science [e-journal], 101, pp. 486–502. DOI: 10.1016/j.ces.2013.07.011.

3. Ching-Fang Lee, Hsien-Ter Chou and Capart, H., 2013. Granular segregation in narrow rotational drums with different wall roughness: Symmetrical and asymmetrical patterns. Powder Technology [e-journal], 233, pp. 103–115. DOI: 10.1016/j.powtec.2012.08.034.

4. East, R. D. P., McGuinness, P., Box, F. and Mullin, T., 2014. Granular segregation in a thin drum rotating with periodic modulation. Physical Review E [e-journal], 90(5). DOI: 10.1103/PhysRevE.90.052205.

5. Finger, T., Schroter, M. and Stannarius, R., 2015. The mechanism of long-term coarsening of granular mixtures in rotating drums. New Journal of Physics, 17. DOI: 10.1088/1367-2630/17/9/093023.

6. Hsiu-Po Kuo, Wei-Ting Tseng and An-Ni Huang, 2016. Controlling of  segregation in rotating drums by independent end wall rotations. KONA Powder and Particle Journal [e-journal], 33, pp. 239–248. DOI: 10.14356/kona.2016004.

7. Zimber, F., Kollmer, J. E. and Poschel, T., 2013. Polydirectional stability of granular matter. Physical Review Letters [e-journal], 111(16). DOI: 10.1103/PhysRevLett.111.168003.

8. Liu, P. Y., Yang, R. Y. and Yu, A. B., 2013. Particle scale investigation of flow and mixing of wet particles in rotating drums. AIP Conference Proceedings [e-journal], 1542(1), pp. 963–966. DOI: 10.1063/1.4812093.

9. Yang, H., Li, R., Kong, P., Sun, Q. C., Biggs, M. J. and Zivkovic, V., 2015. Avalanche dynamics of granular materials under the slumping regime in a rotating drum as revealed by speckle visibility spectroscopy. Physical Review E [e-journal], 91(4). DOI: 10.1103/PhysRevE.91.042206.

10. Yang, H., Jiang, G. L., Saw, H. Y., Davies, C., Biggs, M. J. and Zivkovic, V., 2016. Granular dynamics of cohesive powders in a rotating drum as revealed by speckle visibility spectroscopy and synchronous measurement of forces due to avalanching.Chemical Engineering Science [e-journal], 146, pp. 1–9. DOI: 10.1016/j.ces.2016.02.023.

11. Yang, H., Zhang, B. F., Li, R., Zheng, G. and Zivkovic, V., 2017. Particle dynamics in avalanche flow of irregular sand particles in the slumping regime of a rotating drum.Powder Technology [e-journal], 311, pp. 439–448. DOI: 10.1016/j.powtec.2017.01.064.

12. Wang, Z. and Zhang, J., 2015. Fluctuations of particle motion in granular avalanches – from the microscopic to the macroscopic scales. Soft Matter [e-journal], 11(27), pp. 5408–5416. DOI: 10.1039/c5sm00643k.

13. Wang, Z. and Zhang, J., 2015. Spatiotemporal chaotic unjamming and jamming in granular avalanches. Scientific Reports [e-journal], 5. DOI: 10.1038/srep 08128.

14. Maghsoodi, H. and Luijten, E., 2016. Chaotic dynamics in a slowly rotating drum. Fevista Cubana de Fisica, 33(1), pp. 50–54.

15. Sack, A. and Poschel, T., 2016. Dissipation of energy by dry granular matter in a rotating cylinder. Scientific Reports [e-journal], 6. DOI: 10.1038/srep26833.

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ISSN (print) 2071-2227,
ISSN (online) 2223-2362.
Journal was registered by Ministry of Justice of Ukraine.
Registration number КВ No.17742-6592PR dated April 27, 2011.

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