Vibrations in engineering and technology

Kupchuk Ihor – Candidate of Technical Sciences, Associate professor, Associate Professor of the Department of Engineering Mechanics and Technological Processes in the Agricultural Industry, Faculty of Engineering and Technology, Vinnytsia National Agrarian University (3, Sonychna St., Vinnytsia, 21008, Ukraine, e-mail: kupchuk.igor@i.ua; http://orcid.org/0000-0002-2973-6914).

The article proposes a systemic descriptor-based approach to describing the mixing processes of multicomponent feed materials, within which the process is interpreted as the evolution of a state in a multidimensional space of physically interpretable coordinates. The relevance of the study is determined by the complex rheological nature of feed mixtures, which combine frictional, cohesive, fibrous, and viscoplastic components characterized by spatial and temporal heterogeneity of stress–strain states. The traditional mechanistic description through individual mixing mechanisms does not provide a coherent coordinate-based language for analyzing process regimes and transitions between them.

The aim of the research is to establish a generalized state space of the mixing process and to structure the interrelationships between descriptors as coordinates of this space. The methodological foundation of the research includes conceptual modeling, structural-logical analysis, and physical interpretation of constraints arising from the mechanics of granular and viscoplastic media. The process state is represented as a vector of descriptors D₁-D₉, grouped into four functional blocks: material–microstructural, deformation–energetic, flow–topological, and evolutionary–stabilization.

It is shown that mixing regimes should be interpreted as subregions of an admissible state space formed with consideration of physical, rheological, and structural–topological constraints. Three types of structural relationships between descriptors are identified: physical incompatibility, conditional dominance, and hierarchical subordination. The combined effect of these relationships shapes the internal geometry of the state space, determines the stratification of regime domains, and defines the admissibility of coordinate combinations. It is substantiated that the “narrowing” of the space of possible states is a consequence of the physical properties of the material medium rather than a formal modeling assumption.

The proposed approach enables a transition from an empirically oriented description of the mixing process to its systemic interpretation as a structured set of physically admissible states. This provides a methodological basis for further quantitative parameterization of process evolution, the development of regime stability criteria, and the optimization of operating modes of mixing apparatuses of various design types.

1. Paul, E. L., Atiemo-Obeng, V. A., & Kresta, S. M. (2004). Handbook of industrial mixing: Science and practice. John Wiley & Sons. https://doi.org/10.1002/0471451452

2. Ottino, J. M., & Khakhar, D. V. (2000). Mixing and segregation of granular materials. Annual Review of Fluid Mechanics, 32, 55–91. https://doi.org/10.1146/annurev.fluid.32.1.55

3. Khakhar, D. V., McCarthy, J. J., & Ottino, J. M. (1999). Mixing and segregation of granular materials in chute flows. Chaos, 9(3), 594–610. https://doi.org/10.1063/1.166433

4. Marigo, M., & Stitt, E. (2015). Discrete Element Method (DEM) for industrial applications. KONA Powder and Particle Journal, 32, 236–252. https://doi.org/10.14356/kona.2015016

5. Bonn, D., Denn, M. M., Berthier, L., Divoux, T., & Manneville, S. (2017). Yield stress materials in soft condensed matter. Reviews of Modern Physics, 89(3), 035005. https://doi.org/10.1103/RevModPhys.89.035005

6. Kupchuk, I. (2025). Physical principles and classification of feed material mixing processes. Техніка, енергетика, транспорт АПК, (4(131)), 55–68. https://doi.org/10.37128/2520-6168-2025-4-6

7. Ottino, J. M. (1989). The kinematics of mixing: Stretching, chaos, and transport. Cambridge University Press.

8. Haken, H. (2004). Synergetics: Introduction and advanced topics. Springer. https://doi.org/10.1007/978-3-662-10184-1

9. Bertalanffy, L. (1973). General system theory: Foundations, development, applications. George Braziller.

10. Jaeger, H. M., Nagel, S. R., & Behringer, R. P. (1996). Granular solids, liquids, and gases. Reviews of Modern Physics, 68, 1259–1273. https://doi.org/10.1103/RevModPhys.68.1259

11. Forterre, Y., & Pouliquen, O. (2008). Flows of dense granular media. Annual Review of Fluid Mechanics, 40, 1–24. https://doi.org/10.1146/annurev.fluid.40.111406.102142

12. Schulze, D. (2021). Powders and bulk solids: Behavior, characterization, storage and flow. Springer. https://doi.org/10.1007/978-3-030-76720-4

13. Divoux, T., Fardin, M. A., Manneville, S., & Lerouge, S. (2016). Shear banding in complex fluids. Annual Review of Fluid Mechanics, 48, 81–103. https://doi.org/10.1146/annurev-fluid-122414-034416

14. Gray, J. M. N. T., & Thornton, A. R. (2005). A theory for particle size segregation in shallow granular free-surface flows. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461, 1447–1473. https://doi.org/10.1098/rspa.2004.1420

15. Chhabra, R. P., & Richardson, J. F. (2008). Non-Newtonian flow and applied rheology (2nd ed.). Butterworth-Heinemann. https://doi.org/10.1016/B978-0-7506-8532-0.X0001-7

16. Polievoda, Yu. A., Volynets, Ye. O., & Bystrytskyi, O. P. (2022). Matematychna model rukhu chastynok v tsylindrychnomu konteineri vibrozmishuvacha [Mathematical model of particle motion in a cylindrical container of a vibratory mixer]. Vibrations in Engineering and Technology, (4(107)), 20–25. https://doi.org/10.37128/2306-8744-2022-4-3

17. Stadnik, M., Solona, O., & Burlaka, S. (2025). Doslidzhennia vplyvu heometrii robochykh orhaniv zmishuvacha na kinetiku zmishuvannia sypkykh seredovyshch [Investigation of the influence of working body geometry on the mixing kinetics of granular media]. Herald of Khmelnytskyi National University. Technical Sciences, (2(349)), 23–27. https://doi.org/10.31891/2307-5732-2025-349-2

18. Stadnik, M., Babyn, I., Luts, P., & Ripa, S. (2024). Analiz teoretychnykh doslidzhen prystroiv dozuvannia ta zmishuvannia zhyvylnykh rozchyniv [Analysis of theoretical studies of dosing and mixing devices for nutrient solutions]. Vibrations in Engineering and Technology, (3(114)), 90–96. https://doi.org/10.37128/2306-8744-2024-3-11