TECHNICAL MECHANICS
ISSN (Print): 1561-9184, ISSN (Online): 2616-6380

HOME    ABOUT THIS JOURNAL    EDITORIAL BOARD    GUIDE FOR AUTHORS    CURRENT    JOURNAL ISSUES    CONTACTS

UDC 531.395

Technical mechanics, 2014, 2, 73 - 78

ANALYTICAL SOLUTION OF PROBLEM ON FALL OF SPHERE THE RADIUS OF WHICH DECREASES FROM LINEAR-FRACTIONAL LAW

Olshansky V. P., Olshansky S. V.

      The paper deals with building a mathematical model of motion of the sphere with variable radius and mass. The analytical method for solving the Cauchy problem for a nonlinear equation of motion with variable coeffi-cients is the research method. For the first time a closed analytic solution of a nonlinear differential equation of a vertical fall of a spherical variable-mass body is built in cylindrical functions when its radius is reduced fraction-ally and linearly in time and quadratic resistance of the air environment. The asymptotic behavior of solutions is investigated.

variable-mass sphere, aerodynamic resistance, Riccati equation, vertical displacement of Bessel function

1. Meshchersky I. V. Works on Mechanics of Variable-Mass Bodies (in Russian) / I. V. Meshchersky. – Mos-cow: GITTL, 1952. – 276 p.

2. Tsiolkovsky K. E. Collected Works. Vol. 1, 2 (in Russian) / K. E. Tsiolkovsky. – Moscow: Publishing House of AS USSR, 1954. – 453 p.

3. Timoshenko B. I. Gas Dynamics of High-Temperature of Technological Processes (in Russian) / V. I. Ti-moshenko. – Dniepropetrovsk: Institute of Technical Mechanics, NASU&NSAU, 2003. – 460 p.

4. Zhukovsky N. Ye. Collected Works. Vol. 3 (in Russian) / N. Ye. Zhukovsky. – Moscow: ONTI – NKTP, 1936. – 380 p.

5. Olshansky S. V. Effect of hovering a fine variable-mass particle in air medium (in Russian) / S. V. Olshansky // Vestnik NTU “KhPI”. Issue: Dinamika i Prochnost Mashin. – is. 30. – Kharkov: NTU ‘KhPI”. – 2009. – P. 125 – 129.

6. Kucherenko S. I. Ballistics of Drops Evaporated in the Flight (in Ukrainian) / S. I. Kucherenko, V. P. Ol-shansky, S. V. Olshansky, L. M. Tishchenko. – Kharkiv; KhNTUSG, 2007. – 304 p.

7. Olshansky V. P. Conditions of extremal fall velocity of spherical variable-mass body (in Russian) / V. P. Olshansky, S. V. Olshansky // Vestnik NTU “KhPI”. Issue: Sistemnyi Analiz, Upravlenie i Informatsionnye Tekhnologii. Is. 26. – Kharkov: NTU “KhPI”. – 2008. – P. 67 – 78.

8. Olshansky V. P. Maximal fall velocity of spherical decreasing-mass body 9In Russian) / V. P. Olshansky, S. V. Olshansky // Mekhanika i Mashinostroyenie. – 2007. – No 1. – p. 25 – 29.

9. Olshansky V. P. Analytical Solutions of Meshchersky equation for vertical motion of decreasing-mass sphere (in Russian0 / V. P. Olshansky, S. V. Olshansky // Tekhnichskaya Mekhanika. – 2009. – No 4. – P. 36 – 42.

10. Olshansky V. P. Closed solutions of Meshchersky equation for various laws of decreasing radius of flying sphere (in Russian) / V. P. Olshansky, K. V. Avramov, S. V. Olshansky // Mekhanika Tverdogo Tela. – 2—9. – Is. 39. – P. 207 – 214.

11. Sagitov M. N. Some cases of motion of rotating variable-mass sphere with horizontal axis (in Russian) / M. G. Sagitov // Izvestiya AN Khazakhskoy SSR. Seriya Fiziko-Matematicheskikh Nauk. Matematika i Mek-hanika. – 1963. – Is. 15. – P. 88 – 99.

12. Kamke E. Reference Book on Conventional Differential Equations (in Russian) / E. Kamke. – Moscow: Nau-ka, 1976. – 576 p.

13. Abramovits A. Reference Book on Special Functions (Formula, Plots and Mathematical Tables) (in Russian) / A. Abramovits, I. Stigan. – Moscow: Nauka, 1979. – 832 p.

14. Yanke E. Special Functions (in Russian) / E. Yanke, F. Emde, F. Lesh. – Moscow: Nauka, 1977. – 344 p.

15. Sevrikov V. V. Automatic High-Speed Systems of Fire Safety (in Russian) / V. V. Sevrikov, V. A. Karpenko, I. V. Sevrikov. – SevGTU, 1996. – 260 p.

Copyright (©) 2014 Olshansky V. P., Olshansky S. V.

Copyright © 2014-2018 Technical mechanics