Elements of mathematical phenomenology: 1. Mathematical and qualitative analogies

Теоретическая механика


Авторы

Hedrih (Stevanović) K. R.

Mathematical Institute Serbian Academy of Sciences and Sts (MI SANU), and Faculty of Mechanical Engineering University of Niš Serbia, 36/000ul. Knyazya Mikhaila, 11000, Belgrade, Serbia

e-mail: khedrih@sbb.rs,khedrih@eunet.rs

Аннотация

Paper starts with short description of Element of Mathematical Phenomenology and Phenomenological Mappings published in Petrović’s theory. The biographical data of Mihailo Petrović (1868-1943) is presented. Petrović was a important Serbian mathematician, one of three Poincare’s doctoral students. Some of basic elements of mathematical phenomenology in mechanics are elements of non-linear-functional transformations of coordinates from one to other functional curvilinear coordinate system. Some of these elements, as it is basic vectors of tangent space of kinetic point vector position and their changes (velocity of their magnitude extensions and component angular velocities of rotations), are presented in different functional coordinate systems. Mihailo Petrović’s theory contains two types of analogies: mathematical and qualitative, and in this paper third type — structural analogy is described. Taking into account large possibility for applications of all three types of analogies, numerous original examples are presented using, between other, fractional system dynamics with one degree of freedom, finite number of degrees of freedom as well as multi-body discrete continuum hybrid fractional order system dynamics. Mathematical analogies between vector models in local area of stress state, strain stare of the point in stressed and deformed deformable body as well as with vector model of the mass inertia moment state at point of rigid body, used mass inertia moment vectors coupled for pole and axis, are presented, also.

A number of theorems of energy fractional order dissipation presented in corresponding Tables, also. It is shown that applications of qualitative, structural and mathematical analogies in analysis of fractional order modes appear in analogous mechanical, electrical and biological fractional order chains, and that is very power, suitable and useful tools to reduce research models to corresponding minimal numbers, and, in same time, develop power of analysis use mathematical and qualitative analogies.

Ключевые слова:

Mihailo Petrović (1868-1943), mathematical phenomenology, fractional order system, generalized functions of fractional order dissipation of system energy, fractional order modes, theorem mass moment vectors, vector model, stress; strain

Библиографический список

  1. Goroško O.A. and Hedrih (Stevanovic) K. R.., (2001), Analitička dinamika (mehanika) diskretnih naslednih sistema, (Analytical Dynamics (Mechanics) of Discrete Hereditary Systems), University of Niš, 2001, Monograph, p. 426, YU ISBN 86-7181-054-2.

  2. Hedrih, N. A., Hedrih(Stevanovvić) K. , (2013), Modeling Double DNA Helix Main Chains of the Free and Forced Fractional Order Vibrations, Chapter in Book Advanced topics on fractional calculus on control problem, modeling, system stability and modeling, Editor M. Lazarević, (2013), pp. 145-183 and Appendix pp. 192-200. . WORLD SCIENTIFIC PUBLISHING COMPANY PTE LTD

  3. Hedrih A. N. and Hedrih (Stevanović K.,, (2014), Analysis of energy state of discrete fractional order spherical net of mouse zona pellucida before and after fertilization, Special issue of International Journal of Mechanics, Dedicated to the 100th Anniversary of the Russian Academician Yury Rabotnov, 2014, Vol. 8, pp. 365-370. ISSN: 1998-4448. (Paper submitted in January 2014), Journal indexed in SCOPUS (http://www.naun.org/cms.action?id=2828 ). (u stampi).

  4. Hedrih (Stevanović) K. R., (1992), On some interpretations of the rigid bodies kinetic parameters, XVIIIth ICTAM HAIFA, Apstracts, pp. 73-74.

  5. Hedrih (Stevanović) K. R., (1993), Same vectorial interpretations of the kinetic parameters of solid material lines, ZAMM. Angew.Math. Mech. 73(1993) 4-5, T153- T156.

  6. Hedrih (Stevanović) K. R., (1993), The mass moment vectors at n-dimensional coordinate system, Tensor, Japan, Vol 54 (1993), pp. 83-87.

  7. Hedrih (Stevanović) K. R., (2001), Vector Method of the Heavy Rotor Kinetic Parameter Analysis and Nonlinear Dynamics, University of Niš 2001, Monograph, p. 252. (in English), YU ISBN 86 7181-046-1.

  8. Hedrih (Stevanović) K. R., (1998), Vectors of the Body Mass Moments, Monograph paper, Topics from Mathematics and Mechanics, Mathematical institute SANU, Belgrade, Zbornik radova 8(16), 1998, pp. 45-104. published in 1999. http://elib.mi.sanu.ac.rs/files/journals/zr/16/n016p045.pdf, http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=zr&rbr=16

  9. Hedrih (Stevanović) K., (1991), The analogy between the stress state model, the strain state model and the mass inertia state model, Facta Universitatis Series Mechanics, Automatic Control and Robotics, 1991, Vol. 1, No. 1, pp. 105-120.

  10. Hedrih (Stevanović) K., (2001), Derivatives of the Mass Moment Vectors at the Dimensional Coordinate System N, dedicated to memory of Professor D. Mitrinović, Facta Universitatis Series Mathematics and Informatics, 13 (1998), pp. 139-150. (1998, published in 2001. Edited by G. Milovanović).

  11. Hedrih (Stevanović) K. R.: Modes of the Homogeneous Chain Dynamics, Signal Processing, Elsevier, 86, 2678-2702, 2006.

  12. Hedrih (Stevanović) K. R.: Energy interaction between linear and nonlinear oscillators (Energy transient through the subsystems in the hybrid system), Ukr. Mat. Zhurn., 60, 796-814, 2008.

  13. Hedrih (Stevanovic) K. R., (2007), Energy analysis in the nonlinear hybrid system containing linear and nonlinear subsystem coupled by hereditary element, Nonlinear Dynamics, Springer, 30.01.2007, vol. 51, no. 1, pp. 127-140. DOI 10.1007/s11071-007-9197-2.

  14. Hedrih (Stevanovic) K. R., (2009), Energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system), Special Issue, Arch. of Applied Mech., (2009) vol.79, No.6-7 pp. 529-540. DOI 10.1007/s00419-008-0285-7.

  15. Hedrih (Stevanović K., (2008), Dynamics of coupled systems, Nonlinear Analysis: Hybrid Systems, Volume 2, Issue 2, June 2008, Pages 310-334.

  16. Hedrih (Stevanović K., (2014), Generalized function of fractional order dissipation of system energy and extended Lagrange differential Lagrange equation in matrix form, Dedicated to 86th Anniversary of Radu MIRON’S Birth., Tensor, Vol. 75, No. 1. pp. 35-51. Tensor Society.

  17. Hedrih (Stevanović K., (2014), Linear and nonlinear (functional) transformations, To memory of academician Vladimir Metodievich Matrosov (May 8, 1932-April 17,2011) President of Academy of nonlinear Sciences. Tensor, Vol. 75, No. . 75, No. 1. pp. 71-93. Tensor Society

  18. Hedrih (Stevanoviċ) R. Katica, (2013), Vector method based on mass moment vectors and vector rotators applied to rigid-body multi-coupled rotations around no intersecting axes, International Journal of Structural Stability and Dynamics, Vol. 13, No. 7 (2013) 1340007 (20 pages), #.c World Scientificc Publishing Company, ISSN: 0219-4554, DOI: 10.1142/S0219455413400075 http://dk.doi.org/10.1142/S0219455413400075

  19. Hedrih (Stevanović) K., (213), LINEAR AND NONLINEAR DYANAMICS OF HYBRID SYSTEM, Invited Plenary Lecture, (To memory of my Serbian professors: Draginja Nikoliċ, Danilo P. Raškoviċ and Tatomir P. Andjeliċ and to academicians supported my international scientific acctivity in nonlinear dynamics: Yuri Alekseevich Mitropolskiy, Vladimir Metodievich Matrosov and Valentin Vitalevich Rumяantsev and President of IFNA Professor dr V. Lakshminatham ), Proceedings of Fourth Serbian (29th Yu) Congress on Theoretical and Applied Mechanics, Vrnjačka Banja, Serbia, 4-7 June 2013, pp. 43-58. ISBN 978-86-909973-5-0. ISBN 978-86-909973-5-0. COBISS.SR-ID 198308876.

  20. Hedrih (Stevanović) K.,, (201`3), Fractional order hybrid system dynamics, PAMM, Proc. Appl. Math. Mech. 13, 25 — 26 (2013) / DOI 10.1002/pamm.201310008. http://onlinelibrary.wiley.com/doi/10.1002/pamm.v13.1/issuetoc

  21. Hedrih (Stevanović) K. R., (2005), Partial Fractional Order Differential Equations of Transversal Vibrations of Creep-connected Double Plate Systems, Chap in in Monograph — Fractional Differentiation and its Applications, Edited by Alain Le Mahaute, J. A. Tenreiro Machado, Jean Claude Trigeassou and Jocelyn Sabatier, p. 780, U-Book, Printed in Germany, pp. 289-302.

  22. Hedrih (Stevanović) K., Stamenković B. M., (213), Mass moment vector applications to the milti body dynamics with coupled rotation about no intersecting axes, Proceedings in Applied Mathematics and Mechanics (PAMM), 24 May 2013, GAMM 2013, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics Novi Sad, Serbia, March 18-22, 2013. http://gamm2013.cats-host.com/cms/

  23. Hedrih (Stevanović K.,R., (2014), Elements of mathematical phenomenology in dynamics of multi-body system with fractional order discrete continuum layers, Dedicated to the 100th Anniversary of the Russian Academician Yury Rabotnov, Special issue of International Journal of Mechanics, 2014, Vol. 8, pp. 339- 346, ISSN: 1998-4448 .(Paper submitted in January 2014), Journal indexed in SCOPUS (http://www.naun.org/cms.action?id=2828 ). (u stampi).

  24. Hedrih (Stevanović K.,, Vector method: Kinetic parameter Analysis and impacts to bearings of the coupled rotor dynamics. dedicated in memory of Academician Konstantin Vasiljevich Frolov (1932-2007), Вестник научно-технического развития, Национальная Технологическая Группа, № 9 (25), 2009 г. pp. 42 −59. www.vntr.ru , www.ntgcom.com, www.vntr.ru www.ntgcom.com

  25. Hedrih (Stevanović) R. Katica , (2010), Visibility or appearance of nonlinearity, Tensor, N.S. Vol. 72, No. 1 (2010), pp. 14-33, #3. Tensor Society, Chigasaki, Japan, ISSN 0040-3504.

  26. Hedrih (Stevanović) Katica, (2012), Tangent space extension of the position vectors of a discrete rheonomic mechanical system, Professor N. R. Sen Memorial Lecture, Bulletin of the Calcutta Mathematical Society Volume 104, No.2(2012) pp. 81-102. Bull.Cal.Math. 104 (2) 81-102 (2012).

  27. Hedrih (Stevanović) K., (2005), Nonlinear Dynamics of a Heavy Material Particle Along Circle which Rotates and Optimal Control, Chaotic Dynamics and Control of Systems and Processes in Mechanics (Eds: G. Rega, and F. Vestroni), p. 37-45. IUTAM Book, in Series Solid Mechanics and Its Applications, Editerd by G.M.L. Gladwell, Springer. 2005, XXVI, 504 p., Hardcover ISBN: 1-4020-3267-6.

  28. Hedrih (Stevanović) K., (2004), A Trigger of Coupled Singularities, MECCANICA, Vol.39, No. 3, 2004., pp. 295-314. , DOI: 10.1023/B:MECC.0000022994.81090.5f,

  29. Hedrih (Stevanović), K., (200), Nonlinear Dynamics of a Gyro-rotor, and Sensitive Dependence on initial Conditions of a Heav Gyro-rotor Forced Vibration/Rotation Motion, Semi-Plenary Invited Lecture, Proceedings: COC 2000, Edited by F.L. Chernousko and A.I. Fradkov, IEEE, CSS, IUTAM, SPICS, St. Petersburg, Inst. for Problems of Mech. Eng. of RAS, 2000., Vol. 2 of 3, pp. 259-266.

  30. Hedrih (Stevanović K., and Hedrih A.N., (2014), Phenomenological mapping and dynamical absorptions in chain systems with multiple degrees of freedom, Journal of Vibration and Control 1077546314525984, first published on March 19, 2014 as doi:10.1177/1077546314525984

  31. Hedrih (Stevanović K.,, J. Tenreiro Machado, (2013), Discrete fractional order system vibrations, International Journal Non-Linear Mechanics, Volume 73, July 2015, Pages 2-11. NLM2407, DOI: 10.1016/j.ijnonlinmec.2014.11.009 (January 6, 2014). http://authors.elsevier.com/authorforms/NLM2407/7c32b6b4f19f2471fb24556142da3cd1

  32. Hedrih (Stevanović K.R. and Simonović J., (2013), Structural analogies on systems of deformable bodies coupled with non-linear layers, International Journal Non-Linear Mechanics, Volume 73, July 2015, Pages 18–24. NLM2402, DOI: 10.1016/j.ijnonlinmec.2014.11.004

  33. Hedrih (Stevanović K.R. and Knežević R., «Structural stability of the planetary reductor nonlinear dynamics phase portrait,» Facta Universitatis Seris Mechanics, Automatic Control and Robotics, vol. 1, no. 7, pp. 911–923, 2000.

  34. Hedrih (Stevanović K.R., Knežević R. and Cvetkovi´c, R. «Dynamics of planetary reductoe with turbulent damping,» International Journal of Nonlinear Sciences and Numerical Simulations, vol. 2, no. 3, pp. 265–262, 2001.

  35. Hedrih (Stevanović K.R., and Lj. Veljovic , (2011), Vector Rotators of a Rigid Body Dynamics with Coupled Rotations around Axes without Intersection, International Journal Mathematical Problems in Engineering, MPE/351269, Hindawi Publishing Corporation, Volume 2011, Article ID 351269.

  36. Hedrih (Stevanović) K., Veljović Lj., New Vector Description of Kinetic Pressures on Shaft Bearings of a Rigid Body Nonlinear Dynamics with Coupled Rotations around No Intersecting Axes, Acta Polytechnica Hngarica — Journal of Applied Sciences, Special Issue on Applied Mathematics, Guest Editors: Aurél Galántai and Péter T. Nagy, Volume 10, Issue Number 7, 2013, pp. 151-170. ISSN 1785-8860. http://uni-obuda.hu/journal/Issue45.htm, http://www.uni-obuda.hu/journal/Issue45.htm

  37. Hedrih (Stevanović) K R., Raičević V. and Jović S., Phase Trajectory Portrait of the Vibro-impact Forced Dynamics of Two Heavy Mass Particles Motions along Rough Circle, Communications in Nonlinear Science and Numerical Simulations, ISSN 1007-5704, 2011 16 (12):4745-4755. DOI 10.1016/j.cnsns.2011.05.027.

  38. Hedrih (Stevanović) K R.,, Vibrations of a Heavy Mass Particle Moving along a Rough Line with Friction of Coulomb Type, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 11, No.3 March 2010, pp. 203-210. ISSN 1565-1339. http://www.reference-global.com/toc/ijnsns/11/3

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