Elements of mathematical phenomenology: II. Phenomenological approximate mappings

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

Mihailo Petrović (1868-1943) is an important Serbian mathematician and in general scientist, one of three Poincare’s doctoral students. Paper starts with short citation of Element of Mathematical Phenomenology and Phenomenological Mappings published in Petrović’s theory. Some of basic elements of mathematical phenomenology as it is elements of non-linear-functional transformations of coordinates from one to other functional curvilinear coordinate system and are presented in our previous published paper. Structural analogy between multi deformable system ais presented. Phenomenological approximate mappings on nonlinear phenomena, in local area around stationary points or stationary states, are presented. For obtaining approximate differential equations and approximate solutions in local area around singular points, linear and non-liner approximations are used. Method of local analysis based on phenomenological approximate mappings between local linear as well as nonlinear phenomena is power to obtain information of all local nonlinear phenomena in the nonlinear dynamics of the system for completing kinetic elements for global analysis of the system nonlinear dynamics and stability and to use different analogies.

Ключевые слова:

Mihailo Petrović (1868-1943), mathematical phenomenology; structural analogy, phenomenological approximate mappings; fractional order multi-deformable body system; fractional order modes; non-linear phenomena; local approximation; singular points; analogies; theorem

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(Stevanovvić) K. , (2015), Elements of mathematical phenomenology: I. Mathematical and qualitative analogies, Электронный журнал «Труды МАИ». (submitted, to be appear) www.mai.ru/science/trudy/

  3. 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

  4. 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).

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

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

  7. 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.

  8. Hedrih (Stevanović) K., (2012), Energy and Nonlinear Dynamics of Hybrid Systems, Chapter in Book: Edited by A. Luo, Dynamical Systems and Methods, Springer, 2012, Part 1, 29-83, DOI: 10.1007/978-1-4614-0454-5_2.

  9. 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

  10. 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.

  11. 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).

  12. Hedrih (Stevanović), K., (2002), Discrete Continuum Method, Symposium, Recent Advances in Analytical Dynamics Control, Stability and Differential Geometry, Proceedings Mathematical institute SANU Edited by Vladan Djordjević, p.151, 2002, pp.30-57. ISBN 86-80593-32-X. http://www.mi.sanu.ac.yu/publications.htm

  13. Hedrih (Stevanović) K., (2004), Discrete Continuum Method, COMPUTATIONAL MECHANICS, WCCM VI in conjunction with APCOM’04, Sept. 5-10, 2004, Beijing, China, © 2004 Tsinghua University Press & Springer-Verlag, pp. 1-11, CD. IACAM International Association for Computational Mechanics — www. iacm.info

  14. Hedrih (Stevanović) K., (2004), A model of railway track of sandwich type and its dynamics excited by moving load, Scientific-expert conference on Rayways — RAILCON’04, 2004, Faculty of Mechanical Engineering University of Niš, Proceedings, pp. 149-154.

  15. Hedrih (Stevanović) K., (2004), Partial fractional order differential equations of transversal vibrations of creep connected double plates systems, Workshop Preprints/Proceedings No 2004-1 IFAC FDA 04, ENSEIRB, Bordeaux, France, July 19-21, 2004., pp. 299-304.

  16. Hedrih (Stevanović) K., (2005), Partial Fractional Order Differential Equations of Transversal Vibrations of Creep-connected Double Plate Systems, 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.

  17. Hedrih (Stevanovic) Katica, (2007), Double plate system with a discontinuity in the elastic bonding layer (Article), Acta Mechanica Sinica, (2007) vol.23 br.2 str. 221-229. DOI 10.1007/s10409-007-0061-x. (hard cower and on line).

  18. Hedrih (Stevanović K., (2007), Transversal forced vibrations of an axially moving sandwich belt system, ARCHIVE OF APPLIED MECHANICS, Springer, 22.11.2007, vol. 78, no. 9, pp. 725-735.

  19. Hedrih (Stevanovic) Katica, (2007), Transversal vibrations of the axially moving sandwich belts (Article), ARCHIVE OF APPLIED MECHANICS, (2007 ) vol.77 br.7 str. 523-539.DOI10.1007/s00419-006-0105-x

  20. Hedrih (Stevanović K., (2008), The fractional order hybrid system vibrations, Monograph, Advances in Nonlinear Sciences, ANN, 2008, Vol. 2, pp. 226-326.

  21. Hedrih (Stevanovi´c) K., (2009), Energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system), Special Issue, ARCHIVE OF APPLIED MECHANICS, DOI 10.1007/s00419-008-0285-7. Archive of Applied Mechanics, (2009) vol.79, No.6-7 pp. 529-540.

  22. Hedrih (Stevanović), K., (1988), Изаbрана поглавља из Теорије еластичности, (Selected Chapters form Theory of Elasticity (in Serbian), Faculty of Mechanical Engineering, Niš., First Edition 1977, Second Edition 1988, pp. 424.

  23. Hedrih (Stevanovi´c) K., (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, Dedicated to Centennial Jubilee of Russian Academician Yury N. 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

  24. Hedrih (Stevanović) K., (214), Multi membrane fractional order system vibrations, Theoretical and Applied mechanics, Series: Special Issue — Dedicated to memory of Anton D. Bilimovič (1879-1970), Guest Editors: Katica R. (Stevanović) Hedrih and Dragoslav Šumarac, 2014, Vol. 41 (S1), pp. 43-61.

  25. Hedrih (Stevanovi´c) 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

  26. 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

  27. 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 http://authors.elsevier.com/authorforms/NLM2407/7c32b6b4f19f2471fb24556142da3cd1

  28. 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

  29. Hedrih (Stevanović K.R. and Simonović J., (2012), «Multi-frequency analysis of the double circular plate system non-linear dynamics», NONLINEAR DYNAMICS, (2012), vol. 67 No. 3, pp. 2299-2315, 2012 Springer.

  30. Hedrih (Stevanović K.R. and Simonović J., ( (2010), Non-linear dynamics of the sandwich double circular plate system, Int. J. Non-Linear Mech, Volume 45, Issue 9, November 2010, Pp. 902-918.

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  35. Petrović M.: Fenomenološko preslikavanje (Phenomenological mapping), Srpska kraljevska akademija, Beograd, 1933, pp. 33. (In Serbian)

  36. Petrović M.: Mecanismes communs aux phenomenes disparates, Paris 1921.

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