Dynamics of multi-pendulum systems with fractional order creep elements

• Autorzy: Hedrih (Stevanovic) K. R.

Warianty tytułu

PL

Dynamika układów wielowahadłowych z efektem pełzania opisanym elementami ułamkowego rzędu

• Języki publikacji: EN

Abstrakty

EN

A survey as a short review of author’s research results in area of dynamics of hybrid systems and analytical dynamics of discrete material particle system containing creep elements described by fractional order derivatives, is presented. Free vibrations of a multi-pendulum system intercoupled by standard light elements and different properties are considered. The corresponding system of an ordinary fractional order as well as integro-differential equations, described dynamics of the multi-pendulum system, are derived and analytically solved. For the case of one pendulum and two pendulum systems containing standard light creep elements with the stress-strain constitutive relation expressed by a fractional order derivative, ordinary differential equations are analytically solved. From the analytical solutions, for the case of the homogeneous two-pendulum system, it is visible that free vibrations under arbitrary initial conditions contain three modes, one pure periodic and two aperiodic expressed by time series expansions. The obtained analytical solution modes are numerically analysed and characteristic vibration modes for different kinetic parameters are graphically presented.

PL

W pracy zaprezentowano krótki przegląd rezultatów badań autora nad dynamiką układów hybrydowych i dyskretnych, złożonych z punktów materialnych sprzęgniętych standardowymi elementami odpowiadającymi za pełzanie w materiale i opisywanych pochodną ułamkowego rzędu. Rozważono drgania swobodne układów wielowahadłowych z elementami o różnych właściwościach zdefiniowanych równaniem pomiędzy stanem naprężenia a odkształcenia. Wyprowadzone równania różniczkowo-całkowe ułamkowego rzędu rozwiązano analitycznie. Przedstawiono szczegółowo przypadek układu z pojedynczym wahadłem i układu dwuwahadłowego zawierającego elementy pełzania opisane równaniem konstytutywnym stanu naprężenia i odkształcenia o rzędzie ułamkowym. Na podstawie otrzymanych rozwiązań analitycznych zauważono, że drgania swobodne wykazują charakter okresowy i nieokresowy, przy czym te ostatnie mają dwa różne przebiegi (w tym przypadku rozwiązanie podano w postaci rozwinięć w szeregi potęgowe). Wyniki badań teoretycznych i numerycznych różnego rodzaju drgań przy zmiennych parametrach kinetycznych tych układów przedstawiono graficznie.

Słowa kluczowe

EN

coupled subsystems; coupled dynamics; multi-pendulum system

PL

drgania; układy wielowahadłowe

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2008
• Tom: Vol. 46 nr 3
• Strony: 483--509
• Opis fizyczny: Bibliogr. 34 poz., rys.

Twórcy

autor

Hedrih (Stevanovic) K. R.

• Faculty od Mechanical Engineering University of Nis, Mathematical Institute SANU, Nis, Serbia,

Bibliografia

• 1. Enelund M., 1996, Fractional Calculus and Linear Viscoelasticity in Structural Dynamics, Division of Solid Mechanics, Chalmers Tekniska Hogskola, Goteborg, Sweden, 1-27
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• 3. Goroshko O.A., Puchko N.P., 1997, Lagrangian equations for the multibodies hereditary systems, Facta Universitatis, Series Mechanics, Automatic Control and Robotics, 2, 7, 209-222
• 4. Goroˇsko O.A., Hedrih (Stevanović) K., 2001, Analitiˇcka dinamika (mehanika) diskretnih naslednih sistema (Analytical Dynamics (Mechanics) of Discrete Hereditary Systems), University of Niˇs, Monograph, p. 426, YU ISBN 86-7181-054-2
• 5. Goroˇsko O.A., Hedrih (Stevanović) K., 2007a, Construction of the Lagrange’s mechanics of the hereditary systems, APM Saint Petersburg 2007 pp., Minisymposium Oppening Lecture, The International Summer School APM – Advanced Problem in Mechanics, Saint Petersburg, 133-156
• 6. Goroˇsko O.A., Hedrih (Stevanović) K., 2007b, Construction of the Lagrange’s mechanics of the hereditary systems, Facta Universitatis Series Mechanics, Automatic Control and Robotics, 6, 1, 1-23
• 7. Hedrih (Stevanović) K., 1999, Thermorheological hereditary pendulum, In: Thermal Stresses 99, Edited by J.J. Skrzypek and R.B. Hetnarski, Cracow, 199-202
• 8. Hedrih (Stevanović) K., 2001, Differential equations of two mass particles, constrained with a piezo-thermo-rheological hereditary element, dynamics, Proceedings of full papers, 5th International Conference on Applied Electromagnetics, PES 2001, Edited by D. Veliˇcković, 77-80
• 9. Hedrih (Stevanović) K., 2002a, The dissipation function of a nonconservative system of mass particles, Tensor, N.S., 63, 2, 176-186, ISSN 0040-3504
• 10. Hedrih (Stevanović) K., 2002b, Transversal creep vibrations of a beam with fractional derivative constitutive relation order. I – Partial fractional-differential equation. II – Stochastic stability of the beam dynamic shape, under axial bounded noise excitation, Proceedings of Forth International Conference on Nonlinear Mechanics (ICNM-IV), Edited by Wei Zang Chien et al., Shanghai, P.R. China, 584-595
• 11. Hedrih (Stevanović) K., 2003a, Discrete continuum’s models and thermorheological elements – basic idea and tensor equations, homogeneous linear chain and plane/space material nets, Proceedings of full papers, 6 th International Conference on Applied Electromagnetics, PES 2003, Edited by D. Veliˇcković, 127-130, 131-134
• 12. Hedrih (Stevanović) K., 2003b, The longitudinal and transversal creep vibrations of a fractional order constitutive relation beams, Scientific Bulletin of the Politehnica, University of Timisoara, Transaction on Mechanics, 48, 62, 5-12, 13-22, ISSN 1224-6077, http://www.utt.ro/english/pbseng.shtml
• 13. Hedrih (Stevanović) K., 2004a, Creep vibrations of a fractional derivative order constititive relation deformable bodies, Proceedings Eighth American Congress of Applied Mechanics PACAM VIII, La Habana, Cuba, Series Applied Mechanics in Americas, 10, 548-551, ISBN 959-7056-20-8
• 14. Hedrih (Stevanović) K., 2004b, Discrete continuum method, Computational Mechanics, WCCM VI in conjunction with APCOM’04, Beijing, China, Tsinghua University Press & Springer-Verlag, 1-11, CD
• 15. Hedrih (Stevanović) K., 2004c, 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, 299-304
• 16. Hedrih (Stevanović) K., 2005a, Eigen amplitude vectors and functions extended orthogonality of small oscillations mixed systems of the coupled discrete and continuous subsystems, Facta Universitatis, Series Mechanics, Automatic Control and Robotics, 4, 17, 225-243, YU ISSN 0534-2009
• 17. Hedrih (Stevanović) K., 2005b, Integrity of dynamical systems, Journal Nonlinear Analysis, 63, 854-871
• 18. Hedrih (Stevanović) K., 2005c, Partial Fractional order differential equations of transversal vibrations of creep-connected double plate systems, In: Fractional Differentiation and its Applications, Edited by A. Le Mahaute, J.A.T. Machado, J.C. Trigeassou and J. Sabatier, U-Book, 289-302
• 19. Hedrih (Stevanović) K., 2006a, Modes of the homogeneous chain dynamics, Signal Processing, 86, 2678-2702, ISSN: 0165-1684, www.sciencedirect.com/science/journal/01651684
• 20. Hedrih (Stevanović) K., 2006b, The frequency equation theorem of small oscillations of a hybrid system containing coupled discrete and continuous subsystems, Facta Universitatis Series: Mechanics, Automatic Control and Robotics, 5, 1, 25-41, http://facta.junis.ni.ac.yu/facta/
• 21. Hedrih (Stevanović) K., 2006c, The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam, Mathematical Problems in Engineering, Special issue: Nonlinear Dynamics and their Applications in Engineering Sciences, Edit.: J.M. Barhesar, 2006, 5, 61-78, www.hindawi.com
• 22. Hedrih (Stevanović) K., 2006d, Transversal forced vibrations of an axially moving sandwich belt system, Archive of Applied Mechanics, Springer, http://springerlink.com/content/?k=Hedrih
• 23. Hedrih (Stevanović) K., 2006e, Transversal vibrations of double-plate systems, Acta Mechanica Sinica, 22, 487-501
• 24. Hedrih (Stevanović) K., 2006f, Transversal vibrations of the axially moving sandwich belts, Archive of Applied Mechanics, 77, 7, 523-539, http://springerlink.com/content/?k=Hedrih
• 25. Hedrih (Stevanović) K., 2007a, Double plate system with discontinuity in the elastic bonding layer, Acta Mechanica Sinica, 23, 2, 221-229
• 26. Hedrih (Stevanović) K., 2007b, Energy analysis in the nonlinear hybrid system containing linear and nonlinear subsystem coupled by hereditary element, Nonlinear Dynamics, 51, 1, 127-140
• 27. Hedrih (Stevanović) K., 2007c, Energy analysis of the double plate system, Acta Mechanica Sinica, DOI 10.1007/s10409-007-0124-z
• 28. Hedrih (Stevanović) K., 2007d, Hybrid systems and hybrid dynamics: theory and applications, Invited Plernary Lecture, 8th HSTAM International Congress on Mechanics, Patras, Greece, Edited by N. Bazwos, D.L. Karabalis, D. Polyzos, D.E. Beskos and J.T. Katsikadelis, I, 77-86
• 29. Hedrih (Stevanović) K., 2007e, Stochastic dynamics of hybrid systems with thermo-rheological hereditary elements, Proceedings of IX International conference on Dynamical Systems – Theory and Application, Edited by K. Awrejcewicz, P. Olejnik and Mrozowski, Łódź, Poland, University of Łódź, 193-202
• 30. Hedrih (Stevanović) K., Filipovski A., 2002, Longitudinal vibration of a fractional derivative order rheological rod with variable cross section, Facta Universitatis, Series Mechanics, Automatic Control and Robotics, 3, 12, 327-350, YU ISSN 0534-2009, http://facta.junis.ni.ac.yu/facta/macar/macar2002/macar2002-02.html http://facta.junis.ni.ac.yu/facta/macar/macar200501/macar200501-04.html
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