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Energy and Nonlinear Dynamics of Hybrid Systems

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Dynamical Systems and Methods

Abstract

The transmission of energy between subsystems coupled in hybrid system is very important for different applications. For first as an introduction, by using the author’s previously published references and that of her students, a short survey of an analytical study of the energy transfer between coupled subsystems is presented as a basis of this chapter. An analytical study of the mechanical energy transfer between two coupled subsystems, as well as, between two or more coupled rotation motions is presented. For starting, an analytical analysis of the mechanical energy transfer between a linear and a nonlinear oscillators of a hybrid system (see Refs. by Hedrih (Stevanović) 2002 Hedrih [2006, 2005, 2006,?, 2003], Katica [2005], Hedrih [1997, 2005]) in the free, as well as forced, vibrations of a different types of interconnections between subsystems is presented. Coupling element between subsystems of the considered hybrid systems are standard light elements with elastic, viscoelastic, hereditary, or creeping properties as well as dynamical constrain element realized by rolling element with inertia properties. Using Krilov–Bogolyubov–Mitropolskiy’s asymptotic method, both the solutions in the first approximation and the system of nonlinear-coupled differential equations for the corresponding number of excited amplitudes and phases of multifrequency free as well as forced regimes are derived. By means of this asymptotic approximation of differential equations for the amplitudes and phases for forced vibrations of the coupled oscillators, the mutual influence of the nonlinear harmonics and energy transient were analyzed. The Lyapunov exponents corresponding to the each of two eigen like nonlinear modes are expressed by using energy of the corresponding eigen time components. A generalization of an analytical analysis of the transfer energy between linear and nonlinear oscillators for forced vibrations with different type constraints as a couple between two subsystems, each of them with one degree of freedom, is done. A mathematical analogy between discrete and complex discrete-continual hybrid systems is pointed out.In the second part, an analytical analysis is extended to the transfer energy between plates for free and forced transversal vibrations of a visco and nonlinear elastically connected double plate system. The analysis showed that the visco- and nonlinear elastic connection between plates caused the appearance of two-frequency like regime of time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are coupled, as well as energy transfer between plates in one eigen mode appear.Next, as an author’s new research result, an analytical study of the energy transfer between two coupled-like string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system in the free vibrations is presented.

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References

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

    Google Scholar 

  2. Goroško OA, Hedrih (Stevanović) K (2007) Construction of the Lagrange’s mechanics of the hereditary systems. Facta Universitatis Series Mechanics, Automatic Control and Robotics 6(1):1–23

    Google Scholar 

  3. Goroško OA, Hedrih (Stevanović) K (2007) Construction of the Lagrange’s mechanics of the hereditary systems, APM Saint Petersburg 2007 pp., Minisymposium Opening Lecture, The International Summer School APM –Advanced Problem in Mechanics, Saint Petersburg pp. 133–156.

    Google Scholar 

  4. Goroško OA, Hedrih (Stevanović) K (2008) Advances in development of the analytical dynamics of the hereditary discrete systems. J Phys: Conf Ser, 96: 012143. http://www.iop.org/EJ/main/-list=current/

  5. Hedrih (Stevanović) K (1985) Multifrequency forced vibrations of thin elastic shells with positive Gaussćurvature and finite deformations. Theor Appl Mech (11):59-72. http://www.ssm.org.rs/WebTAM/arhives.html, http://www.ssm.org.rs/WebTAM/_private/arhiva/tam11.jpg

  6. Hedrih (Stevanović) K R. and Simonović J., Multi-frequency analysis of the double circular plate system non-linear dynamics, Nonlinear Dynamics, Springer, NODY1915R2, (in press) DOI: 10.1007/s11071-011-0147-7

    Google Scholar 

  7. Hedrih (Stevanović) K (1975) Selected Chapters from the Theory of Nonlinear Vibrations [Preprint in Serbian], Niš, Yugoslavia

    Google Scholar 

  8. Hedrih (Stevanović) K (2007), Energy analysis in the nonlinear hybrid system containing linear and nonlinear subsystem coupled by hereditary element. Nonlinear Dynam Springer 51(1):127–140. doi: 10.1007/s11071-007-9197-2

    Google Scholar 

  9. Hedrih (Stevanović) K (2007) Transversal vibrations of the axially moving sandwich belts. Arch Appl Mech Springer 78(9):725–735. doi: 10.1007/s00419-006-0105-x

    Google Scholar 

  10. Hedrih (Stevanović) K (2006) The frequency equation theorems of small oscillations of a hybrid system containing coupled discrete and continuous subsystems. Facta Univ. Ser Mech, Automatic Control Robot 5(1):25–41. http://facta.junis.ni.ac.yu/facta/

  11. 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. ISBN 3-86608-025-3 BARCOD 9-783866-0802063

    Google Scholar 

  12. Hedrih (Stevanović) K., (2006), Transversal Vibrations of Double-Plate Systems, Acta Mechanica Sinica, Springer, (2006) 22, pp. 487-501 (hard cower and on line). ISSN 0567-7718

    Google Scholar 

  13. Hedrih (Stevanović) K (2007) Double plate system with discontinuity in the elastic bonding layer. Acta Mech Sinica 23(2):221–229. (hard cower and on line)

    Google Scholar 

  14. Hedrih (Stevanović) K (2005) Integrity of dynamical systems. J Nonlinear Anal 63:854–871

    Google Scholar 

  15. Hedrih (Stevanović) K (2006) Modes of the homogeneous chain dynamics. Signal Process Elsevier 86:2678–2702

    Google Scholar 

  16. Hedrih (Stevanović) K (2006) Transversal vibration of a parametrically excited beam: influence of rotatory inertia and transverse shear on stochastic stability of deformable forms and processes. { ©}Freund Publishing House Ltd. Int J Nonlinear Sci Numerical Simulation 7(1):117–124

    Google Scholar 

  17. Hedrih (Stevanović) K (2003) Frequency equations of small oscillations mixed systems of the coupled discrete and continuous subsystems, Mehanika tverdogo tela (Rigid Body Mechanics), Donetsk, UDC 531.1:534.012:534.013, ISSN 0321-1975, vip. 33, pp. 174–189. Ukraine

    Google Scholar 

  18. Hedrih (Stevanović) K (2005) Eigen amplitude vectors and functions extended orthogonality of small oscillations mixed systems of the coupled discrete and continuous subsystems. Facta Univ Ser Mech Autom Control Robot 4(17):225–243. http://facta.junis.ni.ac.yu/facta/macar/macar200501/macar200501-04.html

  19. Hedrih (Stevanović) K (1996) Interpretation of the transfer energy from high-frequenccy to low-frequenccy modes by averaging asymptotic method Krilov-Bogolyubo-Mitropolsky. Book of abstracts, the second international conference “Asymptotics in Mechanics” St. Petersburg Marine Technical University, Russia, 13–16 October 1996, p 29

    Google Scholar 

  20. Hedrih (Stevanović) K (1997) Interpretation of the transfer of energy from high-frequency to low-frequency modes by averaging asymptotic method Krilov-Bogolybov-Mitropolsky. In: Nayfeh AH, Rozdestvensky KV (ed) AiM 96, Asymptotics in mechanics, Proceedings of the Second International Conference, p. 302, Saint-Petersburg State Marine Technological University, Saint Petersburg, 1997, pp. 97–104. ISBN 5-88303-069-6, Izdatelskiy centr SPbGMTU, 1997

    Google Scholar 

  21. Hedrih (Stevanović) K (1995) Energijska analiza kinetike konstrukcija za razlicite modele materijala (Energy analysis of the sytucture kinetics for different models of the materials), Kratki prikazi radova, “Mehanika, materijali i konstrukcije”, SANU, Beograd, 1995, pp. 106–107. (in Serbian)

    Google Scholar 

  22. Hedrih (Stevanović) K (1997) Interpretation of the transfer of energy from high-frequency to low-frequency modes by averaging asymptotic method Krilov-Bogolybov-Mitropolsky. In: Nayfeh AH, Rozdestvensky KV (ed) AiM 96, Asymptotics in mechanics, Proceedings of the Second International Conference, p. 302, Saint-Petersburg State Marine Technological University, Saint Petersburg, 1997, pp. 97–104

    Google Scholar 

  23. Hedrih (Stevanović) K (1997) Energetic analysis of oscillatory processes and the modes in nonlinear systems, solid mechanics, serbian-Greek symposium 1997, scientific meetings of the Serbian Academy of Sciences and Arts, Department of Technical Sciences, SANU Belgrade, vol LXXVII, Book 3, pp 137–146

    Google Scholar 

  24. Hedrih (Stevanović) K (2005) Transversal vibrations of creep connected multi plate homogeneous systems. In: van Campen DH, Lazurko MD, van den Over WPJM (ed) CD Proceedings, Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven University of Technology, ID 11-428, pp. 1445–1454

    Google Scholar 

  25. Hedrih (Stevanović) K (2008) Energy transfer in double plate system dynamics. Acta Mechanica Sinica 24(3):331–344. doi: 10.1007/s10409-007-0124-z, Springer Berlin/Heidelberg, ISSN (567-7718 (Print) 1614-3116 (Online)

    Google Scholar 

  26. Katica (Stevanović) H (2008) Energy interaction between linear and nonlinear oscillators (Energy transient through the subsystems in the hybrid system), ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6, pp. 796–814

    Google Scholar 

  27. Hedrih (Stevanović) K (1997) Energetic analysis of oscillatory processes and the modes in nonlinear systems. Solid mechanics, Serbian-Greek symposium 1997, scientific meetings of the Serbian Academy of Sciences and Arts, Department of Technical Sciences, SANU Belgrade, vol LXXVII, Book 3, pp 137–146

    Google Scholar 

  28. Hedrih (Stevanović) K (2004) A trigger of coupled singularities. MECCANICA 39(3), 295–314. International Journal of the Italian Association of Theoretical and Applied echanics, Meccanica, Publisher: Springer

    Google Scholar 

  29. Hedrih (Stevanović) K (2004) Contribution to the coupled rotor nonlinear dynamics, advances in nonlinear sciences, Monograph, Belgrade, Academy of Nonlinear Sciences, pp 229–259. (engleski). ISBN 86-905633-0-X UDC 530-18299(082) 51–73:53(082) UKUP

    Google Scholar 

  30. Hedrih (Stevanović) K (2005b) Multifrequency forced vibrations of thin elastic Shell. In: van Campen DH, Lazurko MD, van den Over WPJM (eds) CD proceedings, fifth EUROMECH nonlinear dynamics conference, Eindhoven University of Technology. ID of contribution 21-439, pp 2417–2426. ISBN 90 386 2667 3, www.enoc2005.tue.nl

  31. Hedrih (Stevanović) K (2002) The dissipation function of a nonconservative system of mass particles. Tensor NS 63(2):176–186. Tensor Society, Japan. ISSN 0040-3504

    Google Scholar 

  32. Hedtrih (Stevanović) K., (2008), Dynamics of coupled systems, Journal Nonlinear Analysis: Hybrid Systems, Volume 2, Issue 2, June 2008, Pages 310–334. 10.1016/j.nahs.2006.06.003 doi: in line at http://www.sciencedirect.com/science/journal/1751570X

  33. Hedrih (Stevanović) K (2006) Dynamics of sandwich structures, invited plenary lecture, recent advances in civil engineering. In: Proceeding of papers – Faculty of Civil Engineering, Subotica, 2006, E.III, vol 15, Abstract, pp 20, and CD – Full papers 30–39. YU ISSN 0352-6852. COBISS SR.ID. 14404098

    Google Scholar 

  34. Hedrih (Stevanović) K (2005) Mathematical Analogy and Phenomenological Mapping: Vibrations of Multi Plate and Multi Beam Homogeneous Systems. In: Bereteu L, Cioara T, Toth-Tascau M, Vigaru C (eds) Scientific Buletletin of the “Politehnica” University of Timisoara, Romania, Transaction on Mechanics, Editura Politehnica, Tom 50(64), Special Issue, 11–18, ISSN 1224-6077, 2005

    Google Scholar 

  35. Hedrih (Stevanović) K (2005c) Partial fractional order differential equations of transversal vibrations of creep-connected double plate systems. In: Le Mahaute A, Tenreiro Machado JA, Trigeassou JC, Sabatier J (ed) Monograph - Fractional differentiation and its applications, U-Book, pp. 289–302

    Google Scholar 

  36. Hedrih (Stevanović) K (2003) Frequency equations of small oscillations mixed systems of the coupled discrete and continuous subsystems, Mehanika tverdogo tela (Rigid Body Mechanics), Donetsk, UDC 531.1:534.012:534.013, ISSN 0321–1975, vip. 33, pp. 174–189. Ukraine

    Google Scholar 

  37. Hedrih (Stevanović) K (2005) Eigen amplitude vectors and functions extended orthogonality of small oscillations mixed systems of the coupled discrete and continuous subsystems. Facta Univer Ser Mech Automatic Control Robot 4(17):225–243. YU ISSN 0534-2009. UDC 534.1:534.012:534.013. http://facta.junis.ni.ac.yu/facta/macar/macar200501/macar200501-04.html

  38. Hedrih (Stevanović) K (2004) Forced oscillations of the coupled discrete and continuous subsystems, the first international conference on computational mechanics (CM’04), In memory of Prof.dr Mladen Berkovic, 46 MMM, © 2004, BELGRADE, SERBIA AND MONTENEGRO, 15–17 NOVEMBER 2004, BOOK OF PAPERS, Printing Faculty of Mathematics, University of Belgrade 1–16 ISBN 86-7589-042-7

    Google Scholar 

  39. Hedrih (Stevanović) K (2004) A model of railway track of sandwich type and it’s dynamics excited by moving load. In: Proceedings of scientific-expert conference on Railways - XII RAILKON‘04, Faculty of Mechanical Engineering University of Niš, pp 149–154. 21 i 22 oktobra 2004

    Google Scholar 

  40. Hedrih (Stevanović) K (2006) Transversal vibrations of the axially moving double belt system with creep layer, Preprints, 2nd IFAC workshop on fractional differentiation and its applications, 19–21 July 2006, Porto, Portugal, pp 230–235. + CD. IFAC WS 2006 0007 PT, ISBN 972-8688-42-3, 978-972-8688-42-4.ISBN. http:/www.iser.ipp.pl

  41. Hedrih (Stevanović) K (2005) Transveral vibrations of creep connected multi plate homoheneous systems. In: van Campen DH, Lazurko MD, van den Over WPJM (eds) CD Proceedings, Fifth EUROMECH nonlinear dynamics conference, eindhoven university of technology. ID of contribution 11–428, pp 1445–1454. ISBN 90 386 2667 3, www.enoc2005.tue.nl

    Google Scholar 

  42. Hedrih (Stevanović) K (2004) Discrete continuum method, computational mechanics. In: Zao ZH, Zuang MW, Zhong WX (eds) WCCM VI in conjunction with APCOM’04. { ©} 2004 Tsinghua University Press & Springer, pp 1–11. CD. IACAM International Association for Computational Mechanics – www. iacm.info, ISBN 7-89494-512-9

    Google Scholar 

  43. Hedrih (Stevanović) K (2004) Creep vibrations of a fractional derivative order constititive relation deformable bodies, PACAM VIII. La Habana, 2004. Appl Mech Americas 10:548–551. ISBN 959-7056-20-8

    Google Scholar 

  44. Hedrih (Stevanović) K (2004) Phase portraits and homoclinic orbits visualization of nonlinear dynamics of multiple step reductor/multiplier, proceedings, vol 2, The eleventh world congress in Mechanism and machine Sciences, IFToMM, China Machine press, Tianjin, 1–4 April 2004, pp 1508–1512. ISBN 7-111-14073-7/TH-1438. http://www.iftomm2003.com Publisher: China Machine press, Tianjin, China

  45. Hedrih (Stevanović) K (2004) Contribution to the coupled rotor nonlinear dynamics, Advances in nonlinear Sciences, Monograph, Belgrade, Academy of Nonlinear Sciences, pp. 229–259. (engleski). ISBN 86-905633-0-X UDC 530-18299(082) 51–73:53(082) UKUP.

    Google Scholar 

  46. Hedrih (Stevanović) K (2004) On rheonomic systems with equivalent holonomic conservative systems applied to the nonlinear dynamics of the watt’s regulator, proceedings, vol 2, The eleventh world congress in Mechanism and machine Sciences, IFToMM, China Machine press, Tianjin, 1–4 April 2004, pp 1475–1479. ISBN 7-111-14073-7/TH-1438. http://www.iftomm2003.com

  47. Hedrih (Stevanović) K (1995) Energijska analiza kinetike konstrukcija za razlicite modele materijala, Kratki prikazi radova, “Mehanika, materijali i konstrukcije”, SANU, Beograd, 1995, pp. 106–107

    Google Scholar 

  48. Hedrih (Stevanović) K (1996) Interpretation of the transfer energy from high-frequenccy to low-frequenccy modes by averaging asymptotic method Krilov-Bogolyubo-Mitropolsky. Book of abstracts, the second international conference “Asymptotics in Mechanics” St. Petersburg Marine Tech.Univ., Russia, 13–16 October 1996, pp 29

    Google Scholar 

  49. Hedrih (Stevanović) K (1997) Energetic analysis of oscillatory processes and the modes in nonlinear systems. Solid mechanics, serbian-Greek symposium 1997, scientific meetings of the Serbian Ac of Sci and Arts, Dep of Techn Sciences, Belgrade, vol LXXVII., Book. 3, pp 137–146

    Google Scholar 

  50. Hedrih (Stevanović) K, Knežević R, Cvetković R (2001) Dynamics of planetary reductoe with turbulent damping. Int J Nonlinear Sci Numerical Simulations 2(3):265–277. ISSN 1565-1339. (in English), Freund Publishing House L.T.D. http://www.cs.uky.edu/\simjzhang/NSNS/2001-02-03.html

  51. Hedrih (Stevanović) K, Hedrih AN (2009) Transfer of energy of oscillations trough the double DNA chain helix. In: Ambrósio J, et al (eds) 7th EUROMECH solid mechanics conference. Lisbon, Portugal, 7–11 September 2009, CD –MS-24, Paper 315, pp 1–15

    Google Scholar 

  52. Hedrih (Stevanović) K, Hedrih AN (2009) Transfer of energy of oscillations trough the double DNA chain helix. In: Ambrósio J, et al. (eds) 7th EUROMECH solid mechanics conference, Lisbon, Portugal, September 7–11, 2009, ESMC 2009 Book of Abstracts ‘Mini’Szmposia, iydanje Instituto Superior Tecnico, Lisbon, APMTAC, pp. 591’592.ISBN 9789899 626423

    Google Scholar 

  53. Hedrih (Stevanović) K, Simonovic’ JD (2010) Non-linear dynamics of the sandwich double circular plate system. Int J Nonlinear Mech. doi:10.1016/j.ijnonlinmec.2009.12.007, Int J Nonlinear Mech 2009. 12. 007

    Google Scholar 

  54. Hedrih (Stevanivic) K, Simonovic J (2007) Transversal vibrations of a non-conservative double circular plate system. Facta Univ Series: Mech Automatic Control Robot 6(1):51–64

    Google Scholar 

  55. Hedrih (Stevanović) K, Simonović J (2008) Transversal vibrations of a double circular plate system with visco-elastic layer excited by a random temperature field. Int J Nonlinear Sci Numerical Simulation 9(1):47–50. http://www.ijnsns.com/2008/TOC9.1.doc

    Google Scholar 

  56. Hedrih (Stevanović) K, Simonović J (2007) Dynamical absorption and resonances in the sandwich double plate system vibration with elastic layer. Sci Tech Rev LVII(2):1–10

    Google Scholar 

  57. Hedrih (Stevanović) K, Simonović J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambrósio J, et al (eds) 7th EUROMECH solid mechanics conference. Lisbon, Portugal, 7–11 September 2009, CD –MS-24, Paper 294, pp 1–16

    Google Scholar 

  58. Hedrih (Stevanović) K, Simonović J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambrósio J, et al (eds), 7th EUROMECH solid mechanics conference. Lisbon, Portugal, 7–11 September 2009, ESMC 2009 Book of Abstracts ‘Mini’Szmposia, iydanje Instituto Superior Tecnico, Lisbon, APMTAC, pp 589–590, ISBN 9789899 626423

    Google Scholar 

  59. Hedrih (Stevanović) K, Simonović J (2006) Characteristic eigen numbers and frequencies of the transversal vibrations of sandwich system, first South-East European conference on computational mechanics, SEECCM-06, pp 90–94

    Google Scholar 

  60. Hedrih (Stevanović) K, Simonović J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambrósio J, et al (eds), 7th EUROMECH Solid Mechanics Conference. Lisbon, Portugal, 7–11 September 2009, CD –MS-24, Paper 294, pp 1–16

    Google Scholar 

  61. Katica (Stevanović) H, Simonović J (2009) Energy transfer through double curcular plate nonconservative system dynamics. In: Ambrósio J, et al. (eds) 7th EUROMECH solid mechanics conference, Lisbon, Portugal, September 7–11, 2009, ESMC 2009 Book of Abstracts‘Mini’ Symposia, Instituto Superior Tecnico, Lisbon, APMTAC, pp 589–590, ISBN 9789899 626423

    Google Scholar 

  62. Hedrih (Stevanović) K, Kozić P, Pavlović R (1984) O uzajamnom uticaju harmonika u nelinearnim sistemima s malim parametrom, Recueil des travaux de l’Institut Mathem. Nouvell serie, Tome 4(12), 1984, pp. 91–102. Beograd

    Google Scholar 

  63. Hedrih (Stevanović) K, Kozić P, Pavlović R (1985) Stashionarniy i nestashionarniy R-chastotniy analiz kolebaniy sistem s konechnim chislom stepeni svobodu kolebaniy i vzaimnoe vliyanie garmonikov. Theoretical and Applied Mechanics 11, 73–84, 1985, Beograd

    Google Scholar 

  64. Hedrih (Stevanović) K, Pavlovic R, Kozic P, Mitic Sl (1986) Stashionarniy i nestashionarniy chetiri-chastotniy analiz vinuzhdenih kolebaniy tonkoy uprugoy oboloscchki s nachalynimi nepravilynostyami. Theoretical and Applied Mechanics 12, 33–40, 1986, Beograd

    Google Scholar 

  65. Hedrih (Stevanović) K, Mitić Sl, Pavlović R, Kozić P (1986) Mnogochastotnie vinuzhdenie kolebaniya tonkoy uprugoy oboloschki s nachalynimi nepravilynostyami, (Analiticheskiy analiz). Theoretical and Applied Mechanics 12, 41–58, 1986, Beograd

    Google Scholar 

  66. Kozmin A., Mikhlin Yu. and C. Pierre, Transient in a two DOF nonlinear Systems, Nonlinear Dynamics, 2006; Nonlinear Dynamics, Volume 51, Numbers 1–2 / January, 2008, DOI: 10.1007/s11071-007-9198-1

    Google Scholar 

  67. Lacarbonara W, Rega G, Nayfeh AH (2003) Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems. Int J Non-Linear Mech 38(6):851–872

    Article  MathSciNet  MATH  Google Scholar 

  68. Lacarbonara W, Rega G (2003) Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems. Int J Non-Linear Mech 38(6):873–887

    MathSciNet  MATH  Google Scholar 

  69. Lazarov BS, Thomsen JJ (2009) Using high-frequency vibrations and non-linear inclusions to create metamaterials with adjustable effective properties. Int J Non-Linear Mech 44(1):90–97

    Article  Google Scholar 

  70. Lenci S, Rega G (2007) Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique. Int J Non-Linear Mech 42(3):515–528

    Article  MATH  Google Scholar 

  71. Lenci S, Rega G (2008) Control of the homoclinic bifurcation in buckled beams: Infinite dimensional vs reduced order modeling. Int J Non-Linear Mech 43(6):474–489, Jul 2008

    Article  MATH  Google Scholar 

  72. Malatkar P, Nayfeh AH (2003) On the transfer of energy between widely spaced modes in structures. Nonlinear Dyn 31(2):225–242(18)

    Google Scholar 

  73. Mitropolyskiy YuA (1955), Nestashionarnie proshesi v nelinyeynih sistemah, AN USSR, Kiev. (in Russian)

    Google Scholar 

  74. Mitropolyskiy YuA, Mosseenkov BI (1968), Lekciyi po primenyeniyu metodov k recheniyu uravnyeniy v chastnih proizvodnih, Int. Math. AN USSR, Kiev. (in Russian)

    Google Scholar 

  75. Mitropolyskiy YuA (1964) Problemi asimptoti-cheskoy teorii nestashionarnih kolebaniy, Nauka Moskva. (in Russian)

    Google Scholar 

  76. Mitropolyskiy YuA, Mosseenkov BI (1976) Assimptoticheskie recheniya uravnyeniya v chastnih proizvodnih, Vichaya chkola Kiev. (in Russian)

    Google Scholar 

  77. Mitropolskiy YuA, Nguyen Van Dao (2003) Lectures on asymptotic methods of nonlinear dynamics. Vietnam National University Publishing House, Hanoi, p 494

    Google Scholar 

  78. Mitropolskiy YuA (1995) Some problems in the development in nonlinear mechanics theory and applications. Facta Univ Ser Mech Automatic Control Robot 1(5):539–560

    MATH  Google Scholar 

  79. Mitropolskiy YuA (1996) On application of asymptotic methods of nonlinear mechanics for solving some problems of oscillation theory. Facta Univ, Ser Mech Automatic Control Robot 2(6):1–9

    Google Scholar 

  80. Mitropolskiy YuA (1983) Adiabatic processes in nonlinear oscillation systems (Adiabaticheskie processi v nelinyeynih kolebatelynih sistemah), Simpozijum’83 – Nelinearni problemi dinamike, Arandjelovac, 23–25 Novembar 1983. IT-Društvp za mehaniku Srbije, pp. I-1–I-12. (in Russian)

    Google Scholar 

  81. Nayfeh AH, Balachandran B (1990) Experimental investigation of resonantly forced oscillations of a two-degree-of-freedom structure. Int J Non-Linear Mech 25(2–3):199–209

    Article  Google Scholar 

  82. Nayfeh AH, Balachandran B (1989) Modal interactions in dynamical and structural systems. Appl Mech Rev, 42(11), Part 2:175–202. Review article

    Google Scholar 

  83. Nayfeh AH (1993) Transfer of energy from high-frequency to low frequency modes. In: Paper and poster abstracts, IUTAM symposium nonlinearity and chaos in engineering dynamics. Centre NDA UCL, London

    Google Scholar 

  84. Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics. Wiley, New York

    Book  MATH  Google Scholar 

  85. Nayfeh AH (1996) Transfer of energy from high-frequency to low-frequency modes, book of abstracts. In: The second international conference “Asymptotics in Mechanics.” St. Petersburg Marine Technical University, Russia, p 44, 13–16 October 1996

    Google Scholar 

  86. Nayfeh AH (2000) Nonlinear interactions: analytical, computational and experimental methods. Wiley, New York

    MATH  Google Scholar 

  87. Nayfeh AH, Pai PF (2004) Linear and nonlinear structural mechanics. Wiley, New York

    Book  MATH  Google Scholar 

  88. Rašković D (1965) Teorija oscilacija, (theory of oscillations). In: Nauna knjiga (ed) p. 503 (in Serbian)

    Google Scholar 

  89. Rašković D (1985) Teorija elastičnosti (Theory of Elasticity), Naučna knjiga, 1985, 414

    Google Scholar 

  90. Rega G (2004) Nonlinear vibrations of suspended cables–Part I: modeling and analysis. ASME Appl Mech Rev 57(6):443–478

    Article  Google Scholar 

  91. Rega G (2004) Nonlinear vibrations of suspended cables–Part II: deterministic phenomena. ASME Appl Mech Rev 57(6):479–514

    Article  Google Scholar 

  92. Rega G, Lenci S (2005) Identifying evaluating and controlling dynamical integrity measures in nonlinear mechanical oscillators. Nonlinear Anal 63(5–7):902–914. Springer

    Google Scholar 

  93. Stevanović (Hedrih) K (1975) Application of the asymptotic method for the investigation of the nonlinear oscillations of elastic bodies – energy analysis of the oscillatory motions of elastic bodies [in Serbian], Doctoral Dissertation, Faculty of Mechanical engineering, Niš, p 331. (in Serbian) – Supervision: Prof. dr. Ing. Math Danilo P. Rašković (Yugoslavia) and academician Yuriy Alekseevich Mitropolskiy (Ukraine)

    Google Scholar 

  94. Stevanović (Hedrih) K, Rašković D (1974) Investigation of multi-frequencies vibrations in single-frequency regime in nonlinear systems with many degrees of the freedom and with slowchanging parameters. J Nonlinear Vibrat Probl 15:201–202

    Google Scholar 

  95. Stevanović (Hedrih) K (1972) Two-frequency nonstationary forced vibrations of the beams. Math Phys 12:32

    Google Scholar 

  96. Zakrzhevsky M (2008) New concepts of Nonlinear Dynamics: complete bifurcation groups protuberances unstable periodic infinitums and rare attractors. J vibroengineering 10(4):421–441. Latvia

    Google Scholar 

Appendix – References:

  1. Hedrih (Stevanović) K (1975) Selected chapters from theory of nonlinear vibrations (in Serbian), Faculty of Mechanical Engineering, Niš, First Edition 1975, p 180

    Google Scholar 

  2. Hedrih (Stevanović) K (1972) Study of methods of nonlinear vibrations theory (in Serbian), Poligraphy, Faculty of Mechanical Engineering, Niš, Preprint, p 500

    Google Scholar 

  3. Hedrih (Stevanović) K (1972) Teorija nelinearnih oscilacija i primene na nelinearne sisteme automatskog upravljanja (Theory of non-linear oscillations and applications to nonlinear system automatic control), [in Serbian], Faculty of Technical Sciences in Niš,Supervisor D. Rašković

    Google Scholar 

  4. Hedrih (Stevanović) K (1972) Rešrnje jednačone transverzalnih oscilacija jednoraspone grede asimptotskim metodama nelinearne mehanike (Solution of the equation of transversal oscillation one span beam by asymptotic method of nonlinear mechanics), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Technical Sciences in Niš, Supervision: Prof. dr. Ing. Math Danilo P. Rašković (Yugoslavia) and academician Yuriy Alekseevich Mitropolskiy (Ukraine)

    Google Scholar 

  5. Hedrih (Stevanović) K (1975) Application of the Asymptotic Method for the Investigation of the Nonlinear Oscillations of Elastic Bodies – Energy Analysis of the Oscillatory Motions of Elastic Bodies, [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervision: Prof. dr. Ing. Math Danilo P. Rašković (Yugoslavia) and academician Yuriy Alekseevich Mitropolskiy (Ukraine)

    Google Scholar 

  6. Kozić P (1982) Izučavanje nelinearnih torzijskih oscilacija vratila asimptotskom metodom (Investigation of non-linear torsion oscillations of shafts by asymptotic method), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Belgrade, Working Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  7. Kozić P (1990) Stabilnost diskretnih mehaničkih sistema pri dejstvu slučajne pobude (Stability of discrete mechanical systems subjected by random excitations), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  8. Pavlović R (1982) Prilog nelinearnim oscilacijama plitkih cilindričnih ljuski (Contribution to the non-linear oscillations of the shallow shells), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih(Stevanović)

    Google Scholar 

  9. Pavlović RR (1990) Dinamička stabilnost kontinualnih sistema od kompozitnih materaijala pod dejstvu slučajnih pobuda (Dynamic stability of continuous systems made from composite materials subjected to random excitation), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  10. Mitić Sl (1985) Višefrekventna analiza oscilovanja tankih elastišnih ljuski sa konstantim krivinama i početnim nepravilnostima (Multi-frequency analysis of the vibrations of thin shells with constant curvature and initial inperfetions), [in Serbian], Magistar of Sciences Degree thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih(Stevanović)

    Google Scholar 

  11. Mitić Sl (1989) Stabilnost determinističkih i stohastičkih procesa u vibroudarnim sistemima Stability of desterministic and stochastic processes in vibroimpact systems), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  12. Jovanović D (1989) Analiza naponskog i defprmacionog stanja ravno napregnutih ploča sa primenom na eliptično prstenastu ploču (Stress and strain analysis of plane loaded plate with applications to elliptical annular plate), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  13. Mitić Sn (1989) Analiza naponskog i defprmacionog stanja ravno napregnutih ploča (Stress and strain analysis of plane loaded plates), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  14. Pavlov B (1985) Novootkriveni fenomeni u nelunearnim dinamičkim sistemima sa analogijom na sisteme opisane Mayhieu-ovom diferencijalnom jednačinom (New phenomena in non-linear dynamical systems with analogy tio the systems described by Matheu differential equations), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović).

    Google Scholar 

  15. Filipovski A (1995) Energijska analiza longitudinalnih oscilacija štapova promenljivog preseka (Energy analysis longitudinal oscillations of rods with changeable cross sections), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  16. Perić Lj (2004) Prostorna analiza stanja napona i stanja deformacije napregntog piezokeramičkog materijala (Space analysis of stress and straon state of stressed piezoceramiv materials), Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  17.  ∗ Janevski G (2003) Nelinearne oscilacije ploča od kompoyitnih materijala (Nonlinear vibrations of the composite plates), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor R. Pavlović

    Google Scholar 

  18. Simonović J (2008) Fenomeni dinamike mehaničkih sistema složenih struktura (Phenomen of Dynamics of Complex Structure Mecnanical Systems), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  19. Jović S (2009) Energijska analiza dinamike vibroudarnih sistema (Energy analysis of vibroimpact system dynamics), [in Serbian], Magistar of Sciences Degree Thesis, Faculty of Technical Sciences in Kosovska Mitrovica, University of Priština, Supervisor V. Raičević, Project ON144002 Leader K. Hedrih (Stevanović)

    Google Scholar 

  20. Perić Lj (2005) Spregnuti tenzori stanja piezoeletričnih materijala (Coupled tensors of the piesoelectric material states), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  21. Knežević R (2000) Nelinearni fenomeni u dinamici planetnih prenosnika (Nonlinear phenomena in dynamics of planetary gear transmission), [in Serbian], Doctor’s Degree Thesis, Niš, Yugoslavia, Faculty of Mechanical Engineering in Niš, (200). Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  22. Jovanović D (2009) Potencijalna energija i stanje napona u materijalu sa prslinom (Potential energy and stress state in material with crack), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  23.  ∗  Janevski G (2010) Dinamička stabilnost mehaničkih sistema pri dejstvu slučajnih opterećenja (Dynamic stability of mechanical system;oaded by random excitation), Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor Kozić P.

    Google Scholar 

  24. Jović S (2011) Energijska analiza dinamike vibroudarnih sistema sa krivolinijskim putanjama i neidealnim vezama (Energy analysis of vibroimpact system dynamics with curvilinear paths and no ideal constraints), [in Serbian], Doctor’s Degree Thesis, Faculty of Technical Sciences in Kosovska Mitrovica, University of Priština, Supervisor V. Raičević, Projects ON144002 and OI174001 Leader K. Hedrih (Stevanović)

    Google Scholar 

  25. Simonović J (2011) Dinamika i stabilnost hibridnih dinamičkih sistema (Dynamics and Stability of Dynamics Hybrid Systems), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  26. Veljović Lj (2011) Nelinearne oscilacije giro-rotora (Non-linear oscillations of Gyro-rotors), [in Serbian], Doctor’s Degree Thesis, Faculty of Mechanical Engineering in Niš, Supervisor K. Hedrih (Stevanović)

    Google Scholar 

  27. Oscillations of the System with many degree of Freedom and Elastic Bodies with Nonlinear Properties, Basic Scientific Found of Region Niš (1979-1981). Mechanical Engineering Faculty University of Niš

    Google Scholar 

  28. Oscillations of the Special Elements and Systems, B Basic Scientific Found of Region Niš (1981-1986). Some research results included in two Magistar of sciences theses of P. Kozić and R. Pavlović. Mechanical Engineering Faculty University of Niš

    Google Scholar 

  29. Stochastic Processes in Dynamical Systems-Applications on the Mechanical Engineering Systems, Basic Scientific Found of Region Niš (1986-1989). Some research results included in Magistar of sciences theses of Sl. Mitić and in two doctoral dissertations of P. Kozić and R. Pavlović. Mechanical Engineering Faculty University of Niš

    Google Scholar 

  30. Nonlinear Deterministic and Stochastic Processes with Applications in Mechanical Engineering Systems, Ministry of Science and Technology Republic of Serbia, (1990-1995). Some research results included in two Magistar of sciences theses of Blagoj Pavlov and Alekdsandar Filipovski and in a doctoral dissertation of Sl. Mitić. Mechanical Engineering Faculty University of Niš

    Google Scholar 

  31. Sub-Projects 5.1. Thema: Stress and Strain State of the Deformable Bodies and 5.2. Theme: Vector Interpretation of Body Kinetic Parameters, as a part of Project: Actual Problems on Mechanic (1990-1995), Project Leader prof. dr Mane Šašić Ministry of Sciences, Technology and Development of Republic Serbia. Some research results included in three Magistar of sciences theses of Ljubiša Perić, Dragan Jovanović and Snežana Mitić. Mathematical Institute SANU

    Google Scholar 

  32. Sub-Project: 04M03A Current Problems on Mechanics and Applications (1996-2000), as a part of Project: Methods and Models in Theoretical, Industrial and Applied Mathematics, Project Leader prof. dr Gradimir Milovanović, Ministry of Sciences, Technology and Development of Republic Serbia. Mathematica Institute SANU

    Google Scholar 

  33. Project 1616 Real Problems on Mechanics (2002-2004), Basic Science-Mathematics and Mechanics, Ministry of Sciences, Technology and Development of Republic Serbia. Some research results included in two doctoral dissertations of Ljubiša Perić and Dragan Jovanović. Mathematical Institute SANU and Mechanical Engineering Faculty University of Niš.

    Google Scholar 

  34. Project ON1828 Dynamics and Control of active Structures (2001-2005), Basic Science-Mathematics and Mechanics, Ministry of Sciences, Technology and Development of Republic Serbia. Some research results included in two doctoral dissertations of Ljubiša Perić and Dragan Jovanović. Mechanical Engineering Faculty University of Niš.

    Google Scholar 

  35. Project ON144002 -Theoretical and Applied Mechanics of the Rigid and Solid Bodies. Mechanics of Materials (2006-2010). Support: Ministry of Sciences and Environmental Protection of Republic of Serbia. Some research results included in two Magistar of sciences theses of Srdjan Jović, and Julijana Simonović and in four doctoral dissertations of Dragan Jovanović, Srdjan Jović, Ljiljana Veljović and Julijana Simonović. Institution Coordinator: Mathematical Institute Serbian Academy of Sciences and Arts and Mechanical Engineering Faculty University of Niš.

    Google Scholar 

  36. ON174001 - Dynamics of hybrid systems with complex structures. Mechanics of materials. (2011-2014), Ministry of Sciences and Technology of Republic of Serbia. Some research results included in thre doctoral dissertations of Srdjan Jović, Ljiljana Veljović and Julijana Simonović. Institution Coordinator: Mathematical Institute Serbian Academy of Sciences and Arts and Mechanical Engineering Faculty University of Niš.

    Google Scholar 

  37. Project 1409 Stability and nonlinear oscillations of viscoelastic and composite continuous systems, (2002–2005). Support: Ministry of Sciences and Environmental Protection of Republic of Serbia. Some research results included in a Magistar of sciences these of Goran Janevski. Mechanical Engineering Faculty University of Niš.

    Google Scholar 

  38. OI 144023 Deterministic and stochastic stability of mechanical systems (2006–2010.), Ministry of Sciences and Technology of Republic of Serbia. Some research results included in a doctoral dissertations of Goran Janevski (supervisor Predrag Kozić). Mechanical Engineering Faculty University of Niš.

    Google Scholar 

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Acknowledgments

This paper is dedicated to the memory of my Professors Dr. Ing. Math Danilo P. Rašković (1910–1985, Serbia) and Academician Yuriy Alekseevich Mitropolskiy (1917–2008, Ukraine). Parts of this research were supported by the Ministry of Sciences of Republic Serbia through the Mathematical Institute SANU Belgrade Grants ON144002 “Theoretical and Applied Mechanics of Rigid and Solid Body. Mechanics of Materials” and OI 174001” Dynamics of hybrid systems with complex structures. Mechanics of materials”, and Faculty of Mechanical Engineering University of Niš.

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  1. Head of Department for Mechanics and Faculty of Mechanical Engineering, Mathematical Institute SANU Belgrade, University of Niš, Knez Mihailova 36/III, 11009, Belgrade, Serbia

    Katica R. (Stevanović) Hedrih

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Hedrih, K.R.(. (2012). Energy and Nonlinear Dynamics of Hybrid Systems. In: Luo, A., Machado, J., Baleanu, D. (eds) Dynamical Systems and Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0454-5_2

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