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Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system)

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Ukrainian Mathematical Journal Aims and scope

The analysis of the energy transfer between subsystems coupled in a hybrid system is an urgent problem for various applications. We present an analytic investigation of the energy transfer between linear and nonlinear oscillators for the case of free vibrations when the oscillators are statically or dynamically connected into a double-oscillator system and regarded as two new hybrid systems, each with two degrees of freedom. The analytic analysis shows that the elastic connection between the oscillators leads to the appearance of a two-frequency-like mode of the time function and that the energy transfer between the subsystems indeed exists. In addition, the dynamical linear constraint between the oscillators, each with one degree of freedom, coupled into the hybrid system changes the dynamics from single-frequency modes into two-frequency-like modes. The dynamical constraint, as a connection between the subsystems, is realized by a rolling element with inertial properties. In this case, the analytic analysis of the energy transfer between linear and nonlinear oscillators for free vibrations is also performed. The two Lyapunov exponents corresponding to each of the two eigenmodes are expressed via the energy of the corresponding eigentime components.

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References

  1. K. Hedrih (Stevanović), “Interpretation of the transfer of energy from high-frequency to low-frequency modes by averaging asymptotic method Krylov-Bogolyubov-Mitropolsky,” in: Ali H. Nayfeh and K. V. Rozdestvenskii (editors), Asymptotics in Mech.: Proc. Second Int. Conf., St. Petersburg (1997), pp. 97–104.

  2. K. Hedrih (Stevanović), “Energetic analysis of oscillatory processes and the modes in nonlinear systems,” in: Solid Mech., Serbian-Greek: Symp., 77, Book 3, Sci. Meetings Serb. Acad. Sci. Arts, SANU Belgrade (1997), pp. 137–146.

    Google Scholar 

  3. K. Hedrih (Stevanović), “Multifrequency forced vibrations of thin elastic shell,” in: D. H. van Campen, M. D. Lazurko, and W. P. J. M. van den Over (editors), CD Proc. Fifth EUROMECH Nonlinear Dynamics Conf., Univ. Technology, Eindhoven (2005), pp. 2417–2426.

    Google Scholar 

  4. K. Hedrih (Stevanović), “The dissipation function of a nonconservative system of mass particles,” Tensor, 63, No. 2, 176–186 (2002).

    MATH  MathSciNet  Google Scholar 

  5. K. Hedrih (Stevanović), “Mathematical analogy and phenomenological mapping: vibrations of multi-plate and multi-beam homogeneous systems,” in: L. Bereteu, T. Cioara, M. Toth-Tascau, and C. Vigaru (editors), Sci. Bul. “Politekhnica” Univ. Timisoara, 50, Spec. Issue, Romania (Trans. Mech.) (2005), pp. 1224–6077.

  6. K. Hedrih (Stevanović), “Partial fractional order differential equations of transversal vibrations of creep-connected double plate systems,” in: A. Le Mahaute, J. A. Tenreiro Machado, J.-C. Trigeassou, and J. Sabatier (editors), Fract. Different. Appl. (2005), pp. 289–302.

  7. K. Hedrih (Stevanović), “Frequency equations of small oscillations in mixed systems of coupled discrete and continuous subsystems,” Mekh. Tverd. Tela, 33, 174–189 (2003).

    MATH  Google Scholar 

  8. K. Hedrih (Stevanović), “Eigen amplitude vectors and functions extended orthogonality of small oscillations in mixed systems of the coupled discrete and continuous subsystems,” Facta Univ., Ser. Mech., Automat. Contr. Rob., 4, No. 17, 225–243 (2005).

    Google Scholar 

  9. K. Hedrih (Stevanović), “Transversal vibrations of the axially moving double belt system with creep layer,” in: Preprints, 2nd IFAC Workshop Fract. Different. Appl., (Porto, July 19–21, 2006), Porto, Portugal (2006), pp. 230–235; http:/www.iser.ipp.pl.

  10. K. Hedrih (Stevanović), “Transversal vibrations of creep-connected multi-plate homogeneous systems,” in: D. H. van Campen, M. D. Lazurko, and W. P. J. M. van den Over (editors), CD Proc. Fifth EUROMECH Nonlinear Dynamics Conf,. Univ. Technology, Eindhoven (2005), pp. 1445–1454.

    Google Scholar 

  11. K. Hedrih (Stevanović), “Discrete continuum method,” in: Z. H. Zao, M. W. Zuang, and W. X. Zhong (editors), Computational Mechanics, WCCM VI in Conjunction with APCOM'04 (2004), pp. 1–11.

  12. K. Hedrih (Stevanović), “Creep vibrations of fractional derivative order constitutive relation deformable bodies,” Appl. Mech. Amer., 10, 548–551 (2004).

    Google Scholar 

  13. K. Hedrih (Stevanović), “On rheonomic systems with equivalent holonomic conservative systems applied to the nonlinear dynamics of the Watt's regulator,” in: Proc.of the Eleventh World Congr. in Mechanism and Machine Sci. (IFToMM):., Vol. 2, China Machine Press, Tianjin, China, (2004), pp. 1475–1479.

    Google Scholar 

  14. Ali H. Nayfeh, “Transfer of energy from high-frequency to low-frequency modes,” in: Second Internat. Conf. “Asymptotics in Mech.” St.-Petersburg Marine Techn. Univ., (St. Petersburg, October 13–16, 1996), St. Petersburg, Russia (1996), p. 44.

  15. K. Stevanović (Hedrih), 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, Niš, Yugoslavia (1975).

  16. D. Rašković, Theory of Oscillations [in Serbian], Naučna Knjiga, Yugoslavia (1965).

    Google Scholar 

  17. K. Hedrih (Stevanović), “Transversal vibrations of double-plate systems,” Acta Mech. Sinica, 22, 487–501 (2006).

    Article  Google Scholar 

  18. K. Hedrih (Stevanović), “Modes of the homogeneous chain dynamics,” Signal Proc., 86, 2678–2702 (2006).

    Article  Google Scholar 

  19. K. Hedrih (Stevanović), “Integrity of dynamical systems,” J. Nonlin. Anal., 63, 854–871 (2005).

    Article  MathSciNet  Google Scholar 

  20. K. Stevanović (Hedrih) and D. Rašković, “Investigation of multi-frequencies vibrations in single-frequency regime in nonlinear systems with many degrees of the freedom and with slowchanging parameters,” J. Nonlin. Vibrat. Probl., No. 15, 201–202 (1974).

  21. Yu. A. Mitropolskii, Nonstationary Processes in Nonlinear Systems [in Russian], Izd. Akad. Nauk Ukr. SSR, Kiev (1955).

    Google Scholar 

  22. Yu. A. Mitropolskii, Problems of the Asymptotic Theory of Nonstationary Oscillations [in Russian], Nauka, Moscow (1964).

    Google Scholar 

  23. Yu. A. Mitropolskii, “Some problems in the development of nonlinear mechanics, theory and applications,” Facta Univ., Ser. Mech., Automat. Contr. Rob., 1, No. 5, 539–560 (1995).

    Google Scholar 

  24. Yu. A. Mitropolskii, “On the application of asymptotic methods of nonlinear mechanics for solving some problems of oscillation theory,” Facta Univ., Ser. Mech., Automat. Contr. Rob., 2, No. 6, 1–9 (1996).

    Google Scholar 

  25. K. Hedrih (Stevanović), “One-frequency nonlinear forced vibrations of uniform beams,” Theor. Appl. Mech., No. 4, 39–50 (1978).

  26. K. Hedrih (Stevanović), “One-frequency proper nonlinear vibration of thin plate,” Theor. Appl. Mech., No. 4, 51–65 (1978).

  27. K. Hedrih (Stevanović) and Sl. Mitić, “Dvofrekventne oscilacije plitke ljuske sa konacnim deformacijama i uzajamni uticaj harmonica,” in: Nelinearni Problemi Dinamike [in Serbian], Arandjelovac (1983), pp. 197–203.

  28. K. Hedrih (Stevanović), P. Kozić, and R. Pavlović, “O uzajamnom uticaju harmonika u nelinearnim sistemima s malim parametrom,” Rec. Trav. Inst. Math. Nouv. Ser., 4, 91–102 (1984).

    Google Scholar 

  29. K. Hedrih (Stevanović), “Multifrequency forced vibrations of thin elastic shells with positive Gauss' curvature and finite deformations,” Theor. Appl. Mech., No. 11, 59–72 (1985).

    Google Scholar 

  30. K. Hedrih (Stevanović) and B. Pavlov, “Strange attractors of the phase portrait of motion of a heavy material point along the circle with an oscillating centre and under the influence of two-frequency couple,” in: Proc. Second Internat. Conf. on Nonlinear Mech., 514, Abstrts, Beijing (1993), pp. 938–944.

    Google Scholar 

  31. K. Stevanović (Hedrih), “Two-frequency nonstationary forced vibrations of the beams,” Math. Phys., 12 (1972).

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

  33. K. Hedrih (Stevanović), “A trigger of coupled singularities,” Meccanica, 39, No. 3, 295–314 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  34. K. Hedrih (Stevanović), Homoclinic Orbits Layering in the Coupled Rotor Nonlinear Dynamics and Chaotic Clock Models, Springer (2005).

  35. K. Hedrih (Stevanović), Contribution to the Coupled Rotor Nonlinear Dynamics, Nonlinear Sci., Acad. Nonlinear Sci., Belgrade (2004).

    Google Scholar 

  36. K. Hedrih (Stevanović), “Phase portraits and homoclinic orbits visualization of nonlinear dynamics of multiple step reductor/multiplier,” in: Eleventh World Congr. in Mechanism and Machine Sci. (IFToMM): Proc., Vol. 2, China Machine Press, Tianjin, China (2004), pp. 1508–1512.

    Google Scholar 

  37. K. Hedrih (Stevanović) and Lj. Veljović, Nonlinear dynamics of the heavy gyro-rotor with two rotating axes,” Facta Univ., Ser. Mech., Automat. Contr. Rob., 4, No. 16, 55–68 (2004).

    Google Scholar 

  38. K. Hedrih (Stevanović) and J. Simonović, “Nonlinear phenomena in the dynamics of a car model,” Facta Univ., Ser. Mech., Automat. Contr. Rob., 3, No. 14, 865–879 (2003).

    Google Scholar 

  39. K. Hedrih (Stevanović), “Nonlinear dynamics of a heavy material particle along circle which rotates and optimal control,” in: G. M. L. Gladwell (editor), Solid. Mech. Appl., 26 (2005).

  40. K. Hedrih (Stevanović), Vector Method of the Heavy Rotor Kinetic Parameter Analysis and Nonlinear Dynamics, Niš, Serbia (2001).

  41. K. Hedrih (Stevanović), “Energy transfer in the double-plate system dynamics,” Acta Mech. Sinica, 24, No. 3, 331–344 (2008).

    Article  Google Scholar 

  42. K. Hedrih (Stevanović) and J. Simonović, “Transversal vibrations of a double circular plate system with viscoelastic layer excited by a random temperature field,” Int. J. Nonlin. Sci. Numer. Simulation, 9, No. 1, 47–50 (2008).

    Google Scholar 

  43. K. Hedrih (Stevanović) and J. Simonović, “Transversal vibrations of a nonconservative double circular plate system,” Facta Univ., Ser. Mech., Automat. Contr. Rob., 6, No. 1, 1–64 (2007).

    MATH  Google Scholar 

  44. G. Janevski, “Two-frequency nonlinear vibration of the antis metric laminated and angle-play plate,” Facta Univ., Ser. Mech., Automat. Contr. Rob., 4, No. 17, 345–358 (2005).

    MATH  Google Scholar 

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Authors and Affiliations

  1. University of Niš, Niš, Serbia

    K. Hedrih (Stevanović)

  2. Mathematical Institute of SANU, Belgrade, Serbia

    K. Hedrih (Stevanović)

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Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 796–814, June, 2008.

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Hedrih (Stevanović), K. Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system). Ukr Math J 60, 927–949 (2008). https://doi.org/10.1007/s11253-008-0101-0

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  • DOI: https://doi.org/10.1007/s11253-008-0101-0

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