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Parametrically Excited Vibrations in a Nonlinear Damped Triple-Well Oscillator with Resonant Frequency

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Abstract

Purpose

This paper deals with the parametrically excited vibrations and mode transitions of a nonlinear damped triple-well oscillator in detail. The multiple timescale structure of the triple-well oscillator with resonant frequency is revealed. We present the correct predictions of the region of parameter space where the periodic orbits alternate between epochs of fast and slow motions in such excited vibrations.

Methods

Typically, one can treats the cosinoidal part of the perturbation as a singular variable of the whole system, which only remains autonomous form. We study the critical manifold and bifurcation structure of this system. Since the attracting manifold terminated at a fold leading to the change of local stability, the corresponding singular orbit would jump to another stable branch. Characteristic features of this method are folded singularities and structure of the critical manifold.

Result

Parametrically excited vibrations with resonant frequency could reflect multiple timescale structure. A singular periodic orbit can be constructed, which consists of distinct segments. Starting along the critical manifold with small damped oscillations, then there is a fast transition that flows to another attracting branch for the folded singularity. As the fifth stiffness value varies, mixed-mode oscillations with more jumps are produced. Moreover, the jump-doubling behaviors can be presented under the flip of excitation frequency ratio.

Conclusion

As we show in the paper, damping and external excitations with resonant frequency ratio in nonlinear vibration system could construct singular periodic orbits. It is possible for the trajectory to jump away from the folded singularity with finite speed. Then, it performs small damped oscillations about one attracting branch of stable foci. By taking advantage of time scale separation, we demonstrate the origin of such bursting behaviors.

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References

  1. Guckenheimer J, Holmes PJ (1983) Nonlinear oscillations, dynamical systems and bifurcation of vector field. Springer, New York

    Book  Google Scholar 

  2. Guckenheimer J (2008) Singular Hopf bifurcation in systems with two slow variables. SIAM J Appl Dyn Syst 7(4):1355–1377

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  4. Kovacic I, Radomirovic D (2017) Mechanical vibrations: fundamentals with solved examples. John Wiley & Sons, Chichester

    Google Scholar 

  5. Hartzell S, Bartlett MS, Virgin L, Porporato A (2015) Nonlinear dynamics of the CAM circadian rhythm in response to environmental forcing. J Theor Biol 368:83–94

    Article  Google Scholar 

  6. Meng H, Sun XT, Xu J, Wang F (2020) The generalization of equal-peak method for delay-coupled nonlinear system. Physica D 403:132340

    Article  MathSciNet  Google Scholar 

  7. Wang YQ, Ye C, Zu JW (2019) Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets. Aerosp Sci Technol 85:359–370

    Article  Google Scholar 

  8. Di Giovanni G, Bernardini D (2020) Vibration damping performances of buildings with moving facades under harmonic excitation. J Vib Eng Technol

  9. Mechri C, Scalerandi M, Bentahar M (2019) Separation of damping and velocity strain dependencies using an ultrasonic monochromatic excitation. Phys Rev Appl 11(5):054050

    Article  Google Scholar 

  10. Csernak G, Stepan G (2006) On the periodic response of a harmonically excited dry friction oscillator. J Sound Vib 295(3–5):649–658

    Article  MathSciNet  Google Scholar 

  11. Huera-Huarte FJ (2018) Dynamics and excitation in a low mass-damping cylinder in cross-flow with side-by-side interference. J Fluid Mech 850:370–400

    Article  Google Scholar 

  12. Mora K, Gottlieb O (2017) Parametric excitation of a microbeam-string with asymmetric electrodes: multimode dynamics and the effect of nonlinear damping. J Vib Acoust Transact ASME 139(4):040903

    Article  Google Scholar 

  13. Leninger M, Marsiglia WM, Jerschow A, Traaseth NJ (2018) Multiple frequency saturation pulses reduce CEST acquisition time for quantifying conformational exchange in biomolecules. J Biomol NMR 71(1):19–30

    Article  Google Scholar 

  14. Kempitiya T, Sierla S, De Silva D, Yli-Ojanpera M, Alahakoon D, Vyatkin V (2020) An artificial intelligence framework for bidding optimization with uncertainty in multiple frequency reserve markets. Appl Energy 280:115918

    Article  Google Scholar 

  15. Bronckers LA, Roc’h A, Smolders AB (2019) A new design method for frequency-reconfigurable antennas using multiple tuning components. IEEE Trans Antennas Propag 67(12):7285–7295

    Article  Google Scholar 

  16. Aquino GP, Mendes LL (2020) Sparse code multiple access on the generalized frequency division multiplexing. EURASIP J Wirel Commun Netw 1:212

    Article  Google Scholar 

  17. Lin SD, Butler JE, Boswell-Ruys CL, Hoang P, Jarvis T, Gandevia SC, McCaughey EJ (2019) The frequency of bowel and bladder problems in multiple sclerosis and its relation to fatigue: a single centre experience. PLoS ONE 14(9):e0222731

    Article  Google Scholar 

  18. Nemtsov I, Aviv H, Mastai Y, Tischler YR (2019) Polarization dependence of low-frequency vibrations from multiple faces in an organic single crystal. Curr Comput-Aided Drug Des 9(8):425

    Google Scholar 

  19. Lee H, Poguluri SK, Bae YH (2018) Performance Analysis of multiple wave energy converters placed on a floating platform in the frequency domain. Energies 11(2):406

    Article  Google Scholar 

  20. Farjami S, Alexander RPD, Bowie D, Khadra A (2020) Bursting in cerebellar stellate cells induced by pharmacological agents: sequential spike adding. Plos Comput Biol 16(12):e1008463

    Article  Google Scholar 

  21. Leutcho GD, Kengne J, Kengne LK, Akgul A, Pham VT, Jafari S (2020) A novel chaotic hyperjerk circuit with bubbles of bifurcation: mixed-mode bursting oscillations, multistability, and circuit realization. Phys Scr 95(7):075216

    Article  Google Scholar 

  22. Kasthuri P, Unni VR, Sujith RI (2019) Bursting and mixed mode oscillations during the transition to limit cycle oscillations in a matrix burner. Chaos 29(4):043117

    Article  MathSciNet  Google Scholar 

  23. Vijay SD, Kingston SL, Thamilmaran K (2019) Different transitions of bursting and mixed-mode oscillations in Lienard system. AEÜ-Int J Electron Commun 111:152898

    Article  Google Scholar 

  24. Kingston SL, Thamilmaran K (2017) Bursting oscillations and mixed-mode oscillations in driven Lienard system. Int J Bifurcation Chaos 27(7):1730025

    Article  MathSciNet  Google Scholar 

  25. Teka W, Tabak J, Vo T, Wechselberger M, Bertram R (2011) The dynamics underlying pseudo-plateau bursting in a pituitary cell model. J Math Neurosci 1(12)

  26. Teka W, Tsaneva-Atanasova K, Bertram R, Tabak J (2011) From plateau to pseudo-plateau bursting: making the transition. Bull Math Biol 73(6):1292–1311

    Article  MathSciNet  Google Scholar 

  27. McCormack G, Nath R, Li WB (2020) Nonlinear dynamics of Rydberg-dressed Bose-Einstein condensates in a triple-well potential. Phys Rev A 102(6):063329

    Article  MathSciNet  Google Scholar 

  28. Goodman RH (2017) Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential. Phys D-Nonlinear Phenomena 359:39–59

    Article  MathSciNet  Google Scholar 

  29. Siewe MS, Cao HJ, Sanjuán MAF (2009) On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potential. Chaos Solitons Fractals 41(2):772–782

    Article  Google Scholar 

  30. Miwadinou CH, Monwanou AV, Hinvi LA, Koukpemedji AA, Ainamon C, Orou JBC (2016) Melnikov chaos in a modified Rayleigh-Duffing oscillator with phi(6) potential. Int J Bifurcation Chaos 26(5):1650085

    Article  Google Scholar 

  31. Alih SC, Vafaei M, Ismail N, Pabarja A (2018) Experimental study on a new damping device for mitigation of structural vibrations under harmonic excitation. Earthq Struct 14(6):567–576

    Google Scholar 

  32. Tsiatas GC, Charalampakis AE, Tsopelas P (2020) A comparative study of linear and nonlinear mass damping systems under seismic excitation. Eng Struct 219:110926

    Article  Google Scholar 

  33. Siewe MS, Tchawoua C, Woafo P (2010) Melnikov chaos in a periodically driven Rayleigh-Duffing oscillator. Mech Res Commun 37(4):363–368

    Article  Google Scholar 

  34. Jeevarathinam C, Rajasekar S, Sanjuán MAF (2011) Theory and numerics of vibrational resonance in duffing oscillators with time-delayed feedback. Phys Rev E 83(6):066205

    Article  Google Scholar 

  35. Delnavaz A, Mahmoodi SN, Jalili N, Zohoor H (2010) Linear and nonlinear approaches towards amplitude modulation atomic force microscopy. Curr Appl Phys 10(6):1416–1421

    Article  Google Scholar 

  36. Shimizu K, Sekikawa M, Inaba N (2011) Mix-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation. Phys Lett A 375(14):1566–1569

    Article  Google Scholar 

  37. Roberts A, Widiasih E, Wechselberger M, Jones CKRT (2015) Mixed mode oscillations in a conceptual climate model. Phys D-Nonlinear Phenomena 292:70–83

    Article  MathSciNet  Google Scholar 

  38. Wechselberger M, Weckesser W (2009) Bifurcations of mixed-mode oscillations in a stellate cell model. Phys D-Nonlinear Phenomena 238(16):1598–1614

    Article  Google Scholar 

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Acknowledgements

The work is supported by the National Natural Science Foundation of China (Grant No. 12072165).

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Authors and Affiliations

  1. School of Science, Nantong University, Nantong, 226019, People’s Republic of China

    Daomin Chen, Ning Wang & Yue Yu

  2. Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon Tong, Hong Kong, Special Administrative Region, People’s Republic of China

    Zhenyu Chen & Yue Yu

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  1. Daomin Chen
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  2. Ning Wang
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  3. Zhenyu Chen
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  4. Yue Yu
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Correspondence to Yue Yu.

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Chen, D., Wang, N., Chen, Z. et al. Parametrically Excited Vibrations in a Nonlinear Damped Triple-Well Oscillator with Resonant Frequency. J. Vib. Eng. Technol. 10, 781–788 (2022). https://doi.org/10.1007/s42417-021-00408-5

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  • DOI: https://doi.org/10.1007/s42417-021-00408-5

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