Skip to main content
SpringerLink
Log in

Detecting the weak high-frequency character signal by vibrational resonance in the Duffing oscillator

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The weak high-frequency character signal (HCS) cannot be detected substantially by the traditional vibrational resonance (VR) theory. In this paper, by introducing the scale transformation, the HCS in the original system can be transformed to the low-frequency character signal in the rescaled system. As we know, the two systems are equivalent and the VR can occur at low frequency in the rescaled system. Then, the VR can also occur at high frequency in the original system. We take the underdamped bistable system and the overdamped bistable system of the Duffing oscillator as examples. The method proposed in this paper is verified by both theoretical analysis and numerical simulations. The results obtained by the two ways are in good agreement. The results in this paper provide a tool to detect the weak character signal with arbitrary frequency in the engineering problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Landa, P.S., McClintock, P.V.E.: Vibrational resonance. J. Phys. A Math. Gen. 33, 433–438 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A Math. Gen. 14, L453–L457 (1981)

    Article  MathSciNet  Google Scholar 

  3. Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998)

    Article  Google Scholar 

  4. Anishchenko, V.S., Safonova, M.A., Chua, L.O.: Stochastic resonance in Chua’s circuit. Int. J. Bifurc. Chaos 2, 397–401 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duan, F., Chapeau-Blondeau, F., Abbott, D.: Double-maximum enhancement of signal-to-noise ratio gain via stochastic resonance and vibrational resonance. Phys. Rev. E 90, 860–877 (2014)

    Article  Google Scholar 

  6. Chizhevsky, V.N., Giacomelli, G.: Experimental and theoretical study of the noise-induced gain degradation in vibrational resonance. Phys. Rev. E 70, 062101 (2004)

    Article  Google Scholar 

  7. Chizhevsky, V.N., Giacomelli, G.: Improvement of signal-to-noise ratio in a bistable optical system Comparison between vibrational and stochastic resonance. Phys. Rev. E 71, 011801(R) (2005)

    Article  Google Scholar 

  8. Chizhevsky, V.N., Giacomelli, G.: Experimental and theoretical study of vibrational resonance in a bistable system with asymmetry. Phys. Rev. E 73, 022103 (2006)

    Article  Google Scholar 

  9. Deng, B., Wang, J., Wei, X.L., Tsang, K.M., Chan, W.L.: Vibrational resonance in neuron populations. Chaos 20, 3–4 (2010)

    Article  Google Scholar 

  10. Sun, J.B., Deng, B., Liu, C., Yu, H.T., Wang, J., Wei, X.L., Zhao, J.: Vibrational resonance in neuron populations with hybrid synapses. Appl. Math. Model. 37, 6311–6324 (2013)

    Article  MathSciNet  Google Scholar 

  11. Yu, H.T., Guo, X.M., Wang, J., Deng, B., Wei, X.L.: Vibrational resonance in adaptive small-world neuronal networks with spike-timing-dependent plasticity. Phys. A 436, 170–179 (2015)

    Article  MathSciNet  Google Scholar 

  12. Yang, J.H.: Vibrational resonance in fractional-order anharmonic oscillators. Chin. Phys. Lett. 29, 104501 (2012)

    Article  Google Scholar 

  13. Yang, J.H., Zhu, H.: Vibrational resonance in Duffing systems with fractional-order damping. Chaos 22, 013112 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yang, J.H., Sanjuán Miguel, A.F., Tian, F., Yang, H.F.: Saddle-node bifurcation and vibrational resonance in a fractional system with an asymmetric bistable potential. Int. J. Bifurcat. Chaos 25, 1550023 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, C.J., Yang, K.L.: Vibrational resonance in bistable gene transcriptional regulatory system. Chin. J. Phys. 50, 607–618 (2012)

    Google Scholar 

  16. Yang, L.J., Liu, W.H., Yi, M., Wang, C.J., Zhu, Q.M., Zhang, X., Jia, Y.: Vibrational resonance induced by transition of phase-locking modes in excitable systems. Phys. Rev. E 86, 016209 (2012)

    Article  Google Scholar 

  17. Wang, C.J., Yang, K.L., Qu, S.X.: Vibrational resonance in a discrete neuronal model with time delay. Int. J. Mod. Phys. B 28, 124–127 (2014)

    MathSciNet  Google Scholar 

  18. Jeevarathinam, C., Rajasekar, S., Sanjuán Miguel, A.F.: Vibrational resonance in groundwater-dependent plant ecosystems. Ecol. Complex. 15, 33–42 (2013)

    Article  Google Scholar 

  19. Jeevarathinam, C., Rajasekar, S., Sanjuán Miguel, A.F.: Effect of multiple time-delay on vibrational resonance. Chaos 23, 013136 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Daza, A., Wagemakers, A., Rajasekar, S., Sanjuán Miguel, A.F.: Vibrational resonance in a time-delayed genetic toggle switch. Commun. Nonlinear Sci. Numer. Simul. 18, 411–416 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Coccolo, M., Litak, G., Seoane, J.M., Sanjuán Miguel, A.F.: Energy harvesting enhancement by vibrational resonance. Int. J. Bifurc. Chaos 24, 1430019 (2014)

    Article  Google Scholar 

  22. Yang, J.H., Sanjuán Miguel, A.F., Liu, H.G.: Vibrational subharmonic and superharmonic resonances. Commun. Nonlinear Sci. Numer. Simul. 30, 362–372 (2015)

    Article  MathSciNet  Google Scholar 

  23. Balachandran, B., Magrab, E.B., Balachandran, A.P.: Vibrations. Cengage Learning, Melbourne (2008)

    Google Scholar 

  24. Tan, J.Y., Chen, X.F., Wang, J.Y., Chen, H.X., Cao, H.R., Zi, Y.Y., He, Z.J.: Study of frequency-shifted and re-scaling stochastic resonance and its application to fault diagnosis. Mech. Syst. Signal Process. 23, 811–822 (2009)

    Article  Google Scholar 

  25. Li, J.M., Chen, X.F., He, Z.J.: Multi-stable stochastic resonance and its application research on mechanical fault diagnosis. J. Sound Vib. 332, 5999–6015 (2013)

    Article  Google Scholar 

  26. Lu, S.L., He, Q.B., Kong, F.R.: Effects of underdamped step-varying second-order stochastic resonance for weak signal detection. Digit. Signal Process. 36, 93–103 (2015)

    Article  MathSciNet  Google Scholar 

  27. He, Q.B., Wang, J., Liu, Y.B., Dai, D.Y., Kong, F.R.: Multiscale noise tuning of stochastic resonance for enhanced fault diagnosis in rotating machines. Mech. Syst. Signal Process. 28, 443–457 (2012)

    Article  Google Scholar 

  28. Thomsen, J.J.: Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  29. Blekhman, I.I.: Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach. Applications. World Scientific, Singapore (2000)

    Book  Google Scholar 

  30. Blekhman, I.I., Landa, P.S.: Conjugate resonances and bifurcations in nonlinear systems under biharmonical excitation. Int. J. Nonlinear Mech. 39, 421–426 (2004)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    H. G. Liu, X. L. Liu, J. H. Yang & G. Cheng

  2. Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou, 221116, China

    H. G. Liu & X. L. Liu

  3. Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933, Móstoles, Madrid, Spain

    Miguel A. F. Sanjuán

Authors
  1. H. G. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. X. L. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. H. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Miguel A. F. Sanjuán
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. G. Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. G. Liu.

Additional information

We acknowledge financial supports by the Fundamental Research Funds for the Central Universities (Grant No. 2015XKMS023), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Top-notch Academic Programs Project of Jiangsu Higher Education Institutions, and the Spanish Ministry of Economy and Competitiveness (Grant No. FIS2013-40653-P).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H.G., Liu, X.L., Yang, J.H. et al. Detecting the weak high-frequency character signal by vibrational resonance in the Duffing oscillator. Nonlinear Dyn 89, 2621–2628 (2017). https://doi.org/10.1007/s11071-017-3610-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3610-2

Keywords

Search

Navigation