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Chaotification of quasi-zero-stiffness system with time delay control

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Abstract

Since the control energy for chaotification of Duffing-type dynamics system depends on the linear stiffness of the system, a quasi-zero-stiffness (QZS) isolated system with time delay control is introduced for line spectrum reconstruction of acoustic noise of machinery vibration of underwater vehicles. To investigate the critical condition for chaotification, the stability analysis of the dynamic system with time delay is conducted by the generalized Sturm criterion, and the analytical expression for critical control parameters is obtained. By converting the time-delayed differential equations of the critical condition into discrete mappings through a finite difference scheme, the Lyapunov criterion for chaotification of the QZS system is acquired, which yields the same critical time delay control as the results obtained by the generalized Sturm criterion. Effective chaotification by setting control parameters in the unstable region is demonstrated by numerical examples, which proves the accurate guidance of the stability analysis on chaotification design.

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References

  1. Adomaitienė, E., Mykolaitis, G., Bumelienė, S., Tamaše-vičius, A.: Adaptive nonlinear controller for stabilizing saddle-type steady states of dynamical systems. Nonlinear Dyn. 82, 1743–1753 (2015)

  2. Aghababa, M., Aghababa, H.: Chaos suppression of rotational machine systems via finite-time control method. Nonlinear Dyn. 69, 1881–1888 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Li, Z., Fang, J.-A., Zhang, W., Wang, X.: Delayed impulsive synchronization of discrete-time complex networks with distributed delays. Nonlinear Dyn. 82, 2081–2096 (2015)

    Article  MathSciNet  Google Scholar 

  4. Wang, H., Wu, J.-P., Sheng, X.-S., Wang, X., Zan, P.: A new stability result for nonlinear cascade time-delay system and its application in chaos control. Nonlinear Dyn. 80, 221–226 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo, L., Hu, M., Xu, Z., Hu, A.: Synchronization and chaos control by quorum sensing mechanism. Nonlinear Dyn. 73, 1253–1269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Howard, C.Q.: Recent developments in submarine vibration isolation and noise control. In: Conference 2011 Presentations & Proceedings on USB (2011)

  7. Liu, S., Yu, X., Zhu, S.: Study on the chaos anti-control technology in nonlinear vibration isolation system. J. Sound Vib. 310, 855–864 (2008)

    Article  Google Scholar 

  8. Lou, J.J., Zhu, S.J., He, L., He, Q.W.: Experimental chaos in nonlinear vibration isolation system. Chaos Solitons Fract. 40, 1367–1375 (2009)

    Article  Google Scholar 

  9. Tang, K.S., Man, K.F., Zhong, G.Q., Chen, G.R.: Generating chaos via \(\text{ x }{\vert }\text{ x }{\vert }\). IEEE Trans. Circuits Syst. 48, 636–641 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Konishi, K.: Generating chaotic behavior in an oscillator driven by periodic forces. Phys. Lett. A 320, 200–206 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang, L., Liu, Z., Chen, G.: Chaotifying a continuous-time system via impulsive input. Int. J. Bifurc. Chaos 12, 1121–1128 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yu, X., Zhu, S., Liu, S.: A new method for line spectra reduction similar to generalized synchronization of chaos. J. Sound Vib. 306, 835–848 (2007)

    Article  MathSciNet  Google Scholar 

  13. Wen, G., Lu, Y., Zhang, Z., Ma, C., Yin, H., Cui, Z.: Line spectra reduction and vibration isolation via modified projective synchronization for acoustic stealth of submarines. J. Sound Vib. 324, 954–961 (2009)

    Article  Google Scholar 

  14. Zhang, J., Xu, D., Zhou, J., Li, Y.: Chaotification of vibration isolation floating raft system via nonlinear time-delay feedback control. Chaos Solitons Fract. 45, 1255–1265 (2012)

    Article  MathSciNet  Google Scholar 

  15. Cho, S.J., Jin, M.L., Kuc, T.Y., Lee, J.S.: Control and synchronization of chaos systems using time-delay estimation and supervising switching control. Nonlinear Dyn. 75, 549–560 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, X.F., Chen, G.R., Man, K.F.: Making a continuous-time minimum phase system chaotic by time-delay feedback. IEEE Trans. Circuit Syst. 48, 641–645 (2001)

    Article  MATH  Google Scholar 

  17. Li, Y., Xu, D., Fu, Y., Zhou, J.: Stability and chaotification of vibration isolation floating raft systems with time-delayed feedback control. Chaos Interdiscip. J. Nonlinear Sci. 21, 033110–033115 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhou, J., Xu, D., Li, Y.: An active–passive nonlinear vibration isolation method based on optimal time-delay feedback control. J. Vib. Eng. 24, 639–645 (2011)

    MathSciNet  Google Scholar 

  19. Zhou, J., Xu, D., Zhang, J., Liu, C.: Spectrum optimization-based chaotification using time-delay feedback control. Chaos Solitons Fract. 45, 815–824 (2012)

    Article  Google Scholar 

  20. Carrella, A., Brennan, M., Kovacic, I., Waters, T.: On the force transmissibility of a vibration isolator with quasi-zero-stiffness. J. Sound Vib. 322, 707–717 (2009)

    Article  Google Scholar 

  21. Georg, A.G., Ian, M.: On the implementation of the 0–1 test for chaos. SIAM J. Appl. Dyn. Syst. 8, 129–145 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hassard, B.D.: Counting roots of the characteristic equation for linear delay-differential systems. J. Differ. Equ. 136, 222–235 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kolmanovski, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic Press, London (1986)

    Google Scholar 

  24. Wang, Z.H., Hu, H.Y.: Stability switches of time-delayed dynamic systems with unknown parameters. J. Sound Vib. 233, 215–233 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhou, J.X., Xu, D.L., Li, Y.L.: Chaotifing duffing-type system with large parameter range based on optimal time-delay feedback control. In: Zhou, J.X. (ed.) Proceedings of 2010 International Workshop on Chaos-Fractal Theories and Applications, pp. 121–126. Kunming, Yunnan, China (2010)

  26. Xu, D., Zhang, Y., Zhou, J., Lou, J.: On the analytical and experimental assessment of performance of a quasi-zero-stiffness isolator. J. Vib. Control 1077546313484049 (2013)

  27. Hartman, P.: A lemma in the theory of structural stability of differential equations. Proc. Am. Math. Soc. 11, 610–610 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chicone, C.: Ordinary Differential Equations with Applications. Springer, Berlin (2006)

    MATH  Google Scholar 

  29. Li, Y.L., Xu, D.L., Fu, Y.M., Zhou, J.X.: Stability and chaotification of vibration isolation floating raft systems with time-delayed feedback control. Chaos 21, 033115 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ruan, S., Wei, J.: On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion. IMA J. Math. Appl. Med. Biol. 18, 41–52 (2001)

    Article  MATH  Google Scholar 

  31. Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Heidelberg (2002)

    Book  MATH  Google Scholar 

  32. Forde, J., Nelson, P.: Applications of Sturm sequences to bifurcation analysis of delay differential equation models. J. Math. Anal. Appl. 300, 273–284 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The funding from the National Science Foundation of China (No. 11402082) and the National Science Fund for Distinguished Young Scholars in China (No. 11225212) are acknowledged.

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, People’s Republic of China

    Yingli Li & Daolin Xu

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  1. Yingli Li
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  2. Daolin Xu
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Correspondence to Daolin Xu.

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Li, Y., Xu, D. Chaotification of quasi-zero-stiffness system with time delay control. Nonlinear Dyn 86, 353–368 (2016). https://doi.org/10.1007/s11071-016-2893-z

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  • DOI: https://doi.org/10.1007/s11071-016-2893-z

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