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Power flow behaviour and dynamic performance of a nonlinear vibration absorber coupled to a nonlinear oscillator

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Abstract

The vibrational power flow characteristics of a two-degree-of-freedom system are investigated to examine the performance of nonlinear absorbers in vibration attenuation of nonlinear primary oscillators. The nonlinearities in the oscillator and those in the absorber are both characterised by cubic restoring and damping forces. Both analytical approximations and numerical integrations are used to obtain time-averaged power flow variables, as well as kinetic energies of the system. Power absorption ratio and the kinetic energy of the nonlinear oscillator are proposed to quantitatively evaluate the effectiveness of nonlinear absorbers with respect to the existing nonlinearities in the oscillator. Comparing with linear absorbers, it is found that softening (hardening) stiffness absorber provides benefits for vibration mitigation of a hardening (softening) stiffness primary oscillator by enhancing power absorption efficiency and reducing the kinetic energy of the oscillator so that the functioning frequency range of the absorber can be enlarged. Nonlinear cubic damping in the absorber is shown beneficial for vibration suppression as the power absorption ratio becomes large at resonance frequencies so that the peak power flow and kinetic energy levels are reduced. The developed model can be conveniently extended to study other types of nonlinearities in the absorber/oscillator. Conclusions and suggestions are provided for applications.

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References

  1. Frahm, H.: A device for damping vibrations of bodies. US Patent, 989958 (1911)

  2. Ormondroyd, J., Den Hartog, J.P.: The theory of the dynamic vibration absorber. J. Appl. Mech. 50, 9–22 (1928)

    Google Scholar 

  3. Den Hartog, J.P.: Mechanical Vibrations, 4th edn. McGraw-Hill, New York (1956)

    MATH  Google Scholar 

  4. Snowdon, J.C.: Vibration of simply supported rectangular and square plates to which lumped masses and dynamic vibration absorbers are attached. J. Acoust. Soc. Am. 57, 646–654 (1975)

    Article  Google Scholar 

  5. Ozer, M.B., Rosyton, T.J.: Extending Den Hartog’s vibrations absorber technique to multi-degree-of-freedom systems. J. Vib. Acoust. 127, 1307–1320 (2005)

    Article  Google Scholar 

  6. Roberson, R.E.: Synthesis of a nonlinear dynamic vibration absorber. J. Frankl. Inst. 254, 205220 (1952)

    Article  MathSciNet  Google Scholar 

  7. Pipes, L.A.: Analysis of a non-linear vibration absorber. J. Appl. Mech. 20, 515–518 (1953)

    MATH  Google Scholar 

  8. Arnold, F.R.: Steady state behaviour of systems provided with non-linear dynamic vibration absorbers. J. Appl. Mech. 22, 487–492 (1955)

    MATH  MathSciNet  Google Scholar 

  9. Hunt, J.B., Nissen, J.-C.: The broadband dynamic vibration absorber. J. Sound Vib. 83, 573–578 (1982)

    Article  Google Scholar 

  10. Rice, H.J.: Combinational instability of the non-linear vibration absorber. J. Sound Vib. 108, 526–532 (1986)

    Article  Google Scholar 

  11. Jordanov, I.N., Cheshankov, B.I.: Optimal design of linear and non-linear dynamic vibration absorbers. J. Sound Vib. 123, 157–170 (1988)

    Article  MATH  Google Scholar 

  12. Lazer, A.C., McKenna, P.J.: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kim, G., Singh, R.: A study of passive and adaptive hydraulic engine mount systems with emphasis on non-linear characteristics. J. Sound Vib. 179, 427–453 (1995)

    Article  Google Scholar 

  14. Natsiavas, S.: Steady state oscillations and stability of non-linear dynamic vibration absorbers. J. Sound Vib. 156, 227–245 (1992)

    Article  MATH  Google Scholar 

  15. Bonsel, J.H., Fey, R.H.B., Nijimeijer, H.: Applications of a dynamic vibration absorber to a piecewise linear beam system. Nonlinear Dyn. 37, 227–243 (2004)

    Article  MATH  Google Scholar 

  16. Jo, H., Yabuno, H.: Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling. Nonlinear Dyn. 55, 67–78 (2009)

    Article  MATH  Google Scholar 

  17. Viguié, R., Kerschen, G.: Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology. J. Sound Vib. 326, 780–793 (2009)

    Article  Google Scholar 

  18. Ji, J.C., Zhang, N.: Suppression of the primary resonance vibrations of a forced nonlinear systems using a dynamic vibration absorber. J. Sound Vib. 329, 2044–2056 (2010)

    Article  Google Scholar 

  19. Oueini, S.S., Nayfeh, A.H., Pratt, J.R.: A nonlinear vibration absorber for flexible structures. Nonlinear Dyn. 15, 259–282 (1998)

    Article  MATH  Google Scholar 

  20. Febbo, M., Machado, S.P.: Nonlinear dynamic vibration absorbers with saturation. J. Sound Vib. 332, 1465–1483 (2013)

    Article  Google Scholar 

  21. Goyder, H.G.D., White, R.G.: Vibrational power flow from machines into built-up structures. J. Sound Vib. 68, 59–117 (1980)

    Article  MATH  Google Scholar 

  22. Pinnington, R.J., White, R.G.: Power flow through machine isolators to resonant and non-resonant beams. J. Sound Vib. 75, 179–197 (1992)

    Article  Google Scholar 

  23. Xing, J.T., Price, W.G.: A power-flow analysis based on continuum dynamics. Philos. Trans. R. Soc. Lond. A 455, 401–436 (1999)

    MATH  MathSciNet  Google Scholar 

  24. Mace, B.R., Shorter, P.J.: Energy flow models from finite element analysis. J. Sound Vib. 233, 369–389 (2000)

    Article  Google Scholar 

  25. Xiong, Y.P., Xing, J.T., Price, W.G.: Power flow analysis of complex coupled systems by progressive approaches. J. Sound Vib. 239, 275–295 (2001)

    Article  Google Scholar 

  26. Xiong, Y.P., Xing, J.T., Price, W.G.: A general linear mathematical model of power flow analysis and control for integrated structure-control systems. J. Sound Vib. 267, 301–334 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Xiong, Y.P., Xing, J.T., Price, W.G.: A power flow mode theory based on a system’s damping distribution and power flow design approaches. Philos. Trans. R. Soc. Lond. A 461, 3381–3411 (2005)

    MATH  MathSciNet  Google Scholar 

  28. Zilletti, M., Elliott, S.J., Rustighi, E.: Optimisation of dynamic vibration absorbers to minimise kinetic energy and maximise internal power dissipation. J. Sound Vib. 331, 4093–4100 (2012)

  29. Royston, T.J., Singh, R.: Vibratory power flow through a nonlinear path into a resonant receiver. J. Acoust. Soc. Am. 101, 2059–2069 (1997)

    Article  Google Scholar 

  30. Xing, J.T., Price, W.G.: A substructure method for power flow analysis of non-linear systems consisting of linear substructures and non-linear controllers. Proceedings of the Eleventh International Congress on Sound and Vibration, St. Petersburg. pp. 2657–2664 (2004)

  31. Xiong, Y.P., Xing, J.T., Price, W.G.: Interactive power flow characteristics of an integrated equipment-nonlinear isolator-travelling flexible ship excited by sea waves. J. Sound Vib. 287, 245–276 (2005)

    Article  Google Scholar 

  32. Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Springer, New York (2008)

    MATH  Google Scholar 

  33. Xiong, Y., Cao, Q.: Power flow characteristics of coupled linear and nonlinear oscillators with irrational nonlinear stiffness. Proceedings of the Seventh European Nonlinear Dynamics Conference (ENOC), Rome, CDROM (2011)

  34. Yang, J.: Power Flow Analysis of Nonlinear Dynamical Systems, PhD thesis, University of Southampton, Southampton (2013)

  35. Yang, J., Xiong, Y.P., Xing, J.T.: Dynamics and power flow behaviour of a nonlinear vibration isolation system with a negative stiffness mechanism. J. Sound Vib. 332, 167–183 (2013)

    Article  Google Scholar 

  36. Yang, J., Xiong, Y.P., Xing, J.T.: Nonlinear power flow analysis of the Duffing oscillator. Mech. Syst. Signal Process. 45, 563–578 (2014)

    Article  Google Scholar 

  37. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Willey, New York (1979)

    MATH  Google Scholar 

  38. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (1986)

    Google Scholar 

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Authors and Affiliations

  1. Fluid Structure Interactions Research Group, Faculty of Engineering and the Environment, University of Southampton, Boldrewood Campus, SO16 7QF, Southampton, UK

    J. Yang, Y. P. Xiong & J. T. Xing

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  1. J. Yang
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  2. Y. P. Xiong
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  3. J. T. Xing
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Correspondence to Y. P. Xiong.

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Yang, J., Xiong, Y.P. & Xing, J.T. Power flow behaviour and dynamic performance of a nonlinear vibration absorber coupled to a nonlinear oscillator. Nonlinear Dyn 80, 1063–1079 (2015). https://doi.org/10.1007/s11071-014-1556-1

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  • DOI: https://doi.org/10.1007/s11071-014-1556-1

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