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Applied Mathematics and Mechanics

, Volume 33, Issue 6, pp 817–828 | Cite as

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

  • Bo Wang (王 波)Email author
Article

Abstract

The weakly forced vibration of an axially moving viscoelastic beam is investigated. The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved. The nonlinear equations governing the transverse vibration are derived from the dynamical, constitutive, and geometrical relations. The method of multiple scales is used to determine the steady-state response. The modulation equation is derived from the solvability condition of eliminating secular terms. Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation. The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.

Key words

axially moving beam weakly forced vibration standard linear solid model method of multiple scales steady-state response 

Chinese Library Classification

O326 

2010 Mathematics Subject Classification

74G10 74H10 74K10 

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References

  1. [1]
    Evans, T. Nonlinear Dynamics, INTECH, Croatia, 145–172 (2010)CrossRefGoogle Scholar
  2. [2]
    Pasin, F. Ueber die stabilität der beigeschwingungen von in laengsrichtung periodisch hin und herbewegten stäben. Ingenieur-Archiv, 41, 387–393 (1972)zbMATHCrossRefGoogle Scholar
  3. [3]
    Öz, H. R., Pakdemirli, M., and Özkaya, E. Transition behavior from string to beam for an axially accelerating material. Journal of Sound and Vibration, 215, 571–576 (1998)CrossRefGoogle Scholar
  4. [4]
    Öz, H. R. On the vibrations of an axially traveling beam on fixed supports with variable velocity. Journal of Sound and Vibration, 239, 556–564 (2001)CrossRefGoogle Scholar
  5. [5]
    Suweken, G. and van Horssen, W. T. On the transversal vibrations of a conveyor belt with a low and time-varying velocity, part II: the beam like case. Journal of Sound and Vibration, 267, 1007–1027 (2003)CrossRefGoogle Scholar
  6. [6]
    Pakdemirli, M. and Öz, H. R. Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations. Journal of Sound and Vibration, 311, 1052–1074 (2008)CrossRefGoogle Scholar
  7. [7]
    Chen, L. Q., Yang, X. D., and Cheng, C. J. Dynamic stability of an axially accelerating viscoelastic beam. European Journal of Mechanics-A/Solids, 23, 659–666 (2004)zbMATHCrossRefGoogle Scholar
  8. [8]
    Chen, L. Q. and Yang, X. D. Stability in parametric resonances of an axially moving viscoelastic beam with time-dependent velocity. Journal of Sound and Vibration, 284, 879–891 (2005)CrossRefGoogle Scholar
  9. [9]
    Yang, X. D. and Chen, L. Q. Stability in parametric resonance of axially accelerating beams constituted by Boltzmann’s superposition principle. Journal of Sound and Vibration, 289, 54–65 (2006)CrossRefGoogle Scholar
  10. [10]
    Chen, L. Q. and Yang, X. D. Vibration and stability of an axially moving viscoelastic beam with hybrid supports. European Journal of Mechanics-A/Solids, 25, 996–1008 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Maccari, A. The asymptotic perturbation method for nonlinear continuous systems. Nonlinear Dynamics, 19, 1–18 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Boertjens, G. J. and van Horssen, W. T. On interactions of oscillation modes for a weakly nonlinear undamped elastic beam with an external force. Journal of Sound and Vibration, 235, 201–217 (2000)CrossRefGoogle Scholar
  13. [13]
    Chen, L. Q., Lim, C. W., Hu, Q. Q., and Ding, H. Asymptotic analysis of a vibrating cantilever with a nonlinear boundary. Science in China Series G-Physics Mechanics Astronomy, 52, 1414–1422 (2009)CrossRefGoogle Scholar
  14. [14]
    Chen, L. Q. and Yang, X. D. Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models. International Journal of Solids and Structures, 42, 37–50 (2005)zbMATHCrossRefGoogle Scholar
  15. [15]
    Yang, T. Z., Fang, B., Chen, Y., and Zhen, Y. X. Approximate solutions of axially moving viscoelastic beams subject to multi-frequency excitations. International Journal of Non-Linear Mechanics, 44, 230–238 (2009)CrossRefGoogle Scholar
  16. [16]
    Ding, H. and Chen, L. Q. Nonlinear models for transverse forced vibration of axially moving viscoelastic beams. Shock and Vibration, 18, 281–287 (2011)Google Scholar
  17. [17]
    Chen, L. Q. and Ding, H. Steady-state transverse response in coupled planar vibration of axially moving viscoelastic beams. Journal of Vibration and Acoustics, 132(1), 011009 (2010)CrossRefGoogle Scholar
  18. [18]
    Mockensturm, E. M. and Guo, J. Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings. Journal of Applied Mechanics, 72, 374–380 (2005)zbMATHCrossRefGoogle Scholar
  19. [19]
    Chen, L. Q. and Yang, X. D. Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos, Solitons and Fractals, 23, 249–258 (2005)zbMATHCrossRefGoogle Scholar
  20. [20]
    Hou, Z. and Zu, J.W. Nonlinear free oscillations of moving belts. Mechanism and Machine Theory, 37, 925–940 (2002)zbMATHCrossRefGoogle Scholar
  21. [21]
    Fung, R. F., Huang, J. S., Chen, Y. C., and Yao, C. M. Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed. Computers and Structures, 66, 777–784 (1998)zbMATHCrossRefGoogle Scholar
  22. [22]
    Ha, J. L., Chang, J. R., and Fung, R. F. Nonlinear dynamic behavior of a moving viscoelastic string undergoing three-dimensional vibration. Chaos, Solitons and Fractals, 33, 1117–1134 (2007)zbMATHCrossRefGoogle Scholar
  23. [23]
    Chen, L. Q. and Chen, H. Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model. Journal of Engineering Mathematics, 67, 205–218 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Chen, L. Q. and Ding, H. Steady-state responses of axially accelerating viscoelastic beams: approximate analysis and numerical confirmation. Science in China Series G-Physics Mechanics Astronomy, 51, 1701–1721 (2008)Google Scholar
  25. [25]
    Ding, H. and Chen, L. Q. On two transverse nonlinear models of axially moving beams. Science in China Series E-Technological Sciences, 52, 743–751 (2009)zbMATHCrossRefGoogle Scholar
  26. [26]
    Chen, L. Q. and Jean, W. Z. Solvability condition in multi-scale analysis of gyroscopic continua. Journal of Sound and Vibration, 309, 338–342 (2008)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Mechanical EngineeringShanghai Institute of TechnologyShanghaiP. R. China

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