Applied Mathematics and Mechanics

, Volume 22, Issue 3, pp 294–303 | Cite as

Dynamical Stability of Viscoelastic Column with Fractional Derivative Constitutive Relation

  • Gen-guo Li
  • Zheng-you Zhu
  • Chang-jun Cheng


The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The governing equation of motion was derived as a weakly singular Volterra integro-partial-differential equation, and it was simplified into a weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. In terms of the averaging method, the dynamical stability was analyzed. A new numerical method is proposed to avoid storing all history data. Numerical examples are presented and the numerical results agree with the analytical ones.

viscoelastic column fractional derivative constitutive relation averaging method weakly singular Volterra integro-differential equation dynamical stability 


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  1. [1]
    Cederbaum G, Mond M. Stability properties of a viscoelastic column under a periodic force [J]. J Appl Mech, 1992,59(16):16–19.Google Scholar
  2. [2]
    CHENG Chang-jun, ZHANG Neng-hui. Chaotic and hyperchaotic behaviors of viscoelastic rectangular plates under transvese periodic load [J]. Acta Mechanica Sinica, 1998,30(6):690–699.(in Chinese)Google Scholar
  3. [3]
    Bagley R L, Torvik P J. Fractional calculus—a different approach to the analysis of viscoelastically damped structures [J]. AIAA J, 1983,21(5):741–748.Google Scholar
  4. [4]
    Bagley R L. A theoretical basis for the application of fractional calculus to viscoelasticity [J]. J Rheol, 1983,27(3):201–210.Google Scholar
  5. [5]
    HUANG Wen-hu, WANG Xing-qing, ZHANG Jing-hui, et al. Some advances in the vibration control of aerospace flexible structures [J]. Advances in Mechanics, 1997,27(1):5–18.(in Chinese)Google Scholar
  6. [6]
    Enelund M, Mahler L, Runesson K, et al. Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws [J]. Int J Solids Strut, 1999,36(7):1417–1442.Google Scholar
  7. [7]
    Drozdov A D. Fractional differential models in finite viscoelasticity [J]. Acta Mech, 1997,124(1):155–180.Google Scholar
  8. [8]
    Samko S G, Kilbas A A, Marricher O Z. Fractional Integrals and Derivatives: Theory and Application[M]. New York: Gordon and Breach Science Publishers, 1993.Google Scholar
  9. [9]
    LIU Yan-zhu, CHEN Wen-liang, CHEN Li-qun. Mechanics of Vibrations [M]. Beijing: Advanced Educational Press, 1998.(in Chinese)Google Scholar
  10. [10]
    Rossikhin Y A, Shitikova M V. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solid [J]. Appl Mech Rev, 1997,50(1):15–67.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Gen-guo Li
    • 1
    • 2
  • Zheng-you Zhu
    • 1
    • 2
  • Chang-jun Cheng
    • 1
    • 2
  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghaiP R China
  2. 2.Department of MathematicsShanghai UniversityShanghaiP R China

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