Skip to main content

Almost-Sure Stability of Fractional Viscoelastic Systems Driven by Bounded Noises

Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

The almost-sure stochastic stability of fractional viscoelastic systems, characterized by parametric excitation of bounded noises, is investigated. The viscoelastic material is modelled using a fractional Kelvin–Voigt constitutive relation, which results in a stochastic fractional equation of motion. The method of stochastic averaging, together with the Fokker–Plank equation of the averaged Itô stochastic differential equation, is used to determine asymptotically the top Lyapunov exponent of the system for small damping and weak excitation. It is found that the parametric noise excitation can have a stabilizing effect in the resonant region. The effects of various parameters on the stochastic stability of the system are discussed. The approximate analytical results are confirmed by numerical simulation.

Keywords

  • Bounded noises
  • Fractional differential equations
  • Viscoelastic material
  • Fractional Kelvin–Voigt model
  • Stochastic stability
  • Lyapunov exponents
  • Parametric excitation
  • Engineering

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-1-4614-7385-5_14
  • Chapter length: 21 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-1-4614-7385-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.00
Price excludes VAT (USA)
Hardcover Book
USD   109.99
Price excludes VAT (USA)
Fig. 14.1
Fig. 14.2
Fig. 14.3
Fig. 14.4
Fig. 14.5
Fig. 14.6
Fig. 14.7

References

  1. Ahmadi, G., Glocker, P.G.: J. Eng. Mech. 109(4), 990–999 (1983)

    Google Scholar 

  2. Ariaratnam, S.T.: Stochastic stability of viscoelastic systems under bounded noise excitation. In: Naess, A., Krenk, S. (eds.) IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, pp. 11–18. Kluwer Academic, Dordrecht (1996)

    CrossRef  Google Scholar 

  3. Bagley, R.L., Torvik, P.J.: J. Rheol. 27(3), 201–210 (1983)

    Google Scholar 

  4. Debnath, L.: Int. J. Math. Math. Sci. 54, 3413–3442 (2003)

    MathSciNet  CrossRef  Google Scholar 

  5. Di Paola, M., Pirrotta, A.: Meccanica dei Materiali e delle Strutture 1(2), 52–62 (2009)

    Google Scholar 

  6. Findley, W.N., Lai, J.S., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials with an Introduction to Linear Viscoelasticity. North-Holland, New York (1976)

    MATH  Google Scholar 

  7. Floris, C.: Mech. Res. Comm. 38, 57–61 (2011)

    CrossRef  MATH  Google Scholar 

  8. Khasminskii, R.Z.: Theor Probab. Appl. (English translation) 11, 390–406 (1966)

    Google Scholar 

  9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)

    MATH  Google Scholar 

  10. Koh, C.G., Kelly, J.M.: Earthquake Eng. Struct. Dynam. 19, 229–241 (1990)

    CrossRef  Google Scholar 

  11. Larionov, G.S.: Mech. Polymers (English translation) 5, 714–720 (1969)

    Google Scholar 

  12. Lin, Y.K., Cai, Q.G.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, New York (1995)

    Google Scholar 

  13. Mainardi, R.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)

    CrossRef  Google Scholar 

  14. Onu, K.: Stochastic averaging for mechanical systems, Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, Illinois (2010)

    Google Scholar 

  15. Papoulia, K.D., Panoskaltsis, V.P., Kurup, N.V., Korovajchuk, I.: Rheol. Acta 49(4), 381–400 (2010)

    CrossRef  Google Scholar 

  16. Pfitzenreiter, T.: ZAMM J. Appl. Math. Mech. 84(4), 284–287 (2004)

    MATH  Google Scholar 

  17. Pfitzenreiter, T.: ZAMM J. Appl. Math. Mech. 88(7), 540–551 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  19. Potapov, V.D.: J. Sound Vib. 173, 301–308 (1994)

    CrossRef  MATH  Google Scholar 

  20. Potapov, V.D.: Appl. Numer. Math. 24, 191–201 (1997)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Roberts, J.B., Spanos, P.D.: Int. J. NonLinear Mech. 21, 111–134 (1986)

    MathSciNet  CrossRef  MATH  Google Scholar 

  22. Rosenstein, M.T., Collins, J.J., De Luca, C.J.: Phys. D 65, 117–134 (1993)

    MathSciNet  CrossRef  MATH  Google Scholar 

  23. Rouse, Jr., P.E.: J. Chem. Phys. 21(7), 1272–1280 (1953)

    Google Scholar 

  24. Sri Namachchivaya, N., Ariaratnam, S.T.: Mech. Struct. Mach. 15(3), 323–345 (1987)

    Google Scholar 

  25. Stratonovich, R.L.: Topics in the Theory of Random Noise. Gordon and Breach Science Publishers, New York (1963)

    Google Scholar 

  26. Wolf, A., Swift, J., Swinney, H., Vastano, A.: Phys. D 16, 285–317 (1985)

    MathSciNet  CrossRef  MATH  Google Scholar 

  27. Xie, W.C.: Dynamic Stability of Structures. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

Download references

Acknowledgements

The research for this paper was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jian Deng .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Deng, J., Xie, WC., Pandey, M.D. (2013). Almost-Sure Stability of Fractional Viscoelastic Systems Driven by Bounded Noises. In: d'Onofrio, A. (eds) Bounded Noises in Physics, Biology, and Engineering. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7385-5_14

Download citation