Skip to main content
SpringerLink
Log in

Solution and Stability of a Linear Fractionally Damped Oscillator

  • Conference paper
IUTAM Symposium on Dynamics Modeling and Interaction Control in Virtual and Real Environments

Part of the book series: IUTAM Bookseries ((IUTAMBOOK,volume 30))

Abstract

This paper presents some recent advances in the generalized Bagley-Torvik equation that uses fractional-order derivative to describe the damping force. It addresses the conventional form of solution that is easy for engineers to understand, as well as the stability analysis of the fractionally damped oscillator.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    MATH  Google Scholar 

  2. Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behavior of real materials. ASME Journal of Applied Mechanics 51, 294–298 (1984)

    Article  MATH  Google Scholar 

  3. Eldred, L.B., Baker, W.P., Palazotto, A.N.: Kelvin-Voigt vs fractional derivative model as constitutive relations for viscoelastic materials. AIAA Journal 33(3), 547–550 (1995)

    Article  MATH  Google Scholar 

  4. Podlubny, I.: Fractional-order systems and PI λ D μ-controllers. IEEE Transactions on Automatic Control 44(1), 208–214 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Luo, Y., Chen, Y.-Q.: Fractional order [proportional derivative] controller for a class of fractional order systems. Automatica 45(10), 2446–2450 (2009)

    Article  MATH  Google Scholar 

  6. Farshad, M.B., Masoud, K.G.: An efficient numerical algorithm for stability testing of fractional-delay systems. ISA Transactions 48, 32–37 (2009)

    Article  Google Scholar 

  7. Coronado, A., Trindade, M.A., Sampaio, R.: Frequency-dependent viscoelastic models for passive vibration isolation systems. Shock and Vibration 9, 253–264 (2002)

    Google Scholar 

  8. Wang, Z.H., Zheng, Y.G.: The optimal form of the fractional-order difference feedbacks in enhancing the stability of a sdof vibration system. Journal of Sound and Vibration 326(3-5), 476–488 (2009)

    Article  Google Scholar 

  9. Sun, K.H., Wang, X., Sprott, J.C.: Bifircation and chaos in fractional-order simplified Loreenz system. International Journal of Bifurcation and Chaos 20(4), 1209–1219 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Narahari Achar, B.N., Hanneken, J.W., Clarke, T.: Response characteristics of a fractional oscillator. Physica A 309, 275–288 (2002)

    Article  MATH  Google Scholar 

  11. Suarez, L.E., Shokooh, A.: An eigenvector expansion method for the solution of motion containing fractional derivatives. ASME Journal of Applied Mechanics 64, 629–635 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, Y.L.: Solving a nonlinear fractional differential equation using Chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation (2010), doi:10.1016/j.cnsns.2009.09.020

    Google Scholar 

  13. Bonilla, B., Rivero, M., Trujillo, J.J.: Linear differential equations of fractional orders. In: Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.) Advances in Fractional Calculus, pp. 77–91. Springer, Dordrecht (2007)

    Chapter  Google Scholar 

  14. Wang, Z.H., Wang, X.: General solution of the Bagley-Torvik equation with fractional-order derivative. Communications in Nonlinear Science and Numerical Simulation 15, 1279–1285 (2010)

    Article  MathSciNet  Google Scholar 

  15. Wang, Z.H., Wang, X.: General solution of a vibration system with damping force of fractional-order derivative. In: Luo, A.C. (ed.) Dynamics and Vibrations with Discontinuity, Stochasticity and Time-Delay Systems, pp. 1–8. Springer, Heidelberg (2010)

    Google Scholar 

  16. Naber, M.: Linear fractionally damped oscillator. International Journal of Differential Equations, 2010, 12, Article ID 197020 (2010), doi:10.1155/2010/197020

    Google Scholar 

  17. Wang, Z.H., Du, M.L.: Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system. Shock and Vibration (2010), doi:10.3233/SAV-2010-0566.

    Google Scholar 

  18. Rossikhin, Y.A., Shitikova, M.V.: New approach for the analysis of damped vibrations of fractional oscillators. Shock and Vibration 16, 365–387 (2009)

    Google Scholar 

  19. Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008)

    MATH  Google Scholar 

  20. Buslowicz, M.: Stability of linear continuous-time fractional order systems with delays of the retarded type. Bulletin of the Polish Academy of Sciences: Technical Sciences 56, 319–324 (2008)

    Google Scholar 

  21. Wang, Z.H., Hu, H.Y.: Stability of a linear oscillator with damping force of fractional-order derivative. SCIENCE CHINA: Physics, Mechanics & Astronomy 53(2), 345–352 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, China

    Z. H. Wang

Authors
  1. Z. H. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Applied Mechanics, Budapest University of Technology and Economics, Muegyetem rkp. 5, 1111, Budapest, Hungary

    Gábor Stépán  & László L. Kovács  & 

  2. Department of Manufacturing Science and Engineering, Budapest University of Technology and Economics, Egry J. u. 1 T/46.b, 1111, Budapest, Hungary

    András Tóth

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Dordrecht Heidelberg London New York

About this paper

Cite this paper

Wang, Z.H. (2011). Solution and Stability of a Linear Fractionally Damped Oscillator. In: Stépán, G., Kovács, L.L., Tóth, A. (eds) IUTAM Symposium on Dynamics Modeling and Interaction Control in Virtual and Real Environments. IUTAM Bookseries, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1643-8_12

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-1643-8_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-1642-1

  • Online ISBN: 978-94-007-1643-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation