Skip to main content
SpringerLink
Log in

Parametric Identification of the Fractional-Derivative Order in the Bagley–Torvik Model

  • Published:
Mathematical Models and Computer Simulations Aims and scope

Abstract

We consider a second-order differential equation that contains a fractional derivative (the Bagley–Torvik equation); here, the order of the derivative is in the range from 1 to 2 and is not known in advance. This model is used for describing oscillation processes in a viscoelastic medium. In order to study the equation, we use the Laplace transform; this makes it possible to obtain (in the explicit form) the image of the solution of the corresponding Cauchy problem. Numerical solutions for the various values of the parameter are constructed. Based on this solution, we propose a numerical technique for the parametric identification of an unknown order of the fractional derivative from the available experimental data. In the range of possible values of the parameter, the deviation function is determined by the least-squares method. The minimum of this function determines the search value of the parameter. The developed technique is tested on the experimental data for samples of polymer concrete, the fractional-derivative parameter in the model is determined, the theoretical and experimental curves are compared, and the accuracy of the parametric identification and the adequacy of the technique are established.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Similar content being viewed by others

REFERENCES

  1. A. A. Chirkii and I. I. Matichin, “On linear conflict-controlled processes with fractional derivatives,” Tr. Inst. Mat. Mekh. UrO RAN 17, 256–270 (2011).

    Google Scholar 

  2. D. Ingman and J. Suzdalnitsky, “Control of damping oscillations by fractional differential operator with time-dependent order,” Comput. Methods Appl. Mech. Eng. 193, 5585–5595 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Ingman and J. Suzdalnitsky, “Iteration method for equation of viscoelastic motion with fractional differential operator of damping,” Comput. Methods Appl. Mech. Eng. 190, 5027–5036 (2001).

    Article  MATH  Google Scholar 

  4. G. K. Koh and J. Kelly, “Application of fractional derivatives to seismic analysis of base-isolated models,” Earthquake Eng. Struct. Dyn. 19, 229–241 (1990).

    Article  Google Scholar 

  5. Gh. E. Draganescu, N. Cofan, and D. L. Rujan, “Nonlinear vibrations of a nano-sized sensor with fractional damping,” J. Optoelectron. Adv. Mater. 7, 877–884 (2005).

  6. A. Fenlander, “Modal synthesis when modeling damping by use of fractional derivatives,” AIAA J. 34, 1051–1058 (1996).

    Article  MATH  Google Scholar 

  7. R. P. Meilanov and M. S. Yanpolov, “Features of the phase trajectory of a fractal oscillator,” Tech. Phys. Lett. 28, 30 (2002).

    Article  Google Scholar 

  8. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of their Applications (Nauka Tekhnika, Minsk, 1987) [in Russian].

    MATH  Google Scholar 

  9. R. L. Bagley and P. J. Torvik, “A thoretical basis for the application of fractional calculus to viscoelasticity,” J. Rheolog. 27, 201–203 (1983).

    Article  MATH  Google Scholar 

  10. R. L. Bagley and P. J. Torvik, “Fractional calculus – a different approach to the analysis of viscoelastically damped structures,” J. AIAA 21, 741–748 (1983).

    Article  MATH  Google Scholar 

  11. B. Ibrahim, Q. Dong, and Z. Fan, “Existence for boundary value problems of two-term Caputo fractional differential equations,” J. Nonlin. Sci. Appl., 511–520 (2017).

  12. E. R. Kekharsaeva and V. G. Pirozhkov, “Modeling changes in the deformation-strength characteristics of asphalt concrete under loading using fractional calculus,” in Proceedings of the I. F. Obraztsov and Yu. G. Yanovskii 6th All-Russia Conference with International Participation on Mechanics of Composite Materials and Structures, Complex and Heterogeneous Media (IPRIM RAN, Moscow, 2016), pp. 104–109.

  13. C. M. A. Vasques, R. A. S. Moreira, and J. Dias Rodrigues, “Experimental identification of GHM and ADF parameters for viscoelastic damping modeling,” in Proceedings of the 3rd European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, 2006, Ed. by C. A. Mota Soares et al.

  14. V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus (Nauka, Moscow, 1974; Pergamon, Oxford, 1965).

  15. S. V. Erokhin, T. S. Aleroev, and L. Yu. Frishter, “Sturm-Liouville problem for the equationviscoelastic damping oscillator,” Int. J. Comput. Civil Struct. Eng. 11, 77–81 (2015).

    Google Scholar 

  16. E. R. Kekharsaeva and T. S. Aleroev, “Model of deformation-strength characteristics of chlorine-containing polyesters based on fractional derivatives,” Plast. Massy, No. 3, 35 (2001).

    Google Scholar 

  17. V. A. Man’kovskii and V. T. Sapunov, “Nomographic properties of fractional exponential E-functions when describing linear viscoelasticity,” Zavod. Lab. Diagn. Mater. 66 (3), 45–50 (2000).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State (National Research) University of Civil Engineering, 129337, Moscow, Russia

    T. S. Aleroev & S. V. Erokhin

Authors
  1. T. S. Aleroev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Erokhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. S. Aleroev.

Additional information

Translated by L. Kartvelishvili

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aleroev, T.S., Erokhin, S.V. Parametric Identification of the Fractional-Derivative Order in the Bagley–Torvik Model. Math Models Comput Simul 11, 219–225 (2019). https://doi.org/10.1134/S2070048219020030

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070048219020030

Keywords:

Search

Navigation