The Fractional Derivative as a Complex Eigenvalue Problem

  • Masaharu Kuroda
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 30)


For the dynamics described by an equation of motion including fractional-order-derivative terms, the fractional-order-derivative responses cannot be measured directly through experiments. In the present study, three solutions are proposed that enable the fractional-order-derivative responses to be measured by a combination of signals obtained by existing sensors. Specialized sensors or complicated signal processing are not necessary. Fractional-order-derivative responses at a certain point on a structure can be expressed through linear combinations of the displacement signal and the velocity signal at each point on the structure. Although their calculation processes are different, all three methods eventually reach the same result.


Fractional Derivative Mass Point Fractional Calculus System Matrix Fractional Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Oldham, K.B., Spanier, J.: The Fractional Calculus, pp. 1–15. Dover, New York (2002)Google Scholar
  2. 2.
    Motoishi, K., Koga, T.: Simulation of a Noise Source with 1/f Spectrum by Means of an RC Circuit. IEICE Trans. J65-A(3), 237–244 (1982) (in Japanese) Google Scholar
  3. 3.
    Chen, Y., Vinagre, B.M., Podlubny, I.: A New Discretization Method for Fractional Order Differentiators via Continued Fraction Expansion. In: Proc. ASME IDETC/CIE 2003, DETC2003/VIB 48391, pp. 761–769 (2003)Google Scholar
  4. 4.
    Kuroda, M., Kikushima, Y., Tanaka, N.: Active Wave Control of a Flexible Structure Formulated Using Fractional Calculus. In: Proc. 74th Annual Meeting of JSME, vol. (I), pp. 331–332 (1996) (in Japanese) Google Scholar
  5. 5.
    Kuroda, M.: Active Vibration Control of a Flexible Structure Formulated Using Fractional Calculus. In: Proc. ENOC 2005, pp. 1409–1414 (2005)Google Scholar
  6. 6.
    Kuroda, M.: Formulation of a State Equation Including Fractional-Order State-Vectors. In: Proc. ASME IDETC/CIE 2007, DETC2007-35273, pp. 1–10 (2007)Google Scholar
  7. 7.
    Kuroda, M.: Active Wave Control for Flexible Structures Using Fractional Calculus. In: Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.) Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, pp. 435–448. Springer, Dordrecht (2007)Google Scholar
  8. 8.
    Kuroda, M.: Formulation of a State Equation Including Fractional-Order State Vectors. J. Computational and Nonlinear Dynamics 3, 021202-1–021202-8 (2008)MathSciNetGoogle Scholar
  9. 9.
    Podlubny, I.: Fractional Differential Equations. Academic Press Inc., San Diego (1999)zbMATHGoogle Scholar
  10. 10.
    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)zbMATHGoogle Scholar
  11. 11.
    West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003)Google Scholar
  12. 12.
    Kilbas, A.A., Trujillo, J.J.: Differential Equations of Fractional Order: Methods, Results and Problems. II. Applicable Analysis 81(2), 435–493 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  14. 14.
    Yang, D.L.: Fractional State Feedback Control of Undamped and Viscoelastically damped Structures. Master’s Thesis, AD-A-220-477, Air Force Institute of Technology, pp. 1–98 (1990)Google Scholar
  15. 15.
    Sorrentino, S., Fasana, A.: Finite element analysis of vibrating linear systems with fractional derivative viscoelastic models. J. Sound and Vibration 299, 839–853 (2007)CrossRefGoogle Scholar
  16. 16.
    Nagamatsu, A., et al. (eds.): Dynamics Handbook, pp. 111–112. Asakura, Tokyo (1993) (in Japanese)Google Scholar

Copyright information

© Springer Dordrecht Heidelberg London New York 2011

Authors and Affiliations

  • Masaharu Kuroda
    • 1
  1. 1.National Institute of Advanced Industrial Science and Technology (AIST)TsukubaJapan

Personalised recommendations