Abstract
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.
Keywords
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Oldham, K.B., Spanier, J.: The Fractional Calculus, pp. 1–15. Dover, New York (2002)
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)
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)
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)
Kuroda, M.: Active Vibration Control of a Flexible Structure Formulated Using Fractional Calculus. In: Proc. ENOC 2005, pp. 1409–1414 (2005)
Kuroda, M.: Formulation of a State Equation Including Fractional-Order State-Vectors. In: Proc. ASME IDETC/CIE 2007, DETC2007-35273, pp. 1–10 (2007)
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)
Kuroda, M.: Formulation of a State Equation Including Fractional-Order State Vectors. J. Computational and Nonlinear Dynamics 3, 021202-1–021202-8 (2008)
Podlubny, I.: Fractional Differential Equations. Academic Press Inc., San Diego (1999)
Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003)
Kilbas, A.A., Trujillo, J.J.: Differential Equations of Fractional Order: Methods, Results and Problems. II. Applicable Analysis 81(2), 435–493 (2002)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
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)
Sorrentino, S., Fasana, A.: Finite element analysis of vibrating linear systems with fractional derivative viscoelastic models. J. Sound and Vibration 299, 839–853 (2007)
Nagamatsu, A., et al. (eds.): Dynamics Handbook, pp. 111–112. Asakura, Tokyo (1993) (in Japanese)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Dordrecht Heidelberg London New York
About this paper
Cite this paper
Kuroda, M. (2011). The Fractional Derivative as a Complex Eigenvalue Problem. 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_13
Download citation
DOI: https://doi.org/10.1007/978-94-007-1643-8_13
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-1642-1
Online ISBN: 978-94-007-1643-8
eBook Packages: EngineeringEngineering (R0)