Nonlinear Dynamics

, Volume 38, Issue 1–4, pp 61–68 | Cite as

Impulse Responses of Fractional Damped Systems

  • Ingo Schäfer
  • Siegmar Kempfle


Considered are systems of single-mass oscillators with different fractional damping behaviour. Similar to the classical model, where the damping terms are represented by first derivatives, the eigensystem can be used to decompose the fractional system in frequency domain, if mass, stiffness and damping matrices are linearly dependent. The solution appears as a linear combination of single-mass oscillators. This is true even in the general case such that stability and causality are insured by the same argumentation as used in the linear dependent case.

Key words:

fractional calculus modal analysis viscoelasticity 


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  1. 1.
    Kempfle, S., Schäfer, I., and Beyer, H., ‘Fractional calculus via functional calculus: Theory and applications’, Nonlinear Dynamics29, 2002, 99–127.Google Scholar
  2. 2.
    Kempfle, S. and Beyer, H., ‘The scope of a functional calculus approach to fractional differential equations’, in Progress in Analysis (Proceedings of ISAAC’01), World Scientific, Singapore, 2003, pp. 69–81.Google Scholar
  3. 3.
    Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, University Press, Princeton, New Jersey, 1971.Google Scholar
  4. 4.
    Walter, W., Einführung in die Theorie der Distributionen. 3. Aufl. Bibliographisches Institut Mannheim, Germany, 1994.Google Scholar
  5. 5.
    Gel’fand, I. M. and Shilov, G. E., Generalized Functions, Vol. 1. Academic Press, New York, 1964.Google Scholar
  6. 6.
    Grubb, G., ‘Pseudodifferential boundary problems and applications’, Deutsche Mathematiker Vereinigung, Jahresbericht99(3), 1997, 110–121.Google Scholar
  7. 7.
    Kempfle, S. and Schäfer, I., ‘Fractional differential equations and initial conditions’, Fractional Calculus & Applied Analysis3(4), 2000, 387–400.Google Scholar
  8. 8.
    Beyer, H. and Kempfle, S., ‘Definition of physically consistent damping laws with fractional derivatives’, Zentralblatt für angewandte Mathematik und Mechanik75(8), 1995, 623–635.Google Scholar
  9. 9.
    Kempfle, S. and Beyer, H., ‘Global and causalsolutions of fractional differential equations’, in Proceedings of the 21st International Workshop on Transform Methods & Special Functions, Varna’96,IMI-BAS, Sofia, Bulgaria, P. Rusev, I. Dimovski, and V. Kiryakowa (eds.), 1998, pp. 210–226.Google Scholar
  10. 10.
    Bathe, K. J. and Wilson, E., Numerical Methods in Finite Element Analysis, Prentice Hall, New Jersey, 1976.Google Scholar
  11. 11.
    Schmidt, N., ‘Impulsantworten von fraktional gedämpften Zwei- und Dreimassenschwingern’, Studienarbeit, UniBwH, Hamburg, Germany, 2003 [in German].Google Scholar
  12. 12.
    Conway, J. B., Functions of One Complex Variable, Springer-Verlag, New York, 1978.Google Scholar
  13. 13.
    Henrici, P., Applied and Computational Complex Analysis, Vol. 1, Wiley, New York, 1974.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringHelmut Schmidt UniversitätHamburgGermany

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