Advertisement

Fractional Calculus of Hydraulic Drag in the Free Falling Process

Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

A new approach to describe the hydraulic drag received by a falling body has been developed through fractional calculus, and the analytical solution has been given. This new method treats the measurement of the hydraulic drag as the fractional derivative of the falling body’s displacement. The existing methods could be classified into two categories. The first ones assume the drag could be described by a quadratic equitation of the body’s velocity and use the classical Newton law to describe the falling process. The second ones do introduce the fractional calculus to describe the dynamic process but still treat the drag as the quadratic of velocity, which make the physics meaning of parameters are obscure. Compared to existing methods, the new approach introduced in this paper is original from the perspectives of basic hypothesis and modeling. To evaluate the performance of this method, series experiments have been conducted with the help of high speed camera. The data fit the new method successfully, and compared to the existing approaches, the new one has the overall better performance on the accuracy to describe the dynamic process of the falling body and owns intuitive physical explanation of its parameters.

Keywords

fractional calculus hydraulic drag falling process 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. 1.
    Loverro A., Fractional Calculus: History, Definitions and Applications for the Engineer, 2004Google Scholar
  2. 2.
    Mainardi F., Applications of Fractional Calculus in Mechanics, Transform Methods and Special Functions, Varna’96, SCT Publishers, Singapore, 1997.Google Scholar
  3. 3.
    Rossikhin Y.A, Shitikova M.V., Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids, Appl. Mech. Rev. 50,15-67, 1997.Google Scholar
  4. 4.
    Bagley R.L. and Calico R.A., Fractional Order State Equations for the Control of Viscoelastically Damped Structures, J.Guidance, vol. 14, no. 5 pp.304-311, 1991CrossRefGoogle Scholar
  5. 5.
    Makroglou A., Miller R.K. and Skkar S., Computational Results for a Feedback Control for a Rotating Viscoelastic Beam, J of Guidance, Control and Dynamics, vol. 17, no. 1, pp. 84–90, 1994.CrossRefGoogle Scholar
  6. 6.
    Oldham K.B., A Signal Independent Electro-analytical Method, Anal. Chem., vol.72 pp.371-378, 1976Google Scholar
  7. 7.
    Goto M. and Ishii D., Semi-differential Electro-analysis, J. Electro anal. Chem. and Interfacial Electrochemical., vol.61, pp.361-365, 1975CrossRefGoogle Scholar
  8. 8.
    Wiggins, C. H., and Goldstein R.E. "Flexive and Propulsive Dynamics of Elastica at Low Reynolds Number." Physical Review Letters 80.17 3879–3882, 1997Google Scholar
  9. 9.
    Fa, Kwok Sau, A falling body problem through the air in view of the fractional derivative approach, Physica A, Volume 350, Issue 2–4, p. 199–206. May 2005.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Davis H.T., Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962MATHGoogle Scholar
  11. 11.
    Eiben, A. E. et al, Genetic Algorithms with Multi-parent Recombination". PPSN III: Proceedings of the International Conference on Evolutionary Computation. The Third Conference on Parallel Problem Solving from Nature: 78–87, 1994Google Scholar
  12. 12.
    Caputo M., Linear Model of Dissipation Whose Q is Almost Frequency Independent - II, Geophys. J.R. Astr. Soc., vol. 13, pp.529-539, 1967Google Scholar
  13. 13.
    El-Sayed A. M. A., Multivalued Fractional Differential Equations, Applied. Math and Computer, vol. 80, pp. 1–11, 1994Google Scholar
  14. 14.
    Gorenflo, R. and Mainardi F., Fractional Calculus, Integral and Differential Equations of Fractional Order, Fractals and Fractional Calculus in Continuum Mechanics, 223–276, Springer Verlag. New York, 1997Google Scholar
  15. 15.
    Erdelyi A., Higher Transcendental Functions, vol.3, McGrawHill, New York, 1955.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mechanical Engineering TechnologyPurdue UniversityWest LafayetteUSA
  2. 2.Department of Mechanical Engineering TechnologyPurdue UniversityWest LafayetteUSA

Personalised recommendations