Nonlinear Dynamics

, Volume 38, Issue 1–4, pp 191–206 | Cite as

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

  • Om Prakash AgrawalEmail author


This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.

Key words:

fractional derivatives fractional power polynomials thermal analysis of disk brakes 


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  1. 1.
    Du, S. ‘Thermoelastic Effects in Automotive Brakes’, Ph.D. Thesis, Mechanical Engineering, University of Michigan, Ann Arbor, 1997.Google Scholar
  2. 2.
    Du, S., Zagrodzki, P., Barber, J. R., and Hulbert, G. M. ‘Finite element analysis of frictionally excited thermoelastic instability’, Journal of Thermal Stresses20, 1997, 185–201.Google Scholar
  3. 3.
    Agrawal, O. P. ‘Fractional derivatives and its applicationin thermal analysis and properties measurements of disk brakes’, Quarterly Report, Center for Advanced Friction Studies, SIUC, May 2003.Google Scholar
  4. 4.
    Samko, S. G., Kilbas, A. A., and Marichev, O. I. Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach, Longhorne, Pennsylvania, 1993.Google Scholar
  5. 5.
    Oldham, K. B. and Spanier, J. The Fractional Calculus, Academic Press, New York, 1974.Google Scholar
  6. 6.
    Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Google Scholar
  7. 7.
    Gorenflo, R. and Mainardi, F. ‘Fractional calculus: Integral and differential equations of fractional order’, in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, Vienna,1997, pp. 223–276.Google Scholar
  8. 8.
    Mainardi, F. ‘Fractional calculus: Some basic problems in continuum and statistical mechanics’, in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, Vienna, 1997, pp. 291-348.Google Scholar
  9. 9.
    Rossikhin, Y. A. and Shitikova, M. V. ‘Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids’, Applied Mechanics Reviews50, 1997, 15–67.Google Scholar
  10. 10.
    Podlubny, I. Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
  11. 11.
    Butzer, P. L. and Westphal, U. ‘An introduction tofractional calculus’, in Applications of Fractional Calculus in Physics, R. Hilfer (ed.), World Scientific,River Edge, New Jersey, 2000, 1–85.Google Scholar
  12. 12.
    Kulish, V. V. and Lage, J. L. ‘Fractional diffusion solutions for transient local temperature and heat-flux’, ASME Journal of Heat Transfer122, 2000, 372–377.Google Scholar
  13. 13.
    Kulish, V. V., Lage, J. L., Komarov, P. L., and Raad, P. E. ‘A fractional-diffusion theory for calculating thermal propertiesof thin films from surface transient thermoreflectance measurements’, ASME Journal of Heat Transfer123, 2001, 1133-1138.Google Scholar
  14. 14.
    Battaglia, J. L., Cois, O., Puigsegur, L., Oustaloup, A. ‘Solving an inverse heat conduction problem using a non-integer identified model’, International Journal of Heat and Mass Transfer44, 2001, 2671–2680.Google Scholar
  15. 15.
    Carslaw, H. S. and Jaeger, J. C. Conduction of Heat In Solids, Oxford University Press, Oxford, 1959.Google Scholar
  16. 16.
    Reddy, J. N. and Gartling, D. K. The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, New York, 2001.Google Scholar
  17. 17.
    Beck, J. V., Blackwell, B., and St. Clair, Jr., C. R. Inverse Heat Conduction: Ill-Posed Problems, Wiley, New York, 1985.Google Scholar
  18. 18.
    Wriggers, P. and Miehe, C. ‘Contact constraints within coupled thermomechanical analysis – A finite element model’, Computer Methods in Applied Mechanics and Engineering113, 1994, 301–319.Google Scholar
  19. 19.
    Marx, D.T., Policandriotes, T., Zhang, S., Scott, J., Dinwiddie, R. B., and Wang, H. ‘Measurement of interfacial temperaturesduring testing of a subscale aircraft brake’, Journal of Physics D: Applied Physics34, 2001, 976–984.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Mechanical Engineering and Energy ProcessesSouthern Illinois UniversityCarbondaleU.S.A.

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