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Application of fractional derivatives in thermal analysis of disk brakes. (English) Zbl 1142.74302
Summary: 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.

##### MSC:
 74A15 Thermodynamics in solid mechanics 74F05 Thermal effects in solid mechanics 26A33 Fractional derivatives and integrals 80A20 Heat and mass transfer, heat flow (MSC2010)
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##### References:
 [1] Du, S. ?Thermoelastic Effects in Automotive Brakes?, Ph.D. Thesis, Mechanical Engineering, University of Michigan, Ann Arbor, 1997. [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. · doi:10.1080/01495739708956098 [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. [4] Samko, S. G., Kilbas, A. A., and Marichev, O. I. Fractional Integrals and Derivatives ? Theory and Applications, Gordon and Breach, Longhorne, Pennsylvania, 1993. · Zbl 0818.26003 [5] Oldham, K. B. and Spanier, J. The Fractional Calculus, Academic Press, New York, 1974. · Zbl 0292.26011 [6] Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. · Zbl 0789.26002 [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. [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. · Zbl 0917.73004 [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. · doi:10.1115/1.3101682 [10] Podlubny, I. Fractional Differential Equations, Academic Press, New York, 1999. · Zbl 0924.34008 [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. · Zbl 0987.26005 [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. · doi:10.1115/1.521474 [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. · doi:10.1115/1.1416688 [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. · Zbl 0981.80007 · doi:10.1016/S0017-9310(00)00310-0 [15] Carslaw, H. S. and Jaeger, J. C. Conduction of Heat In Solids, Oxford University Press, Oxford, 1959. · Zbl 0029.37801 [16] Reddy, J. N. and Gartling, D. K. The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, New York, 2001. · Zbl 0855.76002 [17] Beck, J. V., Blackwell, B., and St. Clair, Jr., C. R. Inverse Heat Conduction: Ill-Posed Problems, Wiley, New York, 1985. · Zbl 0633.73120 [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. · Zbl 0847.73069 · doi:10.1016/0045-7825(94)90051-5 [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. · doi:10.1088/0022-3727/34/6/320
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