A new brachistochrone problem of the motion of a material point on a transcendental surface between two given points in a vertical uniform field of gravity is considered. The surface is a cut horizontal cylinder whose directrix is a cycloid, and generatrix is straight lines parallel to the horizontal axis. Energy dissipation is neglected. A time functional is obtained and a differential equation is derived for a single parameter that describes the form of brachistochrons in the problem. Spatial parametric equations of the brachistochronic motion of a material point on a surface are obtained. The fast response time is determined.
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References
P. K. Aravind, “Simplified approach to brachistochrone problem,” American J. Phys., 49, No. 9, 884–886 (1981).
N. Ashby, W. E. Britting, W. F. Love, and W. Wyss, “Brachistochrone with Coulomb friction,” American J. Phys., 43, No. 10, 902–906 (1975).
V. Covic, and M. Veskovic, “Brachistochrone on a surface with Coulomb friction,” Int. J. Non-Lin. Mech., 43, 437–450 (2008).
H. H Denman, “Remarks on brachistochrone-tautochrone problem,” American J. Phys., 53, No. 3, 224–227 (1985).
W. Dunham, Journey Through Genius, Penguin Books, New York (1991).
L. P. Eltsgolts, Differential Equations and Variational Calculus [in Russian], Nauka, Moscow (1974).
H. Erlichson, “Johann Bernoulli’s brachistochrone solution using Fermat’s principle of least time,” European J. Phys., 20, 299–304 (1999).
I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Inc., New York (1963).
J. Gemmer, R. Umble, and M. Nolan, “Generalizations of the brachistochrone problem,” Pi Mu Epsilon J., 13, No. 4, 207–218 (2011).
H. F. Goldstein and C. M. Bender, “Relativistic brachistochrone,” J. Math. Phys., 27, No. 2, 507–511 (1986).
J. C. Hayen, “Brachistochrone with Coulomb friction,” Int. J. of Non-Linear Mechanics, 40, No. 8, 1057–1075 (2005).
V. P. Legeza, “Quickest-descent curve in the problem of rolling of a homogeneous cylinder,” Int. Appl. Mech., 44, No. 12, 1430–1436 (2008).
V. P. Legeza, “Brachistochrone for a rolling cylinder,” Mechanics of Solids, 45, No. 1, 27–33 (2010).
V. P. Legeza, “Cycloidal pendulum with a rolling cylinder,” Mechanics of Solids, 47, No. 4, 380–384 (2012).
S. Lipp, “Brachistochrone with Coulomb friction,” SIAM J. Control Optim., 35, No. 2, 562–584 (1997).
S. Mertens and S. Mingramm, “Brachistochrones with loose ends,” European J. of Physic, 29, 1191–1199 (2008).
A. Obradovic, S. Salinic, and R. Radulovic, “The brachistochronic motion of a vertical disk rolling on a horizontal plane without slip,” Theor. Appl. Mech., 44, No. 2, 237–254 (2017).
D. Palmieri, The Brachistochrone Problem, a New Twist to an Old Problem, Undergraduate Honors Thesis. Millersville University (1996).
A. S. Parnovsky, “Some generalisations of the brachistochrone problem,” Acta Physica Polonica, Ser. A: General Physics, 93, No. 145, 5–55 (1998).
E. Rodgers, “Brachistochrone and tautochrone curves for rolling bodies,” American J. Phys., 14, 249–252 (1946).
G. M. Scarpello and D. Ritelli, “Relativistic brachistochrone under electric or gravitational uniform field,” ZAMM., 86, No. 9, 736–743 (2006).
Srikanth Sarma Gurram, Sharan Raja, Pallab Sinha Mahapatra, and Mahesh V. Panchagnula, “On the brachistochrone of a fluid-filled cylinder,” J. Fluid Mech., 865, 775–789 (2019).
G. Tee, “Isochrones and brachistochrones,” Neural, Parallel Sci. Comput., 7, 311–342 (1999).
A. M. A. Van der Heijden and J. D. Diepstraten, “On the brachistochrone with dry friction,” Int. J. Non-Lin. Mech., 10, 97–112 (1975).
G. Venezian, “Terrestrial brachistochrone,” American J. Phys., 34, No. 8, 701 (1966).
B. Vratanar and M. Saje, “On analytical solution of the brachistochrone problem in a non-conservative field,” Int. J. Non-Lin. Mech., 33, No. 3, 437–450 (1998).
H. A. Yamani and A. A. Mulhem, “A cylindrical variation on the brachistochrone problem,” American J. Phys., 56, No. 5, 467–469 (1988). 366
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Translated from Prikladnaya Mekhanika, Vol. 56, No. 3, pp. 112–1213, May–June 2020.
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Legeza, V.P. Brachistochronic Motion of a Material Point on a Transcendental Surface. Int Appl Mech 56, 358–366 (2020). https://doi.org/10.1007/s10778-020-01019-5
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DOI: https://doi.org/10.1007/s10778-020-01019-5