Abstract
The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate x of the particle, i.e., u=d 2 y/dx 2. Applying Pontryagin’s maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved.
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References
Elsgolc, L.E.: Calculus of Variations. Pergamon Press, Oxford (1963)
Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice Hall, Englewood Cliffs (1964)
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New Jersey (1962)
Gabasov, R., Kirillova, F.M.: Singular Optimal Controls. Nauka, Moscow (1973)
Erlichson, H.: Johann Bernoulli’s brachistochrone solution using Fermat’s principle of least time. Eur. J. Phys. 20, 299–304 (1999)
Čović, V., Lukačević, M., Vesković, M.: On Brachistochronic Motions. Budapest University of Technology and Economics, Budapest (2007)
Parnovsky, A.S.: Some generalisations of brachistochrone problem. Acta Phys. Pol. A 93, S55–S64 (1998)
Ashby, N., Brittin, W.E., Love, W.F., Wyss, W.: Brachistochrone with Coulomb friction. Am. J. Phys. 43(10), 902–906 (1975)
Gershman, M.D., Nagaev, R.F.: O frikcionnoj brakhistokhrone. Izv. Akad. Nauk SSSR, Meh. Tverd. Tela 4, 85–88 (1976)
Hayen, J.C.: Brachistochrone with Coulomb friction. Int. J. Non-Linear Mech. 40, 1057–1075 (2005)
Šalinić, S.: Contribution to the brachistochrone problem with Coulomb friction. Acta Mech. 208(1–2), 97–115 (2009)
Van der Heijden, A.M.A., Diepstraten, J.D.: On the brachistochrone with dry friction. Int. J. Non-Linear Mech. 10, 97–112 (1975)
Vratanar, B., Saje, M.: On the analytical solution of the brachistochrone problem in a non-conservative field. Int. J. Non-Linear Mech. 33(3), 489–505 (1998)
von Kleinschmidt, W., Schulze, H.K.: Brachistochronen in einem zentralsymmetrischen Schwerefeld. Z. Angew. Math. Mech. 50, T234–T236 (1970)
Shevchenko, K.N.: Time-optimal motion of a point acted upon by a system of central forces. Mech. Solids 19(6), 25–31 (1984)
Shevchenko, K.N.: Brachistochrone and the principle of least action. Mech. Solids 21(2), 36–42 (1986)
Singh, B., Kumar, R.: Brachistochrone problem in nonuniform gravity. Indian J. Pure Appl. Math. 19(6), 575–585 (1988)
Ivanov, A.I.: On the brachistochrone of a variable mass point with constant relative rates of particle throwing away and adjoining. Dokl. Akad. Nauk Ukr. SSR Ser. A 683–686 (1968)
Russalovskaya, A.V., Ivanov, G.I., Ivanov, A.I.: On brachistochrone of the variable mass point during motion with friction with an exponential rule of mass rate flow. Dokl. Akad. Nauk Ukr. SSR Ser. A 1024–1026 (1973)
Djukić, Dj.: The brachistochronic motion of a material point on surface. Riv. Mat. Univ. Parma 4(2), 177–183 (1976)
Čović, V., Vesković, M.: Brachistochrone on a surface with Coulomb friction. Int. J. Non-Linear Mech. 43(5), 437–450 (2008)
Maisser, P.: Brachystochronen als zeitkrzeste Fahrspuren von Bobschlitten. Z. Angew. Math. Mech. 78(5), 311–319 (1998)
Djukić, Dj., Atanacković, T.M.: A note on the classical brachistochrone. Z. Angew. Math. Phys. 27, 677–681 (1976)
Dooren, R.V., Vlassenbroeck, J.: A new look at the brachistochrone problem. Z. Angew. Math. Phys. 31, 785–790 (1980)
Lipp, S.C.: Brachistochrone with Coulomb friction. SIAM J. Control Optim. 35(2), 562–584 (1997)
Hennessey, M.P., Shakiban, Ch.: Brachistochrone on a 1D curved surface using optimal control. J. Dyn. Syst. Meas. Control 132, 034505 (2010)
Gershman, M.D., Nagaev, R.F.: The oscillation brachistochrone problem. Mech. Solids 14(2), 9–17 (1979)
Razzaghi, M., Sepehrian, B.: Single-term Walsh series direct method for the solution of nonlinear problems in the calculus of variations. J. Vib. Control 10, 1071–1081 (2004)
Cruz, P.A.F., Torres, D.F.M.: Evolution strategies in optimization problems. Proc. Est. Acad. Sci., Phys. Math. 56(4), 299–309 (2007)
Julstrom, B.A.: Evolutionary algorithms for two problems from the calculus of variations. In: Lecture Notes in Computer Science, Genetic and Evolutionary Computation-GECCO, pp. 2402–2403. Springer, Berlin (2003)
Wensrich, C.M.: Evolutionary solutions to the brachistochrone problem with Coulomb friction. Mech. Res. Commun. 31, 151–159 (2004)
Djukic, Dj.: The brachistochronic motion of a gyroscope mounted on the gimbals. Theor. Appl. Mech. 2, 37–40 (1976)
Legeza, P.V.: Quickest-descent curve in the problem of rolling of a homogeneous cylinder. Int. Appl. Mech. 44(12), 1430–1436 (2008)
Akulenko, L.D.: The brachistochrone problem for a disc. J. Appl. Math. Mech. 73(4), 371–378 (2009)
Legeza, P.V.: Conditions for pure rolling of a heavy cylinder along a brachistochrone. Int. Appl. Mech. 46(6), 730–735 (2010)
Legeza, V.P.: Brachistochrone for a rolling cylinder. Mech. Solids 45(1), 27–33 (2010)
Čović, V., Lukačević, M.: Extension of the Bernoulli’s case of a brachistochronic motion to the multibody system in the form of a closed kinematic chain. Facta Univ., Mech. Autom. Control Robot. 2(9), 973–982 (1999)
Čović, V., Vesković, M.: Extension of the Bernoulli’s case of brachistochronic motion to the multibody system having the form of a kinematic chain with external constraints. Eur. J. Mech. A, Solids 21, 347–354 (2002)
Čović, V., Vesković, M.: Brachistochronic motion of a multibody system with Coulomb friction. Eur. J. Mech. A, Solids 28(9), 882–890 (2009)
Zekovic, D.: On the brachistochronic motion of mechanical systems with non-holonomic, non-linear and rheonomic constraints. J. Appl. Math. Mech. 54(6), 931–935 (1990)
Zekovic, D., Covic, V.: On the brachistochronic motion of mechanical systems with linear nonholonomic nonhomogeneous constraints. Mech. Res. Commun. 20(1), 25–35 (1993)
Obradović, A., Čović, V., Vesković, M., Dražić, M.: Brachistochronic motion of a nonholonomic rheonomic mechanical system. Acta Mech. 214(3–4), 291–304 (2010)
Djukić, Dj.: On the brachistochronic motion of a dynamic system. Acta Mech. 32, 181–186 (1979)
Stoer, J., Bulirsch, J.: Introduction to Numerical Analysis. Springer, Berlin (1993)
McDanell, J.P., Powers, W.F.: Necessary conditions for joining optimal singular and nonsingular subarcs. SIAM J. Control 9(2), 161–173 (1971)
Stork, D.G., Yang, J.: The general unrestrained brachistochrone. Am. J. Phys. 56(1), 22–26 (1988)
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Šalinić, S., Obradović, A., Mitrović, Z. et al. Brachistochrone with limited reaction of constraint in an arbitrary force field. Nonlinear Dyn 69, 211–222 (2012). https://doi.org/10.1007/s11071-011-0258-1
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DOI: https://doi.org/10.1007/s11071-011-0258-1