On the varying brachistochrone shape with allowance for chute loading limitation

S. O. Gladkov   ; S. B. Bogdanova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 73 (2021), pp. 60-70

Résumé

In this paper, the problem of the effect of stress limit on the chute shape is analyzed for the first time. Here, the dynamic equations for the motion of the material body rolling down the chute are formulated neglecting the friction forces. It is shown that if the stress limit for the chute material is taken into account, the shape of the chute varies greatly as a function of the parameter. Four possible cases are analyzed when the parameter is: equal to zero, more than unity, less than unity, and equal to unity. It is found that if the parameter is more than unity, the chute shape represents almost horizontal and vertical segments of a trajectory, which is clear from a physical point of view, since for this type of the trajectory the chute is least affected by the body moving along. If the parameter is equal to unity, the chute takes a specific loop-like shape. If the parameter is equal to zero, the system of equations describes a classical brachistochrone. The solution to the problem is applicable in practice for predicting the shape of the chute withstanding high loads when the stress limit for the material is known.

Détail

Keywords: dynamic equations, reaction force, stress limit.
Mots-clés : chute shape

Bibliographie

[1] Gladkov S. O., Bogdanova S. B., “Geometricheskii fazovyi perekhod v zadache o brakhistokhrone”, Uchenye zapiski fizicheskogo fakulteta MGU, 2016, no. 1, 161101, 5 pp.

[2] Gladkov S. O., “O traektorii dvizheniya tela, vkhodyaschego v zhidkost pod proizvolnym uglom”, Uchenye zapiski fizicheskogo fakulteta MGU, 2016, no. 4, 164002, 5 pp.

[3] Gladkov S. O., Bogdanova S. B., “Obobschennye dinamicheskie uravneniya ploskogo krivolineinogo dvizheniya materialnogo tela po zhelobu s uchetom sil treniya (ikh chislennyi analiz v nekotorykh chastnykh sluchayakh)”, Uchenye zapiski fizicheskogo f-ta MGU, 2017, no. 1, 171101, 5 pp.

[4] Gladkov S. O., Bogdanova S. B., “K teorii dvizheniya sharika po vraschayuscheisya brakhistokhrone s uchetom sil treniya”, Uchenye zapiski fizicheskogo f-ta MGU, 2017, no. 2, 172101, 6 pp.

[5] Gladkov S. O., Bogdanova S. B., “Analiticheskoe i chislennoe reshenie zadachi o brakhistokhrone v nekotorykh obshdkh sluchayakh”, Itogi nauki i tekhniki. Sovremennaya matematika i ee prilozheniya. Tematicheskie obzory, 145, 2018, 114–122

[6] Gladkov S. O., Bogdanova S. B., “K teorii dvizheniya tel s peremennoi massoi”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2020, no. 65, 83–91

[7] Smirnov V. I., Kurs vysshei matematiki, v. 2, Nauka, M., 1967, 655 pp. | MR

[8] Elsgolts L. E., Differentsialnye uravneniya i variatsionnoe ischislenie, Nauka, M., 1969, 424 pp.

[9] Yang L., Lektsii po variatsionnomu ischisleniyu i teorii optimalnogo upravleniya, Mir, M., 1974, 488 pp.

[10] Gladkov S. O., Bogdanova S.B., “K teorii prostranstvennoi brakhistokhrony”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2020, no. 68, 53–60

[11] Boltyanskii V. G., Matematicheskie metody optimalnogo upravleniya, Nauka, M., 1969, 408 pp.

[12] Mansimov K. B., Rasumova Sh. M., “Ob optimalnosti osobykh upravlenii v odnoi zadache optimalnogo upravleniya”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2018, no. 54, 17–33 | DOI

[13] Giannoni F., Piccione P., Verderesi J. A., “An approach to the relativistic brachistochrone problem by sub-Riemannian geometry”, J. Math. Phys., 38:12 (1997), 6367–6381 | DOI | MR | Zbl

[14] Scarpello G. M., Ritelli D., “Relativistic brachistochrones under electric or gravitational uniform fields”, Z. Angew. Math. Mech., 86:9 (2006), 736–743 | DOI | MR | Zbl

[15] Ivanov A. I., “O brakhistokhrone chastitsy peremennoi massy s postoyannym otnosheniem kolichestva prisoedinyaemykh i otdelyaemykh chastits”, Dokl. AN USSR. Ser. A, 1968, 683–686 | Zbl

[16] Russalovskaya A. V., Ivanov G. I., Ivanov A. I., “O brakhistokhrone tochki peremennoi massy s treniem i eksponentsialnym zakonom istecheniya massy”, Dokl. AN USSR. Ser. A, 1973, 1024–1026 | MR | Zbl

[17] Jeremic O., Salinic S., Obradovic A., Mitrovic Z., “On the brachistochrone of a variable mass particle in general force fields”, Math. And Computer Modelling, 54 (2011), 2900–2912 | DOI | MR | Zbl

[18] Obradovic A., Salinic S., Jeremic O., Mitrovic Z., “Brachistochronic motion of a variable mass system”, Third Serbian (28th Yu) Congress on Theoretical and Applied Mechanics (Vlasina lake, Serbia, 5–8 July, 2011), 1237–1246

[19] Salinic S., Obradovic A., Mitrovic Z., Rusov S., “On the brachistochronic motion of the Chaplygin sleigh”, Acta Mech., 224:9 (2013) | DOI | MR | Zbl

[20] Lipp S., “Brachistochrone with Coulomb friction”, SIAM J. Control Optim., 35:2 (1997), 562–584 | DOI | MR | Zbl