Geodesic
Ball on a~viscoelastic plane
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 281 (2013), pp. 98-126.

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We consider dynamical problems arising in connection with the interaction of an absolutely rigid ball and a viscoelastic support plane. The support is a relatively stiff viscoelastic Kelvin–Voigt medium that coincides with the horizontal plane in the undeformed state. We also assume that under the deformation the support induces dry friction forces that are locally governed by the Coulomb law. We study the impact appearing when a ball falls on the plane. Another problem of our interest is the motion of a ball “along the plane”. A detailed analysis of various stages of the motion is presented. We also compare this model with classical models of interaction of solid bodies. 
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A. A. Zobova; D. V. Treschev. Ball on a~viscoelastic plane. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 281 (2013), pp. 98-126. http://geodesic.mathdoc.fr/item/TM_2013_281_a8/

[1] Andronov V.V., Zhuravlev V.F., Sukhoe trenie v zadachakh mekhaniki, Regulyarnaya i khaoticheskaya dinamika, Izhevsk, 2010

[2] Galin L.A., Kontaktnye zadachi teorii uprugosti i vyazkouprugosti, Nauka, M., 1980 | MR | Zbl

[3] Goryacheva I.G., Mekhanika friktsionnogo vzaimodeistviya, Nauka, M., 2001

[4] Ishlinskii A.Yu., “Trenie kacheniya”, PMM, 2:2 (1938), 245–260

[5] Ishlinskii A.Yu., “Teoriya soprotivleniya perekatyvaniyu (trenie kacheniya) i smezhnykh yavlenii”, Vsesoyuz. konf. po treniyu i iznosu v mashinakh, T. 2, Izd-vo AN SSSR, M.; L., 1940, 255–264

[6] Ishlinskii A.Yu., “O proskalzyvanii v oblasti kontakta pri trenii kacheniya”, Izv. AN SSSR. Otd. tekhn. nauk, 1956, no. 6, 3–15 | MR

[7] Ishkhanyan M.V., Karapetyan A.V., “Dinamika odnorodnogo shara na gorizontalnoi ploskosti s uchetom treniya skolzheniya, vercheniya i kacheniya”, Izv. RAN. Mekhanika tverdogo tela, 2010, no. 2, 3–14 | MR

[8] Karapetyan A.V., “Dvukhparametricheskaya model treniya”, PMM, 73:4 (2009), 515–519 | MR | Zbl

[9] Karapetyan A.V., “O modelirovanii sil treniya v dinamike shara na ploskosti”, PMM, 74:4 (2010), 531–535 | MR | Zbl

[10] Kireenkov A.A., “Svyazannye modeli treniya skolzheniya i kacheniya”, DAN, 419:6 (2008), 759–762 | Zbl

[11] Kuleshov A.S., Treschev D.V., Ivanova T.B., Naimushina O.S., “Tverdyi tsilindr na vyazkouprugoi ploskosti”, Nelineinaya dinamika, 7:3 (2011), 601–625

[12] Levi-Chivita T., Amaldi U., Kurs teoreticheskoi mekhaniki, T. 2, Izd-vo inostr. lit., M., 1951

[13] Markeev A.P., Dinamika tela, soprikasayuschegosya s tverdoi poverkhnostyu, 2-e izd., ispr. i dop., Regulyarnaya i khaoticheskaya dinamika, Izhevsk, 2011

[14] Negolonomnye dinamicheskie sistemy: Integriruemost, khaos, strannye attraktory, Pod red. A.V. Borisova, I.S. Mamaeva, In-t kompyut. issled., Moskva; Izhevsk, 2002 | MR

[15] Neimark Yu.I., Fufaev N.A., Dinamika negolonomnykh sistem, Nauka, M., 1967

[16] Raus E.Dzh., Dinamika sistemy tverdykh tel, T. 1, 2, Nauka, M., 1983

[17] Sentemova O.S., “O polikomponentnykh modelyakh treniya”, Vestn. Mosk. un-ta. Matematika. Mekhanika, 2011, no. 6, 57–59 | MR

[18] Chaplygin S.A., Issledovaniya po dinamike negolonomnykh sistem, Gostekhteorizdat, M.; L., 1949; 2-е изд., УРСС, М., 2007

[19] Al-Bender F., De Moerlooze K., “Characterization and modeling of friction and wear: an overview”, Sustainable Constr. Des., 2:1 (2011), 19–28

[20] Contensou P., “Couplage entre frottement de glissement et frottement de pivotement dans la théorie de la toupie”, Kreiselprobleme/Gyrodynamics, Proc. Symp., Celerina, 1962, Springer, Berlin, 1963, 201–216 ; Контенсу П., “Связь между трением скольжения и трением верчения и ее учет в теории волчка”, Проблемы гироскопии, Мир, М., 1967, 60–77 | MR

[21] Erismann Th., “Theorie und Anwendungen des echten Kugelgetriebes”, Z. angew. Math. Phys., 5:5 (1954), 355–388 | DOI | MR | Zbl

[22] Hertz H., “Ueber die Berührung fester elastischer Körper”, J. reine angew. Math., 92 (1882), 156–171

[23] Pöschel T., Brilliantov N.V., Zaikin A., “Bistability and noise-enhanced velocity of rolling motion”, Europhys. Lett., 69 (2005), 371–377 | DOI

[24] Pöschel T., Schwager T., Brilliantov N.V., “Rolling friction of a hard cylinder on a viscous plane”, Eur. Phys. J. B., 10 (1999), 169–174 | DOI

[25] Pöschel T., Schwager T., Brilliantov N.V., Zaikin A., “Rolling friction and bistability of rolling motion”, Powders and grains 2005, Proc. 5th Int. Conf. on Micromechanics of Granular Media, Stuttgart, 2005, V. 2, Taylor Francis, London, 2005, 1247–1253