Geodesic
Analysis of interaction between solids of revolution and a beam corresponding to the Kelvin model
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 65 (2020), pp. 92-97 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Various formulations of the problem of a disk rolling on a plane have been studied by different researchers. In this paper, the effect of longitudinal oscillations of a beam, caused by the circular motion of a solid disk along the beam, on the mode of the disk motion is analyzed. Two versions of properties of a beam material are considered: an elastic beam and a viscoelastic beam corresponding to the Kelvin rheological model with relaxation and creep properties. The Fourier method is used as a method of separation of variables in the problem solving. When testing the beam and assuming its hereditary deformation, the rheological response force is introduced, which depends on longitudinal strains and their rate. The obtained result is presented as functions of time, which are adaptable for numerical integration. It is shown that beam oscillations arise from the disk motion and can be considered as self-oscillations.
Keywords: Kelvin model, Laplace integral transform, integro-differential equations.
Mots-clés : longitudinal oscillations 
@article{VTGU_2020_65_a6,
     author = {M. A. Kalmova and O. V. Ratmanova},
     title = {Analysis of interaction between solids of revolution and a beam corresponding to the {Kelvin} model},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {92--97},
     year = {2020},
     number = {65},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2020_65_a6/}
}
TY  - JOUR
AU  - M. A. Kalmova
AU  - O. V. Ratmanova
TI  - Analysis of interaction between solids of revolution and a beam corresponding to the Kelvin model
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2020
SP  - 92
EP  - 97
IS  - 65
UR  - http://geodesic.mathdoc.fr/item/VTGU_2020_65_a6/
LA  - ru
ID  - VTGU_2020_65_a6
ER  - 
%0 Journal Article
%A M. A. Kalmova
%A O. V. Ratmanova
%T Analysis of interaction between solids of revolution and a beam corresponding to the Kelvin model
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2020
%P 92-97
%N 65
%U http://geodesic.mathdoc.fr/item/VTGU_2020_65_a6/
%G ru
%F VTGU_2020_65_a6
M. A. Kalmova; O. V. Ratmanova. Analysis of interaction between solids of revolution and a beam corresponding to the Kelvin model. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 65 (2020), pp. 92-97. http://geodesic.mathdoc.fr/item/VTGU_2020_65_a6/

[1] A. Yu. Ishlinskiy, “Longitudinal vibrations of a rod in the presence of the linear law of consequences and relaxation”, Prikladnaya matematika i mekhanika – Journal of Applied Mathematics and Mechanics, 4:1 (1940), 18–21

[2] F. B. Badalov, Dynamic dampers of vibrations of hereditary-deformable systems, TashGAI, Tashkent, 2003, 81 pp.

[3] Yu. I. Neymark, N. A. Fufaev, Dynamics of nonholonomic systems, Nauka, M., 1967

[4] V. G. Wilke, Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika, 1997, no. 1, 48–55 | MR | Zbl

[5] Yu. A. Chirkunov, “Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin's model”, Journal of Applied Mathematics and Mechanics, 79:5 (2015), 506–513 | DOI | MR | Zbl

[6] A. R. Miftakhova, “Contact Problems for Rolling with Slip for Viscoelastic Solids”, Journal of Friction and Wear, 39:1 (2018), 55–61 | DOI

[7] V. Bogomolov, I. Raznitsyn, “On equivalence of kelvin and maxwell multielement models”, Avtomobil'nyy transport (Kharkov), 37 (2015), 175–181

[8] Yu. N. Rabotnov, Elements of hereditary mechanics of solids, Nauka, M., 1977

[9] V. V. Moskvitin, Resistance of viscoelastic materials, Nauka, M., 1972

[10] M. A. Koltunov, Creep and relaxation, Vysshaya shkola, M., 1976

[11] A. Yu. Ishlinsky, “On equations of spatial deforming of not quite elastic and viscoplastic bodies”, Izvestiya AN SSR, 1945, no. 3, 24–35

[12] M. Kalmova, G. Pavlov, “Analyzing the influence of the disk motion on longitudinal oscillations of a beam with rheological properties”, MATEC Web of Conferences, 196:4 (2018), 01004 | DOI

[13] A. R. Rzhanitsyn, Some issues of mechanics of the systems deforming with time, Gostekhizdat, M., 1949