Jointing of thin elastic rods and generalized Kirchhoff conditions at nodes
DOI:
https://doi.org/10.26577/JMMCS.2020.v106.i2.06Keywords:
elastic curved thin rod, curved rod, Kirchhoff's condition in knots, theory of elasticity, equation of elasticity theory, model of rod connectionAbstract
The paper presents various models of thin elastic curved rods and their joints. Such rods and their connections are important in applied research and have a long history. Nevertheless, the questions of mathematical research in the description of the procedure of transition from a multidimensional model to a one-dimensional one are not fully investigated. This paper presents the results when a thin object is pulled together into a one-dimensional object. In this case, various difficulties arise. For example, a more detailed consideration is required by the problem of describing the gluing conditions at the vertices of the limiting neck. It turns out that the type of specific bonding conditions depends on various physical premises. On the other hand, the unique solvability in various functional spaces depends on the selected gluing conditions. The first part of the paper describes various categories of bonding conditions. In the second part of the work, various examples of the construction of thin elastic curved rods and their joints will be given, when the ultimate task is supplemented by certain gluing conditions. In the conclusion of the second part, a numerical calculation of the natural frequencies of free vibrations of the joints of elastic thin curved rods is presented.
References
[2] Leckhnickii S.G., Kruchenie anizatropnyh I neodnorodnyh sterzhnei [Torsion of anisotropic and heterogeneous rods] (M.: Nauka, 1971).
[3] Kanguzhin B., Zhapsarbaeva L., Madibaiuly Zh., "Lagrange formula for differential operators self-adjoint restrictions of the maximal operator on a tree" , Eurasian Math. J., 10:1 (2019): 16-29.
[4] Zhapsarbaeva L.K., Kanguzhin B.E., Konyrkulzhayeva M.N., "Self-adjoint restrictions of maximal operator on graph" ,
Ufa. Math. J., 9:4 (2017): 35-43.
[5] Kuchment P., "Graph models of wave propagation in this structures" , Waves in Random Media, Vol. 12 (2002): 1-24.
[6] Kuchment P., Zeng H., "Convergence of spectra of mesoscopic systems collapsing onto a graph" , J. Math. Anal. Appl.
258 (2001): 671-700.
[7] Exner P., Post O., "Convergence of spectra of graph - like thin manifolds" , J. Geom. Phys. 54 (2005): 77-115.
[8] Post O., Spectral Analysis on Graph-like Spaces (Lecture Notes in Mathematics 2039. Springer-Verlag Berlin Heidelberg, 2012): 431.
[9] Nazarov S.A., "Obshaya schema osredneniya samosopryazhennyh ellipticheskih system v mnogomernyh oblostyah, v tom chisel [General scheme of averaging of self-adjoint elliptic systems in multidimensional areas, including thin ones]" , Algebra i analiz V. 7, No 5 (1995): 1-92.
[10] Nazarov S.A., Sluckii A.S., "Odnomernye uravneniya deformacii tonkih slaboiskrivlennyh sterzhnei. Asimptoticheskii analiz I obosnovanie. [One-dimensional equations of deformation of thin slightly curved rods. Asymptotic analysis and justification]" , Izvestiya RAN. ser. matem. V. 64, No 3 (2000): 97-131.
[11] Molchanov S., Vainberg B., "Scattering solutions in networks of thin fibers: Small diameter asymptotics" , Comm. Math. Phys. 273, No. 2 (2007): 533-559.
