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Home » Issues » Archive of Issues » 2022 - 4 issue » pp. 528-545

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Russia Voronezh
Year
2022
Volume
32
Issue
4
Pages
528-545
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Section Mathematics
Title Model of deformations of a Stieltjes string system with a nonlinear condition
Author(-s) Zvereva M.B.a
Affiliations Voronezh State Universitya
Abstract In the present paper we study a model of deformations for a system of $n$ Stieltjes strings located along a geometric graph-star with a nonlinear condition at the node. The corresponding boundary value problem has the form $$ \left\{\begin{array}{lll} -\left(p_iu_i^\prime\right)(x)+\displaystyle{\int_0^x}u_idQ_i=F_i(x)-F_i(+0)-(p_iu_i')(+0), \quad i=1,2, \ldots, n,\\ \sum\limits_{i=1}^np_i(+0)u_i'(+0)\in N_{[-m,m]}u(0),\\ u_1(0)=u_2(0)=\ldots=u_n(0)=u(0),\\ (p_iu_i')(l_i-0)+u_i(l_i)\Delta Q_i(l_i)=\Delta F_i(l_i), \quad i=1,2,\ldots, n. \end{array} \right. $$ Here the functions $u_i(x)$ determine the deformations of each of the strings; $F_i(x)$ describe the distribution of the external load; $p_i(x)$ characterize the elasticity of strings; $Q_i(x)$ describe the elastic response of the environment. The jump $\Delta F_i(l_i)$ is equal to the external force concentrated at the point $l_i$; the jump $\Delta Q_i(l_i)$ coincides with the stiffness of the elastic support (spring) attached to the point $l_i$. The condition $\sum\limits_{i=1}^np_i(+0)u_i'(+0)\in N_{[-m,m]}u(0)$ arises due to the presence of a limiter in the node represented by the segment $ [-m,m]$, on the movement of strings under the influence of an external load, thus it is assumed that $|u(0)|\leq m$. Here $N_{[-m,m]}u(0)$ denotes the normal cone to $[-m,m]$ at the point $u(0)$. In the present paper a variational derivation of the model is carried out; existence and uniqueness theorems for solutions are proved; the critical loads at which the strings come into contact with the limiter are analyzed; an explicit formula for the representation of the solution is presented.
Keywords Stieltjes integral, function of bounded variation, measure, geometric graph, energy functional
UDC 517.927.2
MSC 34B37, 34B16
DOI 10.35634/vm220403
Received 14 November 2022
Language Russian
Citation Zvereva M.B. Model of deformations of a Stieltjes string system with a nonlinear condition, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 4, pp. 528-545.
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