Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, Volume 32, Issue 4, Pages 528–545
DOI: https://doi.org/10.35634/vm220403
(Mi vuu824)
 

This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

Model of deformations of a stieltjes string system with a nonlinear condition

M. B. Zvereva

Department of Mathematical Analysis, Voronezh State University, pl. Universitetskaya, 1, Voronezh, 394018, Russia
Full-text PDF (248 kB) Citations (1)
References:
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.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation FZGF-0640-2020-0009
Russian Foundation for Basic Research 20-51-15003 НЦНИ-а
This research was supported by the Ministry of Education of the Russian Federation within the framework of the state task in the field of science (topic number FZGF-0640-2020-0009); by RFBR and CNRS, project number 20-51-15003.
Received: 14.11.2022
Accepted: 06.12.2022
Bibliographic databases:
Document Type: Article
UDC: 517.927.2
MSC: 34B37, 34B16
Language: Russian
Citation: M. B. Zvereva, “Model of deformations of a stieltjes string system with a nonlinear condition”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 32:4 (2022), 528–545
Citation in format AMSBIB
\Bibitem{Zve22}
\by M.~B.~Zvereva
\paper Model of deformations of a stieltjes string system with a nonlinear condition
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2022
\vol 32
\issue 4
\pages 528--545
\mathnet{http://mi.mathnet.ru/vuu824}
\crossref{https://doi.org/10.35634/vm220403}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4534869}
Linking options:
  • https://www.mathnet.ru/eng/vuu824
  • https://www.mathnet.ru/eng/vuu/v32/i4/p528
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
    Statistics & downloads:
    Abstract page:129
    Full-text PDF :55
    References:22
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024