A Model of Deformations of a Rod – Console With a Displacement Limiter
DOI:
https://doi.org/10.52575/2687-0959-2024-56-1-35-49Keywords:
Bounded Variation, Stieltjes Integral, Absolutely Continuous Function, Outward Normal Cone, Boundary Value Problem, Rod Deformation ModelAbstract
In the present paper a model of deformations of a singular rod – console under the influence of an external force is studied. We assume that one of the ends of the rod is hinged, and the displacement of the free end of the console is restricted by a limiter. Depended on the applied external force this end of the console either remains the internal point of the limiter or touches the boundary of the limiter. The corresponding model is implemented in the form of a boundary value problem for an integro-differential equation with the Stieltjes integral and a nonlinear boundary condition. A variational justification of the model is carried out; the necessary and sufficient conditions for the minimum of the corresponding potential energy functional are established; theorems of existence and uniqueness of the solution to the model are proved; a formula for representing of the solution is written out explicitly; the solution dependence on the size of the limiter is studied.
Acknowledgements
The work is supported by the Russian Science Foundation (project number 22-71-10008).
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Покорный Ю.В., Пенкин О.М., Прядиев В.Л., Боровских А.В., Лазарев К.П., Шабров С.А. Дифференциальные уравнения на геометрических графах. М.: Физматлит; 2004. 272 c.
Шабров С.А. Об одной математической модели малых деформаций стержневой системы с внутренними особенностями. Вестник Воронежского государственного университета. Серия: Физика. Математика. 2013;1:232–250.
Шабров С.А., Ткаченко Д.А., Белов Н.А., Ильченко А.Г. Аналог теоремы Штурма для дифференциальных уравнений четвертого порядка с производными по мере. Вестник Воронежского государственного университета. Серия: Физика. Математика. 2022; 2:107–114.
Шабров С.А., Бородина Е.А., Голованева Ф.В., Давыдова М.Б. О числе решений нелинейной граничной задачи четвертого порядка с производными по мере. Вестник Воронежского государственного университета. Серия: Физика. Математика. 2019;3:93–100.
Ben Amara J., Vladimirov A.A., Shkalikov A.A. Spectral and Oscillatory Properties of a Linear Pencil of Fourth-Order Differential Operators. Mathematical Notes. 2013;94(1):49–59.
Borovskikh A.V., Lazarev K.P. Fourth-order differential equations on geometric graphs. Journal of Mathematical Sciences. 2004;119(6):719–738.
Borovskikh A.V., Mustafokulov R., Lazarev K.P., Pokornyi Yu.V. A class of fourth-order differential equations on a spatial net. Doklady Mathematics. 1995;52(3):433–435.
Halmos P.R. Measure theory. Springer – Verlag; 1950. 304 p.
Kulaev R.Ch. On the oscillation property of Green’s function of a fourth-order boundary value problem. Mathematical Notes. 2016;100(3-4):391–402.
Kulaev R.Ch., Urtaeva A.A. Sturm separation theorems for a fourth-order equation on a graph. Mathematical Notes. 2022;111(5-6):977–981.
Kulaev R.Ch., Urtaeva A.A. On the multiplicity of eigenvalues of a fourth-order differential operator on a graph. Differential Equations. 2022;58(7):869–876.
Kunze M., Monteiro Marques M. An introduction to Moreau’s sweeping process. Lecture Notes in Physics. 2000;551:1–60.
Rudin W. Principles of mathematical analysis. McGraw-Hill; 1964. 342 p.
Shabrov S., Ilina O., Shaina E., Chechin D. On the growth speed of own values for the fourth order spectral problem with Radon-Nikodim derivatives. Journal of Physics: Conference Series. 2020;1479:1–12.
Vladimirov A.A., Shkalikov A.A. On oscillation properties of self-adjoint boundary value problems of fourth order. Doklady Mathematics. 2021;103(1):5–9.
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