title: ÐÐ¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð¸ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ñ ÑÐµÐ¿Ð½Ð¾Ð¹ Ð»Ð¸Ð½Ð¸Ð¸ Ð¸ ÐµÐµ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¹
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Modeling and optimization of the catenary and its modifications
Creators: name: ÐÐµÐ³Ð¸Ð»ÐµÐ²Ð¸Ñ, ÐÐ³Ð¾Ñ
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2022
Subjects: Ð¦ÐµÐ¿Ð¸
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ÐÑÐ°Ð²Ð¸ÑÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ðµ Ð¿Ð¾Ð»Ðµ
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ÐºÐ¾Ð½ÐµÑÐ½Ð¾Ð¼ÐµÑÐ½Ð°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ
catenary
optimization
tension in suspension points
finite-dimensional model

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«ÐÐ¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð¸ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ñ ÑÐµÐ¿Ð½Ð¾Ð¹ Ð»Ð¸Ð½Ð¸Ð¸ Ð¸ ÐµÐµ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¹Â». ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð³ÐµÐ¾Ð¼ÐµÑÑÐ¸ÑÐµÑÐºÐ¸Ñ Ð¸ ÑÐ¸Ð»Ð¾Ð²ÑÑ ÑÐ°ÑÐ°ÐºÑÐµÑÐ¸ÑÑÐ¸Ðº ÑÐµÐ¿Ð½Ð¾Ð¹ Ð»Ð¸Ð½Ð¸Ð¸, Ð¼Ð¸Ð½Ð¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ ÑÐ¸Ð» ÑÐµÐ°ÐºÑÐ¸Ð¹ Ð² Ð¾Ð¿Ð¾ÑÐ°Ñ ÐµÐµ ÐºÑÐµÐ¿Ð»ÐµÐ½Ð¸Ñ, ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð¼Ñ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ ÑÐµÐ¿Ð¸, ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ð´Ð¸ÑÐºÑÐµÑÐ½ÑÑ Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹ ÑÐµÐ¿Ð½Ð¾Ð¹ Ð»Ð¸Ð½Ð¸Ð¸ ÑÐ¾ ÑÐ¾ÑÑÐµÐ´Ð¾ÑÐ¾ÑÐµÐ½Ð½ÑÐ¼Ð¸ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ°Ð¼Ð¸ Ð¸ Ð½ÐµÐ¿ÑÐµÑÑÐ²Ð½ÑÑ Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹ Ñ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð½ÑÐ¼Ð¸ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ°Ð¼Ð¸. Ð Ð°ÑÑÐ¼Ð°ÑÑÐ¸Ð²Ð°ÐµÐ¼ÑÐµ Ð·Ð°Ð´Ð°ÑÐ¸: ÐÐµÑÐ°ÑÑÑÐ¶Ð¸Ð¼Ð°Ñ ÑÐµÐ¿Ð½Ð°Ñ Ð»Ð¸Ð½Ð¸Ñ Ð² Ð¾Ð´Ð½Ð¾ÑÐ¾Ð´Ð½Ð¾Ð¼ Ð³ÑÐ°Ð²Ð¸ÑÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¼ Ð¿Ð¾Ð»Ðµ; ÐÐµÑÐ°ÑÑÑÐ¶Ð¸Ð¼Ð°Ñ ÑÐµÐ¿Ð½Ð°Ñ Ð»Ð¸Ð½Ð¸Ñ Ð² Ð½ÑÑÑÐ¾Ð½Ð¾Ð²Ð¾Ð¼ Ð¿Ð¾Ð»Ðµ ÑÑÐ³Ð¾ÑÐµÐ½Ð¸Ñ; Ð Ð°ÑÑÑÐ¶Ð¸Ð¼Ð°Ñ ÑÐµÐ¿Ð½Ð°Ñ Ð»Ð¸Ð½Ð¸Ñ Ð² Ð¾Ð´Ð½Ð¾ÑÐ¾Ð´Ð½Ð¾Ð¼ Ð³ÑÐ°Ð²Ð¸ÑÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¼ Ð¿Ð¾Ð»Ðµ. ÐÑÐ»Ð¸ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ñ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐµ ÑÐ°ÑÑÐµÑÑ Ð´Ð»Ñ Ð½ÐµÑÐºÐ¾Ð»ÑÐºÐ¸Ñ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¹ ÑÐµÐ¿Ð½Ð¾Ð¹ Ð»Ð¸Ð½Ð¸Ð¸. ÐÐ¾Ð½ÐµÑÐ½Ð¾Ð¼ÐµÑÐ½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð±ÑÐ»Ð¸ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ñ Ð² Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð¾Ð¼ ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐµ Adams. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð±ÑÐ»Ð¸ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÐµ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ñ Ð´Ð»Ð¸Ð½Ñ ÑÐµÐ¿Ð½Ð¾Ð¹ Ð»Ð¸Ð½Ð¸Ð¸, Ð¾Ð±ÐµÑÐ¿ÐµÑÐ¸Ð²Ð°ÑÑÐ¸Ðµ Ð¼Ð¸Ð½Ð¸Ð¼Ð°Ð»ÑÐ½ÑÐµ ÑÐ¸Ð»Ñ ÑÐµÐ°ÐºÑÐ¸Ð¹ Ð² Ð¾Ð¿Ð¾ÑÐ°Ñ Ð² Ð·Ð°Ð²Ð¸ÑÐ¸Ð¼Ð¾ÑÑÐ¸ Ð¾Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹. ÐÐ¾ÑÑÑÐ¾ÐµÐ½Ð½ÑÐµ Ð³ÑÐ°ÑÐ¸ÐºÐ¸ Ð´Ð»Ñ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ð½ÑÑ Ð´Ð¸ÑÐºÑÐµÑÐ½ÑÑ ÑÐµÐ¿ÐµÐ¹ Ñ Ð´Ð¾ÑÑÐ°ÑÐ¾ÑÐ½Ð¾Ð¹ ÑÐ¾ÑÐ½Ð¾ÑÑÑÑ ÑÐ¾Ð²Ð¿Ð°Ð»Ð¸ Ñ Ð½ÐµÐ¿ÑÐµÑÑÐ²Ð½ÑÐ¼Ð¸ Ð¼Ð¾Ð´ÐµÐ»ÑÐ¼Ð¸, Ð° ÑÐ°ÐºÐ¶Ðµ Ð½Ð°Ð±Ð»ÑÐ´Ð°Ð»Ð°ÑÑ ÑÑÐ¾Ð´Ð¸Ð¼Ð¾ÑÑÑ Ð¿ÑÐ¸ ÑÐ²ÐµÐ»Ð¸ÑÐµÐ½Ð¸Ð¸ ÑÐ¸ÑÐ»Ð° ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ¾Ð². ÐÐ¾Ð»ÑÑÐµÐ½Ð½ÑÐµ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð²ÑÑÐ°Ð¶ÐµÐ½Ð¸Ñ Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð¼Ð¾Ð³ÑÑ Ð¾ÐºÐ°Ð·Ð°ÑÑÑÑ Ð¿Ð¾Ð»ÐµÐ·Ð½ÑÐ¼Ð¸ Ð¿ÑÐ¸ Ð´Ð°Ð»ÑÐ½ÐµÐ¹ÑÐ¸Ñ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸ÑÑ Ð² ÑÑÐ¾Ð¹ Ð¾Ð±Ð»Ð°ÑÑÐ¸. Ð ÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¼Ð¾Ð¶Ð½Ð¾ Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÑÑ Ð² ÑÑÑÐ¾Ð¸ÑÐµÐ»ÑÑÑÐ²Ðµ Ð´Ð»Ñ Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ñ ÐºÐ¾Ð½ÑÑÑÑÐºÑÐ¸Ð¹, ÑÐ¾Ð´ÐµÑÐ¶Ð°ÑÐ¸Ñ ÑÐµÐ¿Ð¸, ÑÑÐ¾ÑÑ Ð¸ ÑÑÐ¶ÐµÐ»ÑÐµ Ð½Ð¸ÑÐ¸.
description: The topic of the final qualifying work: Â«Modeling and optimization of the catenary and its modificationsÂ». This work is devoted to the study of the geometric and force characteristics of the catenary, minimizing reaction forces in its suspension points, numerical modeling of the chain, comparison of discrete models of the catenary with concentrated parameters and continuous models with distributed parameters. Tasks under consideration: An inextensible catenary in a homogeneous gravitational field; An inextensible catenary in the Newtonian gravitational field; An extensible catenary in a homogeneous gravitational field. Analytical and numerical calculations were carried out for several modifications of the chain line. Finite-dimensional models were implemented in the Adams software package. As a result, optimal values of the length of the chain line were determined, ensuring minimum reaction forces in the supports, depending on the parameters of the models. The pictured graphs for the implemented discrete chains coincided with continuous models with sufficient accuracy, and convergence was also observed with an increase in the number of elements. The obtained analytical expressions and numerical models may be useful in further research in this area. The results of the study can be used when creating structures containing chains, cables and heavy threads.