title: ÐÐ¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ñ Ð¿ÑÐ¾ÑÐµÑÑÐ¾Ð² Ð·Ð°ÑÑÑ Ð°Ð½Ð¸Ñ ÐºÐ¾Ð»ÐµÐ±Ð°Ð½Ð¸Ð¹ Ð² ÑÐ¸ÑÑÐµÐ¼Ð°Ñ Ñ Ð½ÐµÑÐµÐ»ÑÐ¼ ÑÐ¸ÑÐ»Ð¾Ð¼ ÑÑÐµÐ¿ÐµÐ½ÐµÐ¹ ÑÐ²Ð¾Ð±Ð¾Ð´Ñ
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: Optimization of vibration damping processes in systems with a non-integer number of degrees of freedom
Creators: name: ÐÑÑÐ°Ð²ÑÐµÐ², ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ
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: 2021
Subjects: ÐÐ°ÑÑÐ½Ð¸ÐºÐ¸
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Ð½ÐµÑÐµÐ»Ð¾Ðµ ÑÐ¸ÑÐ»Ð¾ ÑÑÐµÐ¿ÐµÐ½ÐµÐ¹ ÑÐ²Ð¾Ð±Ð¾Ð´Ñ
optimization
mechanical systems
maximum stability degree
Vyshnegradskyâs diagram
non-integer degrees of freedom

description: Ð ÑÑÐ¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸ÑÑÑ ÑÐµÑÐµÐ½Ð¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð´Ð»Ñ Ð´Ð²ÑÑ Ð¼ÐµÑÐ°Ð½Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼ Ñ Ð½ÐµÑÐµÐ»ÑÐ¼ ÑÐ¸ÑÐ»Ð¾Ð¼ ÑÑÐµÐ¿ÐµÐ½ÐµÐ¹ ÑÐ²Ð¾Ð±Ð¾Ð´Ñ Ð¿Ð¾ ÐºÑÐ¸ÑÐµÑÐ¸Ñ Ð¼Ð°ÐºÑÐ¸Ð¼ÑÐ¼Ð° ÑÑÐµÐ¿ÐµÐ½Ð¸ ÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸. ÐÐ±Ðµ ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ð½ÑÐµ ÑÐ¸ÑÑÐµÐ¼Ñ Ð¸Ð¼ÐµÑÑ ÑÐ¸ÑÐ»Ð¾ ÑÑÐµÐ¿ÐµÐ½ÐµÐ¹ ÑÐ²Ð¾Ð±Ð¾Ð´Ñ, ÑÐ°Ð²Ð½Ð¾Ðµ 3/2. ÐÑÐ»ÐµÐ´ÑÑÐ²Ð¸Ðµ ÑÑÐ¾Ð³Ð¾ Ð°Ð²ÑÐ¾ÑÐ¾Ð¼ ÑÐ½Ð°ÑÐ°Ð»Ð° Ð¾Ð¿Ð¸ÑÑÐ²Ð°ÐµÑÑÑ Ð¸Ð´ÐµÑ ÑÐµÑÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð´Ð»Ñ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼ Ñ 3/2 ÑÑÐµÐ¿ÐµÐ½ÑÐ¼Ð¸ ÑÐ²Ð¾Ð±Ð¾Ð´Ñ, ÐºÐ¾ÑÐ¾ÑÐ°Ñ Ð´Ð°Ð»ÐµÐµ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·ÑÐµÑÑÑ Ð´Ð»Ñ Ð²ÑÐ±ÑÐ°Ð½Ð½ÑÑ Ð¼ÐµÑÐ°Ð½Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼. ÐÐµÑÐ²Ð°Ñ Ð¸Ð· ÑÐ°ÑÑÐ¼Ð°ÑÑÐ¸Ð²Ð°ÐµÐ¼ÑÑ ÑÐ¸ÑÑÐµÐ¼ - Ð¿Ð¾Ð´Ð¿ÑÑÐ¶Ð¸Ð½ÐµÐ½Ð½ÑÐ¹ Ð¼Ð°ÑÑÐ½Ð¸Ðº, Ñ. Ðµ. Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ð¹ Ð¼Ð°ÑÑÐ½Ð¸Ðº, ÑÐ¾ÑÐºÐ° Ð¿Ð¾Ð´Ð²ÐµÑÐ° ÐºÐ¾ÑÐ¾ÑÐ¾Ð³Ð¾ Ð´Ð¾Ð¿ÑÑÐºÐ°ÐµÑ Ð³Ð¾ÑÐ¸Ð·Ð¾Ð½ÑÐ°Ð»ÑÐ½Ð¾Ðµ ÑÐ¼ÐµÑÐµÐ½Ð¸Ðµ, ÑÑÐµÑÐ½ÐµÐ½Ð½Ð¾Ðµ Ð¿ÑÑÐ¶Ð¸Ð½Ð¾Ð¹. ÐÐ¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð´Ð»Ñ Ð¿Ð¾Ð´Ð¿ÑÑÐ¶Ð¸Ð½ÐµÐ½Ð½Ð¾Ð³Ð¾ Ð¼Ð°ÑÑÐ½Ð¸ÐºÐ° Ð½Ð°ÑÐ¾Ð´Ð¸ÑÑÑ Ñ ÑÑÐµÑÐ¾Ð¼ Ð´ÐµÐ¹ÑÑÐ²Ð¸Ñ Ð² ÑÐ¸ÑÑÐµÐ¼Ðµ Ð²ÑÐ·ÐºÐ¾Ð³Ð¾ ÑÑÐµÐ½Ð¸Ñ. ÐÑÐ¸ ÑÑÐ¾Ð¼ ÑÑÐµÐ½Ð¸Ðµ Ð²Ð²Ð¾Ð´Ð¸ÑÑÑ Ð´Ð²ÑÐ¼Ñ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÐ¼Ð¸ ÑÐ¿Ð¾ÑÐ¾Ð±Ð°Ð¼Ð¸: Ð² Ð¿ÑÑÐ¶Ð¸Ð½Ðµ Ð¼Ð°ÑÑÐ½Ð¸ÐºÐ° Ð¸ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð´ÐµÐ¼Ð¿ÑÐµÑÐ°, Ð²ÑÑÐ°Ð²Ð»ÐµÐ½Ð½Ð¾Ð³Ð¾ Ð¿Ð°ÑÐ°Ð»Ð»ÐµÐ»ÑÐ½Ð¾ Ð¿ÑÑÐ¶Ð¸Ð½Ðµ. ÐÑÐ¾ÑÐ¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ð¾Ð¹, Ð´Ð»Ñ ÐºÐ¾ÑÐ¾ÑÐ¾Ð¹ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸ÑÑÑ ÑÐ¾Ð»ÑÐºÐ¾ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸, ÑÐ²Ð»ÑÐµÑÑÑ Ð¼Ð¾Ð´ÐµÐ»Ñ ÑÑÐ³Ð°ÑÐ° Ñ Ð¿ÑÐ¸ÑÐµÐ¿Ð¾Ð¼, ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ð½Ð°Ñ ÐÐ°Ð½ÑÐ¾Ð¼ Ð¦Ð¸Ð³Ð»ÐµÑÐ¾Ð¼ Ð² ÐºÐ½Ð¸Ð³Ðµ âÐÑÐ½Ð¾Ð²Ñ ÑÐµÐ¾ÑÐ¸Ð¸ ÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸ Ð¸ ÐºÐ¾Ð½ÑÑÑÑÐºÑÐ¸Ð¹â, Ð²ÑÐ¿ÑÑÐµÐ½Ð½Ð¾Ð¹ Ð² 1971 Ð³Ð¾Ð´Ñ. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ ÑÐµÑÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð°Ð²ÑÐ¾ÑÐ¾Ð¼ Ð²ÑÑÐ²Ð»ÑÑÑÑÑ Ð¾ÑÐ¾Ð±ÐµÐ½Ð½Ð¾ÑÑÐ¸ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ Ð´Ð»Ñ ÑÐ¸ÑÑÐµÐ¼ Ñ 3/2 ÑÑÐµÐ¿ÐµÐ½ÑÐ¼Ð¸ ÑÐ²Ð¾Ð±Ð¾Ð´Ñ Ð¸ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸ÑÑÑ Ð·Ð°ÐºÐ»ÑÑÐµÐ½Ð¸Ðµ Ð¾ Ð¿Ð¾Ð»ÑÐ·Ðµ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÑ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð² Ð´Ð»Ñ Ð½Ð°ÑÐºÐ¸ Ð¸ Ð¿ÑÐ°ÐºÑÐ¸ÐºÐ¸.
description: In this work, we solve the optimization problem for two mechanical systems with a non-integer number of degrees of freedom according to the criterion of the maximum degree of stability. Both considered systems have the number of degrees of freedom equal to 3/2. As a result, the author first describes the idea of solving the optimization problem for dynamical systems with 3/2 degrees of freedom, which is then used for the selected mechanical systems. The first of the systems under consideration is a spring-loaded pendulum, that is, a mathematical pendulum, the suspension point of which allows horizontal displacement, constrained by the spring. The optimal solution for a spring-loaded pendulum is found considering the presence of viscous friction. Viscous friction is introduced into the system in two different ways: in the spring of the pendulum and with a damper inserted parallel to the spring The second system, for which only the numerical solution of the optimization problem is carried out, is the model of a tractor with a trailer, considered by Hans Ziegler in the book âÐÑÐ½Ð¾Ð²Ñ ÑÐµÐ¾ÑÐ¸Ð¸ ÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸ Ð¸ ÐºÐ¾Ð½ÑÑÑÑÐºÑÐ¸Ð¹â, published in 1971. As a result of solving optimization problems, the author identifies the features of the optimal solution for systems with 3/2 degrees of freedom and concludes on the benefits of the results obtained for science and practice.