Regular and Chaotic Dynamics
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Regul. Chaotic Dyn., 2014, том 19, выпуск 1, страницы 100–115 (Mi rcd143)  

Эта публикация цитируется в 4 научных статьях (всего в 4 статьях)

On the Variational Formulation of the Dynamics of Systems with Friction

Alexander P. Ivanov

Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700 Russia

Аннотация: We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples are considered.
For systems with ideal constraints the problem under discussion was solved by Lagrange in his "Analytical Dynamics" (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions [2–4].
The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities [5] includes results on the existence and uniqueness, as well as the developed methods of solution.

Ключевые слова: principle of least constraint, dry friction, Painlevé paradoxes

Финансовая поддержка Номер гранта
Российский фонд фундаментальных исследований No. 11- 01-00354_а
Министерство образования и науки Российской Федерации 14.A18.21.0374
This work was partially supported by the Russian Foundation for Basic Research (project No. 11- 01-00354) and by the Russian Ministry of Education and Science (agreement 14.A18.21.0374).


DOI: https://doi.org/10.1134/S1560354714010079

Список литературы: PDF файл   HTML файл

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Тип публикации: Статья
Поступила в редакцию: 06.08.2013
Принята в печать:30.08.2013
Язык публикации: английский

Образец цитирования: Alexander P. Ivanov, “On the Variational Formulation of the Dynamics of Systems with Friction”, Regul. Chaotic Dyn., 19:1 (2014), 100–115

Цитирование в формате AMSBIB
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\paper On the Variational Formulation of the Dynamics
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\pages 100--115
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    Эта публикация цитируется в следующих статьяx:
    1. Alexander P. Ivanov, “On the Impulsive Dynamics of M-blocks”, Regul. Chaotic Dyn., 19:2 (2014), 214–225  mathnet  crossref  mathscinet  zmath
    2. А. П. Иванов, “О применении вариационных неравенств в динамике систем с трением”, Матем. заметки, 98:2 (2015), 300–302  mathnet  crossref  mathscinet  elib; A. P. Ivanov, “On the Application of Variational Inequalities to the Dynamics of Systems with Friction”, Math. Notes, 98:2 (2015), 328–330  crossref  isi
    3. A. P. Ivanov, “The equilibrium of systems with dry friction”, Pmm-J. Appl. Math. Mech., 79:3 (2015), 217–228  crossref  mathscinet  isi  scopus
    4. А. П. Иванов, “Об особых точках уравнений механики”, Докл. РАН, 479:5 (2018), 493–496  mathnet  crossref  zmath  elib; A. P. Ivanov, “On singular points of equations of mechanics”, Dokl. Math., 97:2 (2018), 167–169  crossref  zmath  isi  scopus
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