
On the variational formulation of dynamics of systems with friction
Alexander P. Ivanov^{} ^{} Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700, Russia
Abstract:
We discuss the basic problem of dynamics of mechanical
systems with constraintsfinding acceleration 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 is considered.
For systems with ideal constraints discussed problem has been 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 to obtain the solution as the minimum of a quadratic function
of acceleration, called “constraint”. In 1872 Jellett gaves examples of
nonuniqueness of solutions in systems with static friction, and in 1895
Painlevé showed that in the presence of friction, together with the
nonuniqueness of solutions is possible. Such situations were a serious
obstacle to the development of theories, mathematical models and practical
use of systems with dry friction. An unexpected and beautiful promotion
was work by Pozharitskii, where the author extended the principle of
Gauss on 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.
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
includes the results of existence and uniqueness, as well as the developed
methods of solution.
Keywords:
principle of least constraint, dry friction, Painlevé paradoxes.
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UDC:
531.315
MSC: 70Е55 Received: 26.06.2013 Revised: 09.07.2013
Citation:
Alexander P. Ivanov, “On the variational formulation of dynamics of systems with friction”, Nelin. Dinam., 9:3 (2013), 478–498
Citation in format AMSBIB
\Bibitem{Iva13}
\by Alexander~P.~Ivanov
\paper On the variational formulation of dynamics of systems with friction
\jour Nelin. Dinam.
\yr 2013
\vol 9
\issue 3
\pages 478498
\mathnet{http://mi.mathnet.ru/nd401}
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http://mi.mathnet.ru/eng/nd401 http://mi.mathnet.ru/eng/nd/v9/i3/p478
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