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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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On Application of the Theory of the Controllable Dynamical Systems to Investigation of the Dynamics of Mechanical Objects in Seismically Active Zones

Author(s):

Vladimir Semionovith Metrikin

Research Institute of mechanics of
N.I. Lobachevsky National research Nizhny Novgorod state University
603950, Nizhny Novgorod, Gagarina Avenue, 23,RIM, h.6
Associate Professor, PhD in Physics and Mathematics

v.s.metrikin@mail.ru

Leonid Alekcandrovich Igumnov

Research Institute for Mechanics,
National Research Lobachevsky State University of Nizhny Novgorod,
Director of the Institute
Nizhny Novgorod, Gagarina of Prospect,23
Doctor of Physics, MD, Professor

igumnov@mech.unn.ru

Andrey Vladimirovich Metrikin

Delft University of Technology,
Faculty of Civil Engineering and Geosciences,
Delft, Building 23 Stevinweg 1, 2628 CN Delft, Netherlands
The head of the Department
Doctor of Physics, MD, Professor

andrei.metrikine@gmail.com

Abstract:

Basing on the general theory of the controllable dynamical systems (CDS), we consider vibrations of a rectangular body on a horizontal rigid surface that vibrates with a non-constant amplitude. It is assumed that the friction force between the body and the vibrating surface is sufficiently large to prevent slip and assure that the body rocks about its corner points. The energy of the body is reduced each time when it collides with the surface. This energy dissipation is accounted for by means of a restitution coefficient of the angular velocity of the body vibrations. Using the qualitative theory of the CDS and assuming that the amplitude of the surface vibrations is bounded by a known maximum amplitude, we found the controllability domains in the parameter space of the dynamical system, which assures that the body can be kept in an infinitesimal vicinity of the equilibrium. The safe controllability zone is determined as well. It is proved that the increase in the maximum amplitude leads to the shrinkage of the safe controllability zone. A parametric study of the controllability zone is conducted with the attention focused on the effects of the size of the body and the maximum acceleration of the surface.

Keywords

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