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Scientific Part
Mechanics
On the complex dynamics in simplest vibrational systems with hereditary-type friction
L. A. Igumnov, V. S. Metrikin Research Institute for Mechanics, National Research Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Ave., Nizhniy Novgorod 603950,
Russia
Abstract:
The dynamics of a number of vibrational systems,
accounting for the forces of hereditary-type dry friction and a
vibration limiter, are studied in the paper. The interaction
between the vibration limiter and the vibrational system is
assumed to obey Newton's hypothesis. A general mathematical model
has been developed, which is a strongly nonlinear non-autonomous
system with a variable structure. The dynamics of the mathematical
model is studied numerically-analytically, using the mathematical
apparatus of the point mapping method. The special feature of the
studying approach is that a point map is not formed in a classical
way (mapping Poincare surface into itself), but based on times of
the relative rest of the vibrational system, which considerably
simplified both the point mapping process and its detailed
analysis. The presence of floating boundaries of plates of
sliding motion required an original approach to point mapping and
interpreting the results obtained. The developed investigation
methodology and software product were used to study the
phase-plane portrait of the mathematical model as a function of
the characteristics of sliding friction forces and rest, as well
as of the type and position of the limiter. Based on the character
of the bifurcation diagrams variation, it was possible to find the
main laws of the motion regimes alteration process (the occurrence
of periodic motion regimes of arbitrary complexity and possible
transition to chaos via the period-doubling process) with the
changing parameters of the vibrational system (the amplitude and
frequency of the periodic effect, forms of the functional relation
describing the variation of the friction coefficient value of
relative rest. The results obtained with and without accounting
for a vibration limiter
are also compared in the paper.
Key words:
mathematical model, hereditary-type friction, Poincare
function, relative rest, fixed point, chaos.
Citation:
L. A. Igumnov, V. S. Metrikin, “On the complex dynamics in simplest vibrational systems with hereditary-type friction”, Izv. Saratov Univ. Math. Mech. Inform., 18:4 (2018), 433–446
Linking options:
https://www.mathnet.ru/eng/isu778 https://www.mathnet.ru/eng/isu/v18/i4/p433
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Abstract page: | 169 | Full-text PDF : | 92 | References: | 22 |
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