On the Theory of Rocking Structures in Seismically Active Zones

Vladimir S. Metrikin*, Leonid A. Igumnov, Andrei V. Metrikine

*Corresponding author for this work

Research output: Chapter in Book/Conference proceedings/Edited volumeChapterScientificpeer-review


Vibrations of a rectangular block on a rigid, horizontally vibrating plane are analyzed in the framework of the theory of the controlled dynamical systems (CDS). The friction force between the block and the supporting plane is assumed to be sufficiently high for the block not to slide along the plane. Under this assumption the block may only rotate about its vertices that are in contact with the supporting plane. Energy loss occurs due to impact interactions of the block and the supporting plane. The impacts are modelled using the angular velocity restitution coefficient. Using qualitative CDS methods the controllability regions for the block to return to any arbitrarily small vicinity of point (0, 0), corresponding to zero values of the angular velocity are derived. Dimensions of a safe zone of the controllability region in the phase space of the CDS are also determined. It is shown that the safe zone decreases with the increase of the maximum value of acceleration of the plane. The controllability regions are shown for different geometrical dimensions of the block and the upper limit of the supporting plane acceleration.

Original languageEnglish
Title of host publicationAdvanced Structured Materials
Place of PublicationCham
Number of pages15
ISBN (Electronic)978-3-030-54928-2
ISBN (Print)978-3-030-54927-5
Publication statusPublished - 2021

Publication series

NameAdvanced Structured Materials
ISSN (Print)1869-8433
ISSN (Electronic)1869-8441


  • Bifurcation diagrams
  • Chaos
  • Mathematical model
  • Point mapping
  • Stability


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