Dynamics of a Rectangular Rigid Body on a Movable Base

Abstract

On the basis of general theory of dynamical systems (DS), vibrations of a rigid rectangular body on a horizontally vibrating with acceleration $$\ddot{U}\left( t \right)$$ rigid sup-porting plane are considered. It is assumed that the friction force between the rigid body and the surface is so great that there is no sliding of the base of the solid body along the plane, but oscillations occur relative to the corner points of the body support. The loss of energy occurs due to impact interactions of bodies with the coefficient of “recovery” of the angular velocity. For an arbitrary set of geometrical dimensions of a rigid rectangular body and acceleration parameters $$\ddot{U}\left( t \right)$$ of the supporting plane, the equations of point mappings of Poincaré surfaces are presented by numerical-analytical methods. Based on them, bifurcation diagrams were obtained, which made it possible to establish both the regions of existence in the multidimensional space of DS parameters of symmetric and asymmetric stable periodic motion modes of a rigid body and the scenario for the occurrence of chaotic motions.