Short Range Contact Forces in the Dynamics of a Rigid Body

The contact of a rigid block with a rigid support is modeled through short range contact forces. The motion of the rigid block is governed by the interaction with this force field, instead of being confined to the half-space above the support by kinematical constraints. The contact forces, which are not Lagrangian multipliers, are explicitly given a constitutive law, resembling that of forces with a Lennard-Jones potential. Such a kind of constitutive law is not derived here from a microscopic contact model, but is intended merely to describe a realistic motion, possibly through a macroscopic identification. Rigid body dynamics is a classical subject in structural mechanics \cite{Housener-1963}, where great interest has been devoted to the study of a rigid body on a moving support \cite{Shenton-Jones-1991}. An interesting field of this research area deals with the behaviour of monolithic art objects subjected to several type of environmental excitations \cite{Agbabian-et-al-1988}. A common aspect of almost all of the papers on the subject is the use of a kinematical approach. This consists in deriving the equations of motion by enforcing some kinematical constraints ab initio. In order to describe a motion with unilateral contact in such a way it is necessary to know and to characterize a priori the different phases of the motion that the rigid body will undergo. The transitions among these phases are then described through balances of forces or energy. Nevertheless the kinematical approach shows some problems that arise in particular when the rigid body has an asymmetric shape \cite{Boroscheck-Romo-2004}. For example, it is necessary to use, in the planar rocking problem, two different equations of motion depending on which one of the two corners is the centre of rotation. But the main problems in using the kinematical approach are related to the description of the free flight of the body and to the analysis of both impact and rebound. Indeed it is rather difficult to detect an impact which is about to occur, to describe the incipient motion after the impact and finally, to assign the right momentum to the detaching body. % In this paper a different approach is followed. It consists in assigning a constitutive description for the contact forces between the boundaries of bodies which get close to each other. In particular the interaction between a rigid body and the horizontal surface of a rigid support which stays at rest is modeled through a field of short range forces described by a Lennard-Jones potential. While a microscopic model of surface interaction could be used in a multiscale approach here the description is only macroscopic. % Several are the advantages that this approach reveals. First of all the ability to describe the motion of a rigid body smoothly during all the different phases of the motion (i.e. bouncing, rocking, free flight) by a unique set of equations of motion. It is not necessary any more to classify, to describe and to study in detail each transition between phases. The use of short range forces makes it easy even to model friction and dissipation. % Obviously it is of great importance the choice of the constitutive parameters in order to reproduce a realistic motion. The numerical simulations described in this article are intended just to show how the motion depends on them in a few examples. Finally, it is necessary to point out the great accuracy requested in the time integration close to an impact, when the contact forces reach the greatest intensity. This fact can be correctly explained by observing that the impact, the rebound and the sliding are events which can be observed at a macroscopic scale but born at a microscopic scale.