Simulation of the contact among surfaces in the Dynamics of rigid bodies by using short range force fields

The contact of a rigid block with a rigid support is modelled through short range contact forces. The motion of the rigid block is governed by the interaction with different force fields, instead of being confined to the half-space above the support by kinematical constraints. The contact forces are explicitly given a constitutive law which is not derived from a microscopic contact model, as it could be in a multiscale approach, but is intended here merely to describe a realistic motion, possibly through a macroscopic identification. Rigid body dynamics is a classical subject in structural mechanics [1], where great interest has been devoted to the study of a rigid body on a moving support [2]. 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 [3]. 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 f1ight 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 initially, to assign the right momentum to the detaching body. In the literature several papers are devoted to the study of computational algorithms to detect and better describe contact and impact mechanisms [4-6]. 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 modelled through a field of short range repulsive forces described by a Lennard-Jones potential. This field of forces depends on the distance between rigid body and rigid surface and tends to infinity when the distance tends to zero. With this approach is possible, for example, to model the impact among surfaces and the successive rebound. Friction and dissipation phenomena, that represents other types of interaction between surface in relative motion to each other, can be easily modelled using also short range forces. Two different fields of short range forces, both depending on distance between rigid body and rigid surface and on tangential and orthogonal components of the velocity with respect the fixed surface, are introduced to model respectively friction and dissipation. 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 f1ight, sliding, slide-rock) by a unique set of equations of motion. It is not necessary any more to classify, describe and study in detail each transition between phases. Obviously it is of great importance the choice of the constitutive parameters in order to reproduce a realistic motion. Several numeric simulations are discussed in this article with the aim to understand how the motion depends on the constitutive parameters; both bi-dimensional and three-dimensional systems are investigated in the numerical simulations. 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 are born at a microscopic scale.