Keywords: contact, short range interaction, rigid body dynamics.

The contact of a rigid block with a rigid support is modelled using 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 merely to describe a realistic motion. 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 between these phases are then described through balances of forces or energy. 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 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 [3].

In this paper a different approach is followed. It consists of 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 flat 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 the rigid body boundary and the rigid support and tends to infinity as the distance tends to zero. Within this approach it is possible, for example, to model the impact between surfaces and the subsequent rebound. Even friction and damping, which are to be taken into account when describing the interaction between surfaces in relative motion to each other, can be easily modelled by using 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.

Obviously it is of great importance the choice of the constitutive parameters in order to reproduce a realistic motion. Several numerical simulations are discussed in this paper with the aim to understand how the motion depends on the constitutive parameters. Both two-dimensional and three-dimensional systems are investigated in the numerical simulations.

1

G.W. Housener, "The behaviour of inverted pendulum structures during earthquakes", Bull. Seism. Soc. Am., 53:403-17, 1963.

2

H.W. Shenton, N.P. Jones, "Base excitation of rigid bodies. I: Formulation", ASCE Eng.Mech., 117:2286-306, 1991. doi:10.1061/(ASCE)0733-9399(1991)117:10(2286)

3

M. Greco, H.B. Coda, W.S. Venturini, "An alternative contact/impact identification algorithm for 2d structural problems", Comput. Mech., 34(5):410-422, 2004. doi:10.1007/s00466-004-0586-9

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