Simulation of rigid-body dynamics with impact and friction (original) (raw)

Rigid-body dynamics is the dynamics of bodies that do not deform. While no body is completely rigid, this is a model for a wide range of everyday objects which are stifF on the time and length scales of interest to us. Another that is of great in life are Coulomb (or dry) fricand impact. Whether v;re are walking, or (hitting balls, running, etc.), or operating a car , or picking up objects, impact and friction are commonplace effects. Yet the theoretical. understanding of these is still in its infancy. One of the reasons for this are the discontinuities introduced by standard models of these phenomena. of rigid bodies leads to discontinuities of the A lesser, but still very discontinuity is due to Coulomb friction, where the equations of motion involves discontinuous functions. These discontinuities lead to both in terms of and in terms of computation. are the more severe form of discontinuity: there are impulses in the contact forces which leads to discontinuities in the velocities (which are part of the state vector of the. Such impulsive forces are best modelled mathematically as 1neasures. Vl/hile measure differential have been around since at least the 1950's (see, for example [12]), in this case the strength of the impulse is determined the configuration of the n1echanical and so is not known a priori This leads to measure differential inclusions and complementarity conditions between measures and measurable) functions. Coulomb friction forces are bounded relative to the normal contact forces, but the additional discontinuity is that the direction in which they apply is discontinuous in the relative slip vvhen that is zero. Since "sticking" is fairly common with Coulomb friction, it nwans that this discontinuity has to be dealt with. 1.1" Impact Impacts in mechanical systems are extremely common, difficult to modeL Since for rigid impacts are instantaneous, there needs to be some rule which specifies how the bodies behave in an impact. Consider the difference between tvvo billiard balls colliding and tvlfO lumps of play-dough colliding. The former will bounce in a nearly elastic way, almost conserving the apparent kinetic energy, while the latter will undergo plastic deformation in the impact, and have little kinetic energy available for separating aftenvards. This is commonly modelled using Newton's lavv of impacts which states that the normal uv.uvi'" of the relative velocity after collision is-e times the normal component of relative velocity just prior to collision [28]. As it is commonly applied, it is known to sometimes give an increase in the total energy when friction is involved [34]. The quantity e is called the coefficient of restitution.