Approximating problems

Abstract

Clearly, the assumption that all bodies are perfectly rigid, hence the introduction of singular distributions in the modeling of collisions has the advantage of providing an attractive framework of impact dynamics. Note however that bodies that collide do possess a certain compliance, so that the collision duration is strictly positive (¹) and local deformations occur near the point of impact. Actually global (vibrational) deformations are also created, and they may play a major role in the dynamics. They may even play a much more important role than the local deformations. Consequently, rigid body dynamics may be considered as a limit case only, which however does not preclude its practical as well as theoretical utility (the pros and cons of rigid and flexible body approaches are not the topic of this chapter). One may therefore choose to work with continuous-dynamics models of collision, such that the bodies deform during the impact, and the collision dynamics are treated as continuous time dynamic phenomena (restricted to local deformations in most of the studies). Historically, it has very often been difficult to certain scientists to accept the idea of perfect rigidity [979]. For instance Leibniz himself [509] [510] (and Bernoulli after him [77]) refused this idea because rigidity yields violation of the “law of continuity” in nature. A strong scientific debate motivated by the London Royal Society in 1668 also concerned the concept of “hardness” (which is to be understood as rigidity in this context): is a hard body able to rebound? Or is it necessary that the bodies possess some “springiness”? Wallis and Mariotte concluded that springs are necessary, while Huygens, Wren and Malebranche thought that hardness is sufficient [979]. We know now the difference between a model of nature and nature itself. We also have many more mathematical tools at our disposal to accept perfect rigidity and to study accurately the relationship between compliant and rigid models (²). Moreover the very short collision durations allow one to safely work with two timescales in many practical cases. And it is possible to lump the deformations effects in one single coefficient while keeping the attractiveness of rigid body models, see section 4.2 and subsection 4.2.10.

However in many practical cases it is very short: 4.10^-4 s for a shock between a golf ball and a flat-nosed wooden projectile with a relative speed of 5.334 m/s [119]. see other values of the same order in subsection 4.2.10 for slender rods against a massive steel table.

Although, as we shall see, this still requires advanced mathematical studies.