1 Introduction
The coefficient of restitution remains to be the most controversial, and arguably, the most important constant that is used in the solution of collision problems. Following its inception by Newton (1686), and after undergoing several revisions regarding its definition, widely accepted solutions of impact
problems remain to be incomplete without a proper definition of the restitution coefficient. Investigators in the field have proposed alternative methods of solution that do not necessitate the use of this constant. Methods such as the ones based on Hertz contact theory (Maw et al, 1976 and
Marghitu and Hurmuzlu, 1996), or discrete, lumped spring-dashpot based models (Stoianovici and Hurmuzlu, 1996) are few that fall in this classification. The appeal of the restitution coefficient, however, lies in the remarkably simple way of resolving the interaction between the colliding bodies. This simplicity also enables the use of momentum based methods in the solution, which are significantly less complex than other methods of approach.
The concept of the coefficient of restitution has evolved progressively since its introduction by Newton as velocity ratio. Routh (1897) used Poisson’s Hypothesis and introduced an impulse based coefficient followed by Stronge (1990) who introduced the energetic coefficient of restitution. Hurmuzlu and Marghitu (1995) and Marghitu and Hurmuzlu (1996) conducted thorough comparisons of the outcomes using the three definitions and concluded that, pending further experimental evidence, the energetic coefficient yields the most consistent results.Stoianovici and Hurmuzlu (1996) conducted a set of experiments with slender bars falling on a massive surface. This problem was selected because it was benchmark example that was widely used in the related literature. The bars were dropped from various heights, and the pre-impact conditions
were selected such that the three definitions of the coefficient of restitution yield identical results (i.e. no tangential velocity reversals at the contact point). The study confirmed several key aspects of rigid body theory that is widely used in solving collision problems. For example, it was shown that Coulomb’s law of friction remained generally valid during the collision 2
process. In addition, for the velocity range that was considered in the experiments, the coefficient of restitution did not depend on the incidence velocity.
Yet, the most surprising result was the unusual variation (as high as 80% of the 0-1 interval) in the coefficient of restitution as the inclination of the bars were varied from vertical to horizontal. They attributed this variation to the vibrational energy that was trapped in the bar as a result of the impact event. The authors also developed a discrete model to explain and model
this behavior. They demonstrated that the outcomes predicted by the model matched the experimental results. Their study cast further questions on the practical utility of using a constant coefficient of restitution that depended on local contact properties only. The authors concluded that, in its present form, the coefficient of restitution, had a very limited applicability even when the underlying conditions of the theory were met (relatively rigid colliding bodies and low impact speeds).
The goal of the present study is to generalize the concept of coefficient of restitution such that it can be applied to a wider range of impact problems. Our goal is to amend the definition of the coefficient of restitution such that it incorporates the effect of internal vibrations in planar collisions of slender members. We seek to obtain a simple algebraic form that can be used with the standard momentum based methods. We impose the requirement that the new coefficient is consistent with the classical one when the effect of vibrations diminish.
In the ensuing article we will focus on the free collisions of bars with massive external surfaces. We will propose a new method to accurately predict the post impact velocities of these bars. inally, we will verify the practical utility of the method by comparing the computed outcomes with
the experimental results.
