Horatiu Barbulescu Spatial Impact of a Slender Beam 1 Literature Review

In this paper, the dynamics of the spatial impact of a slender beam is Literature Review Impact is the subject of a wide range of engineering applications and generally, the study of impact is focused on the reaction forces that develop during collision and the dynamic response of structures to these reaction forces. A collision occurs in a very short period of time in which the structures involved instantaneously change their velocities. In the rigid body impact theory, the equations of dynamics are not sufficient to solve for the changes in velocity and an additional relation is required. This relation is provided by the definition of the coefficient of restitution. The coefficient of restitution relates the normal component of the relative velocity at approach and the relative velocity at separation. This coefficient depends on the material properties of the colliding bodies, the direction of impact, the initial impact velocity, etc. The problem of rigid body impact was studied in many publications. In the following, only the recent literature is presented. Kane and Levinson ͓1͔ considered the generalized impulse and momentum equations for impact with friction to form a system that includes the generalized velocities before and after the impact. Using two assumptions involving the velocity of separation and the normal and tangential impulse, the system of equations was solved for the impact of a sphere with a surface. Bhatt and Koechling ͓2͔ used Coulomb's law of friction to analyze the flow in the tangent velocity space of a three-dimensional frictional impact process. They enumerated possible dynamic scenarios for the sliding behavior during impact. Given the initial contact velocity, the mass parameters and the contact interface material properties, a qualitative behavior of the impact was predicted analytically. Yen and Wu ͓3͔ proposed and verified experimentally a method to identify both the impact location and the transverse impact force history from strain responses on a rectangular plate. A numerical verification of the method was performed by randomly generating the impact locations and the force histories, and then performing forward calculations to obtain the corresponding strain responses. These responses were then used as the input data for the identification purposes. An elastic-plastic contact law was proposed by Yigit ͓4͔ for modeling impact of a single flexible link. The contact law allows continuous transition between contact and non-contact phases and is capable of predicting impact force histories. A calculation of the load and coefficient of restitution of impact between two elastic bodies with a liquid lubricant was given in the work of Chang ͓5͔. His work attempted to devise an efficient modeling strategy for lubricated impacts. The impact of an elastic ball on a rigid surface with a lubricant layer was modeled. The main goal of the work done by Stoianovici and Hurmuzlu ͓6͔ was to study the two-dimensional collisions of unconstrained slender rods with a massive external surface. The effect of eccentricity of collision on the post impact dynamics of the bar was considered. The results of a set of experiments showed the variation of the coefficient of restitution with the orientation of the bar. Sinopoli ͓7͔ adopted a variational formulation to describe the dynamics and to analyze, in the case of contact, the connection between dynamically possible motions and actual evolution of the system. A geometric method was proposed to determine the dynamic evolution without any direct evaluation of the contact forces. The dual compression in single-point three-dimensional collisions was a subject of the work of Battle ͓8͔. The jamb ͑self-locking͒ process may lead to a dual compression: a second compression phase develops after the initial compression-expansion phase. Using the incremental Routh's model, Battle analyzed the conditions leading to dual compression in the case of perfectly elastic collisions. For a given general collision configuration and for each value of the friction coefficient greater than the threshold value for jamb, he calculated the domain of directions of the incident velocity leading to dual compression. The method was applied to an example and a numerical simulation was given. In this paper it is studied the spatial impact of a slender beam with a rigid surface. The cases of impact without slipping and with slipping are considered. The post-impact velocity of the beam and the force of impact are predicted. Kinetic Energy of the System A three-dimensional impact between a rigid beam B and a rigid surface S is considered ͑Fig. 1͒. The beam is of length L and of mass m. Two cases are studied in this work: ͑a͒ the impact between the beam and the surface occurs without slipping and the coefficient of static friction is s ; and ͑b͒ the impact between the beam and the surface takes place with slipping and in this case the coefficient of kinetic friction is k . It is assumed that during collision, the point Q on the beam and point Q S on the surface come into contact one with each other ͑Fig. 1͒. Also, at the moment of impact the origin of a mobile reference frame xQyz attached to the beam coincides with the origin of a fixed reference frame x 0 Q S y 0 z 0 . The directions of the reference frames are shown i