A new approach in the study of the collisions with friction using inertances

This paper presents a complete study on the collision with friction of one or two rigid bodies without constraints. The differential formula between the velocities and impulse uses the notion of inertance resulting from the theory of screws (Plückerian coordinates). One thus may calculate the kinematic and dynamic parameters, the velocities and the kinetic energies of the two rigid solids after the collision, and the variation of the kinetic energy. The calculation is detailed for the Newton, Poisson, and energetic variants of the coefficient of restitution. The variation of the kinematic and dynamic parameters in relation to the coefficient of restitution and coefficient of friction for all the three variants are presented and discussed. A numerical example highlights the theory.

On the Applicability of Newton’s, Poisson’s and Stronge’s Collision Hypotheses

This paper deals with collision with friction. Equations governing a one-point collision of planar, simple non-holonomic systems are generated. Expressions for the normal and tangential impulses, the normal and tangential velocities of separation of the colliding points, and the change of the system mechanical energy, are written for three types of collision in connection with Newton’s hypothesis, and for five types of collision in connection with Poisson’s and Stronge’s hypotheses. These, together with Routh’s semi-graphical method and Coulomb’s coefficient of friction, are used to show that the algebraic signs of five, newly-defined, configuration-related parameters, not all independent, span eleven cases of system configuration. For each, the ratio between the tangential and normal components of the velocity of approach, called α, determine the type of collision which, once found, allows the evaluation of the associated normal and tangential impulses and ultimately the changes in the motion variables. The analysis of the eleven cases with Newton’s hypothesis indicates that the calculated mechanical energy may increase if sticking or reverse sliding occur, and that regions of α exist for which there is no solution or there are multiple solutions. Regarding Poisson’s hypothesis, there are regions of α, narrower than with Newton’s hypothesis, for which there is no solution. However, whenever a solution exists it is unique, coherent and energy-consistent. The same applies to Stronge’s hypothesis, however for a narrower range of application. It is thus concluded that Poisson’s hypothesis is superior as compared with Newton’s and Stronge’s hypotheses.