The planar rocking-block: analysis of kinematic restitution laws, and a new rigid-body impact model with friction

In this report we first analyse the capabilities of a generalized kinematic (Newton's like) restitution law for the modeling of a planar rigid block that impacts a rigid ground. The relationships with the classical angular velocity restitution law for rocking motion are examined in detail. Then a new impact law based on the Darboux-Keller's approach with energetic coefficients of restitution (proposed recently in \cite{liu2008a,liu2008b,liu2009a,liu2009b}) and which allows one to incorporate Coulomb's friction, is applied and many cases are studied numerically. The numerical results show that this restitution law is able to reproduce a rich set of block motions with complex energy dispersion effects, that are consistent with everyday life observations. Most importantly it is shown that the kinetic angle between the two unilateral constraints is a parameter that plays a fundamental role in the block's motion, and a critical kinetic angle $\theta_{c}$ is exhibited that allows one to split the blocks into two main classes: slender blocks with a height/width ratio larger than $\theta_{c}$, and flat blocks when the ratio is smaller than $\theta_{c}$. The value of $\theta_{c}$ varies with the friction value. Detailed comparisons with experimental results on free-rocking and with base excitation found in the literature \cite{pena2008,lipscombe1993,elgawadi2010} are proposed and confirm the validity of the proposed model. New results concerning the conditions for the onset or rocking and for the overturning are presented. All the simulations are done with an event-driven algorithm that is available on the INRIA {\sc siconos} open-source software platform.Dans ce rapport nous commen\c cons par l'analyse d'une loi d'impact cinématique généralisée (á la Newton) pour le cas d'un bloc planaire qui impacte un sol rigide. Les relations avec la loi classique de restitution de vitesse angulaire sont examinées en détail. Ensuite une nouvelle loi d'impact proposée dans \cite{liu2008a,liu2008b,liu2009a,liu2009b} est appliquée. Les simulations numériques concernent les dynamique de type free-rocking (avec fondation fixe), ainsi que les dynamiques avec fondation mobile, telle que le retournement (overturning). Les résultats numériques sont comparés á de nombreux résultats expérimentaux qui les validen