## Collision Mechanics

Rigid body mechanics provides a popular and convenient way of modeling many mechanical systems of the real world. When two nominally rigid bodies collide, the collision contact forces are determined by deformations in these bodies. This happens in a way that remains important in the limit of high stiffness and small during-collision displacements. Two balls made of different materials, when dropped from the same moderate height and on to the same ground, can rebound to different extents. These rebounds cannot be predicted from within rigid body mechanics without additional constitutive modeling assumptions, the most famous of them being the coefficient of restitution. These constitutive assumptions or collision laws are mathematical rules that allow us to approximately describe the collision within the world of rigid body mechanics. The dynamics model for the system uses these rules to jump across the short and violent collisional interaction, on to, say, another phase of well-defined smooth motions. To emphasize the difference between collisional and smooth motions, imagine that these two same balls rest on the ground - the normal reaction is now determined by the weight of the balls alone, and can be predicted within rigid body mechanics without additional constitutive assumptions.

The papers below address various issues in modeling collisions within the rigid body dynamics framework, and present a simple model that has several attractive features.

## Papers

Infinite friction and the no-slip boundary condition. Lecture in Banff 2014. 30 min. Video

### A chain that accelerates rather than slows due to collisions: how compression can cause tension (paper and videos)

Grewal, A., Johnson, Philli.p and Ruina, A ,
American Journal of Physics, Volume 79, Issue 7, pp. 723–729, July 2011
This paper shows that a chain falling on a table can have an acceleration greater than g.

### A new algebraic rigid body collision law based on impulse space considerations (PDF)

Chatterjee, A., Ruina, A.
Journal of Applied Mechanics, Vol. 65, #4, Pages 939-951, Dec 1998

This paper discusses a simple two-parameter collision model for general 3D frictional, single-point rigid body collisions.

### Two Interpretations of Rigidity in Rigid Body Collisions (PDF)

Chatterjee, A., Ruina, A.
Journal of Applied Mechanics, Vol. 65, #4, Pages 894-900, Dec 1998

This paper discusses force-response rigidity vs. impulse-response rigidity. A previous version appeared in the Proceedings of ASME Winter Meeting, Dallas, TX, Nov 1997.

### Anomalous Frictional Behavior in Collisions of Thin Disks (PDF)

Calsamiglia, J., Kennedy, S.W., Chatterjee, A., Ruina, A., and J.T.
Journal of Applied Mechanics, Vol. 66, #1, Pages 146-152, Dec 1998

A previous version of this paper appeared in the Proceedings of ASME Winter Meeting, Dallas, TX, Nov 1997.

### Realizability of Arbitrary Local Mass Matrices in Single-Point Rigid Body Collisions (PDF)

Chatterjee, A., Ruina, A.
Proceedings of the International Symposium on Impact and Friction of Solids, Structures and Intelligent Machine, June 1998, Ottawa, Canada

### Rigid Body Collisions: Some General Considerations, New Collision Laws, and Some Experimental Data (PDF)

Chatterjee, A.
Ph.D. Thesis, Jan 1997

This thesis attempts to present a unified view of the subject of rigid body collisions.