<p>Through this research, we put&nbsp;forth a framework to deliver collision models in 3D space.&nbsp; We deliver&nbsp;the framework through the coefficient of restitution, others may use different constitutive models.&nbsp; The framework builds upon a new method in rigid body&nbsp;dynamics to calculate the contact forces, trajectory and translation, and angular velocities after the impact of free bodies in 3D space.&nbsp; The method exploits Cartan’s notion of moving frames and places frames of reference on all moving bodies.&nbsp; Next, the method exploits Lie group theory SO(3) and its associated algebra, so so(3), to relate such frames to each other.&nbsp; Finally, it the method exploits a new notation to streamline the mathematics.&nbsp; We conducted a numerical validation of the results using MSC Adams. &nbsp;We and &nbsp;compared the rotation and translation values of the bodies with what we found using the MFM.&nbsp; We displayed the results on a 3D web page using WebGL. Finally, we pinpointed the aspects that can encompass alternative constitutive collision models.&nbsp; The goal of this paper is: 1) to introduce a new dynamical method to mechanical engineering community; and 2) to apply the MFM to a traditional problem in mechanical engineering: impact analysis.&nbsp; Engineers define impact as the collision between two bodies during a finitely small time increment.&nbsp; This definition is essential to the application of the laws of linear and angular momentum and the evaluation of the average contact forces. If one focuses on the impact of <i>rigid</i> bodies, one may conduct the analysis under the mathematical framework of the laws of rigid body dynamics. These laws of linear and angular momentum are accurate, true and fundamental for the analysis of classical engineering collision problems.&nbsp; However, if one allows for the local deformation of object, existing analytical methods open many approaches. We will touch on just a few, as we do not intend this as a definitive survey. Some approaches to collision analysis, take recourse to finite element methods to assess material deformation during the collision.&nbsp; Others suggest that the goal of collision studies is to obtain a simple model based on known parameters prior to the impact; e.g., the coefficient of restitution.&nbsp; &nbsp;Collision analyses are sometimes difficult because of their geometric complexity.&nbsp; One must account for the geometry and material properties of both bodies and for the smoothness of the colliding surfaces or friction.&nbsp; In many cases, the speed of the collision also plays a role. Other studies, model relationships between collision forces and local deformation under various loading conditions.&nbsp; &nbsp;We deploy a kinetic based analysis in this work, but assess it with an energy analysis.&nbsp; &nbsp;</p>

A Framework for Spatial 3d Collision Models: Theory and Numerical Validation

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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