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# An Implicit Time-Stepping Method for Quasi-Rigid Multibody Systems With Intermittent Contact

[+] Author Affiliations
Nilanjan Chakraborty, Stephen Berard, Srinivas Akella, Jeff Trinkle

Rensselaer Polytechnic Institute, Troy, NY

Paper No. DETC2007-35526, pp. 455-464; 10 pages
doi:10.1115/DETC2007-35526
From:
• ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
• Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C
• Las Vegas, Nevada, USA, September 4–7, 2007
• Conference Sponsors: Design Engineering Division and Computers and Information in Engineering Division
• ISBN: 0-7918-4806-X | eISBN: 0-7918-3806-4

## abstract

We recently developed a time-stepping method for simulating rigid multi-body systems with intermittent contact that is implicit in the geometric information [1]. In this paper, we extend this formulation to quasi-rigid or locally compliant objects, i.e., objects with a rigid core surrounded by a compliant layer, similar to Song et al. [2]. The difference in our compliance model from existing quasi-rigid models is that, based on physical motivations, we assume the compliant layer has a maximum possible normal deflection beyond which it acts as a rigid body. Therefore, we use an extension of the Kelvin-Voigt (i.e. linear spring-damper) model for obtaining the normal contact forces by incorporating the thickness of the compliant layer explicitly in the contact model. We use the Kelvin-Voigt model for the tangential forces and assume that the contact forces and moment satisfy an ellipsoidal friction law. We model each object as an intersection of convex inequalities and write the contact constraint as a complementarity constraint between the contact force and a distance function dependent on the closest points and the local deformation of the body. The closest points satisfy a system of nonlinear algebraic equations and the resultant continuous model is a Differential Complementarity Problem (DCP). This enables us to formulate a geometrically implicit time-stepping scheme for solving the DCP which is more accurate than a geometrically explicit scheme. The discrete problem to be solved at each time-step is a mixed nonlinear complementarity problem.

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