Optimization-based analysis of rigid mechanical systems with unilateral contact and kinetic friction

Abstract

A widely-accepted technique for the analysis of rigid mechanical systems with unilateral contact and Coulomb friction is to formulate the contact forces (or impulses) as the unknowns of a linear complementarily problem (LCP). However, the solution set of an arbitrary LCP may be nonconvex, which poses the following computational and theoretical problems: if an arbitrary solution of the LCP is chosen, the computed force distributions over continuous contact regions may change discontinuously as a function of time. This could lead to erroneous analysis of continuum properties, such as heat or stress. On the other hand, if a minimum-norm solution is sought, the solution set must be fully characterized, which could be computationally prohibitive for complex problems.During this course of research, we have used Gauss' Principle of Least Constraint to formulate the normal forces of the kinetic (sliding) problem as the minimum-Euclidean norm solution of a convex quadratic program. When the feasible region is nonempty, the solution set is convex, regardless of the friction coefficients, which makes it computationally efficient to compute the minimum Euclidean norm solution. To illustrate this point, Painleve's example is characterized and compared with the corresponding LCP solution. Based on these results, we suspect that solution multiplicity in frictional contact problems has no physical basis and that this phenomenon is simply a deficiency of the LCP formulation.We also demonstrate how our formulation can be used in conjunction with explicit Runge-Kutta differential-algebraic solvers for these types of systems. To experimentally validate this approach, a 5th-order implementation has been used to accurately predict the motion of an unlubricated pendulum. By matching individual trajectories, friction coefficients were calibrated for four different material interface types: brass-aluminum, steel-aluminum, Teflon-aluminum, and nylon-aluminum. In order to accurately match trajectories, we have found that two friction coefficients are necessary: a constant Coulomb term and a viscous damping term. We also used this technique to examine the distribution of the normal load as a function of time, and the results are consistent with a qualitative force-balance analysis.