Classification of Solutions to the Plane Extremal Distance Problem for Bodies With Smooth Boundaries

Jochen Damerau, Robert J. Low

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Abstract

The determination of the contact points between two bodies with analytically described boundaries can be viewed as the limiting case of the extremal point problem, where the distance between the bodies is vanishing. The advantage of this approach is that the solutions can be computed efficiently along with the generalized state during time integration of a multibody system by augmenting the equations of motion with the corresponding extremal point conditions. Unfortunately, these solutions can degenerate when one boundary is concave or both boundaries are nonconvex. We present a novel method to derive degeneracy and nondegeneracy conditions that enable the determination of the type and codimension of all the degenerate solutions that can occur in plane contact problems involving two bodies with smooth boundaries. It is shown that only divergence bifurcations are relevant, and thus, we can simplify the analysis of the degeneracy by restricting the system to its one-dimensional center manifold. The resulting expressions are then decomposed by applying the multinomial theorem resulting in a computationally efficient method to compute explicit expressions for the Lyapunov coefficients and transversality conditions. Furthermore, a procedure to analyze the bifurcation behavior qualitatively at such solution points based on the Tschirnhaus transformation is given and demonstrated by examples. The application of these results enables in principle the continuation of all the solutions simultaneously beyond the degeneracy as long as their number is finite.
Original languageEnglish
Article number061015
Number of pages10
JournalJournal of Computational and Nonlinear Dynamics
Volume11
Issue number6
DOIs
Publication statusPublished - 22 Aug 2016

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Degeneracy
Extremal Point
Bifurcation
Transversality Condition
Plane Problem
Multibody Systems
Center Manifold
Nondegeneracy
Contact Problem
Time Integration
Lyapunov
Codimension
Continuation
Point contacts
Equations of Motion
Divergence
Simplify
Limiting
Contacts (fluid mechanics)
Equations of motion

Cite this

Classification of Solutions to the Plane Extremal Distance Problem for Bodies With Smooth Boundaries. / Damerau, Jochen; Low, Robert J.

In: Journal of Computational and Nonlinear Dynamics, Vol. 11, No. 6, 061015, 22.08.2016.

Research output: Contribution to journalArticle

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