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Research Papers

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

[+] Author and Article Information
Jochen Damerau

Bosch Corporation,
3-6-7 Shibuya, Shibuya-ku,
Tokyo 150-8360, Japan
e-mail: jochen.damerau@jp.bosch.com

Robert J. Low

Faculty of Engineering and Computing,
Coventry University,
Coventry CV1 5FB, UK
e-mail: mtx014@coventry.ac.uk

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 6, 2015; final manuscript received July 25, 2016; published online August 22, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 11(6), 061015 (Aug 22, 2016) (10 pages) Paper No: CND-15-1202; doi: 10.1115/1.4034393 History: Received July 06, 2015; Revised July 25, 2016

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.

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References

Figures

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Fig. 1

Visualization of the degeneracy condition

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Fig. 2

Boundary curves and quantities at extremal points

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Fig. 3

Fold bifurcation at points of vanishing curvature. Left: no solution, middle: one degenerate solution, and right: two regular solutions.

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Fig. 4

Codimension 2 degeneracy not unfolded at points of vanishing curvature. Left: one regular solution, middle: one degenerate solution, and right: one regular solution.

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Fig. 6

Swallowtail bifurcation at concave segment. Upper left: two regular solutions, upper right: one degenerate solution, and lower: four regular solutions.

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Fig. 5

Cusp bifurcation in concave region of the boundary. Left: one regular solution, middle: one degenerate solution, and right: three regular solutions.

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