### Abstract

Original language | English |
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Article number | 061015 |

Number of pages | 10 |

Journal | Journal of Computational and Nonlinear Dynamics |

Volume | 11 |

Issue number | 6 |

DOIs | |

Publication status | Published - 22 Aug 2016 |

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

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Classification of Solutions to the Plane Extremal Distance Problem for Bodies With Smooth Boundaries

AU - Damerau, Jochen

AU - Low, Robert J.

PY - 2016/8/22

Y1 - 2016/8/22

N2 - 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.

AB - 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.

U2 - 10.1115/1.4034393

DO - 10.1115/1.4034393

M3 - Article

VL - 11

JO - Journal of Computational and Nonlinear Dynamics

JF - Journal of Computational and Nonlinear Dynamics

SN - 1555-1423

IS - 6

M1 - 061015

ER -