A Variational Integrator for the Chaplygin–Timoshenko Sleigh

Abstract

The paper introduces a mechanically inspired nonholonomic integrator for numerical simulation of the dynamics of a constrained geometrically exact beam that is a field-theoretic analogue of the Chaplygin sleigh. The integrator features an exact constraint preservation, an excellent numerical energy conservation throughout a large number of iterations, while avoiding the use of unnecessary Lagrange multipliers. Simulations of the dynamics of the constrained beam reveal typical for nonholonomic system’s behavior, such as motion reversals and locomotion generation.

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Notes

  1. 1.

    According to our colleague, an expert in partial differential equations, it would be practically impossible to justify existence and uniqueness using the Euler–Lagrange representation of the dynamics of the sleigh as the Euler–Lagrange formalism yields the system of equations of mixed type. The latter is a reflection of an intrinsic strong anisotropy present in the system, which Hamel’s formalism tackles extremely well.

  2. 2.

    An attempt was made by our colleague, an expert on nonlinear finite elements, to simulate the Chaplygin–Timoshenko sleigh using the Euler–Lagrange representation after eliminating the Lagrange multipliers. During the simulation, the stiffness matrix became singular, triggering divergence of the method.

  3. 3.

    For a more general definition of \(\Psi \), see Shi et al. (2020).

  4. 4.

    That scheme for computational electromagnetism was introduced in Yee (1966).

  5. 5.

    Here and in the rest of the section, lower indices are used to label the components of the vectors \(\gamma \), \(\lambda \), and \(\zeta \) when equations are written in the component form.

  6. 6.

    Recall that \({\mathbb {R}}^3\) here is viewed as a covering space of the special Euclidean group \({\text {SE(2)}}\).

  7. 7.

    This beam model was introduced by Simo (1985), Simo and Vu-Quoc (1986), see also Demoures et al. (2015).

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Acknowledgements

We would like to thank Professors Lorena Bociu, Melvin Leok, and Cheng Liu for valuable discussions. The research of DS, SG, and ZA was partially supported by NSFC Grants 11872107. The research of DVZ was partially supported by NSF Grants DMS-0908995 and DMS-1211454. DVZ would like to acknowledge the support and hospitality of the Beijing Institute of Technology, where a part of this work was carried out.

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A The Algorithm

A The Algorithm

For the numerical simulation of the Chaplygin–Timoshenko sleigh, the integrator constructed in Sect. 4.3 with \(\alpha =\beta =1/2\) has been implemented as the following sequence of steps.

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An, Z., Gao, S., Shi, D. et al. A Variational Integrator for the Chaplygin–Timoshenko Sleigh. J Nonlinear Sci 30, 1381–1419 (2020). https://doi.org/10.1007/s00332-020-09611-2

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Keywords

  • Hamel’s equations
  • Field theories
  • Nonholonomic systems
  • Discrete mechanics
  • Geometric integration

Mathematics Subject Classification

  • 70F25
  • 37J60
  • 70H33
  • 70S05