Cooper Dynamics and Control - A Projection-Based Derivation of the Equations of Motion for the Moving Frame Method for Multi-Body Dynamics

Abstract

The moving frame method for multi-body dynamics, established by Murakami in [10] and [11], embodies a consistent notation and mathematical framework that simplifies the derivation of equations of motion of complex systems. The derivation of the equations of motion follows Hamilton’s principle and requires the calculation of virtual angular velocities and the corresponding virtual rotational displacements. The goal of this paper is to present a projection-based approach, which only requires knowledge of Euler’s first and second law, that results in the same equation of motion. The constraints need not satisfy d’Alembert’s principle and the projection is based on a generalization of Gauss’ principle of least constraint [14]. One advantage of the proposed method is that it avoids variational principles and therefore is more accessible to undergraduate students. In addition, the final form of the equation of motion is more easily understood. We motivate our approach using the example of the simple pendulum, derive the main result, and apply the methodology for derivation of the equations of motion for a modified Chaplygin sleigh and a rotary pendulum.

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BibTeX citation:

@misc{m. luchtenburg,  mili shah,  thomas j. impelluso,  thorstein ravneberg rykkje2022,
  author = {M. Luchtenburg, Mili Shah, Thomas J. Impelluso, Thorstein
    Ravneberg Rykkje, Dirk},
  title = {A {Projection-Based} {Derivation} of the {Equations} of
    {Motion} for the {Moving} {Frame} {Method} for {Multi-Body}
    {Dynamics}},
  date = {2022-01-25},
  url = {https://dluchten.github.io/publications/luchtenburg2021asme},
  doi = {10.1115/IMECE2021-72324},
  langid = {en},
  abstract = {The moving frame method for multi-body dynamics,
    established by Murakami in {[}10{]} and {[}11{]}, embodies a
    consistent notation and mathematical framework that simplifies the
    derivation of equations of motion of complex systems. The derivation
    of the equations of motion follows Hamilton’s principle and requires
    the calculation of virtual angular velocities and the corresponding
    virtual rotational displacements. The goal of this paper is to
    present a projection-based approach, which only requires knowledge
    of Euler’s first and second law, that results in the same equation
    of motion. The constraints need not satisfy d’Alembert’s principle
    and the projection is based on a generalization of Gauss’ principle
    of least constraint {[}14{]}. One advantage of the proposed method
    is that it avoids variational principles and therefore is more
    accessible to undergraduate students. In addition, the final form of
    the equation of motion is more easily understood. We motivate our
    approach using the example of the simple pendulum, derive the main
    result, and apply the methodology for derivation of the equations of
    motion for a modified Chaplygin sleigh and a rotary pendulum.}
}

For attribution, please cite this work as:

M. Luchtenburg, Mili Shah, Thomas J. Impelluso, Thorstein Ravneberg Rykkje, Dirk. 2022. “A Projection-Based Derivation of the Equations of Motion for the Moving Frame Method for Multi-Body Dynamics.” ASME IMECE 2021. https://doi.org/10.1115/IMECE2021-72324.