Abstract
The purpose of this final chapter is to illuminate how the kinematic results presented in the earlier chapters are used to formulate the governing equations for the discrete elastic rod. In particular, representations of the kinetic and potential energies for the discrete elastic rod formulation are discussed. The gradients of the elastic energies associated with bending, stretching, and torsion are used to establish expressions for the internal forces in the rod. Assigned forces, including non-conservative forces, can also be incorporated into the discrete elastic rod formulation. To this end, we include examples, such as a force couple and an applied non-conservative force, to illuminate how applied forces and applied moments can be accommodated.
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Notes
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Given a vector-valued function \(\mathbf { a}\left (\mathbf {b}\right )\) where b is a vector, we recall from Sect. 6.2 that \(\frac {\partial \mathbf {a}}{\partial \mathbf {b}} = \sum _{r=1}^3 \sum _{s=1}^3 \frac {\partial a_r}{\partial b_s} \mathbf {E}_r\otimes \mathbf {E}_s\) where \(\mathbf {a} = \sum _{r=1}^3 a_r \mathbf {E}_r\) and \(\mathbf {b} = \sum _{s=1}^3 b_s \mathbf {E}_s\).
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These computations were also invaluable to us while attempting to eliminate typographical errors from the lengthy expressions for the Hessians.
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Jawed, M.K., Novelia, A., O’Reilly, O.M. (2018). Equations of Motion and Energetic Considerations. In: A Primer on the Kinematics of Discrete Elastic Rods. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-319-76965-3_8
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