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Force–displacement characteristics of circular-shaped massless elastica

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Abstract

The deformation of circular-shaped elasticas, subjected to unilateral constraints due to ground contact, is studied under vertical loading. For C-shaped elasticas, we consider the cases where the elastica rolls without slipping due to the presence of friction and rolls while slipping due to lack of friction. The elastica behaves as a hardening spring in the absence of friction; in the presence of friction, it behaves as a softening or hardening spring depending on boundary conditions. For O-shaped elasticas, we consider symmetric and asymmetric loadings. Under symmetric loading, the stiffness of the elastica decreases with increase in the load while point contact is maintained; thereafter, the stiffness remains nearly constant while the elastica undergoes large deformation and makes line contact. Similar behavior is observed for a symmetric mass-elastica system. Asymmetric loading is studied by assuming that the initial point of contact of the elastica with the ground is fixed; vertical loading results in both vertical and horizontal motion of the point of loading. For the different statics and dynamics problems considered, the deformed shapes of the elasticas are obtained iteratively using an algorithm that solves a series of two-point boundary value problems.

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Notes

  1. The loading on the line segment may not be uniform; it is implicitly assumed that the line segment does not buckle.

  2. It can be seen from Eq. (3a) that a constant value of \(\theta \) implies \(({\text {d}\theta }/{\text {d}s}) = 0\), which in turn implies that the curvature is constant and equal to zero. It will be seen later that the curvature is constant but not equal to zero for an initially curved elastica.

  3. Note that the horizontal force p (see Fig. 2) is zero due to the absence of friction between the elastica and the ground.

  4. It should be noted that the value of \(q_{\mathrm{cr}}\) is quite sensitive to the tolerance used to distinguish a line contact from a point contact.

  5. Note that the value of \(\alpha \) is determined using the procedure discussed in Sect. 4.1.

  6. Since the bending moments (horizontal forces) at the tip of the right and left VC-LES should be equal and opposite, the bending moments (horizontal forces) at the tip of the right and transformed left VC-LES are equal.

  7. Note that the value of \(\alpha \) has to be determined at each time step using the procedure discussed in Sect. 4.3.

  8. When the natural frequencies are not integer multiples of one another, the trajectory of the mass traces a path that does not repeat itself and is dense in a bounded region of the Cartesian space.

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Acknowledgements

The first author gratefully acknowledges the support provided by the National Science Foundation graduate research fellowship program, Grant No. DGE-1424871.

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  1. Department of Mechanical Engineering, Michigan State University, East Lansing, MI, USA

    Sheryl Chau & Ranjan Mukherjee

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  1. Sheryl Chau
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  2. Ranjan Mukherjee
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Correspondence to Ranjan Mukherjee.

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Chau, S., Mukherjee, R. Force–displacement characteristics of circular-shaped massless elastica. Acta Mech 231, 4585–4602 (2020). https://doi.org/10.1007/s00707-020-02766-9

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  • DOI: https://doi.org/10.1007/s00707-020-02766-9

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