The elastica can undergo a large deformation while staying within the elastic limit. This has motivated diverse applications from computer graphics to fly fishing and soft robotics. In this work, we investigate the dynamics and control of a number of different systems that entail the elastica. We first investigate the mechanics of pole vaulting, where an athlete uses a long flexible pole to transform kinetic energy to potential energy to cross a high barrier. The pole, which is an elastica, is used to temporarily store the kinetic energy of the athlete and change the direction of motion of the athlete from the horizontal direction to the vertical direction. In this study, the athlete is replaced by a point mass for the sake of simplicity, but the qualitative behavior of the combined system is found to be very similar to that observed in the sport. Pole vaulters are known to achieve a greater height by doing additional work in bending the tip of the pole; similar results are seen when non-conservative work is done on the elastica during the vaulting maneuver. Similar to pole vaulters, runners with lower-limb prostheses, commonly use C shaped elastica legs to temporarily store energy during the contact phase and retrieve it prior to the flight phase. The force displacement characteristics of circular-shaped elasticas are therefore of great interest in applications where elastica elements are used as springs. The deformation of C-shaped and O-shaped elastica springs, subjected to unilateral constraints due to ground contact, is studied under vertical loading; the hardening and softening behaviors of these springs are observed in the absence or presence of friction. Symmetric and asymmetric loadings are also investigated with the objective of better understanding the dynamics of mass-elastica systems; these investigations reveal the possibility of designing robots capable of locomotion through hopping. We explore a variable-structure mass-elastica hopper; the stiffness of the elastica is changed by changing its boundary conditions. The force-displacement characteristics of the variable-structure elastica are used to modulate the energy of the system and control the hopping height. Finally, rolling locomotion is considered using a flexible wheel that is a closed elastica. In addition to its bending stiffness, the elastica is assumed to have mass and the equations of motion are derived using extended Hamilton's principle. The arc-length constraint of the elastica introduces nonlinearity in the equations of motion and the ground contact leads to unilateral constraints. The weak form of the equations of motion are obtained to solve the dynamics using the finite element methods; the penalty method is used to enforce the ground constraint. For the particular case of constant velocity motion, the effects of the non-dimensional weight and velocity parameters are studied. It is observed that the frequency of the rolling elastica reduces as the velocity increases; this resembles the divergence problem in axially moving materials.
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