index notation equilibrium 3d elastic structure

Pour en tirer le meilleur parti, deux problèmes de mécanique des tiges doivent cependant être résolus. Le modèle multi-échelle et hiérarchique utilisé est particulièrement adapté à l'implémentation de simulations interactives. Motivée par de nombreux travaux théoriques et expérimentaux, cette approche innovante a été validée sur plusieurs structures dites en épingle à cheveux. This is a promising first step towards more complex 3D structures.ĭans le cadre de l’approche Biopolymer Chain Elasticity (BCE) développée au Laboratoire Jean Perrin, cette thèse s’intéresse à la modélisation des acides nucléiques par des tiges élastiques. A sphere was lifted by the gripper, and the grasping force was successfully monitored by the sensing finger. We also demonstrate that this material is suitable to build a gripper for small objects. On the other hand, the linear range and the sensitivity have been empirically modeled in sensing mode. The relationship between the output forces, the tip displacements and the driving voltages was highlighted with an experimental setup while actuating. Two working modes, namely actuation and sensing, are identified through modeling and experimental validations. In this work, we are mainly interested in the electromechanical characterization of the material because of its particular behavior.

This class of polymers can be used in open-air, allowing for large displacements under low voltages with a reversible process. We propose extensions of the method to obtain many solutions instead of only one, using numerical continuation or starting from different initial guesses.Ĭonducting interpenetrating polymer networks (C-IPN) are a promising solution for the design of sensing and actuating parts at macro-or microscale. More than 850 000 tests are performed in a large region of configurational space, and in 99.9% of cases the targeted boundary conditions are reached with short computation time and a precision better than 10⁻⁵. The six variables used in the search algorithm, forces and moments at one end of the rod, are particularly adapted due to their unbounded definition domain. We exhibit the symmetries of the problem, which reduces drastically the size of this grid and shortens the time of selection of an initial guess. We find one solution of these equations with a zero-finding algorithm, by taking initial guesses from a grid of potential candidates.

When the second end is also anchored, six additional nonlinear equations must be resolved to obtain parameter values that fit the targeted boundary conditions.

When one end of the rod is anchored, the analytical integration of these equations lead to one unique solution for given values of these six parameters. The mechanical equilibrium of such a rod is described by ordinary differential equations and parametrized by six scalar quantities. For both results, the domains and the frontiers of every class are plotted in the space of the parameters.Ī general-purpose method is presented and implemented to express analytically one stationary configuration of an ideal 3D elastic rod when the end-to-end relative position and orientation are imposed. In addition, the geometry of the envelope provides another characterization of the ideal 3D elastica. This remarkable structure leads to a classification of the general case according to three properties called chirality components. Second, a general geometric structure is exhibited for every ideal 3D elastic rod, where the trajectory winds around a core helix in a tube-shaped envelope.

First, all particular trajectories are completely characterized and located in the space of these parameters. Detailed analysis of solution trajectories yields two main results. Orientation of sections and mechanical loading are also well described analytically with these parameters. Infinite solution trajectories are expressed analytically and classified in terms of only three parameters related to physical quantities. We investigate the equilibrium configurations of the ideal 3D elastica, i.e., inextensible, unshearable, isotropic, uniform, and naturally straight and prismatic rods, with linear elastic constitutive relations.

Index notation equilibrium 3d elastic structure