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Deterministic dynamics
Abstract:
Using the theory of elasticity for large deflections, we obtain the differential equation describing the equilibrium shape of an elastica subjected to several distinct boundary conditions. In our study, one side of the elastica is assumed to be clamped horizontally at a fixed height, and the other side in three different conditions: free, in touch with a horizontal surface beneath, and clamped to that surface. The underlying surface imposes an inequality constraint on all points of the elastica by forbidding it to go below the surface; this is not a simple boundary condition problem. We solve the equations numerically and compare the results with our experimental observations. For the elastic sheet touching the underlying horizontal surface, two regimes of zero or non-zero angle with the surface are distinguished. In the case of two clamped ends, we see three different regimes, depending on the height or the clamping distance. In the regime of low height or small clamping distance, bifurcation happens and there are two possible shapes for the sheet, which have been confirmed both numerically and experimentally. The forces exerted on two ends of the sheet are also calculated in the first and second cases.
