Control of three-dimensional incompressible hyperelastic beams

The problem of control of three-dimensional incompressible hyperelastic cantilever beams is approached from an analytical dynamics perspective. The dynamic equations of motion of the hyperelastic beam are derived using the absolute nodal coordinate formulation, which is a finite element method that accurately describes large-deformation and large-rotation nonlinear motion in structures. The fully parameterized classical ANCF element is used to characterize the displacement field of each finite element in the beam. Nonlinear constitutive models (such as the near-incompressible neo-Hookean material model) are used to describe the rubber-like behavior of the beam. Control of such a hyperelastic beam is approached using the theory of constrained motion, where the control objectives are reformulated as constraints that are imposed on the continuum. The fundamental equation of mechanics is employed to obtain the explicit generalized nonlinear control forces in closed form, which are applied at the nodes of the beam in order to achieve the desired control objectives. No linearizations and/or approximations are made in the dynamics of the nonlinear continuum, and no a priori structure is imposed on the nature of the nonlinear controller. Four numerical simulations demonstrating the control of a highly flexible 30-element hyperelastic cantilever beam are presented to show the efficacy of the control methodology in achieving the desired control objectives.