Development of an Active Curved Beam Model Using a Moving Frame Method

In order to develop an active large-deformation beam model for slender, flexible or soft robots, the d’Alembert principle of virtual work is derived for three-dimensional elastic solids from Hamilton’s principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross-sections of a deforming beam. In the derivation of the beam model, Élie Cartan’s moving frame method is utilized. The resulting large-deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate-form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. Finally, to illustrate the deformation due to internal actuation, an active Timoshenko-beam model is derived by linearizing the nonlinear planar equations. For an active, simply-supported Timoshenko-beam, the analytical solution is presented.