Nonlinear Dynamics and Control of Reissner's 2D Geometrically Exact Beam by Distributed Port-Hamiltonian System

Abstract

Port Hamiltonian systems formalism [1] is proposed for providing the general control theory for finite-dimensional systems, with the models typically used in multibody dynamics (such as rigid components interconnected with flexible joints, or ports). Many present applications require better modeling of the system flexibility (and risk of damage), and one has to consider infinite-dimensional systems. The nonlinear dynamics and control of such a system in terms of Reissner's geometrically exact beam are studied in this work. More precisely, we first present the theoretical formulation for nonlinear dynamics for a 2D Reissner's beam constructed as a port-Hamiltonian system. This results in a highly nonlinear problem due to nonlinear beam kinematics capable of representing finite displacements, rotations, and strains. The port Hamiltonian formulation suitable for the (nonlinear) control problems is then developed by selecting appropriate effort and flow variables, and the model is reformulated as a coupled system of first-order partial differential equations in a structure-preserving format in a continuum setting. We then develop an expanded format required for a nonlinear system and the corresponding variational formulation by using the principle of virtual power, with the boundary conditions defining the port variables that are used in control. The final step is the finite element discretization by using finite element interpolations for such a nonlinear port-Hamiltonian formulation, resulting in a set of nonlinear ordinary differential equations with nodal degrees that count displacements, rotation, linear and angular velocities, forces, and moments, which provides the greatest flexibility in choosing control strategies. This set of differential equations is here integrated by the backward Euler scheme, resulting in a nonlinear system of algebraic equations. The consistent linearization of such a system provides a robust performance for the proposed port Hamiltonian formulation. This is illustrated with the results of several numerical simulations that confirm the improved performance in energy conservation, which is superior to those provided previously by energy-conserving time integration schemes that have been constructed for fully discretized problems [2].