Port-Hamiltonian formulation and structure-preserving discretization of hyperelastic strings

Abstract

Port-Hamiltonian (PH) systems provide a framework for modeling, analysis and control of complex dynamical systems, where the complexity might result from multi-physical couplings, non-trivial domains and diverse nonlinearities. A major benefit of the PH representation is the explicit formulation of power interfaces, so-called ports, which allow for a power-preserving interconnection of subsystems to compose flexible multibody systems in a modular way. In this work, we present a PH representation of geometrically exact strings with nonlinear material behaviour. Furthermore, using structure-preserving discretization techniques a corresponding finite-dimensional PH state space model is developed. Applying mixed finite elements, the semi-discrete model retains the PH structure and the ports (pairs of velocities and forces) on the discrete level. Moreover, discrete derivatives are used in order to obtain an energy-consistent time-stepping method. The numerical properties of the newly devised model are investigated in a representative example. The developed PH state space model can be used for structure-preserving simulation and model order reduction as well as feedforward and feedback control design.

Figures (5)

• Figure 1: Material and spatial string configurations on $\mathbb{R}^2$.
• Figure 2: Initial configuration and snapshots during the motion
• Figure 3: Energy quantities
• Figure 4: Discrete Hamiltonian $\hat{H}_{n}$
• Figure 5: Discrete Hamiltonian increments $| \hat{H}_{n+1} - \hat{H}_{n} |$