A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control

Many boundary controlled and observed Partial Differential Equations can be represented as port-Hamiltonian systems with dissipation, involving a Stokes-Dirac geometrical structure together with constitutive relations. The Partitioned Finite Element Method, introduced in Cardoso-Ribeiro et al. (2018), is a structure preserving numerical method which defines an underlying Dirac structure, and constitutive relations in weak form, leading to finite-dimensional port-Hamiltonian Differential Algebraic systems (pHDAE). Different types of dissipation are examined: internal damping, boundary damping and also diffusion models

Hamiltonian modelling and buckling analysis of a nonlinear flexible beam with actuation at the bottom

The use of beams and similar structural elements is finding increasing application in many areas including micro and nanotechnology devices. For the purpose of buckling analysis and control, it is essential to account for nonlinear terms in the strains while modelling these flexible structures. Further, the Poisson's effect can be accounted in modelling by the use of a two-dimensional stress-strain relationship. This paper studies the buckling effect for a slender, vertical beam (in the clamped-free configuration) with horizontal actuation at the fixed end and a tip-mass at the free end. Including also the inextensibility constraint of the beam, the equations of motion are derived. A preliminary modal analysis of the system has been carried out to describe candidate post-buckling configurations and study the stability properties of these equilibria. The vertical configuration of the beam under the action of gravity is without loss of generality, since the objective is to model a potential field that determines the equilibria. Neglecting the inextensibility constraint, the equations of motion are then casted in port-Hamiltonian form with appropriately defined flows and efforts as a basis for structure-preserving discretization and simulation. Finally, the finite-dimensional model is simulated to obtain the time response of the tip-mass for different loading conditions