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On implicit and explicit representations for 1D distributed port-Hamiltonian systems

Antoine Bendimerad-Hohl, Denis Matignon, Ghislain Haine, Laurent Lefèvre

Abstract

First, two examples of 1D distributed port-Hamiltonian systems with dissipation, given in explicit (descriptor) form, are considered: the Dzekster model for the seepage of underground water and a nanorod model with non-local viscous damping. Implicit representations in Stokes-Lagrange subspaces are formulated. These formulations lead to modified Hamiltonian functions with spatial differential operators. The associated power balance equations are derived, together with the new boundary ports. Second, the port-Hamiltonian formulations for the Timoshenko and the Euler-Bernoulli beams are recalled, the latter being a flow-constrained version of the former. Implicit representations of these models in Stokes-Lagrange subspaces and corresponding power balance equations are derived. Bijective transformations are proposed between the explicit and implicit representations. It is proven these transformations commute with the flow-constraint projection operator.

On implicit and explicit representations for 1D distributed port-Hamiltonian systems

Abstract

First, two examples of 1D distributed port-Hamiltonian systems with dissipation, given in explicit (descriptor) form, are considered: the Dzekster model for the seepage of underground water and a nanorod model with non-local viscous damping. Implicit representations in Stokes-Lagrange subspaces are formulated. These formulations lead to modified Hamiltonian functions with spatial differential operators. The associated power balance equations are derived, together with the new boundary ports. Second, the port-Hamiltonian formulations for the Timoshenko and the Euler-Bernoulli beams are recalled, the latter being a flow-constrained version of the former. Implicit representations of these models in Stokes-Lagrange subspaces and corresponding power balance equations are derived. Bijective transformations are proposed between the explicit and implicit representations. It is proven these transformations commute with the flow-constraint projection operator.
Paper Structure (21 sections, 9 theorems, 77 equations, 1 figure)

This paper contains 21 sections, 9 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Variational derivatives
  4. Operator transposition
  5. Two examples with damping
  6. Dzektser
  7. Nanorod
  8. Explicit representation
  9. Implicit representation
  10. Equivalent representations for classical beam models
  11. Explicit representations
  12. Timoshenko
  13. Euler--Bernoulli
  14. Implicit representations
  15. Timoshenko
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathcal{K}:D(\mathcal{K}) \subseteq L^2(\Omega,\mathbb{R}^n) \rightarrow L^2(\Omega,\mathbb{R}^m)$ be a closed and densely-defined (differential) linear operator, and let be an energy functional. Assume that the following abstract Green's identity holds for all $\alpha \in D(\mathcal{K}), \, \beta \in D(\mathcal{K}^\dag)$ where $\mathcal{K}^\dag:D(\mathcal{K}^\dag) \subseteq L^2(\Omega,\mat

Figures (1)

  • Figure 1: Diagram of the transformations between beam models and representations

Theorems & Definitions (14)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 4 more