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Timoshenko beam under finite and dynamic transformations: Lagrangian coordinates and Hamiltonian structures

Oscar Cosserat, Loïc Le Marrec

TL;DR

This work develops a geometrically exact Timoshenko beam model within a Cosserat, one-dimensional setting, emphasizing Lagrangian (material) coordinates to preserve invariance of material parameters. It derives both strong and weak formulations of finite transformations in the mobile frame and constructs three distinct Hamiltonian formulations with corresponding Poisson brackets, using Legendre transforms and moving-frame variables to obtain compact, structure-preserving dynamics. The paper proves energy conservation, clarifies how kinematics are determined from strains and curvatures, and shows closure relations arise naturally within each Hamiltonian framework. The results offer multiple, equivalent Hamiltonian viewpoints that are well-suited for structure-preserving numerics and for exploring beam instabilities and bifurcations in a geometrically exact setting.

Abstract

In the framework of Timoshenko beam, the material parameters are inherently prescribed on the material moving frame. In this regard, we derive the strong and weak formulations of the dynamics under finite transformation in Lagrangian coordinates. Accordingly, analytical mechanics tools are used to deduce a new Hamiltonian formulation of the model which proves to be remarkably simple and synthetic.

Timoshenko beam under finite and dynamic transformations: Lagrangian coordinates and Hamiltonian structures

TL;DR

This work develops a geometrically exact Timoshenko beam model within a Cosserat, one-dimensional setting, emphasizing Lagrangian (material) coordinates to preserve invariance of material parameters. It derives both strong and weak formulations of finite transformations in the mobile frame and constructs three distinct Hamiltonian formulations with corresponding Poisson brackets, using Legendre transforms and moving-frame variables to obtain compact, structure-preserving dynamics. The paper proves energy conservation, clarifies how kinematics are determined from strains and curvatures, and shows closure relations arise naturally within each Hamiltonian framework. The results offer multiple, equivalent Hamiltonian viewpoints that are well-suited for structure-preserving numerics and for exploring beam instabilities and bifurcations in a geometrically exact setting.

Abstract

In the framework of Timoshenko beam, the material parameters are inherently prescribed on the material moving frame. In this regard, we derive the strong and weak formulations of the dynamics under finite transformation in Lagrangian coordinates. Accordingly, analytical mechanics tools are used to deduce a new Hamiltonian formulation of the model which proves to be remarkably simple and synthetic.
Paper Structure (30 sections, 19 theorems, 150 equations)

This paper contains 30 sections, 19 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Cosserat beam formalism and associated tools
  3. Modeling
  4. Mobile frame notation
  5. Properties of the cross product
  6. Curvature and spin
  7. Derivations in material coordinates
  8. Perturbations in material coordinates
  9. Derivation and perturbation of a quadratic form
  10. Constitutive laws and dynamical behavior
  11. Internal free energy density
  12. Kinetics
  13. Strong formulation of dynamical equations
  14. In the mobile frame
  15. With algebraic formulation
  16. ...and 15 more sections

Key Result

Proposition 2.1

The map $j$ is a linear isomorphism. Furthermore,

Theorems & Definitions (53)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 1
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2
  • ...and 43 more