The Poincaré-Chetayev equations and flexible multibody systems

Abstract

This article is devoted to the dynamics of flexible multi-body systems and to their links with a fundamental set of equations discovered by H. Poincaré one hundred years ago [1]. These equations, called "Poincaré-Chetayev equations", are today known to be the foundation of the Lagrangian reduction theory. Starting with the extension of these equations to a Cosserat medium, we show that the two basic sets of equations used in flexible multi-body dynamics. The generalized Newton-Euler model of flexible multi-body systems in the floating frame approach and the partial differential equations of the nonlinear geometrically exact theory in the Galilean approach, are Poincaré-Chetayev equations.

Poincaré’s Equations for Cosserat Media: Application to Shells

“…From a more geometrical viewpoint, in [9], the original variational calculus of Poincaré is extended from a conguration Lie group to a principal ber bundle to derive a set of covariant Euler-Poincaré equations but with no relation to Cosserat media. In [11], both elds equations and boundary conditions of a bounded multidimensional Cosserat medium are derived in the context of Euler-Poincaré reduction. In this approach, the dynamics of the Cosserat medium are deduced from a unique Lagrangian density left invariant by the transformations of G, the transformations being parameterized by the time and the material coordinates (Lagrangian labels) of the medium D.…”

“…In particular, though the Hamiltonian structure of the geometrically exact balance equations of rods and plates is revealed in [14] through the derivation of an appropriate bracket, these equations are considered as a starting point in [14], and derived from Newton's laws and Euler's theorems. More recently, the relations between Lagrangian reduction and geometrically exact beam theory, have been established and exploited in [11], [15] and [16], with further extension to the case of molecular strands [17], [18]. In [18], several sets of reduced motion equations, ranging from Euler-Poincaré to Lagrange-Poincaré equations, are developed for modelling molecular strands subjected to nonlocal electrostatic forces, while in the same reference, the case of multidimensional media (in this case, molecular membranes), is evoked as a further perspective by the authors.…”

“…To that end, we will restart from the general construction of [11], and will remind how one can derive from a Lagrangian density related to the space of material labels of D, a rst set of Poincaré equations that will be consequently named "Poincaré equations of Cosserat media in the space of material labels", or more concisely, "in the material space D". In this general context, we will consider both the eld equations and the boundary conditions of a Cosserat medium (D, M), with D of arbitrary dimension p, and M a full (non-degenerated) three-dimensional (3D) microstructure.…”

“…As a result, we will need to derive two further sets of Poincaré equations, one related to the reference conguration, and the second, related to the deformed (current) conguration of the medium, both being embedded in geometric space. To derive these new equations, we will start from the equations in the material space D of [11]. Then, lying on the concept of duality, we will identify the intrinsic geometric nature of all the objects they handle, and establish how they transform from material space D to the reference and deformed conguration of the medium.…”

“…Then, lying on the concept of duality, we will identify the intrinsic geometric nature of all the objects they handle, and establish how they transform from material space D to the reference and deformed conguration of the medium. In parallel to this rst approach of derivation, we will show that these two sets of equations can be derived straightforwardly by extending the variational calculus of [11] to Lagrangian densities related to reference and deformed congurations of the Cosserat medium. Along derivation of these equations, we will progressively relate the objects naturally produced by the Poincaré calculus to the physics of the Cosserat media, and will recover in a pure deductive manner, several of the key concepts of the micropolar theory [1], as those related to the material objectivity of their constitutive laws, their kinematic and kinetic models, several models of strain and stress, and their balance equations in the geometrically exact form.…”