Formalisms of rational dynamics of mechanical systems: Newton to Kane

The aim of this article is to present the general equation formulations for mechanical systems with a finite number of degrees of freedom, enabling the parametric representation of mechanisms and structural elements in their dynamic operation. It thus complements the article [A 1 666] "Mécanique générale – Dynamique générale. Form analytique" by Jean-Pierre BROSSARD, which presents examples of solved mechanisms. The systems of equations presented in this article take stock of all the models that can be produced using five successive formalisms.

The first two paragraphs are a memento of analytical mechanics, taking up the classical developments starting with Newton's classical dynamics in its vector approach and culminating in Hamilton's canonical formalism, via Lagrange's equations. The – real and then – virtual power approach gradually leads to a fully scalar and parametric presentation. This is complemented by a presentation of the Kane formalism, a vector-based approach that relies on the parameterization of velocities rather than positions. It applies to rational systems described by a finite number of degrees of freedom, typically systems made up of undeformable solids connected by geometric links or flexible elements with no inertia. The paragraph 2.3 specifically presents Kane's formalism adapted to robotics and system control. All the developments presented concern the most general rational dynamic systems, linear and non-linear, conservative or not, loaded, constrained or free. The paragraph 3