The equations of Lagrange for a continuous deformable body with rigid body degrees of freedom, written in a momentum based formulation

Abstract

The present paper is concerned with Lagrange׳s Equations, applied to a deformable body in the presence of rigid body degrees of freedom. The Lagrange description of Continuum Mechanics is used. An exact version of the Equations is derived first. This version, which represents an identical extension of the Fundamental Law of Dynamics, does involve the idea of virtual motions. The virtual motion is described in the framework of the Ritz-Ansatz, but our derivation does not make use of D׳Alemberts principle, the principle of virtual work, or variational principles. From the exact version, by involving arguments related to the Galerkin approximation technique, we derive an approximate Ritz type version of Lagrange׳s Equations. This approximate version coincides with the traditional one, which is based on the notion of kinetic energy. However, since our derivation stems from the Fundamental Law of Dynamics, we have at our disposal an alternative formulation, which is based on the notion of local momentum. This momentum based version, which is the main topic of the present contribution, can be used for the purpose of performing independent checks of the energy based version of Lagrange׳s Equations. The momentum based version also clarifies that and how certain terms in the energy based version do cancel out. The momentum based version is worked out in the framework of the Floating Frame of Reference Formulation of Multibody Dynamics. Explicit formulas for the single terms of Lagrange׳s Equations are derived for the translational, rotational and flexible degrees of freedom of the deformable body, respectively. Corresponding Lagrange׳s Equations are explained in the light of the relations of Balance of Total Momentum, Balance of Total Moment of Momentum, of the Mean Stress Theorem and the notion of Virial of Forces. An embedding into the literature is given.