Formulation of the 2-Dimensional Beam Defined by the Absolute Nodal Coordinates with the Algebraic Constraints (The 1st Report, Formulation Based on the Canonical Theory)

Abstract

This paper describes the development of the analytical procedure of the flexible beam in the absolute nodal coordinate formulation (ANCF) by using the canonical theory. The elastic force in the ANCF has the complicated expression when the strain energy is defined by the coordinates in the element coordinate system. In this paper, we introduce the artificial algebraic constraints into the analytical procedure and make the constrained system on purpose to simplify the derivation of the elastic force. Those constraints lead to the redundant degree of freedom and the primary constraints caused by the definition of momentum. In order to formulate this system, the Dirac's approach with the Poisson bracket is introduced. As a result, the equations of motion are obtained in the form including the expanded degrees of freedom. Moreover, the reduced form of the equations of motion shows that the elastic force of this system is expressed as the right-hand side of the evolutionary equation for the momentum in the canonical equation. The elastic force obtained by this approach is the general expression for the two dimensional problem of the flexible beam defined in the ANCF. The proposed method is applied to the two dimensional beam element with the assumption of the Bernoulli and Euler. Then, the derived elastic force is compared with the previous research. The equivalence of these formulation is shown by the linear strain model.