Formulation of the 2-Dimensional Beam Defined by the Absolute Nodal Coordinates with the Algebraic Constraints (The 3rd Report, Configuration of the Algebraic Constraints for the Calculation Efficiency)

Abstract

This paper describes the formulation of the two-dimensional Euler-Bernoulli 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 element coordinate system is introduced to derive the strain energy. In this paper, we introduce the different coordinates to express the kinetic and the strain energies. In particular, the kinetic energy is expressed by the absolute nodal coordinates. On the other hand, the element coordinates are introduced to derive the strain energy with Cauchy and Green-Lagrange strain models. The problems caused by introducing the different coordinate systems at once are solved by the canonical theory with the Poisson's bracket formalism. As a result, it is found that the strain energy of this system can be expressed as the simpler forms than the conventional formulation. The calculation results of the dynamic behavior are completely agree with the previous study. Moreover, the calculation times are significantly decreased by this formulation.