Generalized Gibbs–Appell’s equations and two-dimensional finite elements model used in flexible multibody analysis

A planar mechanism represents a mechanism that is frequently used in engineering, and very often, the elasticity of some elements of the mechanism cannot be neglected. Consideration of all rigid elements does not allow the analysis of vibrations or situations of loss of stability of some elements. Gibbs–Appell’s generalized equations are used in this paper to obtain the governing equations for a two-dimensional finite element, which is in plane motion. Using Lagrange’s equations is the most widely used way for researchers to address such a problem. This is mainly due to the familiarity of researchers with this robust calculation method. There are two major advantages of applying this formalism: a smaller number of differentiation operations is needed to be performed and, by eliminating Lagrange multipliers, the number of unknowns decreases significantly. The method is applied for the plane multibody systems with elastic elements. We hope that this method, due to its simplicity, will be interesting for mechanical designers.