Finite element approach to mathematical modeling of high-speed elastic linkages

A general approach is described for deriving the equations of motion of planar linkages in high-speed machinery. Based on the work of several current authors, well-known displacement finite element method is used to develop the mass and stiffness properties of an elastic linkage. To demonstrate the various steps in the analysis, a 4-bar linkage is utilized; however the method is readily extendible to other planar multi-loop linkages. Starting with a typical elastic planar beam element, the nodal displacement and acceleration expressions are derived including the terms coupling the elastic and rigid-body motions. The linkage is modeled as beam elements and its equations of motion are stated in matrix form. Methods are described for systematic assembly of all elements, resulting in the undamped equations of motion of the total system. Conventional forms of structural damping are reviewed and appended to this paper for inclusion in the equations of motion. This paper also includes assumptions made in order to simplify the analyses here as well as facilitate numerical solutions.