Calculation of non-linear vibration of rotating beams by using tetrahedral and solid finite elements

Abstract

A development is presented of the non-linear dynamic equations that govern the motion of the tetrahedral and solid finite elements that undergo large displacements. The development presented is exemplified by using the four-node, 12-degree-of-freedom tetrahedral element and the eight-node, 24-degree-of-freedom solid element. It is shown that the element shape functions used in this investigation can be used to describe large translations and finite rigid body rotations. Accordingly, the non-linear formulation presented in this paper can be used in the analysis of small as well as large deformation. The element configuration is identified by using four co-ordinate systems. These co-ordinate systems are the global, body, element and intermediate element co-ordinate systems. The large displacement of the tetrahedral and solid elements is described by using a set of absolute co-ordinates that define the location and orientation of the deformable body co-ordinate systems. The non-linear differential equations of motion of the tetrahedral and solid finite elements are developed by using the principle of virtual work in dynamics. The use of the non-linear dynamic formulation presented in this investigation is demonstrated by using a flexible single robotic arm manipulator that undergoes large displacements. The results obtained by using the four-node tetrahedral elements and the eight-node solid elements are compared with the results obtained by using the three-dimensional beam element. This comparison shows that the discrepancy between the results obtained by using the solid and tetrahedral elements in beam problems is more significant in the dynamic analysis as compared to the discrepancy of 10% reported in the literature for the static analysis.