A Dynamic Finite Element (DFE) method for free vibrations of bending-torsion coupled beams

Abstract

In this paper, a Dynamic Finite Element (DFE) formulation for the free vibration analysis of bending-torsion coupled beams is presented. First, the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam are found. The employment of these solutions as basis functions leads to the appropriate frequency dependent shape functions which can then be utilized to find the nodal approximations of variables. By exploiting the Principle of Virtual Work (PVW), the elementary Dynamic Stiffness Matrix (DSM) is then obtained which has both mass and stiffness properties. The implementation of the derived DFE matrices in a program is discussed with a particular reference to the Wittrick–Williams algorithm. The application of the theory is demonstrated by illustrative examples wherein a substantial amount of coupling between bending and torsion is highlighted. The correctness of the theory is confirmed, to a high degree of accuracy, by published results and numerical checks.