Coupled bending torsional vibration of rotating shafts using finite element

The coupled bending and torsional vibration of a rotating tapered shaft based on Timoshenko beam theory is presented by means of the finite element technique. The shaft which is assumed to have circular cross-section and linear taper is discretized into a number of finite elements with ten degrees of freedom each. The equation of motion of the rotating tapered shaft finite element is derived using Lagrange equation. Explicit expressions for the finite element mass, stiffness and gyroscopic matrices are derived by using consistent mass formulation. The shaft finite element is integrated into a procedure to calculate the natural frequencies of rotor-bearing systems. The generalized eigenvalue problem is defined and numerical solutions are generated for a wide range of whirl ratios, spin speeds and taper ratios. Comparisons are made with exact solutions and with numerical results available in the literature. The results display high accuracy when compared to other numerical results.