Torsional vibration analysis of shafts with sections of tapered diameter

This paper presents the development of two tapered finite elements for use in torsional vibration analysis of rotor systems. These elements are particularly useful in analysis of systems that have shaft sections with linearly varying diameters. Both elements are defined by two end nodes, and inertia matrices are derived based on a consistent mass formulation. One element assumes a cubic displacement function and has two degrees of freedom at each node: rotation about the shaft's axis and change in angle of rotation with respect to the axial distance along the shaft. The other element assumes a linear displacement function and has one rotational degree of freedom at each node. The elements are implemented in a computer program. Calculated natural frequencies and mode shapes are compared for both tapered shaft sections and constant diameter sections. These results are compared with results from an available constant diameter element. It is shown that the element derived assuming a cubic displacement function offers much better convergence characteristics in terms of calculated natural frequencies, both for tapered sections and constant diameter sections, than either of the other two elements. The finite element code that was developed for implementation of these elements is specifically designed for torsional vibration analysis of rotor systems. Lumped inertia, lumped stiffness, and gear connection elements necessary for rotor system analysis are also discussed, as well as calculation of natural frequencies, mode shapes, and amplitudes of response due to a harmonic torque input.