An efficient approach for whirling speeds and mode shapes of uniform and nonuniform Timoshenko shafts mounted by arbitrary rigid disks

• Main Authors: Jong‐Shyong Wu, Tzu‐Fu Hsu
• Format: Article
• Language: English
• Published: Wiley 2020-07-01
• Series: Engineering Reports
• Subjects: Timoshenko shaft; gyroscopic moment; equivalent mass moment of inertia; whirling speeds and mode shapes
• Online Access: https://doi.org/10.1002/eng2.12183

Abstract In theory, the whirling motion of a shaft‐disk system is three‐dimensional (3D), however, if the transverse displacement in the vertical principal xy‐plane and that in the horizontal principal xz‐plane for the cross‐section of the shaft located at the axial coordinate x are represented by a complex number, then the mass moment of inertia for unit shaft length (at x) and that for each rigid disk i can be combined with their associated gyroscopic moments (GMs) to form the frequency‐dependent equivalent mass moments of inertia, respectively, in the equations of motion for the rotating Timoshenko shaft carrying arbitrary rigid disks. It is found that the above‐mentioned equations for the rotating 3D shaft‐disk system take the same forms as the corresponding ones for the associated stationary two‐dimensional (2D) Timoshenko beam carrying the same number of disks, so that the approaches for the free vibration analyses of the stationary 2D beam‐disk system can be used to solve the title problem for obtaining the whirling speeds and mode shapes of a rotating 3D shaft‐disk system. Since the order of the eigenproblem equation derived from the presented approach is much smaller than that derived from the conventional finite element method (FEM), the computer time consumed by the former is much less than that by the latter. In addition, the solutions obtained from the presented approach are exact and may be the benchmark for evaluating the accuracy of solutions of the other approximate methods. Numerical examples reveal that the presented approach is available for the uniform or nonuniform shaft‐disk system, and the obtained results are very close to those obtained from existing literature or the FEM. The formulation of this article is available for a shaft‐disk system with various boundary conditions (BCs), to save space, only the cases with pinned‐pinned BCs are illustrated.