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# Vibrations of In-Plane Non-Constant Inward and Outward Rotating Beams

Shueei-Muh Lin1
Department of Mechanical Engineering, Kun Shan University, Tainan, Taiwan 710-03, Republic of China. E-Mail: smlin@mail.ksu.edu.tw

Computer Modeling in Engineering & Sciences 2009, 52(1), 105-124. https://doi.org/10.3970/cmes.2009.052.105

### Abstract

In this study, the mathematical model of a non-constant rotating beam is established. It is an in-plane moving mass problem. Due to the effect of non-constant rotation, this model is composed of a governing differential equation with time-dependent coefficients and forcing term and three homogenous boundary conditions and one non-homogeneous boundary condition with time-dependent coefficients and forcing term. It is basically different to the system with constant rotation speed [Lin, 2008] and the linear moving beam system [Lin, 2009]. Obviously, a moving mass problem with time-dependent coefficients and forcing term is very complicated. A new solution method is here developed to derive the analytical solution for two systems with harmonic inward and outward rotations. First, using the balanced method the system with time-dependent coefficients is transformed into one with time-independent coefficients. Further, one derives the exact general solution of the transformed system which is composed of sixteen fundamental solutions and two particular ones. The exact fundamental solutions are derived by the Frobenius method. The effects of several parameters of a moving beam on the frequency spectrum are significant and investigated. It is found that increasing the excitation frequency greatly increases the tip amplitude. However, in a convention, there is no relation between the excitation frequency and the tip amplitude. Moreover, although the inertial force will induces the vibration during the root moving, the effect of tip mass on suppressing vibration is effective.

### Keywords

non-constant rotation, rotating beam, analytical solution, vibration

Lin, S. (2009). Vibrations of In-Plane Non-Constant Inward and Outward Rotating Beams. CMES-Computer Modeling in Engineering & Sciences, 52(1), 105–124. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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