| Summary: | 碩士 === 國立交通大學 === 機械工程系 === 89 === The equations of motion for the doubly symmetric three dimensional rotating Timoshenko beam is derived using a co-rotational total Lagrangian finite element formulation combined with rotating frame method. The natural frequency of the infinitesimal free vibration of the rotating beam measured from the position of the steady-state deformation of rotating beam is investigated.
Both the element deformation nodal forces and inertia nodal forces are systematically derived by consistent linearization of the fully geometrically non-linear beam theory using the d''Alembert principle and the virtual work principle in the current rotating element coordinates. The terms up to the second order of nodal parameters, their spatial derivatives, and the third order term of twist rate corresponding to the steady-state deformations are all retained. However, only the terms up to the first order of nodal parameters, and their spatial derivatives and time derivatives corresponding to the free vibration are retained.
An incremental-iterative method based on the Newton-Raphson method is employed to solve the steady-state deformation. A bisection method is employed to determine natural frequencies of the rotating Timoshenko beam. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of angular velocities, setting angles, cross sections and slenderness ratios of the beam on the natural frequency of the rotating beams.
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