旋轉三維Timoshenko梁之振動分析

Abstract

本研究主要是利用共旋轉法來推導雙對稱三維旋轉Timosheno梁的運動方程式，並探討旋轉梁振動的自然頻率。本文中的振動是指基於旋轉梁穩態的微小振動。 本文利用非線性梁理論的一致線性化、d'Alembert原理和虛功原理在旋轉元素座標上推導梁元素的節點變形力和節點慣性力。保留穩態變形的節點參數和其微分項到二次項以及扭率的三次項，在振動部份僅保留節點參數和其微分項到一次項。 本文利用基於牛頓法的增量迭代法求解穩態變形，並使用二分法來求取Timoshenko旋轉梁的自然頻率。本文最後以數值例題來驗證所提數值方法的正確性和效能，並探討不同斷面、不同長度、不同轉速及不同設定角對旋轉梁自然頻率的影響。

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.