等速旋轉傾斜Timoshenko梁的穩態變形與自由振動分析

Abstract

本研究之主要目的在探討設定角(setting angle)為 及 時具不同傾斜角(inclination angle)之等速旋轉傾斜Timoshenko梁的穩態變形及自由振動。本研究考慮的Timoshenko梁係以一傾斜角剛接在以等角速度旋轉的圓柱上，並將旋轉梁的運動方程式建立在一個以等角速度旋轉的總體座標上。本研究以Timoshenko梁正確的變形機制，利用虛功原理、d□Alembert原理與幾何非線性梁理論之一致性線性化推導旋轉傾斜Timoshenko梁之運動方程式。 當設定角為 時，旋轉傾斜Timoshenko梁的穩態變形僅有軸向位移且其軸向振動與側向振動不互相耦合，其側向振動僅考慮撲翼振動(flapping vibration)，本研究將以級數解求得旋轉傾斜梁的撲翼振動頻率及振態。當設定角為 時，旋轉傾斜Timoshenko梁的穩態變形含軸向和側向變形且其軸向振動與側向振動因科氏力而相互耦合，側向穩態變形不為零，本研究將以級數解求其側向穩態變形。 本研究將梁結構分割成數段，每一段稱為一個元素，然後在每一個元素當前的變形位置上建立一以等角速度旋轉的元素座標，每一個元素的變形、節點內力與運動方程式都是建立在該元素座標上。當設定角為 時，本研究將每一個元素之統御方程式的解表示成含二個獨立係數的級數矩陣，再由旋轉傾斜梁兩端的邊界條件及相鄰元素在共同節點的連續條件求得一組齊次方程式，該組齊次方程式為一個特徵值問題，其係數形成之矩陣的行列式值為零時的根，即為振動的自然頻率。本研究以二分法(bisection method)求旋轉傾斜Timoshenko梁振動的自然頻率，並以逆冪法(inverse power method)求得其振動模態。當設定角為 時，本研究用類似的方法求得一組非齊次方程式，以求得其側向穩態變形。 本研究最後將以無因次化的數值例題探討旋轉傾斜Timoshenko梁之自然頻率的收斂性、準確性，並探討傾斜角、無因次轉速、無因次轉軸半徑及細長比對旋轉傾斜梁無因次自然頻率的影響，本研究還探討旋轉梁的軸向振態與側向振態對應的自然頻率接近時，其振動模態的耦合及特徵值曲線轉向與特徵值曲線轉向交叉的現象。

The steady state deformation and free vibration analysis of a rotating inclined Timoshenko beam with constant angular velocity is studied in this paper. Two different setting angles and are considered. The governing equations for linear vibration of a rotating Timoshenko beam are derived by the d'Alembert principle, the virtual work principle and the consistent linearization of the fully geometrically nonlinear beam theory in a rotating rectangular Cartesian coordinate system which is rigidly tied to the hub. For of the rotating inclined Timoshenko beam, the steady state deformation is only axial deformation. A method based on the power series solution is employed to solve the natural frequency and vibration modes of the axial vibration and flapping vibration. For , a similar method based on the power series solution is proposed to solve the steady lateral deformation and free vibration. Here the rotating inclined beam is divided into several segments. The governing equations for linear vibration of each segment are solved by a power series. Substituting the power series solution of each segment into the corresponding boundary conditions at two end nodes of the rotating beam and the continuity conditions at common node between two adjacent segments, a set of homogeneous equations can be obtained. The natural frequencies may be determined by solving the homogeneous equations using the bisection method. Then the inverse power method is used to find the corresponding vibration modes. Dimensionless numerical examples are studied to verify the accuracy of the proposed method and to investigate the dimensionless natural frequency of rotating inclined beams with different inclined angles, dimensionless angular velocities, dimensionless radius of the hub, and slenderness ratios. The phenomenon of eigenvalue curve crossing and eigenvalue curve veering are also investigated for rotating inclined beams that has two modes with closely spaced natural frequencies.