旋轉傾斜梁之穩態及自由振動分析

Abstract

本研究探討了具不同傾斜角(inclination angle)之等速旋轉傾斜尤拉梁(Euler beam)的穩態變形及自然振動。本研究考慮了兩種情況：設定角β=0°(setting angle) 及90°。當設定角為90°時，本研究探討了其穩態變形及自然振動頻率，當設定角為0°時，本研究僅探討其穩態變形。 本研究以級數解求設定角為90°之旋轉傾斜尤拉梁的側向振動頻率及振態，本研究亦以級數解求設定角為β=0°之旋轉傾斜尤拉梁的側向穩態變形。本研究用d'Alembert原理、虛功原理及幾何非線性梁理論的一致線性化在一旋轉座標系統中推導等速旋轉傾斜尤拉梁的線性運動方程式。本研究考慮了轉動慣量對自然振動頻率的影響。 本研究將旋轉梁分成數段，每一段稱為一個元素。當設定角β=0°時，本研究將每一個元素之側向振動之統御方程式的解表示成一個含四個獨立係數的級數，再由旋轉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，使該組齊次方程式係數形成之矩陣的行列式值為零時的根，即為自然振動頻率。本研究以二分法(bisection method)求使行式值為零的根，當設定角為β=0°時，本研究用類似的方法求得一組非齊次方程式，再以高斯消去法解該非齊次方程式求得側向穩態變形。 本研究以數值例題說明本文方法的正確性，並探討傾斜角、轉速、轉軸半徑及細長比對等速旋轉傾斜尤拉梁之側向穩態變形(β=0°)和自然振動頻率(β=90°)的影響。

The steady state and vibration analysis of rotating inclined Euler beam with constant angular velocity is studied in this paper. Two different setting angles β=0° and 90° are considered. The steady state and free vibration are studied for β=90°. However, only steady state is studied for β=0°. A method based on the power series solution is employed to solve the natural frequency of the rotating inclined Euler beam for β=90°. A similar method based on the power series solution is proposed to solve the steady lateral deformation for β=0°. In this paper the linear equations of motion for a rotating inclined Euler beam are derived by the d'Alembert principle, the virtual work principle and the consistent linearization of the fully geometrically non-linear beam theory in a rotating coordinate system. The effect of rotary inertia on the natural frequency of the rotating inclined beam is considered. Here the rotating beam is divided into several segments. For the case β=90°, the governing equations for the lateral vibration of each segment are solved by a power series with four independent coefficients. 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 simultaneous linear homogeneous equations can be obtained. The natural frequencies may be determined by solving the homogeneous equations using the bisection method. For the case β=0°, a similar procedure is used to determine the steady lateral deformation. However, a set of simultaneous linear nonhomogeneous equations, which can be solved using Gauss Elimination, are obtained. Numerical examples are studied to verify the accuracy of the proposed method and to investigate the effects of inclined angle on the steady lateral deformation (β=0°) and the natural frequency (β=90°) of rotating Euler beams with different angular velocity, radius of the hub, and slenderness ratio of the beam.