具端點彈性支承及質量之旋轉梁的自由振動及挫屈

Abstract

本文主要目的在探討旋轉梁的支承端之剛度及自由端加掛配重其分別對自然振動頻率的影響，還有探討旋轉梁挫屈的臨界轉速。本文中考慮了Euler梁及Timoshenko梁。本文利用d*Alembert原理及虛功原理推導旋轉梁振動的統御方程式。本文中在分析旋轉梁之自然頻率時考慮了轉動慣量及科氏力的影響。本文利用非線性梁理論的一致線性化以得到所有慣性效應及變形間的耦合效應。本文中將旋轉梁均分成多個元素，每個元素的統御方程式都用一個級數來解，並由旋轉梁兩端點及相鄰兩元素在共同節點上的邊界條件可求得自然頻率。 本文最後以數值例題分析了不同轉速下，不同設定角及不同細長比的旋轉梁在不同邊界條件下之自然頻率並分析了旋轉梁挫屈的臨界轉速，同時並探討了科氏力、轉動慣量對自然頻率的影響。

The objective of this paper is to investigate the effects of stiffness of elastically restrained root, tip mass on the natural frequency of vibration of rotating beam, and investigate the critical rotation speed for buckling of rotating off-axis beam. Here, Euler beam and Timoshenko beam are considered. The governing equations for linear vibration of rotating Euler beam and Timoshenko beam are derived by the d*Alembert principle and the virtual work principle. The effect of Coriolis force and rotary inertia on the natural frequency of the rotating beam is investigated. In order to capture all inertia effect and coupling between extensional and flexural deformation, the consistent linearization of the fully geometrically non-linear beam theory is used. A method based on the power series solution is proposed to solve the natural frequency. The rotating beam is divided into serveral segments. The governing equations of each segment are solved by a power series. From the conditions of the rotating beam at two end nodes and common node between two adjacent segments the natural frequency can be calculated. Numerical examples are studied for the nature frequency of rotating beam with different angular velocity, setting angle, and slenderness ratio, and for the critical rotation speed for buckling. The effect of Coriolis force and rotary inertia on the natural frequency of the rotating beam is investigated by numerical examples.