摘 要
本研究以 Timoshenko 樑理論和歐拉(Euler) 樑理論分別來分析旋轉之傾斜樑之自由振動問題。首先，利用漢米頓原理及非線性樑理論推導一旋轉之傾斜樑之耦合統御方程式，經數次微分及變數消去的運算，可將統御方程式非耦合化，並獲得一變係數六階常微分方程式，再以Frobenius方法來獲得該系統自由振動之確切級數解。
本文針對樑之軸向伸長及柯氏力(Coriolis force)所產生的效應研究旋轉梁之動態行為；並對樑之傾斜角效應亦作一探討。在Timoshenko 樑理論中，一無因次旋轉伸長參數用來表示軸向變形及柯氏力對樑自然頻率之影響。我們發現當該無因次旋轉伸長參數很大時，軸向變形及柯氏力對樑自然頻率之影響相當大。除了設置角(setting angle)設為90度固定外，其他參數包括傾斜角、輪轂半徑及轉動速度被考慮來評估對旋轉傾斜樑之影響；在歐拉樑理論中，設置角及上述其他所提之參數皆予以考慮。圖中展示設置角及傾斜角同時對自然頻率之影響。柯氏力效應在此理論中亦被研究。
除了呈現數值結果分析之外，本文並提出旋轉傾斜樑之定性關係，在不須要麻煩複雜的計算下，就可以很清楚地了解該系統傾斜角、輪轂半徑、設置角與自然頻率之間的物理關係。

Abstract
This paper is devoted to studying the free vibration problem of a rotating inclined beam employing both Timoshenko beam and Euler beam theories. First, utilizing the Hamilton principle and the consistent linearization of the fully non-linear beam theory derives the coupled governing differential equations for a rotating inclined beam. After taking a series of differentiation and variable cancellation operations appropriately, the coupled governing differential equations can be decoupled and reduced to a six-order ordinary differential equation with variable coefficients in terms of the transverse displacement. Subsequently, the exact power series fundamental solutions of the system using the Frobenius method are developed.
The extensional deformation and the Coriolis force effects are taken into account to evaluate the dynamic behavior of a rotating beam. Also, the influence of the inclination angle is investigated. In the Timoshenko beam model, a dimensionless rotating extension parameter is introduced to illustrate the influence of the Coriolis force and the extensional deformation on the natural frequencies of the beam. It is shown that both the extensional deformation and the Coriolis force have significant influence on the natural frequencies of the rotating beam when the dimensionless rotating extension parameter is large. Except setting angle, the parameters including inclination angle, hub radius and rotating speed are considered. In the Euler beam model, the influence of setting angle is included apart from the above parameters. The influence of both setting angle and inclination angle on the variation of the natural frequency is clearly illustrated. The Coriolis force effect using Euler beam theory is also investigated.
Besides, the analytically numerical results are shown; several general qualitative relations are explored without numerical analysis. They do not require a wide range of data to achieve this and the physical relations among the inclination angle, the hub radius, the setting angle and the natural frequencies of the beam system are revealed comprehensively.

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