本研究針對自旋、非線性預扭、承受軸向力、非均勻Timoshenko 樑振動問題做一基礎性研究。首先在自旋、預扭座標上應用Hamilton 原理及Timoshenko 理論推導上述問題之統御方程式及邊界條件。其次對其四條互偶、變係數偏微方程式及邊界條件取微分轉換，吾人可求得遞歸型式之代數方程組，進而使用符號運算求解代數方程組，最後將其結果取反微分轉換吾人可求得多項式型式之頻率方程式。 在不同邊界條件下，應用上述論理分析的結果研究並討論非線性預扭角、自旋速率、軸向負載對非均勻Timoshenko 樑自然頻率的影響；並與自旋、線性預扭、承受軸向力、非均勻 Timoshenko 樑的研究報告做比較，了解兩者之差異性。更進一步探研如何利用非線性預扭角的分佈來改善或調整自然頻率。
In this research, a basic study on vibration problems of a spinning, nonlinearly pretwisted non-uniform Timoshenko beam under axial loading is studied. First, on a spinning twist coordinate, applying Hamilton’s principle and Timoshenko beam theory, the governing equations and boundary conditions of the above-mentioned problem are derived. Second, taking differential transform on these four coupled differential Equations with variant coefficient and boundary conditions, we obtain a set of algebraic equations in recursive form. Moreover, we solve the algebraic equations by using symbolic computation. Last, taking inverse differential transform on these reeults, we obtain a frequency equation in polynomial form. Under different boundary conditions, applying the theoretic results mentioned above, we investigate and discuss the effects of non-linearly pretwisted angle, spinning speed and axial loading on the natural frequencies of non-uniform Timoshenko beams. Moreover, we compare the calculated results with the research report on a spinning, linearly pretwisted Timoshenko beam under axial loading and study the difference between them. Furthermore, we investigate how to modify or design natural frequencies by using non-linearly distributed pretwisted angle.