本篇論文研究主要著重於樑結構的具非線性彈性邊界的自由振動做問題探討。只要是邊界問題，都可使用移位函數法(Shifting function method)，將邊界作移位，系統之邊界即可簡化，經此簡化可使系統之運算較易處理；並可配合使用疊代法探討其影響系統之振幅與頻率之關係。本文利用微擾法(Perturbation method) 將非線性邊界系統拆解成多組具遞迴關係之線性邊界子系統，而在其中使用微擾法時可以使其中子系統的邊界可以找出一組具規律性的遞迴關係式(Recurrence Formula)，再搭配移位函數法(Shifting function method)對處理邊界問題相當有效，利用移位函數法求解出各子系統之解，則原本難解之非線性邊界問題，便能相當簡易的求解。

This study discusses the free vibrations of a beam with nonlinear boundary conditions. We can use the shifting function method to simplify and solve the boundary problem. The associated mathematic system is a fourth order partial differential equation with nonlinear boundary conditions. It is shifted and decomposed into five linear differential equations and at most four algebra equations. After finding the roots of the algebra equations, the exact solution of the nonlinear beam system can be reconstructed. One can find the relation between the amplitude and frequency of the system by applying shifting function method and iteration method.
We use perturbation method to decompose the nonlinear boundary conditions system of beam problem into many subsystems with linear boundary condition, and using shifting function to solve the subsystems, finally, the beam system can be reconstructed. During the solving process, we can fine a regular recurrence formula between the subsystems which can reduce the solving process.

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