決定梁的自然頻率的瑞利商及最佳邊界函數正交法

Abstract

傳統方法中，我們使用特徵函數代入瑞利商極值法用以決定真實自然頻率。當梁為均勻條件時，可以直接使用公式快速求解，但是遇到非均勻梁時求解過程將會變得極其繁瑣。 而在本篇論文中，我們採用滿足所有邊界條件的邊界函數，取代傳統的特徵函數代入瑞利商，以此為替代方法。我們可以略去解繁瑣的四階微分函數，只需用正交性代入邊界函數快速得到誤差很小之估計自然頻率。其中，邊界函數為一種最低為四次之多項式，可以用來求一階自然頻率，而k階邊界函數則可用來求解k-3階自然頻率。 最後，比較近似解與真實解

In the traditional method, we use the eigenfunction to replace the Rayleigh quotient method to determine the true natural frequency. When the beam is uniform, you can use the formula to solve the true natural frequency quickly, but the process of solving the nonuniform beam will become extremely diffcult. In this paper, we use the boundary function that satisfies all the boundary conditions, instead of the traditional eigenfunction into Rayleigh quotient, as an alternative. We can skip the cumbersome fourth-order differential function, just use orthogonality into the boundary function to quickly get the error is very small estimate of the natural frequency. Among them, the boundary function is a minimum of four polynomials, can be used to find the first order natural frequency, and k-order boundary function can be used to solve k-3 order natural frequency. Finally, we compare the approximate third order natural frequencies before the solution with the real solution. We find that the error value is very small and also confirms the satisfying upper bound theory.