改进傅里叶方法在梁结构振动特性分析中的应用

研究了一般边界条件下Euler-Bernoulli梁的振动特性。首先基于改进傅里叶法建立了梁结构的位移函数表达式，其中位移函数被表示为傅里叶余弦级数展开式与辅助多项式函数的叠加，其后基于最小势能原理建立拉格朗日方程，并通过Rayleigh-Ritz法进行求解，得到其固有模态及强迫振动响应。通过讨论旋转方向和横向弹簧刚度取值对计算结果收敛性的影响，验证了本方法的数值稳定性，得到用于模拟经典边界条件的弹簧刚度值。通过将本文计算结果与有限元法对比，验证了本方法的有效性。在此基础上对一般边界条件下梁结构受迫振动的响应特性进行了研究，讨论了弹簧刚度值等参数对梁结构振动特性的影响规律。

Application of Improved Fourier Series Method in Vibration Analysis of Beam Structures

The vibration characteristics of Euler-Bernoulli beam under general boundary conditions were studied. Firstly, the displacement function expression of the beam structure is established based on the modified Fourier method. The displacement function is expressed as a superposition of a Fourier cosine series expansion and an auxiliary polynomial function, and then Lagrange is established using the principle of minimum potential energy. The equation is solved by the Rayleigh-Ritz method, and then the characteristic equation of the beam structure is obtained. The natural frequency and mode shape of the beam structure can be obtained by solving it. By discussing the influence of rotation and lateral spring stiffness on the convergence of the calculation results, the numerical stability of the method is verified, and the spring stiffness values used to simulate several classical boundary conditions are obtained. The calculated results are in good agreement with the finite element method, thus verifying the accuracy of the method. On this basis, the response characteristics of the forced vibration of the beam structure under different boundary conditions are compared. Finally, the differences between the effects of the rotation direction and the stiffness of the transverse spring on the vibration characteristics of the beam structure are discussed.