弹性支撑及连接边界的多跨曲梁面内自由振动分析

In-plane free vibration analysis of multi-span curved beams with elastic support and connecting boundary conditions

针对弹性支撑边界曲梁的振动问题,采用一种改进的傅里叶级数方法对多跨曲梁面内自由振动特性进行了求解分析.将曲梁面内径向和切向位移函数表示成傅里叶级数形式,并引入辅助多项式函数用以解决弹性边界的不连续性.采用瑞利-里茨方法求解基于能量原理的哈密顿方程,得到关于未知位移幅值系数的标准特征值问题,求解得到多跨曲梁的固有频率和振型.通过单跨、两跨的自由、简支、固支等传统边界及弹性边界的曲梁模型结果与有限元法结果的对比验证了本文方法的正确性,并分析了两跨固支曲梁中间连接刚度对固有频率的影响.

According to the vibration problem of a curved beam with elastic support boundary conditions, the inplane free vibration characteristics of multi-span curved beams were analyzed using an improved Fourier series method. The transverse and tangential displacement functions were sought as a Fourier cosine series, and an auxiliary polynomial function was introduced to take all the relevant discontinuities of the elastic boundaries. The Ray- leigh-Ritz method was used to solve Hamilton＇s equation, which is based on the energy principle, and a standard eigenvalue problem concerning the unknown displacement amplitudes was derived from which the natural frequencies and mode shapes can be solved. The results of single-span and two-span curved beams with free, simple supported, clamped, and elastic supported boundary conditions were obtained and compared with the results acquired from the finite element method （FEM） to validate the correctness of the presented method. Furthermore, the effect of the connecting stiffnesses between two-span curved beams on the first four frequencies was described.