非线性弹性地基梁在随动载荷作用下的屈曲和振动

Buckling and vibration of nonlinear elastic foundation beams subjected to follower force

基于可伸长梁的几何非线性理论,建立了非线性弹性地基上梁在随动载荷作用下的屈曲问题和振动问题控制方程,分别采用打靶法分析了弹性地基梁的后屈曲行为以及后屈曲构形上的振动问题。给出了不同非线性弹性地基系数下,梁在随动载荷作用下的过屈曲平衡路径曲线以及过屈曲附近前三阶频率随载荷变化的曲线。研究表明:立方刚度系数K_2对梁的屈曲和振动影响较小,而线性刚度系数K_1对梁的过屈曲性态和固有频率都有影响。

In this article, both post-buckling and free vibration in the vicinity of post-buckling of a simply supported beam subjected to a distributed follower force are discussed. Based on the geometrically non-linear theory for Euler beams, considering both linear and non-linear elastic foundation effects, geometrically nonlinear dynamic equations for beams subjected to a distributed tangential follower force along the central axis are established. By assuming the amplitude of a beam vibration is small and its response is harmonic, the non-linear differential equations are reduced to two sets of coupled ordinary differential equations. One is for the post-buckling, and another is for the linear vibration in the vicinity of post-buckling of the beam. By employing the numerical shooting technique to solve the two sets of ordinary differential equations with simply supported boundary conditions, the post-buckling equilibrium paths and characteristic curves of the first three natural frequencies versus the load parameter for different values of non-linear elastic foundation parameters are plotted. The results show that the non-linear cubic stiffness parameter has a slight effect while the linear stiffness parameter has some effects on both the post-buckling and vibration response of the beam.