轴向力作用不连续梁自由振动的广义函数解

FREE VIBRATION ANALYSIS OF AXIAL-LOADED BEAMS WITH DIFFERENT DISCONTINUITIES USING GENERALIZED FUNCTIONS

首先运用分布理论建立了轴向力作用下含任意不连续点的欧拉梁的自由振动的统一微分方程.不连续点的影响由广义函数(Dirac delta函数)引入梁的振动方程.微分方程运用Laplace变换方法求解;与传统方法不同的是,论文方法适用于含任意类型的不连续点和多种不连续点组合情况的梁,求得的模态函数为整个不连续梁的一般解.由于模态函数的统一化以及连续条件的退化,特征值的求解得到了极大的简化.最后,以轴向力作用下多跨梁—弹簧质量块系统模型为例验证了论方法的正确性与有效性.

The general governing differential equation of the vibration of Euler-Bernoulli beams with arbitrary discontinuities subjected to axial force was derived by using generalized functions and was then solved based on Laplace transformation. Unlike the classical solutions of discontinuous beams, the proposed scheme is valid for various kinds of discontinuity conditions. And the generalized solutions are expressed in terms of a single expression of the entire beam. For the specified discontinuity type, eigenvalue matrices were simplified by the degenerated continuity conditions. Finally, the accuracy and efficiency of the pro- posed method was verified by the analysis of the free vibration of a four-span pinned-pinned beam with three spring-mass systems subject to compressive load.