基于不同梁理论的功能梯度悬臂梁自由振动分析

Analysis of Natural Vibration of Functionally Graded Cantilever Beam Based on Different Beam Theories

基于欧拉-伯努利梁、瑞利梁和铁木辛柯梁理论,研究了功能梯度悬臂梁的自由振动问题。在理论分析中采用分离变量法推导出了不同梁理论下功能梯度悬臂梁自由振动的解析解,进而给出了不同跨深比和材料性能梯度变化的悬臂梁前三阶固有频率。将计算结果与ABAQUS有限元模拟结果进行对比,验证了解析解的准确性,并讨论了不同梁理论在功能梯度悬臂梁自由振动分析时的适用范围。结果表明:对于跨深比大于10的细长梁,三种梁理论均适用;而对于跨深比介于5~10之间的短粗梁,铁木辛柯梁理论表现较好;梯度分布指数在0~5之间改变时,材料性质分布形式的变化非常显著,进而导致相应梁的固有频率发生较大变化,可采用改变材料性质梯度分布的方法调控固有频率来避免目标阶次的共振问题。

Based on the Euler-Bernouli beam, Rayleigh beam and Timoshenko beam theories, the free vibration problem of functionally graded cantilever beam is investigated. In the theoretical analysis, the analytical solution of the free vibration of functionally graded cantilever beam under different beam theories is derived by using the separation variable method. The first three natural frequencies of different span-depth ratios and gradient distributions of material property are considered. The calculation results are compared with the simulation results by ABAQUS, and the accuracy of the analytical solutions is verified. The applicable range of different beam theories in the free vibration analysis of functionally gradient cantilever beam is discussed. The results show that it is reliable to calculate the natural frequency of the functionally graded cantilever beam by a suitable beam theory. For the slender beam with a depth ratio greater than 10, the three beam theories are applicable;for short thick beams with a depth ratio between 5~10, the Timoshenko beam theory is better.When the gradient distribution index changes between 0~5, the distribution form of material parameters changes significantly, which leads to a large change in the natural frequency of the corresponding beam. Therefore, the natural frequency can be adjusted by changing the gradient distribution of material properties to avoid the resonance problem of the target order.