本文通过对状态空间变量进行变量替换，求得了沿轴向指数分布的功能梯度Timoshenko梁的状态空间传递矩阵方程。本文通过传递矩阵法计算了多种边界条件下结构固有频率的精确解，并与解析解进行对比。通过分析梯度参数 对结构固有频率与模态振型的影响，本文纠正了部分文献中存在的错误，并且采用有限元法对本文的计算结果进行验证。本文通过对比不同梁理论的计算结果，定量的分析了剪切刚度和转动惯量对结构固有频率的影响。计算结果表明，本文所提出的方法物理概念清晰，降低问题求解难度的同时可以减少计算量。
Abstract
Based on the state space variable replacement,the transfer matrix equation of a Timoshenko beam with axially exponential distributed functional gradation was derived.The exact solution of natural frequencies of the structure with multiple boundary conditions was obtained by the transfer matrix method and compared with the available analytical solution.The results show that the relation curve between the frequency and the gradient of the material is continuous and smooth,and there is no jumping phenomenon.Meanwhile the finite element method was used to verify the results.The effects of shear stiffness and moment of inertia on the natural frequencies of the structure were analyzed by comparing the results according to different beam theories.The calculation results show that the method presented is clear in physical concept and can reduce the computational complexity and the amount of computation.
关键词
状态空间变量 /
传递矩阵法 /
固有频率 /
功能梯度材料 /
指数梯度
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Key words
state space variable /
transfer matrix method /
natural frequency /
functionally graded materials /
exponential gradient
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参考文献
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