沿轴向指数分布的功能梯度Timoshenko梁的频率精确解

Exact solution of the free vibration of exponentially non-uniform functionlly graded Timoshenko beams

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。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,该计算结果表明频率与材料梯度变量之间的关系曲线是连续光滑的,并未出现部分文献中的跳跃现象,并且采用有限元法该计算结果进行验证。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,该方法物理概念清晰,降低问题求解难度的同时可以减少计算量。

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.