非均匀圆膜轴对称振动的离散模型的振动反问题

Inverse problem of discrete model of an inhomogeneous circular membrane with axial symmetric vibration

研究圆膜的振动反问题。首先,采用二阶中心差分格式,导出了圆膜做轴对称振动的差分离散模型。阐明了这一离散模型属于雅可比正系统,进而获得了该系统的振动定性性质。在此基础上,提出了周边固定或周边弹性支承膜的离散系统的模态反问题以及周边固定和周边弹性支承膜的离散系统的频率反问题。借助Jacobi矩阵反问题的已有成果,成功地求解了上述两个新的反问题。最后给出了反问题的三个计算实例,验证了反问题提法和解法的正确性。

An inverse problems in vibration of a membrane was studied here. Firstly, using the second-order celltet difference scheme, the difference discrete model for axial symmetric vibration of a circular membrane with arbitrary supports was established. Secondly, it was presented that this model belongs to a positive Jacobi system and som~： qualitative properties of the axial symmetric vibration of the system are obtained. Furthermore, an inverse frequency problem and an inverse mode problem of the circular membrane with fixed or spring supported boundary were put forward. The two inverse problems were solved successfully with the aid of the results form the inverse problem of Jacobi matrix. Finally, three numerical examples were given. The correctness of the formulation and the solving method was verified.