振动梁离散模型的一类模态反问题

A class of modal inverse problems for discrete models of vibration beams

讨论了一类对系统的刚度主子阵存在一定约束的条件下,部分或缺损的模态信息构造振动梁模型刚度矩阵的模态反问题.由特征信息得到矩阵方程组,根据给定的频率数据是否属于系统,分三种情形求解.分别利用牛顿-康托洛维奇定理和线性方程组的基本理论,得到了模态反问题存在唯一解的充要条件,并给出了解的显式表达式和数值算法;证明了对应算法的收敛性和数值稳定性分析,结合工程实例,通过数值实验表明了理论的正确性和算法的有效性,并分析说明了计算过程是数值稳定的.

The modal inverse problem of constructing the stiffness matrix of a vibrating beam model from partial or missing modal information under certain constraints on the stiffness main submatrix of the system is discussed in this paper.The matrix equations are obtained from the characteristic information and are solved in three cases according to whether the given frequency data belongs to the system.Using the Newton-Kantorovich theorem and the basic theory of linear equations,the necessary and sufficient conditions for the existence of a unique solution to the inverse modal problem are obtained.The explicit expression and numerical algorithm of the solution are given.The convergence and numerical stability analyses of the corresponding algorithm are presented and proven.Combined with an engineering example,numerical experiments indicate the correctness of the theory and the effectiveness of the algorithm.Additionally,the computational process is shown to be numerically stable through further analysis.