Abstract: 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.

Key words: vibration beam, pentadiagonal matrix, inverse modal problem, engineering example