简单连接弹簧-质点系统的逆模态问题

Inverse Mode Problems for Simply Connected Spring-Mass Systems

设(λ,x)为简单连接弹簧-质点系统的一个特征对。假定在原系统的末端附加质量m和/或刚度为k的弹簧,(μ,y)为修改系统的特征对。考虑由这两组特征对、附加质量m和/或附加弹簧的刚度k构造弹簧-质点系统的3类逆模态问题。本文将问题转化为Jacobi矩阵特征值反问题,给出由(λ,x) ,(μ,y) ,m和/或k构造具有正质量和正刚度的真实物理系统的充分必要条件。如果这些条件得到满足,则可惟一地构造简单连接弹簧-质点系统,并给出了构造真实物理系统的一个算法。

Let (λ,x) be an eigenpair of a simply connected spring-mass system. Suppose that a simple oscillator of mass m and/or spring k is attached to one end of the system and let (μ,y) be an eigenpair of the modified system. Three classes of the inverse mode problems for constructing the physical elements of the system from (λ,x),(μ,y), m and/or k are considered. The problems are transferred into inverse eigenvalue problems for Jacobi matrices. The necessary and sufficient conditions for the construction of a physical realizable system with positive mass and stiffness elements are established. If these conditions are satisfied, the simply connected spring-mass system may be constructed uniquely. An algorithm is used to construct the physical realizable system from the data.