【作者】 田霞；

【导师】 戴华；

【作者基本信息】 南京航空航天大学 ， 一般力学与力学基础， 2005， 博士

【摘要】 本文研究了弹簧—质点系统、杆的离散模型、杆的有限元模型、梁的离散模型以及梁的连续模型的模态反问题,全文主要包括以下内容:研究了弹簧—质点系统的模态反问题。考虑简单连接的弹簧—质点系统,假定在原系统的末端附加质量和/或弹簧成为修改系统。已知简单连接弹簧—质点系统及修改系统的各一个特征对、附加质量和附加弹簧的刚度。考虑由这两组特征对、附加质量和/或附加弹簧的刚度构造弹簧—质点系统的三类逆模态问题,将问题转化为Jacobi矩阵特征值反问题。给出由已知数据来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,则可唯一地构造弹簧—质点系统;考虑弹簧—质点系统的部分质量通过弹簧接地的情形,已知两个特征对以及接地的质量,构造部分质量接地的弹簧—质点系统的物理参数,给出由这些数据构造具有正质量和正刚度的真实的物理系统的充分必要条件。研究了杆离散系统的振动反问题。假定杆沿轴向与弹性基础相连,已知杆一端固定、另一端自由时的频率、杆两端固定时的频率、固定—自由杆对应于最低频率的模态以及杆的总质量,考虑由给定的两组频率、一个模态和系统的总质量构造杆的离散系统的参数。将问题转化为Jacobi矩阵的特征值反问题,给出由已知数据构造具有正质量和正刚度的可实现物理系统的充分必要条件,并且证明如果这些条件得到满足,则可唯一地构造杆离散系统。同时提出了一类Jacobi矩阵的广义特征值反问题。给定一个实向量和两列实数,求Jacobi矩阵和正定对角矩阵所组成的矩阵束,使得这两列实数分别为矩阵束和该矩阵束的最大顺序主子阵的特征值、实向量为矩阵束对应于某个特征值的特征向量。给出了问题有解的条件及求解问题的数值方法。研究了杆有限元模型的模态反问题。提出了基于线性元的杆有限元模型的两类模态反问题。一类是假设系统的弹性模量为已知常量,横截面积及密度皆为非均匀的,由系统的两个特征值和相应的特征向量及系统的总质量构造固定—自由杆以及两端固定杆的物理参数。另一类问题是假设系统的弹性更多还原

【Abstract】 This dissertation studies the inverse mode problems for the spring-mass system, the discrete model of the rod, the finite element model of the rod, the discrete model and the continuous model of the beam. The main contribution is as follows.The inverse mode problems of the spring-mass system are studied. The simply connected spring-mass system and the modified systems (a simple oscillator of mass and/or spring is attached to one end of the system) are considered. Given two eigenpairs of a simply connected spring-mass system and the modified system respectively, one additional mass and/or additional spring, three classes of the inverse mode problems constructing the spring-mass systems are presented. The problems are transferred into inverse eigenvalue problems for Jacobi matrices. The necessary and sufficient conditions for the reconstruction of a physical realizable system with positive mass and stiffness elements from the known eigenpairs, the additional mass and /or stiffness are established. If these conditions are satisfied, the simply connected spring-mass system may be constructed uniquely. An algorithm is presented for constructing the physical realizable system from the data. The inverse mode problem of the spring-mass system with the partial masses connected to the ground by the spring is considered. Given two eigenpairs of the system and the masses connected to the ground, construct the physical parameters of the spring-mass system. The necessary and sufficient conditions for the construction of a physical realizable system with positive mass and stiffness elements are given.An inverse vibration problem for the discrete model of a vibrating rod is considered. Suppose that the rod is connected to the elastic basic along its axis. Given the frequencies of the axial vibrating rod, fixed at one end and free at the other, and the frequencies of the rod, fixed at two ends, and a mode corresponding to the lowest frequency of the fixed-free rod and the total mass of the rod, the更多还原