Inverse Eigenvalue Problems In Structural Dynamics

This dissertation studies the inverse eigenvalue problems in structural dynamics, including vibration inverse problems for the spring-mass system, the discrete beam, damped systems and the damage detection in rods. The main contribution is as follows.First of all, the inverse eigenvalue problems of the spring-mass systems are studied. The simply connected spring-mass system of two freedoms and the modified system with a simple oscillator of mass or spring attached to one end of the system are considered. For simply connected spring-mass system of n freedoms, the necessary and sufficient conditions for the reconstruction of a physical realizable system from the known four and five eigenpairs are established. Also, the inverse eigenvalue problems of the hybrid connected spring-mass system are considered. The necessary and sufficient conditions for the solvability of the problems are obtained. Numerical methods and numerical experiments are given.Secondly,an inverse vibration problem for the discrete beam is considered. Given the three frequencies and corresponding modes of the axial vibrating beam, the problem of constructing the structural physical parameters of the discrete model of the beam from the known data is considered. The problem is transferred into inverse eigenvalue problems for real symmetric pentadiagonal matrices. The necessary and sufficient conditions for the solvability of the problem are obtained. Numerical methods and numerical experiments are presented.Afterwards, the inverse quadratic eigenvalue problems in damped vibration system are studied. These problems include the construction of the stiffness matrix and damped matrix of the damped vibration systems from the full frequencies and corresponding modes, the construction of the stiffness matrix and damped matrix of damped vibration systems with proportional damping from the some frequencies and corresponding modes, the construction of the stiffness matrix and damped matrix of the proportional damped vibration systems from the two frequencies and corresponding modes, and the construction of the stiffness matrix and damped matrix of the non-proportional damped vibration systems from the frequencies data. The solvability of the problems is established. The numerical methos and numerical experiments are given.At last, an inverse eigenvalue procedure for damage detection in homogeneous vibration rods is studied. It is shown that a finite element model based on the geometric parameters of the rod can be reconstructed from two eigenpairs, respectively corresponding to the undamaged state and the damaged state. An inverse eigenvalue produre for damage detection of rods is established. The numerical methos and numerical experiments are given.