It has been observed that finite element or finite difference model of order can approximate with fair accuracy less than one-third of the eigenvalues of the underlying continuous system corresponding to the low spectrum. The new discrete model, namely the spectral conforming model, is developed to predict the eigenvalues of continuous systems in both the low and high spectra.
The spectral conforming models are developed using an inverse vibration method. The classical vibration problem is the direct method. In the direct approach, the characteristic behaviors of the system, i.e. frequency response, natural frequency, and steady state response are analyzed and predicted from known physical parameters such as the geometry information and material properties. In a special case of simple geometry and constant material properties of the continuous system, the exact solution can be obtained analytically. In the inverse problem, the systems are reconstructed formulating the stiffness and mass matrices from known behaviors of vibration. In this dissertation, two types of the spectral conforming models are developed using an inverse method that fits the frequency responses of the discrete system to those of the continuous system. Since the eigenvalues alone cannot determine the discrete system uniquely, the necessary requirements divide the models into two categories. The spectral conforming model in chapter 3 adopts the fundamental inverse eigenvalue problems for reconstructing the chain of a mass-spring system with a prescribed spectra based on the element of a fixed-free uniform rod. Another spectral conforming model named the persymmetric model is developed in chapter 4. In the development of the persymmetric model, the parametric stiffness and mass matrices are formulated with prescribed spectra based on a free-free uniform rod. Asymptotic analysis and other useful methods for describing the behaviors of the continuous system are employed for the development of the persymmetric models.
Several applications are examined showing the advantages of the newly developed discrete models. From the speculation of the applications and their results, we may conclude that the use of the spectral conforming model is very practical in most dynamic problems such as the simulation and control of a continuous system. If the problems involve high frequency excitation, it is highly recommended to use the spectral conforming model.