Adaptive Analysis Of Free Vibration Of Planar Non-uniform Members Based On EEP Super-convergent Method

It is of great significance to make efficient and accurate analysis to the free vibration of non-uniform members, which have wide applications in many fields of engineering. With the development of numerical computation, adaptive analysis becomes the goal of various numerical methods. Based on the condensed scheme of Element Energy Projection (EEP) method with optimal super-convergent order, the dissertation proposed an adaptive solution for the free vibration of non-uniform members. The above method is further applied to general 2nd and 4th order Sturm- Liouville eigenvalue problems successfully.Taking the axial free vibration of non-uniform members as the model problem, the dissertation first presented the adaptive analysis, which is used in one-dimensional C 0 problem based on EEP condensed scheme, to get dynamic shape functions of elements. The above analysis is used to check the feasibility of the method in free vibration problems. Based on the in-depth understanding of equivalence between dynamic stiffness method (DSM) and the exact finite element method (FEM), the dissertation carried out the following work:Firstly, the self-adaptive method based on EEP condensed scheme and the Wittrick-Williams algorithm were creatively combined. The dissertation proposed a generalized self-adaptive method, implementation strategy and computational algorithm in dynamic analysis of non-uniform members. Then the method was applied to solve axial free vibration of non-uniform members and flexural free vibration of non-uniform Bernoulli-Euler beams.Secondly, the governing differential equations of free vibration of non-uniform Timoshenko curved beams was derived. The self-adaptive method to solve the flexural free vibration of non-uniform Timoshenko beams and the free vibration of non-uniform Timoshenko curved beams, which is a kind of eigenvalue problems for ordinary differential equations, was also constructed respectively.Finally, through the analysis of mechanics models of the 2nd and 4th SL problems, the dissertation proposed a way to deal with boundary conditions and negative eigenvalues and extended the self-adaptive method of free vibration to solve the 2nd and 4th SL problems.Computer programs have been developed for self-adaptive analysis, and representative numerical examples for various problems were analyzed. The theoretical research and numerical tests show that the method is independent of the initial mesh provided by users, not only can the program produce the frequency (eigenvalues) and vibration mode (eigenfucntion) which are satisfied the error tolerance automatically, but also the displacements and the internal forces of any point gain the equivalent numerical precision with the displacements at the element nodes. With this method, most shortcomings in other existing methods are overcome. A large number of numerical tests prove this algorithm to be accurate, stable, efficient and convenient. The concept of the adaptive method in this dissertation provides a new way to solve the free vibration of skeletal structures constructed by non-uniform members and eigenvalue problems for ordinary differential equations.