Three-Dimensional Vibration Analysis of Structural Elements Using Chebyshev-Ritz Method

This dissertation, "Three-dimensional Vibration Analysis of Structural Elements Using Chebyshev-Ritz Method" by Ding, Zhou, 周叮, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled "Three-dimensional Vibration Analysis of Structural Elements Using Chebyshev-Ritz Method" Submitted by ZHOU Ding for the degree of Doctor of Philosophy at The University of Hong Kong in May 2003 The three-dimensional (3-D) free vibration of structural components with common shapes is analyzed based on the theory of small strain linear elasticity. The Ritz method is applied to derive the eigenfrequency equation. For each displacement component, the kinematically admissible functions are taken as the product of the Chebyshev polynomial series and a boundary characteristic function. The Chebyshev polynomial series forms the main part of the admissible functions while the boundary characteristic function ensures the satisfaction of the geometric boundary conditions. Such a method is called the Chebyshev-Ritz method. The excellent properties of the Chebyshev polynomials in numerical analysis and functional approximation guarantee high accuracy, rapid convergence and numerical robustness. In the analysis, four approaches to the construction of the admissible functions were explored. The Direct-A approach was applied to structures with regular shapes and simple boundary conditions. The Direct-B approach was applied to structures with convex shapes and simple boundary conditions. The Mapping approach, which used the coordinate mapping method, was applied to structures with convex shapes and simple boundary conditions. The R-function approach, the most general method, is suitable for structures with arbitrary shapes and any boundary conditions, and was applied to structures with complex boundary conditions. iThese approaches, if coupled with a suitable coordinate system to describe displacements, stresses and strains in the structure under consideration, allow most common structures to be dealt with. The vibratory characteristics of thirteen kinds of common structures were analyzed in details to illustrate their application. The Direct-A approach was used to study rectangular plates with/without elastic foundation, circular and annular plates with/without elastic foundation, circular and annular cylinders, and solid tori have been studied in Cartesian coordinate system, cylinder coordinate system and toroidal coordinate system respectively. This approach was also coupled with the domain decomposition technique to analyze the built-in circular plates. The Direct-B approach was used to analyze generalized super elliptical plates. The Mapping approach was used to analyze triangular plates/prisms and skew plates/prisms. The R-function method was used to analyze rectangular plates with complex boundary conditions. These approaches obtained results of high accuracy and robustness, indicating that the Chebyshev-Ritz method can be usefully applied in a variety of practical applications. ii DOI: 10.5353/th_b2751120 Subjects: Vibration - Mathematical models Strains and stresses Chebyshev polynomials