Development of Vibration and Sensitivity Analysis Capability Using the Theory of Structural Variations

In the author\u27s previous work entitled General Theorems of Topological Variations of Elastic Structures and the Method of Topological Variation, 1985, some interesting properties of skeletal structures have been discovered. These properties have been described as five theorems and synthesized as a theory, called the theory of structural variations (TSV). Based upon this theory, an innovative analysis tool, called the structural variation method (SVM), has been derived for static analysis of skeletal structures (one-dimensional finite element systems). The objective of this dissertation research is to extend TSV and SVM from one-dimensional finite element systems to multi-dimensional ones and from statics to vibration and sensitivity analysis. Meanwhile, four new interesting and useful properties of finite element systems are also revealed. One of them is stated as the Gradient Orthogonality Theorem of Basic Displacements, based upon which a set of explicit formulations are derived for design sensitivities of displacements, internal forces, stresses and even the inverse of the global stiffness matrix of a statically loaded structure. The other three new properties are described as the Evaluation Theorem of Principal Z-Deformations, the Monotonousness Theorem of Principal Z-Deformations and the Equivalence Theorem of Basic Displacement Vectors and Eigenvectors, based upon which a new approach, called the Z-deformation method, is developed for vibration analysis of finite element systems. This method is superior to the commonly used inverse power iteration method when adjacent eigenvalues are close. Explicit formulations for eigenpair sensitivities are also derived in accordance with the Z-deformation method. The distinct feature of TSV and SVM is that the analysis results for a loaded structure can be obtained without any matrix assembling and inverse operations. This feature gives TSV and SVM an edge over the traditional finite element analysis in many engineering applications, where the repeated analysis is required, such as structural optimization, reliability analysis, elastic-plastic analysis, vibration, contact problems, crack propagation in solids