1 Overview 1.1 Brief History of the Development of the Qualitative Theory in Structural Mechanics 1.2 Areas Covered in the Study of the Qualitative Theory in Structural Mechanics 1.3 Key Theoretical Results and the Approach to Establish Them 1.4 Theoretical and Practical Significance of the Qualitative Theory in Structural Mechanics 1.5 Overview of Qualitative Properties of Vibration Associated with Bars 1.6 Overview of Qualitative Properties of Vibration Associated with Beams 1.7 Overview of Qualitative Properties of Vibration and Static Deformation Associated with Repetitive Structures 1.8 Overview of the Existence of Solutions in Elasticity and Structural Theories as Well as Other Subjects in Fundamental Theories References 2 Oscillatory Matrices and Kernels as Well as Properties of Eigenpairs 2.1 A Few Notations and Definitions 2.2 Some Relationships Among Minors 2.3 Jacobian Matrix 2.4 Oscillatory Matrices 2.5 Perron's Theorem and Compound Matrices 2.6 Eigenpairs of an Oscillatory Matrix 2.7 Integral Equations with Symmetric Kernels and the Concept of Oscillatory Kernels 2.8 Perron's Theorem for Integral Equations and the Concept of Compound Kernels 2.9 Eigenpairs of the Integral Equation with an Oscillatory Kernel 2.10 Relationship Among Oscillatory Properties in Static Deformation, Flexibility Functions (Matrices) Being Oscillatory Kernels (Matrices), and Oscillatory Properties in Vibration 2.11 From Oscillatory Matrices to Oscillatory Kernels References 3 Qualitative Properties of Vibration and Static Deformation Associated with Discrete Systems of Strings and Bars 3.1 Discrete Systems of Strings and Bars 3.2 Basic Qualitative Properties of Vibration and Static Deformation of Spring-Mass Systems 3.3 Necessary and Sufficient Conditions for Mode Shapes of a Spring-Mass System 3.4 Modal Qualitative Properties of the Finite Difference System of a Bar 3.5 Modal Qualitative Properties of the Finite Element System of a Bar 3.6 Modal Qualitative Properties of the System Consisting of a Massless Elastic Bar and a Number of Concentrated Masses
3.7 Modal Qualitative Properties of Discrete Systems of Strings and Bars on Elastic Foundations References Qualitative Properties of Vibration and Static Deformation Associated with Discrete Systems of Beams 4.1 The Finite Difference Model of a Beam and the Corresponding Physical Model 4.2 Qualitative Properties of Vibration and Static Deformation Associated with Finite Difference Systems of Well-Constrained Beams 4.3 Modal Qualitative Properties of the Finite Difference System of an Under-Constrained Beam 4.4 Numbers of Sign Reversals of Various Mode Shapes Associated with the Finite Difference Systems of Beams 4.5 Construction of the Finite Difference System of a Beam Using Modes 4.6 Interlacement of Natural Frequencies of the Finite Difference System Modeling a Beam Subject to Various Boundary Constraints 4.7 Oscillatory Properties of Finite Element Systems of a Beam... 4.8 Modal Qualitative Properties of Discrete Systems of Multi-Span Beams 4.9 Modal Qualitative Properties of Discrete Systems of Overhang Beams References 5 Qualitative Properties of Vibration and Static Deformation of the Sturm-Liouville System 5.1 Natural Vibration of the Sturm-Liouville System 5.2 Green's Function of the Sturm-Liouville System 5.3 Oscillatory Properties in Vibration and Static Deformation of the Sturm-Liouville System 5.4 Number of Independent Modes of a Bar and Additional Properties of Its Modal Shapes 5.5 Interlacement of Natural Frequencies of a Bar Subject to Various Boundary Constraints 5.6 Comparison Between Discrete and Continuous Systems References 6 Qualitative Properties of Vibration and Static Deformation Associated with Continuous Systems of Beams 6.1 Differential Equation and Boundary Constraints of a Vibrating Beam 6.2 Green's Function of a Beam 6.3 Oscillatory Properties in Static Deformation and Vibration of a Well-Constrained Beam 6.4 Oscillatory Properties in Vibration of an Under-Constrained Beam 6.5 Number of Independent Modes of a Beam 6.6 Additional Properties of Natural Frequencies of a Beam 6.7 Modal Qualitative Properties of the Continuous System of an Overhang Beam
References 7 Qualitative Properties of Vibration and Static Deformation of Repetitive Structures 7.1 Modal Qualitative Properties of Symmetric Structures 7.2 Modal Qualitative Properties of Rotationally Periodic Structures 7.3 Modal Qualitative Properties of Linearly Periodic Structures... 7.4 Modal Qualitative Properties of Chain Structures 7.5 Modal Qualitative Properties of Axisymmetric Structures 7.6 Qualitative Properties of Forced Vibration of Repetitive Structures 7.7 Dimension Reduction in Vibration and Shape Control of Repetitive Structures 7.8 Qualitative Properties in Static Deformation of Repetitive Structures References Theory on the Existence of Solutions in Structural Mechanics 8.1 Introduction 8.2 Variational Solutions for Three Categories of Problems in Structural Theories 8.3 Existence of the Extremum of a Functional 8.4 Existence of Solutions of Static Deformation and Vibrational Modes in Elasticity 8.5 Existence of Solutions of Static Deformation and Vibrational Modes in Structural Theories 8.6 Validity of Models in Structural Theories 8.7 Convergence of the Ritz Method for Problem Solving in Structural Theories References Index
[