1 Start of Research-Oriented Study 1.1 Needs for Research-Oriented Study 1.2 Case Studies and Associated Problems 1.2.1 Preliminary Study of a Tethered Satellite 1.2.2 Design of a Hydro-Elastic Vibration Isolation System 1.2.3 Two Kinds of Immovable Points in a Vibration System 1.2.4 Identical Natural Frequencies of Different Structures 1.2.5 Closely Distributed Natural Modes of a Symmetric Structure 1.2.6 Transient Response of a Slender Structure 1.3 Scope and Style of the Book 1.4 Further Reading and Thinking 2 Preparation of Research-Oriented Study 2.1 Briefs of Beauty of Science 2.2 Beautiful Features of Vibration Mechanics 2.2.1 Unity 2.2.2 Simplicity 2.2.3 Regularity 2.2.4 Symmetry 2.2.5 Singularity 2.3 Enlightenment of Beauty of Vibration Mechanics 2.3.1 Methods of Thinking 2.3.2 Aesthetic Literacy 2.4 Concluding Remarks 2.5 Further Reading and Thinking 3 Models of Vibration Systems 3.1 Continuous Systems and Their Discrete Models 3.1.1 Dynamic Models of Continuous Systems 3.1.2 Preconditions of Discretization 3.1.3 Case Studies of Discretization 3.1.4 Concluding Remarks 3.2 A Half Degree of Freedom of Discrete Systems 3.2.1 Degeneration of a Degree of Freedom 3.2.2 Conventional Concept of Degree of Freedom 3.2.3 Degree of Freedom Based on Accessible Manifolds 3.2.4 Demonstrative Systems with a Non-holonomic Constraint 3.2.5 Demonstrative Systems with Two Non-holonomic Constraints 3.2.6 Concluding Remarks 3.3 Structural Damping 3.3.1 Frequency-Invariant Damping and Its Limitations 3.3.2 Frequency-Variant Damping Model and System Response 3.3.3 An Approximate Viscous Damping Model 3.3.4 An Approximate Viscoelastic Damping Model 3.3.5 Concluding Remarks 3.4 Further Reading and Thinking 4 Vibration Analysis of Discrete Systems 4.1 Vibration Systems with Non-holonomic Constraints 4.1.1 Dynamic Analysis in Time Domain 4.1.2 Dynamic Analysis in Frequency Domain 4.1.3 Concluding Remarks 4.2 Node Number of a Natural Mode Shape 4.2.1 Reexamination of Current Results 4.2.2 Rules of Node Numbers of 2-DoF Systems 4.2.3 Design Feasibility of Nodes in a Mode Shape 4.2.4 Concluding Remarks 4.3 Anti-resonances of a Harmonically Excited System 4.3.1 Anti-resonances of 2-DoF Systems 4.3.2 Mechanisms Behind Two Kinds of Anti-resonances 4.3.3 Design Feasibility of Anti-resonances 4.3.4 Concluding Remarks 4.4 Dynamic Modifications of a System 4.4.1 Frequency Response of a Composite System 4.4.2 Adjusting an Anti-resonance of a Primary System 4.4.3 Adjusting Resonances of a Primary System 4.4.4 Concluding Remarks 4.5 Further Reading and Thinking 5 Natural Vibrations of One-Dimensional Structures 5.1 Natural Vibrations of a Tether Pendulum 5.1.1 Dynamic Equation of a Tether Pendulum 5.1.2 Analysis of Natural Vibrations 5.1.3 Comparison Between Continuous Model and Discrete Models 5.1.4 Concluding Remarks 5.2 Duality Analysis of Rods in Natural Vibrations 5.2.1 A Dual of Different Cross-Sections 5.2.2 A Dual of Identical Cross-Sections 5.2.3 A Dual of Two Uniform Rods 5.2.4 A Dual of Two Rods with Axially Varying Material Properties 5.2.5 Concluding Remarks 5.3 Duality Analysis of Beams in Natural Vibrations 5.3.1 A Dual of Different Cross-Sections 5.3.2 A Dual of Identical Cross-Sections 5.3.3 A Dual of Two Uniform Beams 5.3.4 Concluding Remarks 5.4 Further Reading and Thinking 6 Natural Vibrations of Symmetric Structures 6.1 Natural Vibrations of Mirror-Symmetric Structures 6.1.1 Decoupling of Mirror-Symmetric Structures 6.1.2 Free Vibrations of Thin Rectangular Plates 6.1.3 Repeated Natural Freque