This book is a novel tutorial for research-oriented study of vibration mechanics. The book begins with twelve open problems from six case studies of vibration mechanics in order to guide readers in studying the entire book. Then, the book surveys both theories and methods of linear vibrations in an elementary course from a new perspective of aesthetics of science so as to assist readers to upgrade their way of learning. The successive chapters offer a theoretical frame of linear vibrations and waves, covering the models of vibration systems, the vibration analysis of discrete systems, the natural vibrations of one-dimensional structures, the natural vibrations of symmetric structures, and the waves and vibrations of one-dimensional structures. The chapters help readers solve the twelve open problems step by step during the research-oriented study.
The book tries to arouse the interest of graduate students and professionals, who have learnt an elementary course of vibration mechanics of two credits, to conduct the research-oriented study and achieve a helical upgrade understanding to vibration mechanics.
Author(s): Haiyan Hu
Publisher: Springer
Year: 2022
Language: English
Pages: 503
City: Singapore
Preface
Contents
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 Enlightenments 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 Frequencies of a Thin Rectangular Plate
6.1.4 Close Natural Frequencies of a Thin Rectangular Plate
6.1.5 Concluding Remarks
6.2 Vibration Computations of Cyclosymmetric Structures
6.2.1 Decoupling a Cyclosymmetric Structure Without a Central Axis
6.2.2 Decoupling a Cyclosymmetric Structure with a Central Axis
6.2.3 High-Efficient Computation Based on Modal Reduction
6.2.4 Concluding Remarks
6.3 Modal Properties of Cyclosymmetric Structures
6.3.1 Modal Properties in Representative Subspaces of Cyclic Group
6.3.2 Modal Properties in Physical Space
6.3.3 Central Displacements in Mode Shapes
6.3.4 Orthogonality of Mode Shapes with Repeated Frequencies
6.3.5 Modal Test of a Bladed Disc Model
6.3.6 Concluding Remarks
6.4 Further Reading and Thinking
7 Waves and Vibrations of One-Dimensional Structures
7.1 Non-dispersive Waves of Rods
7.1.1 Wave Analysis of an Infinitely Long Rod
7.1.2 Complex Function Analysis of Waves of an Infinitely Long Rod
7.1.3 Harmonic Wave Analysis of a Finitely Long Rod
7.1.4 Harmonic Vibration Analysis of a Finitely Long Rod
7.1.5 Concluding Remarks
7.2 Dispersive Waves of Rods
7.2.1 Dispersive Waves in a Rod Due to Non-uniform Cross-section
7.2.2 Dispersive Waves of a Rod Due to Transverse Inertia
7.2.3 Dispersive Waves of a Rod Due to an Elastic Boundary
7.2.4 Concluding Remarks
7.3 Waves in a Rod Impacting a Rigid Wall
7.3.1 Wave Analysis of an Impacting Uniform Rod
7.3.2 Modal Analysis of an Impacting Uniform Rod
7.3.3 Influence of Non-uniform Cross-section
7.3.4 Influence of Transverse Inertia
7.3.5 Criteria of Modal Truncation
7.3.6 Concluding Remarks
7.4 Free Waves and Vibrations of Beams
7.4.1 Free Waves of an Euler–Bernoulli Beam
7.4.2 Free Waves of a Timoshenko Beam
7.4.3 Natural Vibrations of a Timoshenko Beam
7.4.4 Concluding Remarks
7.5 Forced Waves and Vibrations of Beams
7.5.1 Waves of an Infinitely Long Beam to a Transverse Harmonic Force
7.5.2 Vibrations of a Finitely Long Beam to a Transverse Harmonic Force
7.5.3 Impact Response of a Clamped-Free Beam
7.5.4 Concluding Remarks
7.6 Further Reading and Thinking
Appendix Three-Dimensional Waves in an Elastic Medium
A.1 Description of Three-Dimensional Waves
A.1.1 Elasto-Dynamic Equations of a Three-Dimensional Medium
A.1.1.1 Basic Description
A.1.1.2 Navier’s Equations
A.1.2 Helmholtz Decomposition of a Displacement Field
A.1.2.1 Two Special Cases
A.1.2.2 Decomposition in a General Case
A.1.3 Solutions of Three-Dimensional Wave Equations
A.2 Two Kinds of Simple Waves
A.2.1 Planar Waves
A.2.1.1 Planar Waves Traveling in a Given Direction
A.2.1.2 Planar Waves Traveling in an Arbitrary Direction in a Plane
A.2.1.3 Inhomogeneous Planar Waves
A.2.2 Spherical Waves
A.3 Wave Reflections at Boundary of a Semi-infinite Medium
A.3.1 Reflections of a P-wave and an SV-Wave
A.3.1.1 An Incident P-wave Reflected at a Clamped Boundary
A.3.1.2 An Incident P-wave Reflected at a Free Boundary
A.3.1.3 An Incident SV-Wave Reflected at a Clamped Boundary
A.3.1.4 An Incident SV-Wave Reflected at a Free Boundary
A.3.2 Reflection of an SH-Wave
A.3.2.1 An Incident SH-Wave Reflected at a Clamped Boundary
A.3.2.2 An Incident SH-Wave Reflected at a Free Boundary
A.3.3 Rayleigh Surface Waves
Bibliography
Index
