Preface
Contents
1 Introduction
1.1 A Brief Historical Perspective
1.2 Importance of Vibrations
1.3 Analysis of Vibrating Systems
1.4 About the Book
Reference
2 Spring-Mass Systems and Inerters
2.1 Introduction
2.2 Some Preliminaries
2.2.1 Single Degree-of-Freedom Systems
2.2.2 General Solution for Harmonically Varying Forcing and Base Excitation
2.2.3 Structural Damping
2.2.4 Response in the Time Domain
2.2.5 Maxwell Model
2.2.6 Vibration Isolation: Quasi-zero Stiffness (QZS) Systems and Bio-inspired Designs
2.3 Two Degree-of-Freedom Systems
2.3.1 Introduction
2.3.2 Harmonic Excitation: Natural Frequencies and Frequency–Response Functions
2.3.3 Vibration Absorber
2.3.4 Time-Domain Response: An Example
2.3.5 Pendulum Systems
2.4 Inerters
2.4.1 Introduction
2.4.2 Two Degree-of-Freedom Systems: Vibration Absorbers with Inerters
References
3 Thin Beams: Natural Frequencies and Mode Shapes
3.1 Introduction
3.2 Derivation of Governing Equation and Boundary Conditions
3.2.1 Introduction
3.2.2 Contributions to the Total Energy
3.2.3 Governing Equation
3.2.4 Boundary Conditions
3.2.5 Static Equilibrium Position
3.2.6 Non Dimensional Form of the Governing Equation and Boundary Conditions
3.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section and with In-Span Attachments
3.3.1 Introduction
3.3.2 Solution for Very General Boundary Conditions
3.3.3 General Solution in the Absence of an Axial Force and an Elastic Foundation
3.3.4 Numerical Results
3.4 Beams with Tuned Mass Dampers
3.4.1 Spring-Mass Single Degree-of-Freedom Systems
3.4.2 Pendulum Single Degree-of-Freedom Systems
3.4.3 Two Degrees-of-Freedom System with Translation and Rotation
3.4.4 Inerter-Based Vibration Absorber
3.5 Effects of an Axial Force and an Elastic Foundation on the Natural Frequency
3.6 Beams with a Rigid Extended Mass
3.6.1 Introduction
3.6.2 Cantilever Beam with a Rigid Extended Mass at Its Free End
3.6.3 Beam with an In-Span Rigid Extended Mass
3.7 Beams with Variable Cross Section
3.7.1 Introduction
3.7.2 Continuously Changing Cross Section
3.7.3 Linear Taper
3.7.4 Exponential Taper
3.7.5 Constant Cross Section with a Step Change in Properties
3.7.6 Stepped Beam with an In-Span Rigid Support
3.7.7 Pre-twisted Beams
3.8 Elastically Connected Beams
3.8.1 Introduction
3.8.2 Beams Connected by a Continuous Elastic Spring
3.8.3 Beams with a Concentrated Mass Connected by an Elastic Spring
References
4 Thin Beams: Applications
4.1 Forced Excitation
4.1.1 Boundary Conditions and the Generation of Orthogonal Functions
4.1.2 General Solution
4.1.3 Impulse Response
4.1.4 Traveling Force on a Beam
4.1.5 Time-Dependent Boundary Excitation of a Cantilever Beam with an In-Span Mass
4.1.6 Time-Dependent Response of a Base-Excited Cantilever Beam with a Tuned Mass Damper (TMD) at Its Free End
4.1.7 Forced Harmonic Oscillations
4.1.8 Harmonic Boundary Excitation
4.1.9 Harmonic Boundary Excitation of a Cantilever Beam with an PVID at Its Free End
4.2 Moving Mass on a Beam
4.3 Coupled Bending and Torsion
4.3.1 Introduction
4.3.2 Natural Frequencies
4.3.3 Flutter
4.4 Beam-Fluid Interactions
4.5 Beams Conveying Fluids
4.6 Layered Beam Piezoelectric Energy Harvester
4.6.1 Governing Equation and Boundary Conditions
4.6.2 Power from the Harmonic Oscillations of a Base-Excited Cantilever Beam Energy Harvester
4.7 Finite Metamaterial Beams: An Introduction
References
5 Timoshenko Beams
5.1 Introduction
5.2 Derivation of the Governing Equations and Boundary Conditions
5.2.1 Introduction
5.2.2 Contributions to the Total Energy
5.2.3 Governing Equations
5.2.4 Boundary Conditions
5.2.5 Non Dimensional Form of the Governing Equations and Boundary Conditions
5.2.6 Reduction of the Timoshenko Equations to that of Euler–Bernoulli
5.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section, Elastic Foundation, Axial Force, and In-Span Attachments
5.3.1 Introduction
5.3.2 Solution for Very General Boundary Conditions
5.3.3 Special Cases
5.3.4 Numerical Results
5.4 Natural Frequencies of Beams with Variable Cross Section
5.4.1 Beams with a Continuous Taper: Rayleigh–Ritz Method
5.4.2 Constant Cross Section with a Step Change in Properties
5.4.3 Numerical Results
5.5 Beams Connected by a Continuous Elastic Spring
5.6 Forced Excitation
5.6.1 Boundary Conditions and the Generation of Orthogonal Functions
5.6.2 General Solution
5.6.3 Impulse Response
5.6.4 Traveling Force on a Beam
5.7 Moving Mass on a Beam
5.8 Beams with Functionally Graded Materials
Appendix 5.1 Definitions of the Solution Functions fl and gl and Their Derivatives
Appendix 5.2 Definitions of Solution Functions fli and gli and Their Derivatives
References
6 Thin Plates
6.1 Introduction
6.2 Derivation of Governing Equation and Boundary Conditions: Rectangular Plates
6.2.1 Introduction
6.2.2 Contributions to the Total Energy
6.2.3 Governing Equations
6.2.4 Boundary Conditions
6.2.5 Non Dimensional Form of the Governing Equation and Boundary Conditions
6.3 Governing Equations and Boundary Conditions: Circular Plates
6.4 Natural Frequencies and Mode Shapes of Circular Plates for Very General Boundary Conditions
6.4.1 Introduction
6.4.2 Natural Frequencies and Mode Shapes of Annular and Solid Circular Plates
6.4.3 Numerical Results
6.5 Natural Frequencies and Mode Shapes of Rectangular Plates
6.5.1 Introduction
6.5.2 Natural Frequencies and Mode Shapes of Rectangular Plates Hinged on All Four Edges
6.5.3 Natural Frequencies and Mode Shapes of Rectangular Plates Hinged on Two Opposite Edges
6.5.4 Rectangular Plate Hinged on All Four Edges and Carrying a Concentrated Mass
6.5.5 Natural Frequencies and Mode Shapes of Rectangular Plates with Other Combinations of Boundary Conditions
6.5.6 Numerical Results for Sect. 6.5.5
6.6 Forced Excitation of Circular Plates
6.6.1 General Solution to the Forced Excitation of Circular Plates
6.6.2 Impulse Response of a Solid Circular Plate
6.7 Solid Circular Plate with Concentrated Mass Revisited
6.8 Fluid Loaded Solid Circular Plate
Appendix 6.1 Elements of the Matrices in Eq. (6.143)
References
7 Mindlin-Reissner Plates
7.1 Derivation of Governing Equations and Boundary Conditions: Rectangular Plates
7.1.1 Introduction
7.1.2 Contributions to the Total Energy
7.1.3 Governing Equations
7.1.4 Boundary Conditions
7.1.5 Non Dimensional Form of the Governing Equations and Boundary Conditions
7.2 Natural Frequencies and Mode Shapes of Rectangular Plates
7.2.1 General Solution
7.2.2 Plate Hinged on All Four Edges
7.3 Forced Vibrations of a Rectangular Plate Hinged on All Four Edges
7.4 Natural Frequencies and Mode Shapes of Circular Plates
7.4.1 General Solution
7.4.2 Natural Frequencies of a Clamped Solid Circular Plate
References
8 Cylindrical Shells
8.1 Introduction
8.2 Derivation of Governing Equations and Boundary Conditions: Flügge’s Theory
8.2.1 Introduction
8.2.2 Contributions to the Total Energy
8.2.3 Governing Equations
8.2.4 Boundary Conditions
8.2.5 Boundary Conditions and the Generation of Orthogonal Functions
8.3 Derivation of Governing Equations and Boundary Conditions: Donnell’s Theory
8.3.1 Introduction
8.3.2 Contributions to the Total Energy
8.3.3 Governing Equations
8.3.4 Boundary Conditions
8.4 Natural Frequencies of Clamped, Hinged, and Cantilever Shells
8.4.1 Rayleigh–Ritz Solution
8.4.2 Numerical Results
8.5 Acoustic Radiation from an Infinitely Long Cylindrical Shell in an Infinite Fluid Medium
8.6 Blood Flow in Arteries: A Simple Model
8.6.1 Governing Equations
8.6.2 Solution of the Coupled System
8.6.3 Long Wavelength Approximation
References
Appendix A Strain Energy in Linear Elastic Bodies
Stress–Strain Relations
Appendix B Some Results from Variational Calculus
B.1 System with One Dependent Variable
B.2 System with N Dependent Variables
B.3 Orthogonal Functions
B.3.1 Systems with One Dependent Variable
B.3.2 Systems with N Dependent Variables
B.4 Application to Specific Elastic Systems
B.4.1 Lagrange Equation
B.4.2 One Dependent Variable
B.4.3 Two Dependent Variables
B.4.4 Three Dependent Variables
Appendix C Laplace Transforms and the Solutions to Ordinary Differential Equations
C.1 Definition of the Laplace Transform
C.2 Solution to a Second-Order Equation
C.3 Solution to a Fourth-Order Equation
C.4 Table of Laplace Transform Pairs
Index