Solid Acoustic Waves And Vibration: Theory And Applications

Contents
Preface
Chapter 1. Introduction
	1.1 Rayleigh’s Foundation Works
		1.1.1 The frequency equations
		1.1.2 The thin-plate case
		1.1.3 The thick-plate case
	1.2 Lamb’s Contributions
		1.2.1 The period equations
		1.2.2 The long-wave (i.e., thin-plate) case
		1.2.3 The short-wave (i.e., thick-plate) case
	1.3 Viktorov’s Contributions
		1.3.1 Characteristic equations
		1.3.2 The long-wave and short-wave cases
		1.3.3 Results obtained by numerical calculations
		1.3.4 A discussion about normal modes
	1.4 The Author’s Preliminary Works
		1.4.1 A 3D plot representation method
		1.4.2 Zero-order symmetric transitional mode (ZSTM)
		1.4.3 A space operator introduced to the characteristic equations
		1.4.4 Wave in and vibration of finite plates
		1.4.5 Waves in and vibration of a plate with coating
	1.5 The Author’s Further Works
		1.5.1 Canonical characteristic equations
		1.5.2 Analytical models for limiting wave modes (LWMs)
		1.5.3 A representative description of dispersion curves
		1.5.4 Dispersion model and dispersion theorem
		1.5.5 The TDK Equation and Space-time factor
		1.5.6 Electromechanical transducers
		1.5.7 Micro-load theory for ultrasonic mass resonators (UMRs)
Chapter 2. Acoustic Field in Solids
	2.1 Basic Terminology
		2.1.1 Displacement and velocity vector
		2.1.2 Deformation
		2.1.3 Strain and stress
		2.1.4 Young’s modulus
		2.1.5 Poisson’s ratio
	2.2 Stress–Strain and Strain–Stress Equations
	2.3 Motion Equation of a Volume Element in Solid
	2.4 Acoustic Field and three-Dimensional Wave Equation
		2.4.1 Gradient, divergence and curl
		2.4.2 Three-dimensional wave equation of solids
		2.4.3 Infinite solid and infinite plates
		2.4.4 Finite solids and vibration of plates
	2.5 Solid Mediums: Physical Parameters and Wave Velocities
	2.6 Fluid Mediums: Wave Equations and Wave Velocities
		2.6.1 Three-dimensional wave equation of fluids
		2.6.2 Sound velocities in fluid mediums
Chapter 3. Plane Waves in Infinite Plates
	3.1 Symmetric Vibration and Two-Dimensional Wave Equation
	3.2 Characteristic Equations with Free Boundaries
	3.3 Wavenumbers and Wave Vector Analysis
	3.4 Space-Time Variable and Characteristic Parameter
		3.4.1 Two major functions
		3.4.2 A special function
		3.4.3 Characteristic parameter
		3.4.4 Space–time variable
		3.4.5 Space-time flying velocity
		3.4.6 Characteristic value problem
	3.5 Trigonometric, Hyperbolic and Bessel Functions
		3.5.1 Trigonometric and hyperbolic functions
		3.5.2 Bessel functions
Chapter 4. Approaches to Characteristic Values
	4.1 Canonical Characteristic Equations
	4.2 Characteristic Equations for Three Different Velocity Regions
	4.3 Reference System and Structure Frame of Characteristic Values
		4.3.1 Reference Poisson’s ratio and reference system
		4.3.2 Reference points and structure frame
	4.4 Analytical Approach: The Limiting Wave Modes
	4.5 Computational Approach: The 3D Plot Representation
	4.6 Computational Approach: The 2D Plot Representation
Chapter 5. Low-Frequency Wave Modes
	5.1 Characteristic Equations of Low-Frequency Modes
	5.2 Low-Frequency Symmetric Mode (LSM)
		5.2.1 Characteristic values of LSM
		5.2.2 The gold triangle and middle-C
		5.2.3 The range of LSM
		5.2.4 LSW velocity and frequency
	5.3 Low-Frequency Antisymmetric Mode (LAM)
		5.3.1 Characteristic values of LAM
		5.3.2 The range of LAM
		5.3.3 LAW velocities and frequencies
	5.4 Dynamic Characteristics of LSM
		5.4.1 Thickness compression stiffness
		5.4.2 Longitudinal vibration and wave equation of plates
	5.5 Dynamic Characteristics of LAM
		5.5.1 Bending stiffness
		5.5.2 Transverse vibration and wave equation of plates
Chapter 6. High-Frequency Wave Modes
	6.1 Characteristic Equations of High-Frequency Modes
	6.2 High-Frequency Zero-Order Mode (HZM)
		6.2.1 Characteristic values of HZM
		6.2.2 The range of HZM (i.e., SAW mode)
		6.2.3 Wave velocity and frequencies of HZW
	6.3 High-Frequency Non-zero Modes (HNM)
		6.3.1 Characteristic values of HNMs
		6.3.2 The ranges of HNMs
		6.3.3 Typical examples of HNMs
		6.3.4 The frequencies of HNWs
	6.4 Trend Equation and Trend Curves of Characteristic Values
		6.4.1 Trend curves with variable y
		6.4.2 Trend curves with variable ζ
	6.5 Zero-Order and Non-zero Modes
Chapter 7. Transitional Wave Modes
	7.1 Characteristic Equations of Transitional Modes
	7.2 Zero-Order Symmetric Transitional Mode (ZSTM)
		7.2.1 Characteristic equation of ZSTM
		7.2.2 Characteristic values of ZSTM
		7.2.3 An analytical model for ZSTM
	7.3 Non-zero Symmetric Transitional Modes (NSTMs)
		7.3.1 Characteristic equation of NSTMs
		7.3.2 Characteristic values of NSTMs
		7.3.3 An analytical model for NSTMs
	7.4 Non-zero Antisymmetric Transitional Modes (NATMs)
		7.4.1 Characteristic equation of NATMs
		7.4.2 Characteristic values of NATMs
		7.4.3 An analytical model for NATMs
	7.5 Base Axis Mode (BAM) and Base Point of Dispersion Curves
Chapter 8. Resonance Wave Modes
	8.1 Characteristic Equations of Resonance Modes
	8.2 Symmetric Resonance Modes (SRMs)
		8.2.1 Characteristic values of SRMs
		8.2.2 Solutions with ν = νref = 1/3
		8.2.3 Solutions with ν < 1/3 and ν > 1/3
		8.2.4 Numerical examples of solutions
	8.3 Antisymmetric Resonance Modes (ARMs)
		8.3.1 Characteristic values of ARMs
		8.3.2 Solutions with ν = νref = 1/3
		8.3.3 Solutions with ν < 1/3 and ν > 1/3
		8.3.4 Numerical examples of solutions
	8.4 Longitudinal Modes and Transverse Mode
	8.5 Sequence of Non-zero Symmetric Modes (NS-modes)
	8.6 Sequence of Non-zero Antisymmetric Modes (NA-modes)
Chapter 9. Reference System of Dispersion Curves
	9.1 Limiting Wave Modes and Reference Point Coordinates
	9.2 Reference Structure Frame and Reference System
	9.3 Basic Reference System of the Dispersion Curves
	9.4 Fundamental Period and Dual-Period
	9.5 Standard Dispersion Curves and Six-Plus-Five Structure
Chapter 10. Dispersion Curves
	10.1 The 3D Plot Representation of Dispersion Curves
		10.1.1 The 3D plots using variable pairs (x, y)
		10.1.2 The 3D plots using variable pairs (x, ζ)
	10.2 Dispersion Curves of Zero-Order Modes
	10.3 Dispersion Curves of Non-zero Modes
	10.4 A Representative Description of Dispersion Curves
Chapter 11. Dispersion Theorem of Waves
	11.1 A General Dispersion Model and Dispersion Theorem
	11.2 Physical Basis and Design Rule of Solid-State Transducers
	11.3 Mathematical Models for Ultrasonic Resonators
	11.4 Optimal Designs of Ultrasonic Resonators
		11.4.1 Longitudinal and transverse bulk acoustic resonators (BARs)
		11.4.2 LAW delay-line ultrasonic resonators
		11.4.3 SAW delay-line ultrasonic resonators
	11.5 Dynamic Characteristics of Ultrasonic Resonators
Chapter 12. Longitudinal and Transverse Modes
	12.1 Longitudinal Wave and Vibration Modes
	12.2 Longitudinal Modes: Parallel and Circular Plane Waves
	12.3 Transverse Wave and Vibration Modes
	12.4 Transverse Modes: Parallel and Circular Plane Waves
	12.5 Dynamics of Single Mode and Multi-Mode Plates
Chapter 13. Longitudinal Vibrations
	13.1 Longitudinal Vibration and Wave Equation
	13.2 Longitudinal Vibrations of Plates, Posts and 1D Bars
		13.2.1 Symmetric longitudinal vibration (SLVs) of plates
		13.2.2 Longitudinal vibrations of 1D bars
	13.3 Solutions of Longitudinal Vibration Equations
		13.3.1 F–F boundary conditions
		13.3.2 C–C boundary conditions
		13.3.3 C–F boundary conditions
	13.4 Space–Time Symmetry and Reciprocity of Longitudinal Vibrations
	13.5 Dynamic Characteristics of Longitudinal Vibrations
		13.5.1 Characteristic values and natural frequencies
		13.5.2 Normalized deflection function and mode shapes
		13.5.3 Basic reference values
Chapter 14. Vibrations of Membranes
	14.1 Transverse Vibration Equation of Membranes
	14.2 Rectangular Symmetric Vibrations of Membranes
		14.2.1 A particular case: Transverse vibrations of springs
	14.3 Space–Time Symmetry and Reciprocity of Rectangular Membranes
	14.4 Dynamic Characteristics of Rectangular Membranes
		14.4.1 Characteristic values and natural frequencies
		14.4.2 Normalized deflection function and mode shapes
		14.4.3 Basic reference values
	14.5 Circular Symmetric Vibrations of Membranes
	14.6 Space-Time Symmetry and Reciprocity of Circular Membranes
	14.7 Dynamic Characteristics of Circular Membranes
		14.7.1 Characteristic equation and natural frequencies
		14.7.2 Normalized deflection function and mode shapes
		14.7.3 Basic reference values
Chapter 15. Vibration of Plates
	15.1 Transverse Vibration Equation of Plates
	15.2 Rectangular Symmetric Vibrations of Plates and 1D Bars
		15.2.1 A particular case: T-vibrations of 1D bars
		15.2.2 F–F boundary conditions
		15.2.3 C–C boundary conditions
		15.2.4 C–F boundary conditions
	15.3 Space-Time Symmetry and Reciprocity of Rectangular Plates
	15.4 Dynamic Characteristics of Rectangular Plates and 1D Bar
		15.4.1 Characteristic equations and natural frequencies
		15.4.2 Normalized deflection function and mode shapes
		15.4.3 Basic reference values
	15.5 Circular Symmetric Vibrations of Plates
	15.6 Space–Time Symmetry and Reciprocity of Circular Plates
	15.7 Dynamic Characteristics of Circular Plates
		15.7.1 Characteristic equation and natural frequencies
		15.7.2 Normalized deflection function and mode shapes
		15.7.3 Basic reference values
Chapter 16. The TDK Equation of Plates
	16.1 The TDK Vibration Equation of Plates
	16.2 Particular Cases of the General TDK Equation
		16.2.1 The TD equation
		16.2.2 The DK and D equations
		16.2.3 TK and T equations
		16.2.4 The K equation: Simple harmonic resonator
	16.3 The TDK Equation of Rectangular Plates
	16.4 Rectangular Symmetric Vibrations (RSVs) of Plates
		16.4.1 The solutions with boundary conditions
	16.5 Characteristic Values of RSVs of TDK Rectangular Plates
		16.5.1 Canonical characteristic equation
		16.5.2 Natural frequencies
	16.6 The TDK Equation of Circular Plates
		16.6.1 Clamped TDK circular plates
	16.7 Circular Symmetric Vibrations (CSVs) of Plates
	16.8 Characteristic Values of CSVs of TDK Circular Plates
		16.8.1 Canonical characteristic equation
		16.8.2 Natural frequencies
Chapter 17. Forced Vibrations
	17.1 Forced Vibration Equation of TDK Plates
		17.1.1 The kernel equation
		17.1.2 Particular solution of the kernel equation
		17.1.3 Deflection function and displacement solution
	17.2 Impedance Analysis of Free Vibration TDK Equation
	17.3 Impedance Analysis of Forced Vibration TDK Equation
	17.4 Solutions of the Forced Vibration of TDK and TD Plates
		17.4.1 Forced symmetric vibrations of TDK rectangular plates
		17.4.2 Forced symmetric vibrations of TDK circular plates
	17.5 Forced Symmetric Vibration of Clamped TDK Circular Plates
		17.5.1 Solution of forced symmetric vibration
		17.5.2 A numerical example
Chapter 18. Damped Vibration and Space–Time Factor
	18.1 Free Oscillation of the Damped System
	18.2 Impedance Analysis of the Damped TDK Equation
	18.3 Solution of the Damped TDK Equation
	18.4 Mechanical Sensitivity of Elastic Transduction Elements
		18.4.1 Transduction effects of TDK plates
		18.4.2 Elastic transduction element and its mechanical sensitivity
		18.4.3 Displacement, velocity and acceleration sensitivity
	18.5 Electromechanical Sensitivity of Transducers
		18.5.1 Receiving sensitivity
		18.5.2 Transmitting sensitivity
	18.6 Space–Time Operator and Space–Time Factor
		18.6.1 Space operator
		18.6.2 Time operator
		18.6.3 Space–time operator j
		18.6.4 Space–time factor jω
		18.6.5 Space–time conversion and inversion and dispersion theorem
		18.6.6 The jω factor and reciprocity theorem of mechanical system
		18.6.7 Mechanical impedance expression with jω factor
		18.6.8 The jω factor and reciprocity theorem of electromechanical system
Chapter 19. Electromechanical Transducers
	19.1 Principles of Electromechanical Transducers
	19.2 A Unified Equivalent Circuit of Electromechanical Transducers
	19.3 Reciprocity Calibration of Electromechanical Transducers
		19.3.1 Piezoelectric reciprocity method of vibration calibration
		19.3.2 Electrodynamic reciprocity method of vibration calibration
		19.3.3 Free-field reciprocity method of condenser microphone
		19.3.4 Wave effect correction of reciprocity calibrations
	19.4 Modeling: Motion Equation and Static Deflection Equation
		19.4.1 Motion equation of the general TDK vibration diaphragm
		19.4.2 Static deflection equation of the TD diaphragm
		19.4.3 Transmitting voltage displacement response
	19.5 Modeling: Mechanical Resistance and Lumped Parameters
	19.6 Modelling: Response of mCUTs and First-Order Approximation
Chapter 20. Micro-Load Theory and Ultrasonic Resonators
	20.1 Micro-load Theory and Loading Effects
	20.2 Micro–Nano Plate and Beam Ultrasonic Resonators
	20.3 Film Bulk Acoustic Wave Ultrasonic Resonators (FBARs)
	20.4 LAW and SAW Delay-Line Ultrasonic Resonators
	20.5 A Typical Example: The Sensitivity of SAW Gas Sensors
	20.6 Mass Sensitivity and Resolution of UMRs
		20.6.1 Mass sensitivity and mass resolution
		20.6.2 Trance measurement and environmental monitoring
References
Index