A guide to the application of viscoelastic damping materials to control vibration and noise of structures, machinery, and vehiclesActive and Passive Vibration Damping is a practical guide to the application of passive as well as actively treated viscoelastic damping materials to control vibration and noise of structures, machinery and vehicles. The author -- a noted expert on the topic -- presents the basic principles and reviews the potential applications of passive and active vibration damping technologies. The text presents a combination of the associated physical fundamentals, governing theories and the optimal design strategies of various configurations of vibration damping treatments.The text presents the basics of various damping effective treatments such as constrained layers, shunted piezoelectric treatments, electromagnetic and shape memory fibers. Classical and new models are included as well as aspects of viscoelastic materials models that are analyzed from the experimental characterization of the material coefficients as well as their modeling. The use of smart materials to augment the vibration damping of passive treatments is pursued in depth throughout the book. This vital guide:* Contains numerical examples that reinforce the understanding of the theories presented* Offers an authoritative text from an internationally recognized authority and pioneer on the subject* Presents, in one volume, comprehensive coverage of the topic that is not available elsewhere* Presents a mix of the associated physical fundamentals, governing theories and optimal design strategies of various configurations of vibration damping treatmentsWritten for researchers in vibration damping and research, engineers in structural dynamics and practicing engineers, Active and Passive Vibration Damping offers a hands-on resource for applying passive as well as actively treated viscoelastic damping materials to control vibration and noise of structures, machinery and vehicles.

Preface xviiList of Symbols xxiAbbreviations xxxiPart I Fundamentals of Viscoelastic Damping 11 Vibration Damping 31.1 Overview 31.2 Passive, Active, and Hybrid Vibration Control 31.2.1 Passive Damping 31.2.1.1 Free and Constrained Damping Layers 31.2.1.2 Shunted Piezoelectric Treatments 41.2.1.3 Damping Layers with Shunted Piezoelectric Treatments 51.2.1.4 Magnetic Constrained Layer Damping (MCLD) 51.2.1.5 Damping with Shape Memory Fibers 61.2.2 Active Damping 61.2.3 Hybrid Damping 71.2.3.1 Active Constrained Layer Damping (ACLD) 71.2.3.2 Active Piezoelectric Damping Composites (APDC) 71.2.3.3 Electromagnetic Damping Composites (EMDC) 81.2.3.4 Active Shunted Piezoelectric Networks 81.3 Summary 9References 92 Viscoelastic Damping 112.1 Introduction 112.2 Classical Models of Viscoelastic Materials 112.2.1 Characteristics in the Time Domain 112.2.2 Basics for Time Domain Analysis 122.2.3 Detailed Time Response of Maxwell and Kelvin-Voigt Models 142.2.4 Detailed Time Response of the Poynting-Thomson Model 172.3 Creep Compliance and Relaxation Modulus 202.3.1 Direct Laplace Transformation Approach 222.3.2 Approach of Simultaneous Solution of a Linear Set of Equilibrium, Kinematic, and Constitutive Equations 232.4 Characteristics of the VEM in the Frequency Domain 252.5 Hysteresis and Energy Dissipation Characteristics of Viscoelastic Materials 272.5.1 Hysteresis Characteristics 272.5.2 Energy Dissipation 282.5.3 Loss Factor 282.5.3.1 Relationship between Dissipation and Stored Elastic Energies 282.5.3.2 Relationship between Different Strains 292.5.4 Storage Modulus 292.6 Fractional Derivative Models of Viscoelastic Materials 322.6.1 Basic Building Block of Fractional Derivative Models 322.6.2 Basic Fractional Derivative Models 332.6.3 Other Common Fractional Derivative Models 362.7 Viscoelastic versus Other Types of Damping Mechanisms 382.8 Summary 40References 403 Characterization of the Properties of Viscoelastic Materials 573.1 Introduction 573.2 Typical Behavior of Viscoelastic Materials 573.3 Frequency Domain Measurement Techniques of the Dynamic Properties of Viscoelastic Material 593.3.1 Dynamic, Mechanical, and Thermal Analyzer 603.3.2 Oberst Test Beam Method 643.3.2.1 Set-Up and Beam Configurations 643.3.2.2 Parameter Extraction 663.4 Master Curves of Viscoelastic Materials 683.4.1 The Principle of Temperature-Frequency Superposition 683.4.2 The Use of the Master Curves 713.4.3 The Constant Temperature Lines 713.5 Time-Domain Measurement Techniques of the Dynamic Properties of Viscoelastic Materials 723.5.1 Creep and Relaxation Measurement Methods 733.5.1.1 Testing Equipment 733.5.1.2 Typical Creep and Relaxation Behavior 743.5.1.3 Time-Temperature Superposition 763.5.1.4 Boltzmann Superposition Principle 783.5.1.5 Relationship between the Relaxation Modulus and Complex Modulus 803.5.1.6 Relationship between the Creep Compliance and Complex Compliance 813.5.1.7 Relationship between the Creep Compliance and Relaxation Modulus 833.5.1.8 Alternative Relationship between the Creep Compliance and Complex Compliance 833.5.1.9 Alternative Relationship between the Relaxation Modulus and Complex Modulus 843.5.1.10 Summary of the Basic Interconversion Relationship 853.5.1.11 Practical Issues in Implementation of Interconversion Relationships 863.5.2 Split Hopkinson Pressure Bar Method 943.5.2.1 Overview 943.5.2.2 Theory of 1D SHPB 953.5.2.3 Complex Modulus of a VEM from SHPB Measurements 983.5.3 Wave Propagation Method 1053.5.4 Ultrasonic Wave Propagation Method 1093.5.4.1 Overview 1093.5.4.2 Theory 1093.5.4.3 Measurement of the Phase Velocity and Attenuation Factor 1113.5.4.4 Typical Attenuation Factors 1133.6 Summary 115References 1164 Viscoelastic Materials 1274.1 Introduction 1274.2 Golla-Hughes-McTavish (GHM) Model 1274.2.1 Motivation of the GHM Model 1284.2.2 Computation of the Parameters of the GHM Mini-Oscillators 1324.2.3 On the Structure of the GHM Model 1354.2.3.1 Other Forms of GHM Structures 1354.2.3.2 Relaxation Modulus of the GHM Model 1354.2.4 Structural Finite Element Models of Rods Treated with VEM 1374.2.4.1 Unconstrained Layer Damping 1384.2.4.2 Constrained Layer Damping 1424.3 Structural Finite Element Models of Beams Treated with VEM 1504.3.1 Degrees of Freedom 1504.3.2 Basic Kinematic Relationships 1514.3.3 Stiffness and Mass Matrices of the Beam/VEM Element 1524.3.4 Equations of Motion of the Beam/VEM Element 1534.4 Generalized Maxwell Model (GMM) 1554.4.1 Overview 1554.4.2 Internal Variable Representation of the GMM 1574.4.2.1 Single-DOF System 1574.4.2.2 Multi-Degree of Freedom System 1584.4.2.3 Condensation of the Internal Degrees of Freedom 1594.4.2.4 Direct Solution of Coupled Structural and Internal Degrees of Freedom 1604.5 Augmenting Thermodynamic Field (ATF) Model 1634.5.1 Overview 1634.5.2 Equivalent Damping Ratio of the ATF Model 1644.5.3 Multi-degree of Freedom ATF Model 1654.5.4 Integration with a Finite Element Model 1654.6 Fractional Derivative (FD) Models 1674.6.1 Overview 1674.6.2 Internal Degrees of Freedom of Fractional Derivative Models 1694.6.3 Grunwald Approximation of Fractional Derivative 1694.6.4 Integration Fractional Derivative Approximation with Finite Element 1704.6.4.1 Viscoelastic Rod 1704.6.4.2 Beam with Passive Constrained Layer Damping (PCLD) Treatment 1724.7 Finite Element Modeling of Plates Treated with Passive Constrained Layer Damping 1764.7.1 Overview 1764.7.2 The Stress and Strain Characteristics 1784.7.2.1 The Plate and the Constraining Layers 1784.7.2.2 The VEM Layer 1794.7.3 The Potential and Kinetic Energies 1794.7.4 The Shape Functions 1794.7.5 The Stiffness Matrices 1814.7.6 The Mass Matrices 1814.7.7 The Element and Overall Equations of Motion 1824.8 Finite Element Modeling of Shells Treated with Passive Constrained Layer Damping 1854.8.1 Overview 1854.8.2 Stress-Strain Relationships 1864.8.2.1 Shell and Constraining Layer 1864.8.2.2 Viscoelastic Layer 1874.8.3 Kinetic and Potential Energies 1894.8.4 The Shape Functions 1894.8.5 The Stiffness Matrices 1894.8.6 The Mass Matrices 1904.8.7 The Element and Overall Equations of Motion 1914.9 Summary 192References 1965 Finite Element Modeling of Viscoelastic Damping by Modal Strain Energy Method 2055.1 Introduction 2055.2 Modal Strain Energy (MSE) Method 2055.3 Modified Modal Strain Energy (MSE) Methods 2105.3.1 Weighted Stiffness Matrix Method (WSM) 2105.3.2 Weighted Storage Modulus Method (WSTM) 2115.3.3 Improved Reduction System Method (IRS) 2115.3.4 Low Frequency Approximation Method (LFA) 2135.4 Summary of Modal Strain Energy Methods 2155.5 Modal Strain Energy as a Metric for Design of Damping Treatments 2155.6 Perforated Damping Treatments 2205.6.1 Overview 2205.6.2 Finite Element Modeling 2225.6.2.1 Element Energies 2245.6.2.2 Topology Optimization of Unconstrained Layer Damping 2275.6.2.3 Sensitivity Analysis 2285.7 Summary 234References 2346 Energy Dissipation in Damping Treatments 2436.1 Introduction 2436.2 Passive Damping Treatments of Rods 2436.2.1 Passive Constrained Layer Damping 2436.2.1.1 Equation of Motion 2436.2.1.2 Energy Dissipation 2476.2.2 Passive Unconstrained Layer Damping 2486.3 Active Constrained Layer Damping Treatments of Rods 2516.3.1 Equation of Motion 2516.3.2 Boundary Control Strategy 2536.3.3 Energy Dissipation 2546.4 Passive Constrained Layer Damping Treatments of Beams 2576.4.1 Basic Equations of Damped Beams 2576.4.2 Bending Energy of Beams 2586.4.3 Energy Dissipated in Beams with Passive Constrained Layer Damping 2586.5 Active Constrained Layer Damping Treatments of Beams 2646.6 Passive and Active Constrained Layer Damping Treatments of Plates 2676.6.1 Kinematic Relationships 2686.6.2 Energies of the PCLD and ACLD Treatments 2696.6.2.1 The Potential Energies 2696.6.2.2 The Kinetic Energy 2696.6.2.3 Work Done 2696.6.3 The Models of the PCLD and ACLD Treatments 2706.6.4 Boundary Control of Plates with ACLD Treatments 2706.6.5 Energy Dissipation and Loss Factors of Plates with PCLD and ACLD Treatments 2716.7 Passive and Active Constrained Layer Damping Treatments of Axi-Symmetric Shells 2746.7.1 Background 2756.7.2 The Concept of the Active Constrained Layer Damping 2766.7.3 Variational Modeling of the Shell/ACLD System 2766.7.3.1 Main Assumptions of the Model 2766.7.3.2 Kinematic Relationships 2766.7.3.3 Stress-Strain Relationships 2776.7.3.4 Energies of Shell/ACLD System 2796.7.3.5 The Model 2806.7.4 Boundary Control Strategy 2826.7.4.1 Overview 2826.7.4.2 Control Strategy 2826.7.4.3 Implementation of the Boundary Control Strategy 2836.7.4.4 Transverse Compliance and Longitudinal Deflection 2836.7.5 Energy Dissipated in the ACLD Treatment of an Axi-Symmetric Shell 2876.8 Summary 288References 290Part II Advanced Damping Treatments 3017 Vibration Damping of Structures Using Active Constrained Layer Damping 3037.1 Introduction 3037.2 Motivation for Using Passive and Active Constrained Layer Damping 3037.2.1 Base Structure 3047.2.2 Structure Treated with Unconstrained Passive Layer Damping 3067.2.3 Structure Treated with Constrained Passive Layer Damping 3087.2.4 Structure Treated with Active Constrained Passive Layer Damping 3117.3 Active Constrained Layer Damping for Beams 3167.3.1 Introduction 3167.3.2 Concept of Active Constrained Layer Damping 3167.3.3 Finite Element Modeling of a Beam/ACLD Assembly 3187.3.3.1 The Model 3197.3.3.2 Equations of Motion 3227.3.4 Distributed-Parameter Modeling of a Beam/ACLD Assembly 3287.3.4.1 Overview 3287.3.4.2 The Energies and Work Done on the Beam/ACLD Assembly 3287.3.4.3 The Distributed-Parameter Model 3317.3.4.4 Globally Stable Boundary Control Strategy 3337.3.4.5 Implementation of the Globally Stable Boundary Control Strategy 3337.3.4.6 Response of the Beam/ACLD Assembly 3347.4 Active Constrained Layer Damping for Plates 3367.4.1 Control Forces and Moments Generated by the Active Constraining Layer 3377.4.1.1 The In-Plane Piezoelectric Forces 3377.4.1.2 The Piezoelectric Moments 3387.4.1.3 Piezoelectric Sensor 3387.4.1.4 Control Voltage to Piezoelectric Constraining Layer 3397.4.2 Equations of Motion 3397.5 Active Constrained Layer Damping for Shells 3447.5.1 Control Forces and Moments Generated by the Active Constraining Layer 3447.5.2 Equations of Motion 3447.6 Summary 348References 3518 Advanced Damping Treatments 3618.1 Introduction 3618.2 Stand-Off Damping Treatments 3628.2.1 Background of Stand-Off Damping Treatments 3628.2.2 The Stand-Off Damping Treatments 3628.2.3 Distributed-Parameter Model of the Stand-Off Layer Damping Treatment 3648.2.3.1 Kinematic Equations 3648.2.3.2 Constitutive Equations 3658.2.4 Distributed Transfer Function Method 3698.2.5 Finite Element Model 3708.2.6 Summary 3758.3 Functionally Graded Damping Treatments 3758.3.1 Background of Functionally Graded Constrained Layer Damping 3758.3.2 Concept of Constrained Layer Damping with Functionally Graded Viscoelastic Cores 3768.3.3 Finite Element Model 3778.3.3.1 Quasi-Static Model of the Passive Constrained Damping Layer of Plunkett and Lee (1970) 3778.3.3.2 Dispersion Characteristics of Passive Constrained Damping Layer with Uniform and Functionally Graded Cores 3838.3.4 Summary 3908.4 Passive and Active Damping Composite Treatments 3908.4.1 Passive Composite Damping Treatments 3908.4.2 Active Composite Damping Treatments 3948.4.3 Finite Element Modeling of Beam with APDC 3968.4.3.1 Model and Main Assumptions 3968.4.3.2 Kinematics 3978.4.3.3 Degrees of Freedom and Shape Functions 3988.4.3.4 System Energies 3988.4.3.5 Equations of Motion 4008.4.3.6 Control Law 4008.4.4 Summary 4088.5 Magnetic Damping Treatments 4108.5.1 Magnetic Constrained Layer Damping Treatments 4108.5.2 Analysis of Magnetic Constrained Layer Damping Treatments 4128.5.2.1 Equation of Motion 4128.5.2.2 Response of the MCLD Treatment 4148.5.3 Passive Magnetic Composites 4158.5.3.1 Concept of Passive Magnetic Composite Treatment 4178.5.3.2 Finite Element Modeling of Beams with PMC Treatment 4178.5.4 Summary 4308.6 Negative Stiffness Composites 4308.6.1 Motivation to Negative Stiffness Composites 4318.6.1.1 Sinusoidal Excitation 4318.6.1.2 Impact Loading 4368.6.1.3 Magnetic Composite with Negative Stiffness Inclusions 4388.7 Summary 445References 4459 Vibration Damping with Shunted Piezoelectric Networks 4699.1 Introduction 4699.2 Shunted Piezoelectric Patches 4699.2.1 Basics of Piezoelectricity 4699.2.1.1 Effect of Electrical Boundary Conditions 4719.2.1.2 Effect of Mechanical Boundary Conditions 4719.2.2 Basics of Shunted Piezo-Networks 4729.2.2.1 Resistive-Shunted Circuit 4749.2.2.2 Resistive and Inductive Shunted Circuit 4759.2.2.3 Resistive, Capacitive, and Inductive Shunted Circuit 4779.2.3 Electronic Synthesis of Inductances and Negative Capacitances 4799.2.3.1 Synthesis of Inductors 4799.2.3.2 Synthesis of Negative Capacitances 4809.2.4 Why Negative Capacitance Is Effective? 4809.2.5 Effectiveness of the Negative Capacitance from a Control System Perspective 4829.2.6 Electrical Analogy of Shunted Piezoelectric Networks 4859.3 Finite Element Modeling of Structures Treated with Shunted Piezo-Networks 4879.3.1 Equivalent Complex Modulus Approach of Shunted Piezo-Networks 4879.3.2 Coupled Electromechanical Field Approach of Shunted Piezo-Networks 4919.4 Active Shunted Piezoelectric Networks 4969.4.1 Basic Configurations 4969.4.2 Dynamic Equations 4989.4.2.1 Short-Circuit Configuration 4989.4.2.2 Open-Circuit Configuration 4989.4.2.3 Resistive-Shunted Configuration 4989.4.3 More on the Resistive Shunting Configuration 4989.4.4 Open-Circuit to Resistive Shunting (OC-RS) Configuration 5009.4.4.1 Dynamic Equations 5009.4.4.2 Switching Between OC and RS Modes 5009.4.5 Energy Dissipation of Different Shunting Configurations 5039.4.5.1 Energy Dissipation with Resistive Shunting 5039.4.5.2 Energy Dissipation with OC-RS Switched Shunting 5039.5 Multi-Mode Vibration Control with Shunted Piezoelectric Networks 5049.5.1 Multi-Mode Shunting Approaches 5049.5.2 Parameters of Behrens et al.'s Multi-Mode Shunting Network 5079.5.2.1 Components of the Current Flowing Branches 5079.5.2.2 Components of the Shunting Branches 5079.6 Summary 510References 51110 Vibration Control with Periodic Structures 52310.1 Introduction 52310.2 Basics of Periodic Structures 52410.2.1 Overview 52410.2.2 Transfer Matrix Method 52510.2.2.1 The Transfer Matrix 52510.2.2.2 Basic Properties of the Transfer Matrix 52610.3 Filtering Characteristics of Passive Periodic Structures 53310.3.1 Overview 53310.3.2 Periodic Rods in Longitudinal Vibrations 53410.4 Natural Frequencies, Mode Shapes, and Response of Periodic Structures 53510.4.1 Natural Frequencies and Response 53510.4.2 Mode Shapes 53910.5 Active Periodic Structures 54110.5.1 Modeling of Active Periodic Structures 54310.5.2 Dynamics of One Cell 54310.5.2.1 Dynamics of the Passive Sub-Cell 54310.5.2.2 Dynamics of the Active Sub-Cell 54310.5.2.3 Dynamics of the Entire Cell 54510.5.2.4 Dynamics of the Entire Periodic Structure 54610.6 Localization Characteristics of Passive and Active Aperiodic Structures 54910.6.1 Overview 54910.6.2 Localization Factor 55010.7 Periodic Rod with Periodic Shunted Piezoelectric Patches 55910.7.1 Transfer Matrix of a Plain Rod Element 55910.7.2 Transfer Matrix of a Rod/Piezo-Patch Element 56010.7.3 Transfer Matrix of a Unit Cell 56110.8 Two-Dimensional Active Periodic Structure 56210.8.1 Dynamics of Unit Cell 56210.8.2 Formulation of Phase Constant Surfaces 56610.8.3 Filtering Characteristics 56810.9 Periodic Structures with Internal Resonances 56910.9.1 Dynamics of Conventional Periodic Structure 57010.9.2 Dynamics of Periodic Structure with Internal Resonances 57210.9.2.1 Equivalent Mass. Of the Mass-In-Mass Arrangement 57210.9.2.2 Transfer Matrix of the Mass-In-Mass Arrangement 57210.10 Summary 578References 57811 Nanoparticle Damping Composites 58911.1 Introduction 58911.2 Nanoparticle-Filled Polymer Composites 59011.2.1 Composites with Unidirectional Inclusions 59111.2.2 Arbitrarily Oriented Inclusion Composites 59911.3 Comparisons with Classical Filler Reinforcement Methods 60711.4 Applications of Carbon Black/Polymer Composites 61411.4.1 Basic Physical Characteristics 61411.4.2 Modeling of the Piezo-Resistance of CB/Polymer Composites 61711.4.3 The Piezo-Resistivity of CB/Polymer Composites 61911.5 CB/Polymer Composite as a Shunting Resistance of Piezoelectric Layers 62011.5.1 Finite Element Model 62011.5.2 Condensed Model of a Unit Cell 62411.6 Hybrid Composites with Shunted Piezoelectric Particles 62911.6.1 Composite Description and Assumptions 62911.6.2 Shunted Piezoelectric Inclusions 63111.6.3 Typical Performance Characteristics of Hybrid Composites 63111.7 Summary 636References 63612 Power Flow in Damped Structures 65112.1 Introduction 65112.2 Vibrational Power 65112.2.1 Basic Definitions 65112.2.2 Relationship to System Energies 65212.2.3 Basic Characteristics of the Power Flow 65312.3 Vibrational Power Flow in Beams 65612.4 Vibrational Power of Plates 66112.4.1 Basic Equations of Vibrating Plates 66112.4.2 Power Flow and Structural Intensity 66212.4.3 Control of the Power Flow and Structural Intensity 66812.4.4 Power Flow and Structural Intensity for Plates with Passive and Active Constrained Layer Damping Treatments 67112.5 Power Flow and Structural Intensity for Shells 67912.6 Summary 682References 682Glossary 699Appendix 703Index 715