Active and Passive Vibration Damping

Preface xvii List of Symbols xxi Abbreviations xxxi Part I Fundamentals of Viscoelastic Damping 1 1 Vibration Damping 3 1.1 Overview 3 1.2 Passive, Active, and Hybrid Vibration Control 3 1.2.1 Passive Damping 3 1.2.1.1 Free and Constrained Damping Layers 3 1.2.1.2 Shunted Piezoelectric Treatments 4 1.2.1.3 Damping Layers with Shunted Piezoelectric Treatments 5 1.2.1.4 Magnetic Constrained Layer Damping (MCLD) 5 1.2.1.5 Damping with Shape Memory Fibers 6 1.2.2 Active Damping 6 1.2.3 Hybrid Damping 7 1.2.3.1 Active Constrained Layer Damping (ACLD) 7 1.2.3.2 Active Piezoelectric Damping Composites (APDC) 7 1.2.3.3 Electromagnetic Damping Composites (EMDC) 8 1.2.3.4 Active Shunted Piezoelectric Networks 8 1.3 Summary 9 References 9 2 Viscoelastic Damping 11 2.1 Introduction 11 2.2 Classical Models of Viscoelastic Materials 11 2.2.1 Characteristics in the Time Domain 11 2.2.2 Basics for Time Domain Analysis 12 2.2.3 Detailed Time Response of Maxwell and Kelvin–Voigt Models 14 2.2.4 Detailed Time Response of the Poynting–Thomson Model 17 2.3 Creep Compliance and Relaxation Modulus 20 2.3.1 Direct Laplace Transformation Approach 22 2.3.2 Approach of Simultaneous Solution of a Linear Set of Equilibrium, Kinematic, and Constitutive Equations 23 2.4 Characteristics of the VEM in the Frequency Domain 25 2.5 Hysteresis and Energy Dissipation Characteristics of Viscoelastic Materials 27 2.5.1 Hysteresis Characteristics 27 2.5.2 Energy Dissipation 28 2.5.3 Loss Factor 28 2.5.3.1 Relationship between Dissipation and Stored Elastic Energies 28 2.5.3.2 Relationship between Different Strains 29 2.5.4 Storage Modulus 29 2.6 Fractional Derivative Models of Viscoelastic Materials 32 2.6.1 Basic Building Block of Fractional Derivative Models 32 2.6.2 Basic Fractional Derivative Models 33 2.6.3 Other Common Fractional Derivative Models 36 2.7 Viscoelastic versus Other Types of Damping Mechanisms 38 2.8 Summary 40 References 40 3 Characterization of the Properties of Viscoelastic Materials 57 3.1 Introduction 57 3.2 Typical Behavior of Viscoelastic Materials 57 3.3 Frequency Domain Measurement Techniques of the Dynamic Properties of Viscoelastic Material 59 3.3.1 Dynamic, Mechanical, and Thermal Analyzer 60 3.3.2 Oberst Test Beam Method 64 3.3.2.1 Set-Up and Beam Configurations 64 3.3.2.2 Parameter Extraction 66 3.4 Master Curves of Viscoelastic Materials 68 3.4.1 The Principle of Temperature-Frequency Superposition 68 3.4.2 The Use of the Master Curves 71 3.4.3 The Constant Temperature Lines 71 3.5 Time-Domain Measurement Techniques of the Dynamic Properties of Viscoelastic Materials 72 3.5.1 Creep and Relaxation Measurement Methods 73 3.5.1.1 Testing Equipment 73 3.5.1.2 Typical Creep and Relaxation Behavior 74 3.5.1.3 Time-Temperature Superposition 76 3.5.1.4 Boltzmann Superposition Principle 78 3.5.1.5 Relationship between the Relaxation Modulus and Complex Modulus 80 3.5.1.6 Relationship between the Creep Compliance and Complex Compliance 81 3.5.1.7 Relationship between the Creep Compliance and Relaxation Modulus 83 3.5.1.8 Alternative Relationship between the Creep Compliance and Complex Compliance 83 3.5.1.9 Alternative Relationship between the Relaxation Modulus and Complex Modulus 84 3.5.1.10 Summary of the Basic Interconversion Relationship 85 3.5.1.11 Practical Issues in Implementation of Interconversion Relationships 86 3.5.2 Split Hopkinson Pressure Bar Method 94 3.5.2.1 Overview 94 3.5.2.2 Theory of 1D SHPB 95 3.5.2.3 Complex Modulus of a VEM from SHPB Measurements 98 3.5.3 Wave Propagation Method 105 3.5.4 Ultrasonic Wave Propagation Method 109 3.5.4.1 Overview 109 3.5.4.2 Theory 109 3.5.4.3 Measurement of the Phase Velocity and Attenuation Factor 111 3.5.4.4 Typical Attenuation Factors 113 3.6 Summary 115 References 116 4 Viscoelastic Materials 127 4.1 Introduction 127 4.2 Golla–Hughes–McTavish (GHM) Model 127 4.2.1 Motivation of the GHM Model 128 4.2.2 Computation of the Parameters of the GHM Mini-Oscillators 132 4.2.3 On the Structure of the GHM Model 135 4.2.3.1 Other Forms of GHM Structures 135 4.2.3.2 Relaxation Modulus of the GHM Model 135 4.2.4 Structural Finite Element Models of Rods Treated with VEM 137 4.2.4.1 Unconstrained Layer Damping 138 4.2.4.2 Constrained Layer Damping 142 4.3 Structural Finite Element Models of Beams Treated with VEM 150 4.3.1 Degrees of Freedom 150 4.3.2 Basic Kinematic Relationships 151 4.3.3 Stiffness and Mass Matrices of the Beam/VEM Element 152 4.3.4 Equations of Motion of the Beam/VEM Element 153 4.4 Generalized Maxwell Model (GMM) 155 4.4.1 Overview 155 4.4.2 Internal Variable Representation of the GMM 157 4.4.2.1 Single-DOF System 157 4.4.2.2 Multi-Degree of Freedom System 158 4.4.2.3 Condensation of the Internal Degrees of Freedom 159 4.4.2.4 Direct Solution of Coupled Structural and Internal Degrees of Freedom 160 4.5 Augmenting Thermodynamic Field (ATF) Model 163 4.5.1 Overview 163 4.5.2 Equivalent Damping Ratio of the ATF Model 164 4.5.3 Multi-degree of Freedom ATF Model 165 4.5.4 Integration with a Finite Element Model 165 4.6 Fractional Derivative (FD) Models 167 4.6.1 Overview 167 4.6.2 Internal Degrees of Freedom of Fractional Derivative Models 169 4.6.3 Grunwald Approximation of Fractional Derivative 169 4.6.4 Integration Fractional Derivative Approximation with Finite Element 170 4.6.4.1 Viscoelastic Rod 170 4.6.4.2 Beam with Passive Constrained Layer Damping (PCLD) Treatment 172 4.7 Finite Element Modeling of Plates Treated with Passive Constrained Layer Damping 176 4.7.1 Overview 176 4.7.2 The Stress and Strain Characteristics 178 4.7.2.1 The Plate and the Constraining Layers 178 4.7.2.2 The VEM Layer 179 4.7.3 The Potential and Kinetic Energies 179 4.7.4 The Shape Functions 179 4.7.5 The Stiffness Matrices 181 4.7.6 The Mass Matrices 181 4.7.7 The Element and Overall Equations of Motion 182 4.8 Finite Element Modeling of Shells Treated with Passive Constrained Layer Damping 185 4.8.1 Overview 185 4.8.2 Stress–Strain Relationships 186 4.8.2.1 Shell and Constraining Layer 186 4.8.2.2 Viscoelastic Layer 187 4.8.3 Kinetic and Potential Energies 189 4.8.4 The Shape Functions 189 4.8.5 The Stiffness Matrices 189 4.8.6 The Mass Matrices 190 4.8.7 The Element and Overall Equations of Motion 191 4.9 Summary 192 References 196 5 Finite Element Modeling of Viscoelastic Damping by Modal Strain Energy Method 205 5.1 Introduction 205 5.2 Modal Strain Energy (MSE) Method 205 5.3 Modified Modal Strain Energy (MSE) Methods 210 5.3.1 Weighted Stiffness Matrix Method (WSM) 210 5.3.2 Weighted Storage Modulus Method (WSTM) 211 5.3.3 Improved Reduction System Method (IRS) 211 5.3.4 Low Frequency Approximation Method (LFA) 213 5.4 Summary of Modal Strain Energy Methods 215 5.5 Modal Strain Energy as a Metric for Design of Damping Treatments 215 5.6 Perforated Damping Treatments 220 5.6.1 Overview 220 5.6.2 Finite Element Modeling 222 5.6.2.1 Element Energies 224 5.6.2.2 Topology Optimization of Unconstrained Layer Damping 227 5.6.2.3 Sensitivity Analysis 228 5.7 Summary 234 References 234 6 Energy Dissipation in Damping Treatments 243 6.1 Introduction 243 6.2 Passive Damping Treatments of Rods 243 6.2.1 Passive Constrained Layer Damping 243 6.2.1.1 Equation of Motion 243 6.2.1.2 Energy Dissipation 247 6.2.2 Passive Unconstrained Layer Damping 248 6.3 Active Constrained Layer Damping Treatments of Rods 251 6.3.1 Equation of Motion 251 6.3.2 Boundary Control Strategy 253 6.3.3 Energy Dissipation 254 6.4 Passive Constrained Layer Damping Treatments of Beams 257 6.4.1 Basic Equations of Damped Beams 257 6.4.2 Bending Energy of Beams 258 6.4.3 Energy Dissipated in Beams with Passive Constrained Layer Damping 258 6.5 Active Constrained Layer Damping Treatments of Beams 264 6.6 Passive and Active Constrained Layer Damping Treatments of Plates 267 6.6.1 Kinematic Relationships 268 6.6.2 Energies of the PCLD and ACLD Treatments 269 6.6.2.1 The Potential Energies 269 6.6.2.2 The Kinetic Energy 269 6.6.2.3 Work Done 269 6.6.3 The Models of the PCLD and ACLD Treatments 270 6.6.4 Boundary Control of Plates with ACLD Treatments 270 6.6.5 Energy Dissipation and Loss Factors of Plates with PCLD and ACLD Treatments 271 6.7 Passive and Active Constrained Layer Damping Treatments of Axi-Symmetric Shells 274 6.7.1 Background 275 6.7.2 The Concept of the Active Constrained Layer Damping 276 6.7.3 Variational Modeling of the Shell/ACLD System 276 6.7.3.1 Main Assumptions of the Model 276 6.7.3.2 Kinematic Relationships 276 6.7.3.3 Stress-Strain Relationships 277 6.7.3.4 Energies of Shell/ACLD System 279 6.7.3.5 The Model 280 6.7.4 Boundary Control Strategy 282 6.7.4.1 Overview 282 6.7.4.2 Control Strategy 282 6.7.4.3 Implementation of the Boundary Control Strategy 283 6.7.4.4 Transverse Compliance and Longitudinal Deflection 283 6.7.5 Energy Dissipated in the ACLD Treatment of an Axi-Symmetric Shell 287 6.8 Summary 288 References 290 Part II Advanced Damping Treatments 301 7 Vibration Damping of Structures Using Active Constrained Layer Damping 303 7.1 Introduction 303 7.2 Motivation for Using Passive and Active Constrained Layer Damping 303 7.2.1 Base Structure 304 7.2.2 Structure Treated with Unconstrained Passive Layer Damping 306 7.2.3 Structure Treated with Constrained Passive Layer Damping 308 7.2.4 Structure Treated with Active Constrained Passive Layer Damping 311 7.3 Active Constrained Layer Damping for Beams 316 7.3.1 Introduction 316 7.3.2 Concept of Active Constrained Layer Damping 316 7.3.3 Finite Element Modeling of a Beam/ACLD Assembly 318 7.3.3.1 The Model 319 7.3.3.2 Equations of Motion 322 7.3.4 Distributed-Parameter Modeling of a Beam/ACLD Assembly 328 7.3.4.1 Overview 328 7.3.4.2 The Energies and Work Done on the Beam/ACLD Assembly 328 7.3.4.3 The Distributed-Parameter Model 331 7.3.4.4 Globally Stable Boundary Control Strategy 333 7.3.4.5 Implementation of the Globally Stable Boundary Control Strategy 333 7.3.4.6 Response of the Beam/ACLD Assembly 334 7.4 Active Constrained Layer Damping for Plates 336 7.4.1 Control Forces and Moments Generated by the Active Constraining Layer 337 7.4.1.1 The In-Plane Piezoelectric Forces 337 7.4.1.2 The Piezoelectric Moments 338 7.4.1.3 Piezoelectric Sensor 338 7.4.1.4 Control Voltage to Piezoelectric Constraining Layer 339 7.4.2 Equations of Motion 339 7.5 Active Constrained Layer Damping for Shells 344 7.5.1 Control Forces and Moments Generated by the Active Constraining Layer 344 7.5.2 Equations of Motion 344 7.6 Summary 348 References 351 8 Advanced Damping Treatments 361 8.1 Introduction 361 8.2 Stand-Off Damping Treatments 362 8.2.1 Background of Stand-Off Damping Treatments 362 8.2.2 The Stand-Off Damping Treatments 362 8.2.3 Distributed-Parameter Model of the Stand-Off Layer Damping Treatment 364 8.2.3.1 Kinematic Equations 364 8.2.3.2 Constitutive Equations 365 8.2.4 Distributed Transfer Function Method 369 8.2.5 Finite Element Model 370 8.2.6 Summary 375 8.3 Functionally Graded Damping Treatments 375 8.3.1 Background of Functionally Graded Constrained Layer Damping 375 8.3.2 Concept of Constrained Layer Damping with Functionally Graded Viscoelastic Cores 376 8.3.3 Finite Element Model 377 8.3.3.1 Quasi-Static Model of the Passive Constrained Damping Layer of Plunkett and Lee (1970) 377 8.3.3.2 Dispersion Characteristics of Passive Constrained Damping Layer with Uniform and Functionally Graded Cores 383 8.3.4 Summary 390 8.4 Passive and Active Damping Composite Treatments 390 8.4.1 Passive Composite Damping Treatments 390 8.4.2 Active Composite Damping Treatments 394 8.4.3 Finite Element Modeling of Beam with APDC 396 8.4.3.1 Model and Main Assumptions 396 8.4.3.2 Kinematics 397 8.4.3.3 Degrees of Freedom and Shape Functions 398 8.4.3.4 System Energies 398 8.4.3.5 Equations of Motion 400 8.4.3.6 Control Law 400 8.4.4 Summary 408 8.5 Magnetic Damping Treatments 410 8.5.1 Magnetic Constrained Layer Damping Treatments 410 8.5.2 Analysis of Magnetic Constrained Layer Damping Treatments 412 8.5.2.1 Equation of Motion 412 8.5.2.2 Response of the MCLD Treatment 414 8.5.3 Passive Magnetic Composites 415 8.5.3.1 Concept of Passive Magnetic Composite Treatment 417 8.5.3.2 Finite Element Modeling of Beams with PMC Treatment 417 8.5.4 Summary 430 8.6 Negative Stiffness Composites 430 8.6.1 Motivation to Negative Stiffness Composites 431 8.6.1.1 Sinusoidal Excitation 431 8.6.1.2 Impact Loading 436 8.6.1.3 Magnetic Composite with Negative Stiffness Inclusions 438 8.7 Summary 445 References 445 9 Vibration Damping with Shunted Piezoelectric Networks 469 9.1 Introduction 469 9.2 Shunted Piezoelectric Patches 469 9.2.1 Basics of Piezoelectricity 469 9.2.1.1 Effect of Electrical Boundary Conditions 471 9.2.1.2 Effect of Mechanical Boundary Conditions 471 9.2.2 Basics of Shunted Piezo-Networks 472 9.2.2.1 Resistive-Shunted Circuit 474 9.2.2.2 Resistive and Inductive Shunted Circuit 475 9.2.2.3 Resistive, Capacitive, and Inductive Shunted Circuit 477 9.2.3 Electronic Synthesis of Inductances and Negative Capacitances 479 9.2.3.1 Synthesis of Inductors 479 9.2.3.2 Synthesis of Negative Capacitances 480 9.2.4 Why Negative Capacitance Is Effective? 480 9.2.5 Effectiveness of the Negative Capacitance from a Control System Perspective 482 9.2.6 Electrical Analogy of Shunted Piezoelectric Networks 485 9.3 Finite Element Modeling of Structures Treated with Shunted Piezo-Networks 487 9.3.1 Equivalent Complex Modulus Approach of Shunted Piezo-Networks 487 9.3.2 Coupled Electromechanical Field Approach of Shunted Piezo-Networks 491 9.4 Active Shunted Piezoelectric Networks 496 9.4.1 Basic Configurations 496 9.4.2 Dynamic Equations 498 9.4.2.1 Short-Circuit Configuration 498 9.4.2.2 Open-Circuit Configuration 498 9.4.2.3 Resistive-Shunted Configuration 498 9.4.3 More on the Resistive Shunting Configuration 498 9.4.4 Open-Circuit to Resistive Shunting (OC-RS) Configuration 500 9.4.4.1 Dynamic Equations 500 9.4.4.2 Switching Between OC and RS Modes 500 9.4.5 Energy Dissipation of Different Shunting Configurations 503 9.4.5.1 Energy Dissipation with Resistive Shunting 503 9.4.5.2 Energy Dissipation with OC-RS Switched Shunting 503 9.5 Multi-Mode Vibration Control with Shunted Piezoelectric Networks 504 9.5.1 Multi-Mode Shunting Approaches 504 9.5.2 Parameters of Behrens et al.’s Multi-Mode Shunting Network 507 9.5.2.1 Components of the Current Flowing Branches 507 9.5.2.2 Components of the Shunting Branches 507 9.6 Summary 510 References 511 10 Vibration Control with Periodic Structures 523 10.1 Introduction 523 10.2 Basics of Periodic Structures 524 10.2.1 Overview 524 10.2.2 Transfer Matrix Method 525 10.2.2.1 The Transfer Matrix 525 10.2.2.2 Basic Properties of the Transfer Matrix 526 10.3 Filtering Characteristics of Passive Periodic Structures 533 10.3.1 Overview 533 10.3.2 Periodic Rods in Longitudinal Vibrations 534 10.4 Natural Frequencies, Mode Shapes, and Response of Periodic Structures 535 10.4.1 Natural Frequencies and Response 535 10.4.2 Mode Shapes 539 10.5 Active Periodic Structures 541 10.5.1 Modeling of Active Periodic Structures 543 10.5.2 Dynamics of One Cell 543 10.5.2.1 Dynamics of the Passive Sub-Cell 543 10.5.2.2 Dynamics of the Active Sub-Cell 543 10.5.2.3 Dynamics of the Entire Cell 545 10.5.2.4 Dynamics of the Entire Periodic Structure 546 10.6 Localization Characteristics of Passive and Active Aperiodic Structures 549 10.6.1 Overview 549 10.6.2 Localization Factor 550 10.7 Periodic Rod with Periodic Shunted Piezoelectric Patches 559 10.7.1 Transfer Matrix of a Plain Rod Element 559 10.7.2 Transfer Matrix of a Rod/Piezo-Patch Element 560 10.7.3 Transfer Matrix of a Unit Cell 561 10.8 Two-Dimensional Active Periodic Structure 562 10.8.1 Dynamics of Unit Cell 562 10.8.2 Formulation of Phase Constant Surfaces 566 10.8.3 Filtering Characteristics 568 10.9 Periodic Structures with Internal Resonances 569 10.9.1 Dynamics of Conventional Periodic Structure 570 10.9.2 Dynamics of Periodic Structure with Internal Resonances 572 10.9.2.1 Equivalent Mass. Of the Mass-In-Mass Arrangement 572 10.9.2.2 Transfer Matrix of the Mass-In-Mass Arrangement 572 10.10 Summary 578 References 578 11 Nanoparticle Damping Composites 589 11.1 Introduction 589 11.2 Nanoparticle-Filled Polymer Composites 590 11.2.1 Composites with Unidirectional Inclusions 591 11.2.2 Arbitrarily Oriented Inclusion Composites 599 11.3 Comparisons with Classical Filler Reinforcement Methods 607 11.4 Applications of Carbon Black/Polymer Composites 614 11.4.1 Basic Physical Characteristics 614 11.4.2 Modeling of the Piezo-Resistance of CB/Polymer Composites 617 11.4.3 The Piezo-Resistivity of CB/Polymer Composites 619 11.5 CB/Polymer Composite as a Shunting Resistance of Piezoelectric Layers 620 11.5.1 Finite Element Model 620 11.5.2 Condensed Model of a Unit Cell 624 11.6 Hybrid Composites with Shunted Piezoelectric Particles 629 11.6.1 Composite Description and Assumptions 629 11.6.2 Shunted Piezoelectric Inclusions 631 11.6.3 Typical Performance Characteristics of Hybrid Composites 631 11.7 Summary 636 References 636 12 Power Flow in Damped Structures 651 12.1 Introduction 651 12.2 Vibrational Power 651 12.2.1 Basic Definitions 651 12.2.2 Relationship to System Energies 652 12.2.3 Basic Characteristics of the Power Flow 653 12.3 Vibrational Power Flow in Beams 656 12.4 Vibrational Power of Plates 661 12.4.1 Basic Equations of Vibrating Plates 661 12.4.2 Power Flow and Structural Intensity 662 12.4.3 Control of the Power Flow and Structural Intensity 668 12.4.4 Power Flow and Structural Intensity for Plates with Passive and Active Constrained Layer Damping Treatments 671 12.5 Power Flow and Structural Intensity for Shells 679 12.6 Summary 682 References 682 Glossary 699 Appendix 703 Index 715