Active and Passive Vibration Damping

  • ID: 2866173
  • Book
  • 492 Pages
  • John Wiley and Sons Ltd
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An important work presenting in one volume the application of viscoelastic damping materials to control vibration and noise of structures, machinery, and vehicles

Focuses primarily on the application of passive as well as actively treated viscoelastic damping materials to control vibration and noise of structures, machinery, and vehicles.  Emphasis is placed on presenting the basic principles and potential applications of passive and active vibration damping technologies. The presentation encompasses a mix between the associated physical fundamentals, governing theories and optimal design strategies of various configurations of vibration damping treatments. Utilization of the smart materials to augment the vibration damping of passive treatments is the common thread which is pursued, in depth, among all the chapters of the book.

This book is divided into two parts, the first dealing with materials theory and the second dealing with the practical application to vibrations.  It presents the basics of various damping effective treatments such as constrained layers, shunted piezoelectric treatments, electromagnetic and shape memory fibers. Classical aspects of viscoelastic materials models are also analyzed from the experimental characterization of the material coefficients as well as their modeling.  New models such as Golla–Hughes–McTavish  (GHM ) model, augmented temperature field (ATF) model, fractional derivatives (FD) models. Modal strain energy (MSE) models are also deeply analyzed.  Part II of the book is devoted to the detailed descriptions of advanced damping treatments using all the fundamentals presented in Part I.  Each chapter of the book ends with a number of problems that cover the different aspects of theoretical analysis, design, and applications of vibration damping technologies.

The book has a large number of numerical examples to reinforce the understanding of the theories covered, providing the means for exercising the knowledge gained, and emphasizing the learning of strategies for the design and application of active and passive vibration damping systems.  The examples are supported by a set of MATLAB software modules to enable the designers of vibration damping systems to extend the theories presented to various applications.

Written by an internationally recognized authority and pioneer, it summarizes and presents comprehensive coverage in one volume material that until now appears throughout a selection of references
Presents a mix of the associated physical fundamentals, governing theories and optimal design strategies of various configurations of vibration damping treatments, a comprehensive coverage not available elsewhere
With companion website including MATLAB software, solutions to all problems, and power point slides, the supporting tools enable hands–on experience of the analysis, design, optimization, and application to a wide range of situations

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Preface 3

Glossary of Terms 6

Abbreviations 11

List of Symbols 14

Greek Symbols 19

Subscripts 22

Superscripts 22

Operators 23

Table of Contents 24

PART I FUNDAMENTALS OF VISCOELASTIC DAMPING 39

1 Vibration Damping 41

1.1 Overview 41

1.2 Passive, Active and Hybrid Vibration Control 41

1.2.1 Passive Damping 42

1.2.1.1 Free and Constrained Damping Layers 42

1.2.1.2 Shunted Piezoelectric Treatments 43

1.2.1.3 Damping Layers with Shunted Piezoelectric Treatments 43

1.2.1.4 Magnetic Constrained Layer Damping (MCLD) 44

1.2.1.5 Damping with Shape Memory Fibers 44

1.2.2 Active Damping 45

1.2.3 Hybrid Damping 45

1.2.3.1 Active Constrained Layer Damping (ACLD) 46

1.2.3.2 Active Piezoelectric Damping Composites (APDC) 46

1.2.3.3 Electromagnetic Damping Composites (EMDC) 47

1.2.3.4 Active Shunted Piezoelectric Networks 48

1.3 Summary 48

References 49

2 Viscoelastic Damping 51

2.1 Introduction 51

2.2 Classical Models of Viscoelastic Materials 51

2.2.1 Characteristics in the Time Domain 52

2.2.2 Basics for Time Domain Analysis 53

2.2.3 Detailed Time Response of Maxwell and Kelvin–Voigt Models 55

2.2.4 Detailed Time Response of the Poynting–Thomson Model 59

2.3 Creep Compliance and Relaxation Modulus 64

2.3.1 Direct Laplace Transformation Approach 65

2.3.2 Approach of Simultaneous Solution of a Linear Set of Equilibrium, Kinematic and Constitutive Equations 66

2.4 Characteristics of the VEM in the Frequency Domain 70

2.5 Hysteresis and Energy Dissipation Characteristics of Viscoelastic Materials 73

2.5.1 Hysteresis Characteristics 73

2.5.2 Energy Dissipation 74

2.5.3 Loss Factor 74

2.5.3.1 Relationship Between Dissipation and Stored Elastic Energies 74

2.5.3.2 Relationship Between Different Strains 75

2.5.4 Storage Modulus 76

2.6 Fractional Derivative Models of Viscoelastic Materials 80

2.6.1 Basic Building Block of Fractional Derivative Models 80

2.6.2 Basic Fractional Derivative Models 81

2.6.3 Other Common Fractional Derivative Models 86

2.7 Viscoelastic Versus Other Types of Damping Mechanisms 89

2.8 Summary 91

References 92

Appendix 2.A Initial and Final Value Theorems 94

Appendix 2.B Fractional Calculus 95

2.B.1 Fractional Integration 95

2.B.2 Convolution Theorem 96

2.B.3 Fractional Derivatives 97

2.B.4 Laplace Transform of Fractional Derivatives 98

2.B.5 Grunwald–Letnikov Definition of Fractional Derivatives 99

Problems 102

3 Characterization of the Properties of Viscoelastic Materials 111

3.1 Introduction 111

3.2 Typical Behavior of Viscoelastic Materials 111

3.3 Frequency Domain Measurement Techniques of the Dynamic Properties of Viscoelastic Material 117

3.3.1 Dynamic, Mechanical, and Thermal Analyzer 118

3.3.2 Oberst Test Beam Method 121

3.3.2.1 Set–up and Beam Configurations 121

3.3.2.2 Parameters Extraction 123

3.4 Master Curves of Viscoelastic Materials 127

3.4.1 The Principle of Temperature–Frequency Superposition 127

3.4.2 The Use of the Master Curves 133

3.4.3 The Constant Temperature Lines 133

3.5 Time Domain Measurement Techniques of the Dynamic Properties of Viscoelastic Materials 135

3.5.1 Creep and Relaxation Measurement Methods 135

3.5.1.1 Testing Equipment 135

3.5.1.2 Typical Creep and Relaxation Behavior 136

3.5.1.3 Time–Temperature Superposition 138

3.5.1.4 Boltzmann Superposition Principle 140

3.5.1.5 Relationship between the Relaxation Modulus and Complex Modulus 142

3.5.1.6 Relationship between the Creep Compliance and Complex Compliance 144

3.5.1.7 Relationship between the Creep Compliance and Relaxation Modulus 146

3.5.1.8 Alternative Relationship between the Creep Compliance and Complex Compliance 147

3.5.1.9 Alternative Relationship between the Relaxation Modulus and Complex Modulus 148

3.5.1.10 Summary of the Basic Interconversion Relationship 149

3.5.1.11 Practical Issues in Implementation of Interconversion Relationships 150

3.5.2 Split Hopkinson Pressure Bar Method 161

3.5.2.1 Overview 161

3.5.2.2 Theory of 1–D SHPB 162

3.5.2.3 Complex Modulus of a VEM from SHPB Measurements 168

3.5.3 Wave Propagation Method 174

3.5.4 Ultrasonic Wave Propagation Method 178

3.5.4.1 Overview 178

3.5.4.2 Theory 179

3.5.4.3 Measurement of the Phase Velocity and Attenuation Factor 181

3.5.4.4 Typical Attenuation Factors 184

3.6 Summary 188

References 189

Appendix 3.A Convolution Theorem 192

Problems 193

4 Viscoelastic Materials 200

4.1 Introduction 200

4.2 Golla–Hughes–McTavish (GHM) Model 200

4.2.1 Motivation of the GHM Model 201

4.2.2 Computation of the Parameters of the GHM Mini–Oscillators 207

4.2.3 On the Structure of the GHM Model 211

4.2.3.1 Other Forms of GHM Structures 211

4.2.3.2 Relaxation Modulus of the GHM Model 212

4.2.4 Structural Finite Element Models of Rods Treated With VEM 213

4.2.4.1 Unconstrained Layer Damping 213

4.2.4.2 Constrained Layer Damping 219

4.3 Structural Finite Element Models of Beams Treated with VEM 227

4.3.1 Degrees of Freedom 227

4.3.2 Basic Kinematic Relationships 228

4.3.3 Stiffness and Mass Matrices of the Beam/VEM Element 229

4.3.4 Equations of Motion of the Beam/VEM Element 231

4.4 Generalized Maxwell Model (GMM) 234

4.4.1 Overview 234

4.4.2 Internal Variable Representation of the GMM 235

4.4.2.1 Single Degree of Freedom System 235

4.4.2.2 Multi–Degree of Freedom System 237

4.4.2.3 Condensation of the Internal Degrees of Freedom 238

4.4.2.4 Direct Solution of Coupled Structural and Internal Degrees of Freedom 238

4.5 Augmenting Thermodynamic Field (ATF) Model 242

4.5.1 Overview 242

4.5.2 Equivalent Damping Ratio of the ATF Model 244

4.5.3 Multi–degree of Freedom ATF Model 245

4.5.4 Integration with a Finite Element Model 245

4.6 Fractional Derivative (FD) Models 247

4.6.1 Overview 247

4.6.2 Internal Degrees of Freedom of Fractional Derivative Models 249

4.6.3 Grunwald Approximation of Fractional Derivative 250

4.6.4 Integration Fractional Derivative Approximation with Finite Element 251

4.6.4.1 Viscoelastic Rod 251

4.6.4.2 Beam with Passive Constrained Layer Damping (PCLD) Treatment 254

4.7 Finite Element Modeling of Plates Treated with Passive Constrained Layer Damping 259

4.7.1 Overview 259

4.7.2 The Stress and Strain Characteristics 260

4.7.2.1 The Plate and the Constraining Layers 260

4.7.2.2 The VEM Layer 260

4.7.3 The Potential and Kinetic Energies 261

4.7.4 The Shape Functions 261

4.7.5 The Stiffness Matrices 263

4.7.6 The Mass Matrices 264

4.7.7 The Element and Overall Equations of Motion 264

4.8 Finite Element Modeling of Shells Treated with Passive Constrained Layer Damping 270

4.8.1 Overview 270

4.8.2 Stress–Strain Relationships 270

4.8.2.1 Shell and Constraining Layer 270

4.8.2.2 Viscoelastic Layer 272

4.8.3 Kinetic and Potential Energies 273

4.8.4 The Shape Functions 274

4.8.5 The Stiffness Matrices 274

4.8.6 The Mass Matrices 275

4.8.7 The Element and Overall Equations of Motion 276

4.9 Summary 281

References 282

Problems 285

5 Finite Element Modeling of Viscoelastic Damping by Modal Strain Energy Method 292

5.1 Introduction 292

5.2 Modal Strain Energy (MSE) Method 292

5.3 Modified Modal Strain Energy (MSE) Methods 299

5.3.1 Weighted Stiffness Matrix Method (WSM) 299

5.3.2 Weighted Storage Modulus Method (WSTM) 300

5.3.3 Improved Reduction System Method (IRS) 301

5.3.4 Low Frequency Approximation Method (LFA) 302

5.4 Summary of Modal Strain Energy Methods 306

5.5 Modal Strain Energy as a Metric for Design of Damping Treatments 306

5.6 Perforated Damping Treatments 314

5.6.1 Overview 314

5.6.2 Finite Element Modeling 314

5.6.2.1 Element Energies 317

5.6.2.2 Topology Optimization of Unconstrained Layer Damping 321

5.6.2.3 Sensitivity Analysis 322

5.7 Summary 329

References 330

Problems 332

6 Energy Dissipation in Damping Treatments 340

6.1 Introduction 340

6.2 Passive Damping Treatments of Rods 340

6.2.1 Passive Constrained Layer Damping 340

6.2.1.1 Equation of Motion 341

6.2.1.2 Energy Dissipation 345

6.2.2 Passive Unconstrained Layer Damping 348

6.3 Active Constrained Layer Damping Treatments of Rods 350

6.3.1 Equation of Motion 350

6.3.2 Boundary Control Strategy 352

6.3.3 Energy Dissipation 354

6.4 Passive Constrained Layer Damping Treatments of Beams 359

6.4.1 Basic Equations of Damped Beams 359

6.4.2 Bending Energy of Beams 359

6.4.3 Energy Dissipated in Beams with Passive Constrained Layer Damping 360

6.5 Active Constrained Layer Damping Treatments of Beams 370

6.6 Passive and Active Constrained Layer Damping Treatments of Plates 373

6.6.1 Kinematic Relationships 374

6.6.2 Energies of the PCLD and ACLD Treatments 374

6.6.2.1 The Potential Energies 374

6.6.2.2 The Kinetic Energy 375

6.6.2.3 Work Done 375

6.6.3 The models of the PCLD and ACLD Treatments 376

6.6.4 Boundary Control of Plates with ACLD Treatments 377

6.6.5 Energy Dissipation and Loss Factors of Plates with PCLD and ACLD Treatments 378

6.7 Passive and Active Constrained Layer Damping Treatments of Axi–Symmetric Shells 382

6.7.1 Background 382

6.7.2 The Concept of the Active Constrained Layer Damping 383

6.7.3 Variational Modeling of the Shell/ACLD System 384

6.7.3.1 Main Assumptions of the Model 384

6.7.3.2 Kinematic Relationships 385

6.7.3.3 Stress–Strain Relationships 385

6.7.3.4 Energies of Shell/ACLD System 388

6.7.3.5 The Model 389

6.7.4 Boundary Control Strategy 391

6.7.4.1 Overview 391

6.7.4.2 Control Strategy 392

6.7.4.3 Implementation of the Boundary Control Strategy 393

6.7.4.4 Transverse Compliance and Longitudinal Deflection 393

6.7.5 Energy Dissipated in the ACLD Treatment of an Axi–symmetric Shell 398

6.8 Summary 401

References 402

Appendix 6.A Basic Identities 405

Appendix 6.B Piezoelectricity 406

6.B.1 Piezoelectric Effects 406

6.B.2 Basic Constitutive Equations 407

Problems 409

PART II ADVANCED DAMPING TREATMENTS 416

7 Vibration Damping of Structures Using Active Constrained Layer Damping 418

7.1 Introduction 418

7.2 Motivation For Using Passive and Active Constrained Layer Damping 418

7.2.1 Base Structure 419

7.2.2 Structure Treated with Unconstrained Passive Layer Damping 422

7.2.3 Structure Treated with Constrained Passive Layer Damping 427

7.2.4 Structure Treated with Active Constrained Passive Layer Damping 431

7.3 Active Constrained Layer Damping For Beams 437

7.3.1 Introduction 437

7.3.2 Concept of Active Constrained Layer Damping 437

7.3.3 Finite Element Modeling of a Beam/ACLD Assembly 440

7.3.3.1 The Model 440

7.3.3.2 Equations of Motion 444

7.3.4 Distributed–Parameter Modeling of a Beam/ACLD Assembly 454

7.3.4.1 Overview 454

7.3.4.2 The Energies and Work Done on the Beam/ACLD Assembly 455

7.3.4.3 The Distributed–Parameter Model 456

7.3.4.4 Globally Stable Boundary Control Strategy 458

7.3.4.5 Implementation of the Globally Stable Boundary Control Strategy 459

7.3.4.6 Response of the Beam/ACLD Assembly 460

7.4 Active Constrained Layer Damping For Plates 466

7.4.1 Control forces and Moments Generated by the Active Constraining Layer 466

7.4.1.1 The in–plane Piezoelectric Forces 466

7.4.1.2 The Piezoelectric Moments 466

7.4.1.3 Piezoelectric Sensor 467

7.4.1.4 Control Voltage to Piezoelectric Constraining Layer 467

7.4.2 Equations of Motion 468

7.5 Active Constrained Layer Damping for Shells 474

7.5.1 Control Forces and Moments Generated by the Active Constraining Layer 474

7.5.2 Equations of Motion 475

7.6 Summary 482

References 483

Appendix 7.A Piezoelectric Sensor Basic Equations 485

7.A.1 Basic Equations 485

7.A.2 Basic Sensor Configurations 486

7.A.3 Output Voltage of Sensor 487

Problems 488

8 Advanced Damping Treatments 496

8.1 Introduction 496

8.2 Stand–Off Damping Treatments 497

8.2.1 Background of Stand–off Damping Treatments 497

8.2.2 The Stand–off Damping Treatments 498

8.2.3 Distributed–Parameter Model of the Stand–off Layer Damping Treatment 500

8.2.3.1 Kinematic Equations 500

8.2.3.2 Constitutive Equations 501

8.2.4 Distributed Transfer Function Method 506

8.2.5 Finite Element Model 507

8.2.6 Summary 514

8.3 Functionally Graded Damping Treatments 514

8.3.1 Background of Functionally Graded Constrained Layer Damping 515

8.3.2 Concept of Constrained Layer Damping With Functionally Graded Viscoelastic Cores 515

8.3.3 Finite Element Model 517

8.3.3.1 Quasi–Static Model of the Passive Constrained Damping Layer of Plunket and Lee (1970) 517

8.3.3.2 Dispersion Characteristics of Passive Constrained Damping Layer with Uniform and Functionally Graded Cores 524

8.3.4 Summary 535

8.4 Passive and Active Damping Composite Treatments 536

8.4.1 Passive Composite Damping Treatments 536

8.4.2 Active Composite Damping Treatments 540

8.4.3 Finite Element Modeling of Beam with APDC 543

8.4.3.1 Model and Main Assumptions 543

8.4.3.2 Kinematics 544

8.4.3.3 Degrees of Freedom and Shape Functions 545

8.4.3.4 System Energies 546

8.4.3.5 Equations of Motion 548

8.4.3.6 Control Law 548

8.4.4 Summary 557

8.5 Magnetic Damping Treatments 558

8.5.1 Magnetic Constrained Layer Damping Treatments 558

8.5.2 Analysis of Magnetic Constrained Layer Damping Treatments 560

8.5.2.1 Equation of Motion 560

8.5.2.2 Response of the MCLD Treatment 563

8.5.3 Passive Magnetic Composites 566

8.5.3.1 Concept of Passive Magnetic Composite Treatment 566

8.5.3.2 Finite Element Modeling of Beams with PMC Treatment 568

8.5.4 Summary 582

8.6 Negative Stiffness Composites 584

8.6.1 Motivation to Negative Stiffness Composites 584

8.6.1.1 Sinusoidal Excitation 585

8.6.1.2 Impact Loading 594

8.6.2 Magnetic Composite with Negative Stiffness Inclusions 595

8.6.2.1 Force Between Two Bar Magnets 596

8.6.2.2 Finite Element Model 598

8.7 Summary 604

References 605

Appendix 8.A Matrices of the Models of Passive Stand–Off Layer 610

8.A.1 Distributed Transfer Function Model 610

8.A.2 Finite Element Model 611

Appendix 8.B The Electromechanical Coupling Factor of One Piezoelectric Rod 612 Appendix 8.C Constitutive Equations of APDC 614

Appendix 8.D Magnetic Forces in the Passive Magnetic Composite 616

Appendix 8.E Stiffness and Mass Matrices Passive Magnetic Composite (PMC) 618

8.E.1 Strain Energies 618

8.E.2 Kinetic Energies 620

8.E.3 Work Done by the Magnetic Forces 621

8.E.4 Geometric Stiffness Matrix 621

Problems 622

9 Vibration Damping With Shunted Piezoelectric Networks 631

9.1 Introduction 632

9.2 Shunted Piezoelectric Patches 632

9.2.1 Basics of Piezoelectricity 632

9.2.1.1 Effect of Electrical Boundary Conditions 634

9.2.1.2 Effect of Mechanical Boundary Conditions 635

9.2.2 Basics of Shunted Piezo–Networks 636

9.2.2.1 Resistive Shunted Circuit 638

9.2.2.2 Resistive and Inductive Shunted Circuit 640

9.2.2.3 Resistive, Capacitive, and Inductive Shunted Circuit 642

9.2.3 Electronic Synthesis of Inductances and Negative Capacitances 645

9.2.3.1 Synthesis of Inductors 645

9.2.3.2 Synthesis of Negative Capacitances 646

9.2.4 Why Negative Capacitance is Effective? 647

9.2.5 Effectiveness of the Negative Capacitance From A Control System Perspective 649

9.2.6 Electrical Analogy of Shunted Piezoelectric Networks 653

9.3 Finite Element Modeling of Structures Treated With Shunted Piezo–Networks 655

9.3.1 Equivalent Complex Modulus Approach of Shunted Piezo–Networks 655

9.3.2 Coupled Electromechanical Field Approach of Shunted Piezo–Networks 659

9.4 Active Shunted Piezoelectric Networks 667

9.4.1 Basic Configurations 667

9.4.2 Dynamic Equations 668

9.4.2.1 Short–Circuit Configuration 668

9.4.2.2 Open–Circuit Configuration 668

9.4.2.3 Resistive–Shunted Configuration 669

9.4.3 More on the Resistive Shunting Configuration 669

9.4.4 Open–Circuit to Resistive Shunting (OC–RS) Configuration 671

9.4.4.1 Dynamic Equations 671

9.4.4.2 Switching Between OC and RS Modes 672

9.4.5 Energy Dissipation of Different Shunting Configurations 674

9.4.5.1 Energy Dissipation with Resistive Shunting 675

9.4.5.2 Energy Dissipation with OC/RS Switched Shunting 675

9.5 Multi–Mode Vibration Control With Shunted Piezoelectric Networks 678

9.5.1 Multi–Mode Shunting Approaches 678

9.5.2 Parameters of Behrens et al. Multi–Mode Shunting Network 681

9.5.2.1 Components of the Current Flowing Branches 681

9.5.2.2 Components of the Shunting Branches 681

9.6 Summary 685

References 686

Appendix 9.A Electromechanical Coupling Factor 688

Problems 690

10 Vibration Control With Periodic Structures 702

10.1 Introduction 702

10.2 Basics of Periodic Structures 704

10.2.1 Overview 704

10.2.2 Transfer Matrix Method 704

10.2.2.1 The Transfer Matrix 704

10.2.2.2 Basic Properties of the Transfer Matrix 706

10.3 Filtering Characteristics of Passive Periodic Structures 716

10.3.1 Overview 716

10.3.2 Periodic Rods in Longitudinal Vibrations 716

10.4 Natural Frequencies, Mode Shapes and Response of Periodic Structures 720

10.4.1 Natural Frequencies and Response 720

10.4.2 Mode Shapes 725

10.5 Active Periodic Structures 727

10.5.1 Modeling of Active Periodic Structures 727

10.5.2 Dynamics of One Cell 729

10.5.2.1 Dynamics of the Passive Sub–Cell 729

10.5.2.2 Dynamics of the Active Sub–Cell 729

10.5.2.3 Dynamics of the Entire Cell 731

10.5.2.4 Dynamics of the Entire Periodic Structure 732

10.6 Localization Characteristics of Passive And Active Aperiodic Structures 737

10.6.1 Overview 737

10.6.2 Localization Factor 737

10.7 Periodic Rod with Periodic Shunted Piezoelectric Patches 749

10.7.1 Transfer Matrix of a Plain Rod Element: 749

10.7.2 Transfer Matrix of a Rod/Piezo–Patch Element 750

10.7.3 Transfer Matrix of a Unit Cell: 751

10.8 Two–Dimensional Active Periodic Structure 754

10.8.1 Dynamics of Unit Cell 754

10.8.2 Formulation of Phase Constant Surfaces 757

10.8.3 Filtering Characteristics 759

10.9 Periodic Structures with Internal Resonances 762

10.9.1.1 Dynamics of Conventional Periodic Structure 763

10.9.1.2 Dynamics of Periodic Structure with Internal Resonances 764

10.10 Summary 772

References 773

Appendix 10.A The Wavelet Transform 776

Problems 778

11 Nanoparticle Damping Composites 786

11.1 Introduction 786

11.2 Nanoparticle–Filled Polymer Composites 789

11.2.1 Composites with Unidirectional Inclusions 789

11.2.2 Arbitrarily Oriented Inclusion Composites 798

11.3 Comparisons with Classical Filler Reinforcement Methods 811

11.1 Applications of Carbon Black/Polymer Composites 821

11.1.1 Basic Physical Characteristics 821

11.1.2 Modeling of the Piezo–Resistance of CB/Polymer Composites 823

11.1.3 The Piezo–Resistivity of CB/Polymer Composites 826

11.2 CB/Polymer Composite as a Shunting Resistance Of Piezoelectric–Layers 828

11.2.1 Finite Element Model 828

11.2.2 Condensed Model of a Unit Cell 832

11.3 Hybrid Composites With Shunted Piezoelectric Particles 839

11.3.1 Composite Description and Assumptions 839

11.3.2 Shunted Piezoelectric Inclusions 840

11.3.3 Typical Performance Characteristics of Hybrid Composites 841

11.4 Summary 845

References 846

Appendix 11.A Transformation Matrix 849

Appendix 11.B Reinforcement Mechanics of Particle–Filled Polymers 850

11.B.1 Basics 850

Problems 854

12 Power Flow In Damped Structures 864

12.1 Introduction 864

12.2 Vibrational Power 865

12.2.1 Basic Definitions 865

12.2.2 Relationship to System Energies 866

12.2.3 Basic Characteristics of the Power Flow 866

12.3 Vibrational Power Flow in Beams 870

12.4 Vibrational Power of Plates 877

12.4.1 Basic Equations of Vibrating Plates 877

12.4.2 Power Flow and Structural Intensity 879

12.4.3 Control of the Power Flow and Structural Intensity 887

12.4.4 Power Flow and Structural Intensity for Plates with Passive and Active Constrained Layer Damping Treatments 890

12.5 Power Flow and Structural Intensity for Shells 901

12.6 Summary 905

References 906

Problems 909

Appendix 12.A Calculation of Power Flow in ANSYS 916

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Amr M. Baz
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