Vibration Theory and Applications with Finite Elements and Active Vibration Control

<p>Preface xv</p>
<p>Acknowledgments and Dedication xxi</p>
<p>About the Companion Website xxiii</p>
<p>List of Acronyms xxv</p>
<p>1 Background, Motivation, and Overview 1</p>
<p>1.1 Introduction 1</p>
<p>1.2 Background 1</p>
<p>1.2.1 Units 5</p>
<p>1.3 Our Vibrating World 6</p>
<p>1.3.1 Small–Scale Vibrations 6</p>
<p>1.3.2 Medium–Scale (Mesoscale) Vibrations 8</p>
<p>1.3.3 Large–Scale Vibrations 8</p>
<p>1.4 Harmful Effects of Vibration 9</p>
<p>1.4.1 Human Exposure Limits 9</p>
<p>1.4.2 High–Cycle Fatigue Failure 11</p>
<p>1.4.3 Rotating Machinery Vibration 21</p>
<p>1.4.4 Machinery Productivity 27</p>
<p>1.4.5 Fastener Looseness 28</p>
<p>1.4.6 Optical Instrument Blurring 28</p>
<p>1.4.7 Ethics and Professional Responsibility 29</p>
<p>1.4.8 Lifelong Learning Opportunities 29</p>
<p>1.5 Stiffness, Inertia, and Damping Forces 29</p>
<p>1.6 Approaches for Obtaining the Differential Equations of Motion 34</p>
<p>1.7 Finite Element Method 35</p>
<p>1.8 Active Vibration Control 37</p>
<p>1.9 Chapter 1 Exercises 37</p>
<p>1.9.1 Exercise Location 37</p>
<p>1.9.2 Exercise Goals 37</p>
<p>1.9.3 Sample Exercises: 1.6 and 1.11 38</p>
<p>References 38</p>
<p>2 Preparatory Skills: Mathematics, Modeling, and Kinematics 41</p>
<p>2.1 Introduction 41</p>
<p>2.2 Getting Started with MATLAB and MAPLE 42</p>
<p>2.2.1 MATLAB 42</p>
<p>2.2.2 MAPLE (Symbolic Math) 45</p>
<p>2.3 Vibration and Differential Equations 50</p>
<p>2.3.1 MATLAB and MAPLE Integration 50</p>
<p>2.4 Taylor Series Expansions and Linearization 56</p>
<p>2.5 Complex Variables (CV) and Phasors 60</p>
<p>2.6 Degrees of Freedom, Matrices, Vectors, and Subspaces 63</p>
<p>2.6.1 Matrix Vector Related Definitions and Identities 69</p>
<p>2.7 Coordinate Transformations 75</p>
<p>2.8 Eigenvalues and Eigenvectors 79</p>
<p>2.9 Fourier Series 80</p>
<p>2.10 Laplace Transforms, Transfer Functions, and Characteristic Equations 83</p>
<p>2.11 Kinematics and Kinematic Constraints 86</p>
<p>2.11.1 Particle Kinematic Constraint 86</p>
<p>2.11.2 Rigid Body Kinematic Constraint 90</p>
<p>2.11.3 Assumed Modes Kinematic Constraint 96</p>
<p>2.11.4 Finite Element Kinematic Constraint 98</p>
<p>2.12 Dirac Delta and Heaviside Functions 100</p>
<p>2.13 Chapter 2 Exercises 101</p>
<p>2.13.1 Exercise Location 101</p>
<p>2.13.2 Exercise Goals 101</p>
<p>2.13.3 Sample Exercises: 2.9 and 2.17 101</p>
<p>References 102</p>
<p>3 Equations of Motion by Newton s Laws 103</p>
<p>3.1 Introduction 103</p>
<p>3.2 Particle Motion Approximation 103</p>
<p>3.3 Planar (2D) Rigid Body Motion Approximation 107</p>
<p>3.3.1 Translational Equations of Motion 107</p>
<p>3.3.2 Rotational Equation of Motion 108</p>
<p>3.4 Impulse and Momentum 129</p>
<p>3.4.1 Linear Impulse and Momentum 129</p>
<p>3.4.2 Angular Impulse and Momentum 134</p>
<p>3.5 Variable Mass Systems 138</p>
<p>3.6 Chapter 3 Exercises 140</p>
<p>3.6.1 Exercise Location 140</p>
<p>3.6.2 Exercise Goals 140</p>
<p>3.6.3 Sample Exercises: 3.8 and 3.21 141</p>
<p>References 141</p>
<p>4 Equations of Motion by Energy Methods 143</p>
<p>4.1 Introduction 143</p>
<p>4.2 Kinetic Energy 143</p>
<p>4.2.1 Particle Motion 143</p>
<p>4.2.2 Two–Dimensional Rigid Body Motion 144</p>
<p>4.2.3 Constrained 2D Rigid Body Motion 146</p>
<p>4.3 External and Internal Work and Potential Energy 147</p>
<p>4.3.1 External Work and Potential Energy 150</p>
<p>4.4 Power and Work Energy Laws 151</p>
<p>4.4.1 Particles 151</p>
<p>4.4.2 Rigid Body with 2D Motion 153</p>
<p>4.5 Lagrange Equation for Particles and Rigid Bodies 157</p>
<p>4.5.1 Derivation of the Lagrange Equation 158</p>
<p>4.5.2 System of Particles 160</p>
<p>4.5.3 Collection of Rigid Bodies 162</p>
<p>4.5.4 Potential, Circulation, and Dissipation Functions 168</p>
<p>4.5.5 Summary for Lagrange Equation 182</p>
<p>4.5.6 Nonconservative Generalized Forces and Virtual Work 183</p>
<p>4.5.7 Effects of Gravity for the Lagrange Approach 195</p>
<p>4.5.8 Automating the Derivation of the LE Approach 201</p>
<p>4.6 LE for Flexible, Distributed Mass Bodies: Assumed Modes Approach 211</p>
<p>4.6.1 Assumed Modes Kinetic Energy and Mass Matrix Expressions 212</p>
<p>4.6.2 Rotating Structures 215</p>
<p>4.6.3 Internal Forces and Strain Energy of an Elastic Object 216</p>
<p>4.6.4 The Assumed Modes Approximation 219</p>
<p>4.6.5 Generalized Force for External Loads Acting on a Deformable Body 221</p>
<p>4.6.6 Assumed Modes Model Generalized Forces for External Load Acting on a Deformable Body 221</p>
<p>4.6.7 LE for a System of Rigid and Deformable Bodies 223</p>
<p>4.7 LE for Flexible, Distributed Mass Bodies: Finite Element Approach General Formulation 267</p>
<p>4.7.1 Element Kinetic Energy and Mass Matrix 267</p>
<p>4.7.2 Element Stiffness Matrix 269</p>
<p>4.7.3 Summary 272</p>
<p>4.8 LE for Flexible, Distributed Mass Bodies: Finite Element Approach Bar/Truss Modes 275</p>
<p>4.8.1 Introduction 275</p>
<p>4.8.2 1D Truss/Bar Element 276</p>
<p>4.8.3 1D Truss/Bar Element: Element Stiffness Matrix 276</p>
<p>4.8.4 1D Truss/Bar Element: Element Mass Matrix 277</p>
<p>4.8.5 1D Truss/Bar Element: Element Damping Matrix 277</p>
<p>4.8.6 1D Truss/Bar Element: Generalized Force Vector 278</p>
<p>4.8.7 1D Truss/Bar Element: Nodal Connectivity Array 279</p>
<p>4.8.8 System of 1D Bar Elements: Matrix Assembly 280</p>
<p>4.8.9 Incorporation of Displacement Constraint 285</p>
<p>4.8.10 Modeling of 2D Trusses 289</p>
<p>4.8.11 2D Truss/Bar Element: Element Stiffness Matrix 292</p>
<p>4.8.12 2D Truss/Bar Element: Element Mass Matrix 293</p>
<p>4.8.13 2D Truss/Bar Element: Element Damping Matrix 294</p>
<p>4.8.14 2D Truss/Bar Element: Element Force Vector 295</p>
<p>4.8.15 2D Truss/Bar Element: Element Action Vector 296</p>
<p>4.8.16 2D Truss/Bar Element: Degree of Freedom Connectivity Array and Matrix Assembly 296</p>
<p>4.8.17 2D Truss/Bar Element: Rigid Region Modeling for 2D Trusses 303</p>
<p>4.9 Chapter 4 Exercises 306</p>
<p>4.9.1 Exercise Location 306</p>
<p>4.9.2 Exercise Goals 307</p>
<p>4.9.3 Sample Exercises: 4.43 and 4.52 307</p>
<p>References 308</p>
<p>5 Free Vibration Response 309</p>
<p>5.1 Introduction 309</p>
<p>5.2 Single Degree of Freedom Systems 309</p>
<p>5.2.1 SDOF Eigenvalues (Characteristic Roots) 310</p>
<p>5.2.2 SDOF Initial Condition Response 311</p>
<p>5.2.3 Log Decrement: A Measure of Damping Displacement–Based Measurement 313</p>
<p>5.2.4 Log Decrement: A Measure of Damping Acceleration–Based Measurement 315</p>
<p>5.3 Two–Degree–of–Freedom Systems 319</p>
<p>5.3.1 Special Case I (C=G=KC = 0) (Undamped, Nongyroscopic, and Noncirculatory Case) 322</p>
<p>5.3.2 Special Case IIC=KC = 0 (Undamped, Gyroscopic, and Noncirculatory Case) 332</p>
<p>5.3.3 Special Case III C=G=0 (Undamped, Nongyroscopic, and Circulatory Case) 342</p>
<p>5.4 N–Degree–of–Freedom Systems 346</p>
<p>5.4.1 General Identities 346</p>
<p>5.4.2 Undamped, Nongyroscopic, and Noncirculatory Systems Description 346</p>
<p>5.4.3 Undamped, Nongyroscopic, and Noncirculatory Systems Solution Form 347</p>
<p>5.4.4 Undamped, Nongyroscopic, and Noncirculatory Systems Orthogonality 349</p>
<p>5.4.5 Undamped, Nongyroscopic, and Noncirculatory Systems IC Response 351</p>
<p>5.4.6 Undamped, Nongyroscopic, and Noncirculatory Systems Rigid Body Modes 352</p>
<p>5.4.7 Summary 353</p>
<p>5.4.8 Undamped, Nongyroscopic, and Noncirculatory Systems Response to an IC Modal Displacement Distribution 363</p>
<p>5.4.9 Orthogonally Damped, Nongyroscopic, and Noncirculatory Systems Description 364</p>
<p>5.4.10 Orthogonally Damped, Nongyroscopic, and Noncirculatory Systems Eigenvalues and Eigenvectors 365</p>
<p>5.4.11 Orthogonally Damped, Nongyroscopic, and Noncirculatory Systems IC Response 366</p>
<p>5.4.12 Orthogonally Damped, Nongyroscopic, and Noncirculatory Systems Determination of C0 367</p>
<p>5.4.13 Nonorthogonally Damped System with Symmetric Mass, Stiffness, and Damping Matrices 377</p>
<p>5.4.14 Undamped, Gyroscopic, and Noncirculatory Systems Description 379</p>
<p>5.4.15 Undamped, Gyroscopic, and Noncirculatory Systems Eigenvalues and Eigenvectors 380</p>
<p>5.4.16 Undamped, Gyroscopic, and Noncirculatory Systems Biorthogonality 385</p>
<p>5.4.17 General Linear Systems Description 388</p>
<p>5.4.18 General Linear Systems Biorthogonality 389</p>
<p>5.5 Infinite Dof Continuous Member Systems 390</p>
<p>5.5.1 Introduction 390</p>
<p>5.5.2 Transverse Vibration of Strings and Cables 390</p>
<p>5.5.3 Axial Vibration of a Uniform Bar 394</p>
<p>5.5.4 Torsion of Bars 396</p>
<p>5.5.5 Euler Bernoulli (Classical) Beam Theory 400</p>
<p>5.5.6 Timoshenko Beam Theory 404</p>
<p>5.6 Unstable Free Vibrations 408</p>
<p>5.6.1 Oil Film Bearing–Induced Instability 411</p>
<p>5.7 Summary 418</p>
<p>5.8 Chapter 5 Exercises 418</p>
<p>5.8.1 Exercise Location 418</p>
<p>5.8.2 Exercise Goals 418</p>
<p>5.8.3 Sample Exercises: 5.25 and 5.35 419</p>
<p>References 419</p>
<p>6 Vibration Response Due to Transient Loading 421</p>
<p>6.1 Introduction 421</p>
<p>6.2 Single Degree of Freedom Transient Response 421</p>
<p>6.2.1 Direct Analytical Solution Method 422</p>
<p>6.2.2 Laplace Transform Method 428</p>
<p>6.2.3 Convolution Integral 434</p>
<p>6.2.4 Response to Successive Disturbances 438</p>
<p>6.2.5 Pulsed Excitations 440</p>
<p>6.2.6 Response Spectrum 444</p>
<p>6.3 Modal Condensation of Ndof: Transient Forced Vibrating Systems 451</p>
<p>6.3.1 Undamped and Orthogonally Damped Nongyroscopic, Noncirculatory Systems 452</p>
<p>6.3.2 Unconstrained Structures 465</p>
<p>6.3.3 Base Excitation 475</p>
<p>6.3.4 Participation Factor and Modal Effective Mass 477</p>
<p>6.3.5 General Nonsymmetric, Nonorthogonal Damping Models 488</p>
<p>6.4 Numerical Integration of Ndof Transient Vibration Response 493</p>
<p>6.4.1 Second–Order System NI Algorithms 494</p>
<p>6.4.2 First–Order System NI Algorithms 500</p>
<p>6.5 Summary 521</p>
<p>6.6 Chapter 6 Exercises 522</p>
<p>6.6.1 Exercise Location 522</p>
<p>6.6.2 Exercise Goals 522</p>
<p>6.6.3 Sample Exercises: 6.18 and 6.21 522</p>
<p>References 523</p>
<p>7 Steady–State Vibration Response to Periodic Loading 525</p>
<p>7.1 Introduction 525</p>
<p>7.2 Complex Phasor Approach 525</p>
<p>7.3 Single Degree of Freedom Models 527</p>
<p>7.3.1 Class I (f(t) 0, y(t) = 0): No Support Excitation Only External Forcing 529</p>
<p>7.3.2 Class II (f(t) = 0, y(t) 0): Only Support Excitation No External Forcing 535</p>
<p>7.3.3 Half Power Point Damping Identification 538</p>
<p>7.3.4 Phase Slope Damping Identification 541</p>
<p>7.3.5 Accurate Measurement of n 541</p>
<p>7.3.6 Experimental Parameter Identification: Receptances 543</p>
<p>7.3.7 Steady–State Harmonic Response of a Jeffcott Rotor and Rotor Balancing 545</p>
<p>7.3.8 Single Plane Influence Coefficient Balancing 549</p>
<p>7.3.9 Related American Petroleum Institute Standards for Balancing 551</p>
<p>7.3.10 Response of an SDOF Oscillator to Nonsinusoidal, Periodic Excitation 552</p>
<p>7.3.11 Forced Harmonic Response with Elastomeric Stiffness and Damping 556</p>
<p>7.4 Two Degree of Freedom Response 559</p>
<p>7.4.1 Vibration Absorber: Principles 560</p>
<p>7.4.2 Vibration Absorber: Mass Ratio Effect 563</p>
<p>7.5 N Degree of freedom Steady–State Harmonic Response 566</p>
<p>7.5.1 Direct Approach 566</p>
<p>7.5.2 Modal Approach 570</p>
<p>7.5.3 Receptances 572</p>
<p>7.5.4 Receptance–Based Synthesis 575</p>
<p>7.5.5 Dominance of a Single Mode in the Steady–State Harmonic Response 577</p>
<p>7.5.6 Receptance–Based Modal Parameter Identification: Method I 578</p>
<p>7.5.7 Receptance–Based Modal Parameter Identification: Method II 585</p>
<p>7.5.8 Modal Assurance Criterion for Mode Shape Correlation 590</p>
<p>7.6 Other Phasor Ratio Measures of Steady–State Harmonic Response 591</p>
<p>7.7 Summary 593</p>
<p>7.8 Chapter 7 Exercises 593</p>
<p>7.8.1 Exercise Location 593</p>
<p>7.8.2 Exercise Goals 594</p>
<p>7.8.3 Sample Exercises: 7.9 and 7.23 594</p>
<p>References 594</p>
<p>8 Approximate Methods for Large–Order Systems 595</p>
<p>8.1 Introduction 595</p>
<p>8.2 Guyan Reduction: Static Condensation 596</p>
<p>8.3 Substructures: Superelements 608</p>
<p>8.4 Modal Synthesis 609</p>
<p>8.4.1 Uncoupled System Equations 610</p>
<p>8.4.2 Coupled System Equations: Displacement Compatibility at Junction 611</p>
<p>8.4.3 Coupled System Equations: Subspace Condensation 611</p>
<p>8.5 Eigenvalue/Natural Frequency Changes for Perturbed Systems 620</p>
<p>8.5.1 Undamped, Nongyroscopic, Noncirculatory M and K Type Systems 620</p>
<p>8.5.2 Orthogonally Damped Systems 624</p>
<p>8.5.3 Rayleigh s Quotient 631</p>
<p>8.6 Summary 633</p>
<p>8.7 Chapter 8 Exercises 634</p>
<p>8.7.1 Exercise Location 634</p>
<p>8.7.2 Exercise Goals 634</p>
<p>8.7.3 Sample Exercises 8.4 and 8.13 634</p>
<p>References 635</p>
<p>9 Beam Finite Elements for Vibration Analysis 637</p>
<p>9.1 Introduction 637</p>
<p>9.2 Modeling 2D Frame Structures with Euler Bernoulli Beam Elements 637</p>
<p>9.2.1 2D Frame Element Stiffness Matrix and Strain Energy: Transverse Deflection 639</p>
<p>9.2.2 2D Frame Element Stiffness Matrix and Strain Energy: Axial Deflection 641</p>
<p>9.2.3 2D Frame Element Stiffness Matrix and Strain Energy 642</p>
<p>9.2.4 2D Frame Element Mass Matrix 642</p>
<p>9.2.5 2D Frame Element Force Vector 644</p>
<p>9.2.6 2D Frame Element Stiffness and Mass Matrices and Force Vector: Transformation to Global Coordinates 646</p>
<p>9.2.7 2D Frame: Beam Element Assembly Algorithm 654</p>
<p>9.2.8 Imposed Support Excitation Modeling 666</p>
<p>9.3 Three–Dimensional Timoshenko Beam Elements: Introduction 670</p>
<p>9.4 3D Timoshenko Beam Elements: Nodal Coordinates 672</p>
<p>9.5 3D Timoshenko Beam Elements: Shape Functions, Element Stiffness, and Mass Matrices 679</p>
<p>9.5.1 Strain Displacement Relations and Shear Form Factors 681</p>
<p>9.5.2 x1 x2 Plane, Translational Differential Equations, Shape Functions, and Stiffness and Mass Matrices 684</p>
<p>9.5.3 x1 x3 Plane, Translational Differential Equations, Shape Functions, and Stiffness and Mass Matrices 691</p>
<p>9.5.4 Axial (Longitudinal) x1 Differential Equation, Shape Functions, and Stiffness and Mass Matrices 696</p>
<p>9.5.5 Torsional 1 Differential Equation, Shape Functions, and Stiffness and Mass Matrices 698</p>
<p>9.5.6 Twelve–Dof Element Stiffness Matrix 700</p>
<p>9.5.7 Twelve–Dof Element Mass Matrix 703</p>
<p>9.6 3D Timoshenko Beam Element Force Vectors 704</p>
<p>9.6.1 Axial Loads 706</p>
<p>9.6.2 Torsional Loads 706</p>
<p>9.6.3 x1 x2 Plane Loads 707</p>
<p>9.6.4 x1 x3 Plane Loads 709</p>
<p>9.6.5 Element Load Vector 711</p>
<p>9.7 3D Frame: Beam Element Assembly Algorithm 713</p>
<p>9.8 2D Frame Modeling with Timoshenko Beam Elements 725</p>
<p>9.9 Summary 748</p>
<p>9.10 Chapter 9 Exercises 749</p>
<p>9.10.1 Exercise Location 749</p>
<p>9.10.2 Exercise Goals 749</p>
<p>9.10.3 Sample Exercises: 9.4 and 9.15a 749</p>
<p>References 750</p>
<p>10 2D Planar Finite Elements for Vibration Analysis 751</p>
<p>10.1 Introduction 751</p>
<p>10.2 Plane Strain (P ) 751</p>
<p>10.3 Plane Stress (P ) 753</p>
<p>10.4 Plane Stress and Plane Strain: Element Stiffness and Mass Matrices and Force Vector 754</p>
<p>10.4.1 External Forces 760</p>
<p>10.4.2 Concentrated Forces 760</p>
<p>10.4.3 General Volumetric Loading 761</p>
<p>10.4.4 Edge Loads 762</p>
<p>10.5 Assembly Procedure for 2D, 4–Node, Quadrilateral Elements 763</p>
<p>10.6 Computation of Stresses in 2D Solid Elements 768</p>
<p>10.6.1 Interior Stress Determination 768</p>
<p>10.6.2 Surface Stresses 770</p>
<p>10.7 Extra Shape Functions to Improve Accuracy 774</p>
<p>10.8 Illustrative Example 776</p>
<p>10.9 2D Axisymmetric Model 786</p>
<p>10.9.1 Axisymmetric Model Stresses and Strains 786</p>
<p>10.9.2 4–Node, Bilinear Axisymmetric Element 788</p>
<p>10.10 Automated Mesh Generation: Constant Strain Triangle Elements 801</p>
<p>10.10.1 Element Stiffness Matrix 803</p>
<p>10.10.2 Element Mass Matrix 803</p>
<p>10.10.3 System Matrix Assembly 804</p>
<p>10.11 Membranes 810</p>
<p>10.11.1 Kinetic Energy and Element Mass Matrix 810</p>
<p>10.11.2 Strain Potential Energy and Element Stiffness Matrix 811</p>
<p>10.11.3 Matrix Assembly 812</p>
<p>10.12 Banded Storage 815</p>
<p>10.13 Chapter 10 Exercises 820</p>
<p>10.13.1 Exercise Location 820</p>
<p>10.13.2 Exercise Goals 821</p>
<p>10.13.3 Sample Exercises: 10.5 and 10.8 821</p>
<p>References 822</p>
<p>11 3D Solid Elements for Vibration Analysis 823</p>
<p>11.1 Introduction 823</p>
<p>11.2 Element Stiffness Matrix 825</p>
<p>11.2.1 Shape Functions 825</p>
<p>11.2.2 Element Stiffness Matrix Integral and Summation Forms 828</p>
<p>11.3 The Element Mass Matrix and Force Vector 835</p>
<p>11.3.1 Element Mass Matrix 836</p>
<p>11.3.2 Element External Force Vector 836</p>
<p>11.3.3 Force Vector: Concentrated Nodal Forces 837</p>
<p>11.3.4 Force Vector: Volumetric Loads 838</p>
<p>11.3.5 Force Vector: Face Loading 839</p>
<p>11.4 Assembly Procedure for the 3D, 8–Node, Hexahedral Element Model 842</p>
<p>11.5 Computation of Stresses for a 3D Hexahedral Solid Element 846</p>
<p>11.5.1 Computation of Interior Stress 846</p>
<p>11.5.2 Computation of Surface Stresses 848</p>
<p>11.5.3 Coordinate Transformation 849</p>
<p>11.5.4 Geometry Mapping in Surface Tangent Coordinates 850</p>
<p>11.5.5 Displacements in the Surface Tangent Coordinate System 851</p>
<p>11.5.6 Strains in the Surface Tangent Coordinate System 851</p>
<p>11.5.7 Surface Stresses Obtained from Cauchy s Boundary Formula 852</p>
<p>11.5.8 Surface Stresses Obtained from the Constitutive Law and Surface Strains 853</p>
<p>11.5.9 Summary of Surface Stress Computation 854</p>
<p>11.6 3D Solid Element Model Example 856</p>
<p>11.7 3D Solid Element Summary 864</p>
<p>11.8 Chapter 11 Exercises 865</p>
<p>11.8.1 Exercise Location 865</p>
<p>11.8.2 Exercise Goals 865</p>
<p>11.8.3 Sample Exercises: 11.2 and 11.3 865</p>
<p>References 866</p>
<p>12 Active Vibration Control 867</p>
<p>12.1 Introduction 867</p>
<p>12.2 AVC System Modeling 871</p>
<p>12.3 AVC Actuator Modeling 874</p>
<p>12.4 System Model with an Infinite Bandwidth Feedback Approximation 878</p>
<p>12.4.1 Closed–Loop Feedback Controlled System Model 882</p>
<p>12.5 System Model with Finite Bandwidth Feedback 886</p>
<p>12.6 System Model with Finite Bandwidth Feedback and Lead Compensation 893</p>
<p>12.6.1 Transfer Function Approach 893</p>
<p>12.6.2 State Space Approach 897</p>
<p>12.7 Sensor/Actuator Noncollocation Effect on Vibration Stability 901</p>
<p>12.8 Piezoelectric Actuators 907</p>
<p>12.8.1 Piezoelectric Stack Actuator 908</p>
<p>12.8.2 Piezoelectric Layer (Patch) Actuator 915</p>
<p>12.9 Summary 923</p>
<p>12.10 Chapter 12 Exercises 923</p>
<p>12.10.1 Exercise Location 923</p>
<p>12.10.2 Exercise Goals 923</p>
<p>12.10.3 Sample Exercise: 12.1 923</p>
<p>References 924</p>
<p>Appendix A Fundamental Equations of Elasticity 927</p>
<p>Index 941</p>