Contents

Preface xv

Acknowledgements xvii

C h a p t e r 1 Introduction 1 1.1 The Making of This Book 1 1.2 How to Use This Book 4

Part I Strength of Materials 7

Chapter 2

Chapter 3

Static Analysis of Euler-Bernoulli Beams 2.1 2.2

2.3

2.4 2.5 2.6

Getting Started Beam Theory 2.2.1 Static Problem 2.2.2 Solution Methods 2.2.3 Fundamentals of Euler-Bernoulli Beams Static Analysis by the Toolbox 2.3.1 System Setup 2.3.2 Response to External Forces 2.3.3 Response to Boundary Disturbances 2.3.4 Total Response 2.3.5 Exact Analytical Solutions 2.3.6 Other Useful Functions Moments of Inertia of Beam Cross-Section Area Quick Solution Guide References

Static Analysis of Bars, Shafts, and Strings 3.1 3.2 3.3

Getting Started System Description Static Analysis by the Toolbox 3.3.1 System Setup

9 9

11 11 16 23 25 26 28 33 34 36 40 42 46 51

53 53 55 59 59

VII

3.3.2 Response to External Forces 60 3.3.3 Response to Boundary Disturbances 63 3.3.4 Total Response 65

3.4 Stepped Bars and Shafts 67 3.4.1 System Description 67 3.4.2 Solution by the Toolbox 68

3.5 The Distributed Transfer Function Method 74 3.6 Quick Solution Guide 78 3.7 References 83

C h a p t e r 4 Buckling Analysis of Columns 85 4.1 Getting Started 85 4.2 Uniform Columns 87

4.2.1 Column Buckling Theory 87 4.2.2 Solution by the Toolbox 91 4.2.3 Eccentric Loading 96 4.2.4 Beam-Column Problem 100 4.2.5 Geometric Imperfection 104

4.3 Stepped and Nonuniform Columns 109 4.3.1 Constrained Stepped Columns 109 4.3.2 Nonuniform Columns 116

4.4 The Distributed Transfer Function Method 123 4.5 Quick Solution Guide 127 4.6 References 133

C h a p t e r 5 Stress Analysis in Two-Dimensional Problems 135 5.1 Getting Started 135 5.2 Plane Stress and Strain 136

5.2.1 In-Plane Stresses 136 5.2.2 In-Plane Strains 142 5.2.3 Strain Rosette 147 5.2.4 Hooke's Law 149 5.2.5 Criteria of Failure 151

5.3 Quick Solution Guide 155 5.4 References 156

Part II Structural Mechanics 157

Chapter 6 Static Analysis of Constrained Multispan Beams 6.1 Getting Started 6.2 System Description 6.3 Solution by the Toolbox

6.3.1 System Setup 6.3.2 Response to External Forces 6.3.3 Response to Boundary Disturbances 6.3.4 Response to Support Settlement 6.3.5 Total Response 6.3.6 Influence Lines

6.4 The Distributed Transfer Function Method

159 159 161 166 166 171 176 178 181 183 189

viii Contents

6.5 Quick Solution Guide 6.6 References 200

Chapter 7

Chapter 8

Static Analysis of Plane Trusses 7.1 7.2 7.3

7.4 7.5 7.6

Stati 8.1 8.2 8.3

8.4 8.5 8.6

Getting Started System Description Solution by the Toolbox 7.3.1 System Setup 7.3.2 Response to External Forces 7.3.3 Support Settlement 7.3.4 Fabrication Errors and Thermal Effects 7.3.5 Total Response The Stiffness Method Quick Solution Guide References

c Analysis of Plane Frames Getting Started System Description Solution by the Toolbox 8.3.1 System Setup 8.3.2 Response to External Forces 8.3.3 Response to Settlement of Supports 8.3.4 Total Response 8.3.5 Response of Frame Members The Distributed Transfer Function Method Quick Solution Guide References

201 201 203 205 205 210 213 216 220 224 230 235

237 237 240 243 243 248 252 256 260 263 268 276

Part III Dynamics and Vibrations 277

C h a p t e r 9 Dynamics of Particles and Rigid Bodies 279 9.1 Getting Started 279 9.2 Dynamics of Particles 280

9.2.1 Preliminaries 280 9.2.2 Kinematics 284 9.2.3 Kinetics of Single Particle 286 9.2.4 Particle Motion via Numerical Integration 289 9.2.5 Systems of Particles 297 9.2.6 Central Force Motion 303

9.3 Dynamics of Rigid Bodies in Plane Motion 307 9.3.1 Plane Motion 307 9.3.2 Mass Moments of Inertia 310 9.3.3 Kinetics 314 9.3.4 Simulation of Dynamic Response 317

9.4 Rigid Body Dynamics in Three Dimensions 323 9.4.1 Kinematics 323 9.4.2 Inertia Properties 327 9.4.3 Energy and Momentum 335

Contents

9.4.4 Equations of Motion 336 9.4.5 Rotation of Axisymmetric Bodies 340

9.5 Quick Solution Guide 349 9.6 References 350

C h a p t e r 1 0 Vibration Analysis of One-Degree-of-Freedom Systems 351 10.1 Getting Started 351 10.2 System Description 353 10.3 Time Response 357

10.3.1 Free Response 357 10.3.2 Forced Response 361 10.3.3 Specific Forcing Functions 365 10.3.4 Total Response 374 10.3.5 Delayed Forcing Functions 376 10.3.6 Solution by Superposition 377 10.3.7 Mechanical Energy 379 10.3.8 Plotting Computed Response 382

10.4 Analytical Vibration Solutions 384 10.4.1 Analytical Methods 384 10.4.2 Solution by the Toolbox 389

10.5 Frequency Response 394 10.5.1 Harmonic Excitation 394 10.5.2 Base Excitation 400 10.5.3 Response under Rotating Unbalance 403 10.5.4 Vibration Isolation 406

10.6 Response to Periodic Excitation 410 10.6.1 Steady-State Solution in Fourier Series 410 10.6.2 Solution by the Toolbox 412 10.6.3 Fourier Coefficients of Certain Loads 419

10.7 Nonlinear Vibration 421 10.8 Quick Solution Guide 431 10.9 References 435

C h a p t e r 1 1 Vibration and Control of Multiple-Degree-of-Freedom Systems 437 11.1 Getting Started 437 11.2 System Description 439

11.2.1 Introduction 439 11.2.2 Modes of Vibration 442 11.2.3 Solution by the Toolbox 444

11.3 Dynamic Response 450 11.3.1 Modal Analysis 450 11.3.2 Laplace Transform Method 458 11.3.3 Plotting Analytical Solutions 462 11.3.4 Runge-Kutta Algorithm 465 11.3.5 Solution by Superposition 468 11.3.6 Harmonic Excitation 472

11.4 Dynamic Vibration Absorption 475 11.4.1 Undamped Vibration Absorbers 476 11.4.2 Damped Vibration Absorbers 481

x Contents

11.5 Transfer Function Formulation 11.5.1 Transfer Function and Green's Function 487 11.5.2 Open-Loop Transfer Function 491

11.6 Feedback Control 497 11.6.1 Vibration Control System 497 11.6.2 Position Control System 501 11.6.3 Solution by the Toolbox 506

11.7 Quick Solution Guide 515 11.8 References 518

Part IV Structural Dynamics 519

521 521 523 523 524 528 534 535 537 537 539 540 544 552 554 555 559 559 567 581 581 582 588 594 601 608 615

C h a p t e r 1 3 Dynamic Analysis of Bars, Shafts, and Strings 617 13.1 Getting Started 617 13.2 System Description 619

13.2.1 Governing Equations 619 13.2.2 Eigenvalue Problem 622 13.2.3 System Setup by the Toolbox 624 13.2.4 Display of Modes of Vibration 627

13.3 Dynamic Response 630 13.3.1 Modal Expansion 631 13.3.2 Free Vibration 632

Chapter 12 Dynamics and Control of Euler-Bernoulli Beams 12.1 12.2

12.3

12.4

12.5

12.6 12.7

Getting Started System Description 12.2.1 12.2.2 12.2.3 12.2.4 12.2.5

Governing Equation Eigenvalue Problem System Setup by the Toolbox Animation of Modes of Vibration Distributed Transfer Function Method

Dynamic Response 12.3.1 12.3.2 12.3.3 12.3.4 12.3.5 12.3.6 12.3.7

Modal Expansion Specification of Damping Free Vibration Forced Vibration Total Response Animation of Time Response Frequency Response

Feedback Control 12.4.1 12.4.2 Dynami 12.5.1 12.5.2 12.5.3 12.5.4 12.5.5

Control System Formulation Solution by the Toolbox

cs and Control of Nonuniform Beams Problem Statement Rayleigh-Ritz Discretization Modes of Vibration Time Response Control System Formulation

Quick Solution Guide References

Contents xi

13.3.3 Forced Vibration 635 13.3.4 Total Response 639 13.3.5 Animation of Time Response 642 13.3.6 Frequency Response 643

13.4 Free Vibration of Stepped Systems 647 13.4.1 System Description 647 13.4.2 Free Vibration Analysis 651 13.4.3 Solution by the Toolbox 655

13.5 Quick Solution Guide 661 13.6 References 665

C h a p t e r 1 4 Dynamic Analysis of Constrained, Combined, and Stepped Beams 667 14.1 Getting Started 667 14.2 Constrained Beams 670

14.2.1 System Description 670 14.2.2 System Setup and Eigensolutions 675 14.2.3 Eigenvalue Locus 683 14.2.4 Frequency Response 687 14.2.5 Multispan Beam Structures 689 14.2.6 Transient Response 691

14.3 Combined Beams 691 14.3.1 System Description 692 14.3.2 System Setup and Eigensolutions 696 14.3.3 Transient Response 698 14.3.4 Frequency Response 707 14.3.5 Oscillators for Vibration Absorption 712

14.4 Stepped Beams 715 14.4.1 Free Vibration Analysis 715 14.4.2 Solution by the Toolbox 721

14.5 Quick Solution Guide 729 14.6 References 736

Part V Two-Dimensional Elastic Continua 737

;er 15 Static Analysis of Linearly Elastic Bodies 15.1 15.2

15.3

Getting Started Theory of Linear Elasticity 15.2.1 15.2.2 15.2.3 15.2.4 15.2.5

Stress and Strain Basic Equations of Elasticity Conversion of Elasticity Constants Strain Energy Principal of Minimum Potential Energy

Elasticity Problems in Two Dimensions 15.3.1 15.3.2 15.3.3 15.3.4 15.3.5 15.3.6

Plane Stress and Plane Strain Governing Equations Solution by Stress Function Thermal Stresses Elasticity Problems in Polar Coordinates Stress Concentrations

739 739 742 742 749 753 755 755 759 759 760 761 764 765 769

15.4 Finite Element Method for 2-D Elasticity Problems 770 15.4.1 Finite Element Formulation 770 15.4.2 MATLAB Solutions in Rectangular Regions 777 15.4.3 MATLAB Solutions in Arbitrary-Shaped

Regions 792 15.5 Quick Solution Guide 802 15.6 References 805

C h a p t e r 1 6 Free Vibration of Membranes and Plates 16.1 Getting Started 16.2 Free Vibration of Membranes

16.2.1 Rectangular Membranes 16.2.2 Circular Membranes

16.3 Free Vibration of Rectangular Plates 16.3.1 Plate Theory 16.3.2 Eigenvalue Problem 16.3.3 Solution by the Toolbox

16.4 Free Vibration of Circular Plates 16.4.1 Equations in Polar Coordinates 16.4.2 Modes of Vibration

Appendix A

Appendix B

16.5 16.6

Quick Solution Guide References

Commonly Used Mathematical Formulas A.l A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.ll A.12

Algebraic Formulas Areas and Volumes of Common Shapes Trigonometry Hyperbolic Functions Derivatives and Integration Series Expansion Analytical Geometry Vector Analysis Matrix Theory Complex Numbers and Complex Functions Laplace Transforms Inverse Laplace Transform via Partial Fraction Expansion

MATLAB Basics B.l B.2 B.3 B.4 B.5 B.6 B.7

Getting Started Matrix and Vector Manipulations Graphics M-Files Control Flow Solution of Algebraic and Differential Equations Control System Toolbox

807 807 809 810 816 823 823 827 832 839 839 842 848 851

853 853 855 857 860 860 862 864 868 872 878 879

882

889 889 893 898 901 903 907 910

Appendix C The Distributed Transfer Function Method 913 C.l DTFM for One-Dimensional Continua 913 C.2 Transfer Function Synthesis of Multibody Structures 917

Contents xiii

C.3 DTFM for Two- and Three-Dimensional Problems 919 C.4 References 920

Appendix D Conversion of Units 923

Appendix E Mechanical Properties of Engineering Materials 925

Index 929

xiv Contents