## Preface to the SI Edition

This edition of Mechanical Vibrations: Theory and Applications has been adapted to

incorporate the International System of Units (Le Système International d’Unités or SI)

throughout the book.

Le Systeme International d’ Unites

The United States Customary System (USCS) of units uses FPS (foot-pound-second) units

(also called English or Imperial units). SI units are primarily the units of the MKS (meterkilogram-

second) system. However, CGS (centimeter-gram-second) units are often accepted

as SI units, especially in textbooks.

Using SI Units in this Book

In this book, we have used both MKS and CGS units. USCS units or FPS units used in

the US Edition of the book have been converted to SI units throughout the text and problems.

However, in case of data sourced from handbooks, government standards, and product

manuals, it is not only extremely difficult to convert all values to SI, it also encroaches

upon the intellectual property of the source. Also, some quantities such as the ASTM grain

size number and Jominy distances are generally computed in FPS units and would lose

their relevance if converted to SI. Some data in figures, tables, examples, and references,

therefore, remains in FPS units. For readers unfamiliar with the relationship between the

FPS and the SI systems, conversion tables have been provided inside the front and back

covers of the book.

To solve problems that require the use of sourced data, the sourced values can be converted

from FPS units to SI units just before they are to be used in a calculation. To obtain

standardized quantities and manufacturers’ data in SI units, the readers may contact the

appropriate government agencies or authorities in their countries/regions.

Instructor Resources

A Printed Instructor’s Solution Manual in SI units is available on request. An electronic

version of the Instructor’s Solutions Manual, and PowerPoint slides of the figures from the

SI text are available through http://login.cengage.com.

The readers’ feedback on this SI Edition will be highly appreciated and will help us improve

subsequent editions.

## Preface

Engineers apply mathematics and science to solve problems. In a traditional undergraduate

engineering curriculum, students begin their academic career by taking

courses in mathematics and basic sciences such as chemistry and physics. Students

begin to develop basic problem-solving skills in engineering courses such as statics, dynamics,

mechanics of solids, fluid mechanics, and thermodynamics. In such courses, students

learn to apply basic laws of nature, constitutive equations, and equations of state to develop

solutions to abstract engineering problems.

Vibrations is one of the first courses where students learn to apply the knowledge obtained

from mathematics and basic engineering science courses to solve practical problems. While the

knowledge about vibrations and vibrating systems is important, the problem-solving skills

obtained while studying vibrations are just as important. The objectives of this book are twofold:

to present the basic principles of engineering vibrations and to present them in a framework

where the reader will advance his/her knowledge and skill in engineering problem solving.

This book is intended for use as a text in a junior- or senior-level course in vibrations. It

could be used in a course populated by both undergraduate and graduate students. The latter

chapters are appropriate for use as a stand-alone graduate course in vibrations. The prerequisites

for such a course should include courses in statics, dynamics, mechanics of materials, and

mathematics using differential equations. Some material covered in a course in fluid mechanics

is included, but this material can be omitted without a loss in continuity.

Chapter 1 is introductory, reviewing concepts such as dynamics, so that all readers are

familiar with the terminology and procedures. Chapter 2 focuses on the elements that comprise

mechanical systems and the methods of mathematical modeling of mechanical systems.

It presents two methods of the derivation of differential equations: the free-body diagram

method and the energy method, which are used throughout the book. Chapters 3 through 5

focus on single degree-of-freedom (SDOF) systems. Chapter 6 is focused solely on two

degree-of-freedom systems. Chapters 7 through 9 focus on general multiple degree-of-freedom

systems. Chapter 10 provides a brief overview of continuous systems. The topic of Chapter 11

is the finite-element methods, which is a numerical method with its origin in energy methods,

allowing continuous systems to be modeled as discrete systems. Chapter 12 introduces

the reader to nonlinear vibrations, while Chapter 13 provides a brief introduction to random

vibrations.

The references at the end of this text list many excellent vibrations books that address

the topics of vibration and design for vibration suppression. There is a need for this book,

as it has several unique features:

• Two benchmark problems are studied throughout the book. Statements defining the

generic problems are presented in Chapter 1. Assumptions are made to render SDOF

models of the systems in Chapter 2 and the free and forced vibrations of the systems

studied in Chapters 3 through 5, including vibration isolation. Two degree-of-freedom

system models are considered in Chapter 6, while MDOF models are studied in

Chapters 7 through 9. A continuous-systems model for one benchmark problem is

considered in Chapter 10 and solved using the finite-element method in Chapter 11.

A random-vibration model of the other benchmark problem is considered in Chapter 13.

The models get more sophisticated as the book progresses.

• Most vibration problems (certainly ones encountered by undergraduates) involve the

planar motion of rigid bodies. Thus, a free-body diagram method based upon

D’Alembert’s principle is developed and used for rigid bodies or systems of rigid bodies

undergoing planar motion.

• An energy method called the equivalent systems method is developed for SDOF systems

without introducing Lagrange’s equations. Lagrange’s equations are reserved for

MDOF systems.

• Most chapters have a Further Examples section which presents problems using concepts

presented in several sections or even several chapters of the book.

• MATLAB® is used in examples throughout the book as a computational and graphical

aid. All programs used in the book are available at the specific book website accessible

through www.cengage.com/engineering.

• The Laplace transform method and the concept of the transfer function (or the impulsive

response) is used in MDOF problems. The sinusoidal transfer function is used to

solve MDOF problems with harmonic excitation.

• The topic of design for vibration suppression is covered where appropriate. The design

of vibration isolation for harmonic excitation is covered in Chapter 4, vibration isolation

from pulses is covered in Chapter 5, design of vibration absorbers is considered

in Chapter 6, and vibration isolation problems for general MDOF systems is considered

in Chapter 9.

To access additional course materials, please visit www.cengagebrain.com. At the

cengagebrain.com home page, search for the ISBN of your title (from the back cover of

your book) using the search box at the top of the page. This will take you to the product

page where these resources can be found.

The author acknowledges the support and encouragement of numerous people in the

preparation of this book. Suggestions for improvement were taken from many students

at The University of Akron. The author would like to especially thank former students

Ken Kuhlmann for assistance with the problem involving the rotating manometer in

Chapter 12, Mark Pixley for helping with the original concept of the prototype for the software

package available at the website, and J.B. Suh for general support. The author also

expresses gratitude to Chris Carson, Executive Director, Global Publishing; Chris Shortt,

Publisher, Global Engineering; Randall Adams, Senior Acquisitions Editor; and Hilda

Gowans, Senior Developmental Editor, for encouragement and guidance throughout the

project. The author also thanks George G. Adams, Northeastern University; Cetin

Cetinkaya, Clarkson University; Shanzhong (Shawn) Duan, South Dakota State

University; Michael J. Leamy, Georgia Institute of Technology; Colin Novak, University of

Windsor; Aldo Sestieri, University La Sapienza Roma; and Jean Zu, University of Toronto,

for their valuable comments and suggestions for making this a better book. Finally, the

author expresses appreciation to his wife, Seala Fletcher-Kelly, not only for her support and

encouragement during the project but for her help with the figures as well.

## Contents

CHAPTER 1 INTRODUCTION 1

1.1 The Study of Vibrations 1

1.2 Mathematical Modeling 4

1.2.1 Problem Identification 4

1.2.2 Assumptions 4

1.2.3 Basic Laws of Nature 6

1.2.4 Constitutive Equations 6

1.2.5 Geometric Constraints 6

1.2.6 Diagrams 6

1.2.7 Mathematical Solution 7

1.2.8 Physical Interpretation of Mathematical Results 7

1.3 Generalized Coordinates 7

1.4 Classification of Vibration 11

1.5 Dimensional Analysis 11

1.6 Simple Harmonic Motion 14

1.7 Review of Dynamics 16

1.7.1 Kinematics 16

1.7.2 Kinetics 18

1.7.3 Principle of Work-Energy 22

1.7.4 Principle of Impulse and Momentum 24

1.8 Two Benchmark Examples 27

1.8.1 Machine on the Floor of an Industrial Plant 27

1.8.2 Suspension System for a Golf Cart 28

1.9 Further Examples 29

1.10 Summary 34

1.10.1 Important Concepts 34

1.10.2 Important Equations 35

Problems 37

Short Answer Problems 37

Chapter Problems 41

CHAPTER 2 MODELING OF SDOF SYSTEMS 55

2.1 Introduction 55

2.2 Springs 56

2.2.1 Introduction 56

2.2.2 Helical Coil Springs 57

2.2.3 Elastic Elements as Springs 59

2.2.4 Static Deflection 61

2.3 Springs in Combination 62

2.3.1 Parallel Combination 62

2.3.2 Series Combination 62

2.3.3 General Combination of Springs 66

2.4 Other Sources of Potential Energy 68

2.4.1 Gravity 68

2.4.2 Buoyancy 70

2.5 Viscous Damping 71

2.6 Energy Dissipated by Viscous Damping 74

2.7 Inertia Elements 76

2.7.1 Equivalent Mass 76

2.7.2 Inertia Effects of Springs 79

2.7.3 Added Mass 83

2.8 External Sources 84

2.9 Free-Body Diagram Method 87

2.10 Static Deflections and Gravity 94

2.11 Small Angle or Displacement Assumption 97

2.12 Equivalent Systems Method 100

2.13 Benchmark Examples 106

2.13.1 Machine on a Floor in an Industrial Plant 106

2.10.2 Simplified Suspension System 107

2.14 Further Examples 108

2.15 Chapter Summary 116

2.15.1 Important Concepts 116

2.15.2 Important Equations 117

Problems 119

Short Answer Problems 119

Chapter Problems 123

CHAPTER 3 FREE VIBRATIONS OF SDOF SYSTEMS 137

3.1 Introduction 137

3.2 Standard Form of Differential Equation 138

3.3 Free Vibrations of an Undamped System 140

3.4 Underdamped Free Vibrations 147

3.5 Critically Damped Free Vibrations 154

3.6 Overdamped Free Vibrations 156

3.7 Coulomb Damping 160

3.8 Hysteretic Damping 167

3.9 Other Forms of Damping 171

3.10 Benchmark Examples 174

3.10.1 Machine on the Floor of an Industrial Plant 174

3.10.2 Simplified Suspension System 175

3.11 Further Examples 178

3.12 Chapter Summary 185

3.12.1 Important Concepts 185

3.12.2 Important Equations 186

Problems 188

Short Answer Problems 188

Chapter Problems 194

CHAPTER 4 HARMONIC EXCITATION OF SDOF SYSTEMS 205

4.1 Introduction 205

4.2 Forced Response of an Undamped System Due

to a Single-Frequency Excitation 208

4.3 Forced Response of a Viscously Damped System

Subject to a Single-Frequency Harmonic Excitation 214

4.4 Frequency-Squared Excitations 220

4.4.1 General Theory 220

4.4.2 Rotating Unbalance 222

4.4.3 Vortex Shedding from Circular Cylinders 225

4.5 Response Due to Harmonic Excitation of Support 228

4.6 Vibration Isolation 234

4.7 Vibration Isolation from Frequency-Squared Excitations 238

4.8 Practical Aspects of Vibration Isolation 241

4.9 Multifrequency Excitations 244

4.10 General Periodic Excitations 246

4.10.1 Fourier Series Representation 246

4.10.2 Response of Systems Due to General Periodic Excitation 251

4.10.3 Vibration Isolation for Multi-Frequency and Periodic

Excitations 253

4.11 Seismic Vibration Measuring Instruments 255

4.11.1 Seismometers 255

4.11.2 Accelerometers 256

4.12 Complex Representations 259

4.13 Systems with Coulomb Damping 260

4.14 Systems with Hysteretic Damping 265

4.15 Energy Harvesting 268

4.16 Benchmark Examples 273

4.16.1 Machine on Floor of Industrial Plant 273

4.16.2 Simplified Suspension System 274

4.17 Further Examples 281

4.18 Chapter Summary 289

4.18.1 Important Concepts 289

4.18.2 Important Equations 290

Problems 293

Short Answer Problems 293

Chapter Problems 297

CHAPTER 5 TRANSIENT VIBRATIONS OF SDOF SYSTEMS 313

5.1 Introduction 313

5.2 Derivation of Convolution Integral 315

5.2.1 Response Due to a Unit Impulse 315

5.3 Response Due to a General Excitation 318

5.4 Excitations Whose Forms Change at Discrete Times 323

5.5 Transient Motion Due to Base Excitation 330

5.6 Laplace Transform Solutions 332

5.7 Transfer Functions 337

5.8 Numerical Methods 340

5.8.1 Numerical Evaluation of Convolution Integral 340

5.8.2 Numerical Solution of Differential Equations 344

5.9 Shock Spectrum 350

5.10 Vibration Isolation for Short Duration Pulses 357

5.11 Benchmark Examples 361

5.11.1 Machine on Floor of Industrial Plant 361

5.11.2 Simplified Suspension System 362

5.12 Further Examples 365

5.13 Chapter Summary 370

5.13.1 Important Concepts 370

5.13.2 Important Equations 371

Problems 372

Short Answer Problems 372

Chapter Problems 374

CHAPTER 6 TWO DEGREE-OF-FREEDOM SYSTEMS 383

6.1 Introduction 383

6.2 Derivation of the Equations of Motion 384

6.3 Natural Frequencies and Mode Shapes 388

6.4 Free Response of Undamped Systems 393

6.5 Free Vibrations of a System with Viscous Damping 396

6.6 Principal Coordinates 398

6.7 Harmonic Response of Two Degree-Of-Freedom Systems 401

6.8 Transfer Functions 404

6.9 Sinusoidal Transfer Function 408

6.10 Frequency Response 411

6.11 Dynamic Vibration Absorbers 414

6.12 Damped Vibration Absorbers 420

6.13 Vibration Dampers 424

6.14 Benchmark Examples 425

6.14.1 Machine on Floor of Industrial Plant 425

6.14.2 Simplified Suspension System 427

6.15 Further Examples 432

6.16 Chapter Summary 442

6.16.1 Important Concepts 442

6.16.2 Important Equations 443

Problems 444

Short Answer Problems 444

Chapter Problems 448

CHAPTER 7 MODELING OF MDOF SYSTEMS 459

7.1 Introduction 459

7.2 Derivation of Differential Equations Using the Free-Body

Diagram Method 461

7.3 Lagrange’s Equations 467

7.4 Matrix Formulation of Differential Equations for Linear Systems 478

7.5 Stiffness Influence Coefficients 483

7.6 Flexibility Influence Coefficients 492

7.7 Inertia Influence Coefficients 497

7.8 Lumped-Mass Modeling of Continuous Systems 499

7.9 Benchmark Examples 502

7.9.1 Machine on Floor of an Industrial Plant 502

7.9.2 Simplified Suspension System 506

7.10 Further Examples 508

7.11 Summary 517

7.11.1 Important Concepts 517

7.11.2 Important Equations 518

Problems 519

Short Answer Problems 519

Chapter Problems 523

CHAPTER 8 FREE VIBRATIONS OF MDOF SYSTEMS 533

8.1 Introduction 533

8.2 Normal-Mode Solution 534

8.3 Natural Frequencies and Mode Shapes 536

8.4 General Solution 543

8.5 Special Cases 545

8.5.1 Degenerate Systems 545

8.5.2 Unrestrained Systems 548

8.6 Energy Scalar Products 552

8.7 Properties of Natural Frequencies and Mode Shapes 555

8.8 Normalized Mode Shapes 558

8.9 Rayleigh’s Quotient 560

8.10 Principal Coordinates 562

8.11 Determination of Natural Frequencies and Mode Shapes 565

8.12 Proportional Damping 568

8.13 General Viscous Damping 571

8.14 Benchmark Examples 574

8.14.1 Machine on Floor of an Industrial Plant 574

8.14.2 Simplified Suspension System 576

8.15 Further Examples 578

8.16 Summary 583

8.16.1 Important Concepts 583

8.16.2 Important Equations 584

Problems 585

Short Answer Problems 585

Chapter Problems 588

CHAPTER 9 FORCED VIBRATIONS OF MDOF SYSTEMS 593

9.1 Introduction 593

9.2 Harmonic Excitations 594

9.3 Laplace Transform Solutions 599

9.4 Modal Analysis for Undamped Systems and Systems

with Proportional Damping 603

9.5 Modal Analysis for Systems with General Damping 611

9.6 Numerical Solutions 614

9.7 Benchmark Examples 615

9.7.1 Machine on Floor of Industrial Plant 615

9.7.2 Simplified Suspension System 616

9.8 Further Examples 620

9.9 Chapter Summary 623

9.9.1 Important Concepts 623

9.9.2 Important Equations 624

Problems 625

Short Answer Problems 625

Chapter Problems 627

CHAPTER 10 VIBRATIONS OF CONTINUOUS SYSTEMS 633

10.1 Introduction 633

10.2 General Method 636

10.3 Second-Order Systems: Torsional Oscillations of a Circular Shaft 639

10.3.1 Problem Formulation 639

10.3.2 Free-Vibration Solutions 642

10.3.3 Forced Vibrations 650

10.4 Transverse Beam Vibrations 651

10.4.1 Problem Formulation 651

10.4.2 Free Vibrations 654

10.4.3 Forced Vibrations 662

10.5 Energy Methods 667

10.6 Benchmark Examples 672

10.7 Chapter Summary 676

10.7.1 Important Concepts 676

10.7.2 Important Equations 677

Problems 678

Short Answer Problems 678

Chapter Problems 682

CHAPTER 11 FINITE-ELEMENT METHOD 689

11.1 Introduction 689

11.2 Assumed Modes Method 690

11.3 General Method 693

11.4 The Bar Element 696

11.5 Beam Element 700

11.6 Global Matrices 705

11.7 Benchmark Example 709

11.8 Further Examples 714

11.9 Summary 726

11.9.1 Important Concepts 726

11.9.2 Important Equations 726

Problems 728

Short Answer Problems 728

Chapter Problems 730

CHAPTER 12 NONLINEAR VIBRATIONS 737

12.1 Introduction 737

12.2 Sources of Nonlinearity 738

12.3 Qualitative Analysis of Nonlinear Systems 743

12.4 Quantitative Methods of Analysis 747

12.5 Free Vibrations of SDOF Systems 749

12.6 Forced Vibrations of SDOF Systems

with Cubic Nonlinearities 753

12.7 MDOF Systems 759

12.7.1 Free Vibrations 759

12.7.2 Forced Vibrations 760

12.8 Continuous Systems 760

12.9 Chaos 761

12.10 Chapter Summary 769

12.10.1 Important Concepts 769

12.10.2 Important Equations 769

Problems 770

Short Answer Problems 770

Chapter Problems 775

CHAPTER 13 RANDOM VIBRATIONS 781

13.1 Introduction 781

13.2 Behavior of a Random Variable 782

13.2.1 Ensemble Processes 782

13.2.2 Stationary Processes 783

13.2.3 Ergodic Processes 784

13.3 Functions of a Random Variable 784

13.3.1 Probability Functions 784

13.3.2 Expected Value, Mean, and Standard Deviation 786

13.3.3 Mean Square Value 786

13.3.4 Probability Distribution for Arbitrary Function of Time 787

13.3.5 Gaussian Process 788

13.3.6 Rayleigh Distribution 791

13.3.7 Central Limit Theorem 792

13.4 Joint Probability Distributions 793

13.4.1 Two Random Variables 793

13.4.2 Autocorrelation Function 794

13.4.3 Cross Correlations 797

13.5 Fourier Transforms 797

13.5.1 Fourier Series In Complex Form 797

13.5.2 Fourier Transform for Nonperiodic Functions 798

13.5.3 Transfer Functions 801

13.5.4 Fourier Transform in Terms of f 802

13.5.5 Parseval’s Identity 802

13.6 Power Spectral Density 803

13.7 Mean Square Value of the Response 808

13.8 Benchmark Example 812

13.9 Summary 814

13.9.1 Important Concepts 814

13.9.2 Important Equations 815

13.10 Problems 817

13.10.1 Short Answer Problems 817

13.10.2 Chapter Problems 819

APPENDIX A UNIT IMPULSE FUNCTION AND UNIT STEP FUNCTION 825

APPENDIX B LAPLACE TRANSFORMS 827

B.1 Definition 827

B.2 Table of Transforms 827

B.3 Linearity 827

B.4 Transform of Derivatives 828

B.5 First Shifting Theorem 829

B.6 Second Shifting Theorem 830

B.7 Inversion of Transform 830

B.8 Convolution 831

B.9 Solution of Linear Differential Equations 831

APPENDIX C LINEAR ALGEBRA 833

C.1 Definitions 833

C.2 Determinants 834

C.3 Matrix Operations 835

C.4 Systems of Equations 836

C.5 Inverse Matrix 837

C.6 Eigenvalue Problems 838

C.7 Scalar Products 840

APPENDIX D DEFLECTION OF BEAMS SUBJECT

TO CONCENTRATED LOADS 842

APPENDIX E INTEGRALS USED IN RANDOM VIBRATIONS 846

APPENDIX F VIBES 847

REFERENCES 851

INDEX 853