Mechanical Vibration is core subject of Bachelor and master's degree in mechanical Engineering.There are many books available for vibration theory. This eBook covers all the fundamental and theories very briefly. This is not free to download.

Using computational Technics, book represents theories, aspects and application of vibrations. All the theories covered in very simple language as possible as.For engineers and other professionals who want a clear introduction to vibration engineering.

Automation in Design. Currently, he is on the editorial boards of Engineering Optimization, Reliability Engineering & System Safety, and Microelectronics and Reliability.

## About Mechanical Vibrations by S.S Rao 5th edition

This book covers all basic and fundamental topics, theories and example gratefully. In first chapter history of vibration and interesting facts covered very briefly, That could help reader to make interest in further theories and methods.Using computational Technics, book represents theories, aspects and application of vibrations. All the theories covered in very simple language as possible as.For engineers and other professionals who want a clear introduction to vibration engineering.

**Book information****Title**Mechanical vibrations**Volume**0 of Addison-Wesley series in mechanical engineering**Author**S. S. Rao**Edition**2, illustrated**Publisher**Addison-Wesley Longman, Incorporated, 1990 Original fromthe University of Michigan**Digitized**29 Nov 2007**ISBN**0201501562, 9780201501568**Length**1100 pages**Subjects**Science , Waves & Wave Mechanics

### About the author (1990)

SINGIRESU S. RAO, PhD, is a professor in the School of Mechanical Engineering at Purdue University. He earned his doctorate from Case Western Reserve University and has extensive teaching and research experience at Purdue, San Diego State University, Indian Institute of Technology (Kanpur), and NASA Langley Research Center. Dr. Rao has published more than 125 technical papers in internationally reputed journals and more than 100 papers in conference proceedings in the areas of engineering optimization, reliability-based design, fuzzy systems, active control of structures, concurrent design, and vibration engineering. His previous books include Optimization: Theory and Applications, The Finite Element Method in Engineering, Mechanical Vibrations, and Reliability-Based Design. Dr. Rao has edited several conference proceedings and served as the Conference Chairman and Papers Review Chairman for the ASME Design Automation Committee and as an Associate Editor for the ASME Journal of Mechanisms, Transmissions, andAutomation in Design. Currently, he is on the editorial boards of Engineering Optimization, Reliability Engineering & System Safety, and Microelectronics and Reliability.

#### Chapters that covered in this Edition

Preface xi

Acknowledgments xv

List of Symbols xvi

**CHAPTER 1**

**Fundamentals of Vibration 1**

1.1 Preliminary Remarks 2

1.2 Brief History of the Study of Vibration 3

1.2.1 Origins of the Study of Vibration 3

1.2.2 From Galileo to Rayleigh 6

1.2.3 Recent Contributions 9

1.3 Importance of the Study of Vibration 10

1.4 Basic Concepts of Vibration 13

1.4.1 Vibration 13

1.4.2 Elementary Parts of Vibrating Systems 13

1.4.3 Number of Degrees of Freedom 14

1.4.4 Discrete and Continuous Systems 16

1.5 Classification of Vibration 16

1.5.1 Free and Forced Vibration 17

1.5.2 Undamped and Damped Vibration 17

1.5.3 Linear and Nonlinear Vibration 17

1.5.4 Deterministic and Random Vibration 17

1.6 Vibration Analysis Procedure 18

1.7 Spring Elements 22

1.7.1 Nonlinear Springs 23

1.7.2 Linearization of a Nonlinear Spring 25

1.7.3 Spring Constants of Elastic Elements 27

1.7.4 Combination of Springs 30

1.7.5 Spring Constant Associated with the Restoring Force due to Gravity 39

1.8 Mass or Inertia Elements 40

1.8.1 Combination of Masses 40

1.9 Damping Elements 45

1.9.1 Construction of Viscous Dampers 46

1.9.2 Linearization of a Nonlinear Damper 52

1.9.3 Combination of Dampers 52

1.10 Harmonic Motion 54

1.10.1 Vectorial Representation of

1.10.2 Complex-Number Representationof Harmonic Motion 57

1.10.3 Complex Algebra 58

1.10.4 Operations on Harmonic Functions 59

1.10.5 Definitions and Terminology 62

1.11 Harmonic Analysis 64

1.11.1 Fourier Series Expansion 64

1.11.2 Complex Fourier Series 66

1.11.3 Frequency Spectrum 67

1.11.4 Time- and Frequency-Domain

Representations 68

1.11.5 Even and Odd Functions 69

1.11.6 Half-Range Expansions 71

1.11.7 Numerical Computationof Coefficients 72

1.12 Examples Using MATLAB 76

1.13 Vibration Literature 80

Chapter Summary 81

References 81

Review Questions 83

Problems 87

Design Projects 120

**CHAPTER 2**

**Free Vibration of Single-Degree-of-Freedom Systems 124**

2.1 Introduction 126

2.2 Free Vibration of an Undamped Translational System 129

2.2.1 Equation of Motion Using Newton s Second Law of Motion 129

2.2.2 Equation of Motion Using Other Methods 130

2.2.3 Equation of Motion of a Spring-MassSystem in Vertical Position 132

2.2.4 Solution 133

2.2.5 Harmonic Motion 134

2.3 Free Vibration of an Undamped Torsional System 146

2.3.1 Equation of Motion 147

2.3.2 Solution 148

2.4 Response of First Order Systemsand Time Constant 151

2.5 Rayleigh s Energy Method 153

2.6 Free Vibration with Viscous Damping 158

2.6.1 Equation of Motion 158

2.6.2 Solution 158

2.6.3 Logarithmic Decrement 164

2.6.4 Energy Dissipated in ViscousDamping 166

2.6.5 Torsional Systems with ViscousDamping 168

2.7 Graphical Representation of Characteristic Rootsand 174

2.7.1 Roots of the Characteristic Equation 174

2.7.2 Graphical Representation of Roots and Corresponding Solutions 175

2.8 Parameter Variations and Root LocusRepresentations 176

2.8.1 Interpretations of and in s-plane 176

2.8.2 Root Locus and Parameter Variations 179

2.9 Free Vibration with Coulomb Damping 185

2.9.1 Equation of Motion 186

2.9.2 Solution 187

2.9.3 Torsional Systems with CoulombDamping 190 vn, vd, z, t

2.10 Free Vibration with Hysteretic Damping 192

2.11 Stability of Systems 198

2.12 Examples Using MATLAB 202

Chapter Summary 208

References 209

Review Questions 209

Problems 214

Design Projects 256

**CHAPTER 3**

**Harmonically Excited Vibration 259**

3.1 Introduction 261

3.2 Equation of Motion 261

3.3 Response of an Undamped SystemUnder Harmonic Force 263

3.3.1 Total Response 267

3.3.2 Beating Phenomenon 267

3.4 Response of a Damped System UnderHarmonic Force 271

3.4.1 Total Response 274

3.4.2 Quality Factor and Bandwidth 276

3.5 Response of a Damped SystemUnder 278

3.6 Response of a Damped System Under theHarmonic Motion of the Base 281

3.6.1 Force Transmitted 283

3.6.2 Relative Motion 284

3.7 Response of a Damped System Under RotatingUnbalance 287

3.8 Forced Vibration with Coulomb Damping 293

3.9 Forced Vibration with Hysteresis Damping 298

3.10 Forced Motion with Other Types ofDamping 300

3.11 Self-Excitation and Stability Analysis 301

3.11.1 Dynamic Stability Analysis 301

3.11.2 Dynamic Instability Caused by FluidFlow 305

3.12 Transfer-Function Approach 313

3.13 Solutions Using Laplace Transforms 317

3.14 Frequency Transfer Functions 320

3.14.1 the General Transfer function and the Frequency Transfer Function322

3.14.2 Representation of Frequency-ResponseCharacteristics 323

3.15 Examples Using MATLAB 326

Chapter Summary 332

References 332

Review Questions 333

Problems 336

Design Projects 362

**CHAPTER 4**

**Vibration Under General Forcing Conditions 363**

4.1 Introduction 364

4.2 Response Under a General Periodic Force 365

4.2.1 First-Order Systems 366

4.2.2 Second-Order Systems 372

4.3 Response Under a Periodic Force of Irregular Form 378

4.4 Response Under a Nonperiodic Force 380

4.5 Convolution Integral 381

4.5.1 Response to an Impulse 382

4.5.2 Response to a General Forcing Condition 385

4.5.3 Response to Base Excitation 386

4.6 Response Spectrum 394

4.6.1 Response Spectrum for Base Excitation 396

4.6.2 Earthquake Response Spectra 399

4.6.3 Design Under a Shock Environment 403

4.7 Laplace Transform 406

4.7.1 Transient and Steady-State Responses 406

4.7.2 Response of First-Order Systems 407

4.7.3 Response of Second-Order Systems 409

4.7.4 Response to Step Force 414

4.7.5 Analysis of the Step Response 420

4.7.6 Description of Transient Response 421

4.8 Numerical Methods 428

4.8.1 Runge-Kutta Methods 429

4.9 Response to Irregular Forcing Conditions UsingNumerical Methods 431

4.10 Examples Using MATLAB 436

Chapter Summary 440

References 440

Review Questions 441

Problems 444

Design Projects 465

**CHAPTER 5**

**Two-Degree-of-Freedom Systems 467**

5.1 Introduction 468

5.2 Equations of Motion for Forced Vibration 472

5.3 Free Vibration Analysis of an Undamped System 474

5.4 Torsional System 483

5.5 Coordinate Coupling and Principal Coordinates 488

5.6 Forced-Vibration Analysis 494

5.7 Semidefinite Systems 497

5.8 Self-Excitation and Stability Analysis 500

5.9 Transfer-Function Approach 502

5.10 Solutions Using Laplace Transform 504

5.11 Solutions Using Frequency Transfer Functions 512

5.12 Examples Using MATLAB 515

Chapter Summary 522

References 523

Review Questions 523

Problems 526

Design Projects 552

**CHAPTER 6**

**Multidegree-of-Freedom Systems 553**

6.1 Introduction 555

6.2 Modeling of Continuous Systems as Multidegreeof-Freedom Systems 555

6.3 Using Newton s Second Law to Derive Equationsof Motion 557

6.4 Influence Coefficients 562

6.4.1 Stiffness Influence Coefficients 562

6.4.2 Flexibility Influence Coefficients 567

6.4.3 Inertia Influence Coefficients 572

6.5 Potential and Kinetic Energy Expressions in Matrix Form 574

6.6 Generalized Coordinates and Generalized Forces 576

6.7 Using Lagrange s Equations to Derive Equations of Motion 577

6.8 Equations of Motion of Undamped Systems in Matrix Form 581

6.9 Eigenvalue Problem 583

6.10 Solution of the Eigenvalue Problem 585

6.10.1 Solution of the Characteristic(Polynomial) Equation 585

6.10.2 Orthogonality of Normal Modes 591

6.10.3 Repeated Eigenvalues 594

6.11 Expansion Theorem 596

6.12 Unrestrained Systems 596

6.13 Free Vibration of Undamped Systems 601

6.14 Forced Vibration of Undamped Systems Using Modal Analysis 603

6.15 Forced Vibration of Viscously Damped Systems 610

6.16 Self-Excitation and Stability Analysis 617

6.17 Examples Using MATLAB 619

Chapter Summary 627

References 627

Review Questions 628

Problems 632

Design Project 653

**CHAPTER 7**

**Determination of Natural Frequencies and Mode Shapes 654**

7.1 Introduction 655

7.2 Dunkerley s Formula 656

7.3 Rayleigh s Method 658

7.3.1 Properties of Rayleigh s Quotient 659

7.3.2 Computation of the Fundamental Natural Frequency 661

7.3.3 Fundamental Frequency of Beams and Shafts 663

7.4 Holzer s Method 666

7.4.1 Torsional Systems 666

7.4.2 Spring-Mass Systems 669

7.5 Matrix Iteration Method 670

7.5.1 Convergence to the Highest Natural Frequency 672

7.5.2 Computation of Intermediate Natural Frequencies 673

7.6 Jacobi s Method 678

7.7 Standard Eigenvalue Problem 680

7.7.1 Choleski Decomposition 681

7.7.2 Other Solution Methods 683

7.8 Examples Using MATLAB 683

Chapter Summary 686

References 686

Review Questions 688

Problems 690

Design Projects 698

**CHAPTER 8**

**Continuous Systems 699**

8.1 Introduction 700

8.2 Transverse Vibration of a String or Cable 701

8.2.1 Equation of Motion 701

8.2.2 Initial and Boundary Conditions 703

8.2.3 Free Vibration of a Uniform String 704

8.2.4 Free Vibration of a String with Both Ends Fixed 705

8.2.5 Traveling-Wave Solution 709

8.3 Longitudinal Vibration of a Bar or Rod 710

8.3.1 Equation of Motion and Solution 710

8.3.2 Orthogonality of Normal Functions 713

8.4 Torsional Vibration of a Shaft or Rod 718

8.5 Lateral Vibration of Beams 721

8.5.1 Equation of Motion 721

8.5.2 Initial Conditions 723

8.5.3 Free Vibration 723

8.5.4 Boundary Conditions 724

8.5.5 Orthogonality of Normal Functions 726

8.5.6 Forced Vibration 730

8.5.7 Effect of Axial Force 732

8.5.8 Effects of Rotary Inertia and Shear Deformation 734

8.5.9 Other Effects 739

8.6 Vibration of Membranes 739

8.6.1 Equation of Motion 739

8.6.2 Initial and Boundary Conditions 741

8.7 Rayleigh s Method 742

8.8 The Rayleigh-Ritz Method 745

8.9 Examples Using MATLAB 748

Chapter Summary 751

References 751

Review Questions 753

Problems 756

Design Project 768

**CHAPTER 9**

**Vibration Control 769**

9.1 Introduction 770

9.2 Vibration Nomograph and Vibration Criteria 771

9.3 Reduction of Vibration at the Source 775

9.4 Balancing of Rotating Machines 776

9.4.1 Single-Plane Balancing 776

9.4.2 Two-Plane Balancing 779

9.5 Whirling of Rotating Shafts 785

9.5.1 Equations of Motion 785

9.5.2 Critical Speeds 787

9.5.3 Response of the System 788

9.5.4 Stability Analysis 790

9.6 Balancing of Reciprocating Engines 792

9.6.1 Unbalanced Forces Due to Fluctuations in Gas Pressure 792

9.6.2 Unbalanced Forces Due to Inertia of the Moving Parts 793

9.6.3 Balancing of Reciprocating Engines 796

9.7 Control of Vibration 798

9.8 Control of Natural Frequencies 798

9.9 Introduction of Damping 799

9.10 Vibration Isolation 801

9.10.1 Vibration Isolation System with Rigid Foundation 804

9.10.2 Vibration Isolation System with Base Motion 814

9.10.3 Vibration Isolation System with Flexible Foundation 821

9.10.4 Vibration Isolation System with Partially Flexible Foundation 822

9.10.5 Shock Isolation 824

9.10.6 Active Vibration Control 827

9.11 Vibration Absorbers 832

9.11.1 Undamped Dynamic Vibration Absorber 833

9.11.2 Damped Dynamic Vibration Absorber 840

9.12 Examples Using MATLAB 843

Chapter Summary 851

References 851

Review Questions 853

Problems 855

Design Project 869

**CHAPTER 10**

**Vibration Measurement and Applications 870**

10.1 Introduction 871

10.2 Transducers 873

10.2.1 Variable Resistance Transducers 873

10.2.2 Piezoelectric Transducers 876

10.2.3 Electrodynamic Transducers 877

10.2.4 Linear Variable Differential TransformerTransducer 878

10.3 Vibration Pickups 879

10.3.1 Vibrometer 881

10.3.2 Accelerometer 882

10.3.3 Velometer 886

10.3.4 Phase Distortion 888

10.4 Frequency-Measuring Instruments 890

10.5 Vibration Exciters 892

10.5.1 Mechanical Exciters 892

10.5.2 Electrodynamic Shaker 893

10.6 Signal Analysis 895

10.6.1 Spectrum Analyzers 896

10.6.2 Bandpass Filter 897

10.6.3 Constant-Percent Bandwidth andConstant-Bandwidth Analyzers 898

10.7 Dynamic Testing of Machines and Structures 900

10.7.1 Using Operational Deflection-Shape Measurements 900

10.7.2 Using Modal Testing 900

10.8 Experimental Modal Analysis 900

10.8.1 The Basic Idea 900

10.8.2 The Necessary Equipment 900

10.8.3 Digital Signal Processing 903

10.8.4 Analysis of Random Signals 905

10.8.5 Determination of Modal Data from Observed Peaks 907

10.8.6 Determination of Modal Data from Nyquist Plot 910

10.8.7 Measurement of Mode Shapes 912

10.9 Machine Condition Monitoring and Diagnosis 915

10.9.1 Vibration Severity Criteria 915

10.9.2 Machine Maintenance Techniques 915

10.9.3 Machine Condition Monitoring Techniques 916

10.9.4 Vibration Monitoring Techniques 918

10.9.5 Instrumentation Systems 924

10.9.6 Choice of Monitoring Parameter 924

10.10 Examples Using MATLAB 925

Chapter Summary 928

References 928

Review Questions 930

Problems 932

Design Projects 938

**CHAPTER 11**

**Numerical Integration Methods in Vibration Analysis 939**

11.1 Introduction 940

11.2 Finite Difference Method 941

11.3 Central Difference Method for Single-Degree-of- Freedom Systems 942

11.4 Runge-Kutta Method for Single-Degree-of- Freedom Systems 945

11.5 Central Difference Method for Multidegree-of- Freedom Systems 947

11.6 Finite Difference Method for Continuous Systems 951

11.6.1 Longitudinal Vibration of Bars 951

11.6.2 Transverse Vibration of Beams 955

11.7 Runge-Kutta Method for Multidegree-of- Freedom Systems 960

11.8 Houbolt Method 962

11.9 Wilson Method 965

11.10 Newmark Method 968

11.11 Examples Using MATLAB 972

Chapter Summary 978

References 978

Review Questions 979

Problems 981

**CHAPTER 12**

**Finite Element Method 987**

12.1 Introduction 988

12.2 Equations of Motion of an Element 989

12.3 Mass Matrix, Stiffness Matrix, and Force Vector 991

12.3.1 Bar Element 991

12.3.2 Torsion Element 994

12.3.3 Beam Element 995

12.4 Transformation of Element Matrices and Vectors 998

12.5 Equations of Motion of the Complete System of Finite Elements 1001

12.6 Incorporation of Boundary Conditions 1003

12.7 Consistent- and Lumped-Mass Matrices 1012

12.7.1 Lumped-Mass Matrix for a Bar Element 1012

12.7.2 Lumped-Mass Matrix for a Beam Element 1012

12.7.3 Lumped-Mass Versus Consistent-Mass Matrices 1013

12.8 Examples Using MATLAB 1015

Chapter Summary 1019

References 1019

Review Questions 1020

Problems 1022

**Design Projects 1034**

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