# Fundamentals of Mechanical Vibrations

ISBN: 978-1-119-05012-4
480 pages
June 2016

## Description

This introductory book covers the most fundamental aspects of linear vibration analysis for mechanical engineering students and engineers. Consisting of five major topics, each has its own chapter and is aligned with five major objectives of the book. It starts from a concise, rigorous and yet accessible introduction to Lagrangian dynamics as a tool for obtaining the governing equation(s) for a system, the starting point of vibration analysis. The second topic introduces mathematical tools for vibration analyses for single degree-of-freedom systems. In the process, every example includes a section Exploring the Solution with MATLAB. This is intended to develop student's affinity to symbolic calculations, and to encourage curiosity-driven explorations. The third topic introduces the lumped-parameter modeling to convert simple engineering structures into models of equivalent masses and springs. The fourth topic introduces mathematical tools for general multiple degrees of freedom systems, with many examples suitable for hand calculation, and a few computer-aided examples that bridges the lumped-parameter models and continuous systems. The last topic introduces the finite element method as a jumping point for students to understand the theory and the use of commercial software for vibration analysis of real-world structures.

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Series Preface ix

Preface xi

1 A Crash Course on Lagrangian Dynamics 1

1.1 Objectives 1

1.2 Concept of "Equation of Motion" 1

1.3 Generalized Coordinates 5

1.5 Degrees of Freedom 16

1.6 Virtual Work and Generalized Forces 17

1.7 Lagrangian 24

1.8 Lagrange’s Equation 24

1.9 Procedure for Deriving Equation(s) of Motion 24

1.10 Worked Examples 25

1.10.1 Systems Containing Only Particles 25

1.10.2 Systems Containing Rigid Bodies 38

1.11 Linearization of Equations of Motion 57

1.11.1 Equilibrium Position(s) 58

1.11.2 Linearization 59

1.11.3 Observations and Further Discussions 62

1.12 Chapter Summary 63

2 Vibrations of Single-DOF Systems 81

2.1 Objectives 81

2.2 Types of Vibration Analyses 81

2.3 Free Vibrations of Undamped System 83

2.3.1 General Solution for Homogeneous Differential Equation 83

2.3.2 Basic Vibration Terminologies 85

2.3.3 Determining Constants via Initial Conditions 87

2.4 Free Vibrations of Damped Systems 93

2.5 Using Normalized Equation of Motion 94

2.5.1 Normalization of Equation of Motion 94

2.5.2 Classification of Vibration Systems 95

2.5.3 Free Vibration of Underdamped Systems 96

2.5.4 Free Vibration of Critically Damped System 100

2.5.5 Free Vibration of Overdamped System 102

2.6 Forced Vibrations I: Steady-State Responses 108

2.6.2 Mechanical Significance of Steady-State Solution 110

2.7 Forced Vibrations II: Transient Responses 133

2.7.3 Laplace Transform Method 146

2.7.4 Decomposition Method 150

2.7.5 Convolution Integral Method 158

2.8 Chapter Summary 172

2.8.1 Free Vibrations of Single-DOF Systems 172

2.8.2 Steady-State Responses of Single-DOF Systems 173

2.8.3 Transient Responses of Single-DOF Systems 174

3 Lumped-Parameter Modeling 186

3.1 Objectives 186

3.2 Modeling 186

3.3 Idealized Elements 187

3.3.1 Mass Elements 187

3.3.2 Spring Elements 188

3.3.3 Damping Elements 189

3.4 Lumped-Parameter Modeling of Simple Components and Structures 190

3.4.1 Equivalent Spring Constants 191

3.4.2 Equivalent Masses 204

3.4.3 Damping Models 212

3.5 Alternative Methods 218

3.5.1 Castigliano Method for Equivalent Spring Constants 218

3.5.2 Rayleigh–Ritz Method for Equivalent Masses 223

3.5.3 Rayleigh–Ritz Method for Equivalent Spring Constants 227

3.5.4 Rayleigh–Ritz Method for Natural Frequencies 230

3.5.5 Determining Lumped Parameters Through Experimental Measurements 231

3.6 Examples with Lumped-Parameter Models 233

3.7 Chapter Summary 252

4 Vibrations of Multi-DOF Systems 269

4.1 Objectives 269

4.2 Matrix Equation of Motion 269

4.3 Modal Analysis: Natural Frequencies and Mode Shapes 273

4.4 Free Vibrations 284

4.4.1 Free Vibrations of Undamped Systems 284

4.4.2 Free Vibrations of Undamped Unconstrained Systems 293

4.4.3 Free Vibrations of Systems of Many DOFs 296

4.5 Eigenvalues and Eigenvectors 305

4.5.1 Standard Eigenvalue Problem 305

4.5.2 Generalized Eigenvalue Problem 306

4.6 Coupling, Decoupling, and Principal Coordinates 307

4.6.1 Types of Coupling 307

4.6.2 Principal Coordinates 307

4.7 Forced Vibrations I: Steady-State Responses 319

4.8 Forced Vibrations II: Transient Responses 328

4.8.1 Direct Analytical Method 328

4.8.2 Decoupling Method 331

4.8.3 Laplace Transform Method 347

4.8.4 Convolution Integral Method 349

4.9 Chapter Summary 357

4.9.1 Modal Analyses 357

4.9.2 Free Vibrations of Multi-DOF Systems 357

4.9.3 Steady-State Responses of Multi-DOF Systems 359

4.9.4 Transient Responses of Multi-DOF Systems 359

5 Vibration Analyses Using Finite Element Method 370

5.1 Objectives 370

5.2 Introduction to Finite Element Method 370

5.2.1 Lagrangian Dynamics Formulation of FEM Model 371

5.2.2 Matrix Formulation 374

5.3 Finite Element Analyses of Beams 378

5.3.1 Formulation of Beam Element 379

5.3.2 Implementation Using MATLAB 383

5.3.3 Generalization: Large-Scale Finite Element Simulations 392

5.3.4 Damping Models in Finite Element Modeling 394

5.4 Vibration Analyses Using SOLIDWORKS 395

5.4.1 Introduction to SOLIDWORKS Simulation 396

5.4.2 Static Analysis 398

5.4.3 Modal Analysis 415

5.4.4 Harmonic Vibration Analysis 419

5.4.5 Transient Vibration Analysis 425

5.5 Chapter Summary 428

5.5.1 Finite Element Formulation 428

5.5.2 Using Commercial Finite Element Analysis Software 429

Appendix A Review of Newtonian Dynamics 433

A.1 Kinematics 433

A.1.1 Kinematics of a Point or a Particle 433

A.1.2 Relative Motions 435

A.1.3 Kinematics of a Rigid Body 436

A.2 Kinetics 437

A.2.1 Newton–Euler Equations 437

A.2.2 Energy Principles 438

A.2.3 Momentum Principles 439

Appendix B A Primer on MatLab 440

B.1 Matrix Computations 440

B.1.1 Commands and Statements 440

B.1.2 Matrix Generation 441

B.1.3 Accessing Matrix Elements and Submatrices 442

B.1.4 Operators and Elementary Functions 444

B.1.5 Flow Controls 446

B.1.6 M-Files, Scripts, and Functions 449

B.1.7 Linear Algebra 452

B.2 Plotting 454

B.2.1 Two-Dimensional Curve Plots 454

B.2.2 Three-Dimensional Curve Plots 456

B.2.3 Three-Dimensional Surface Plots 457

Appendix C Tables of Laplace Transform 459

C.1 Properties of Laplace Transform 459

C.2 Function Transformations 459

Index 461

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## Author Information

Liang-Wu Cai, Associate Professor, Department of Mechanical and Nuclear Engineering, Kansas State University
Dr. Cai completed his Sc.D. degree in Applied Mechanics from Massachusetts Institute of Technology in 1998. He has been teaching at Kansas State University since 2001 covering a wide range of topics in both undergraduate and graduate levels, from dynamics, to machine design, mechanical vibration, finite element analysis, to theory of elasticity, and wave dynamics. Dr. Cai is a Fellow of ASME, currently serves as an Associate Editor for the American Society of mechanical Engineering (ASME) Journal of Vibration and Acoustics, and the Vice Chair of ASME’s Noise Control and Acoustics Division.
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