PREFACE
With the recent interest in environmental issues, vibration problems have beena subject of intense interest for many years in modern industrial societies, such asmaintaining high performance level and production efficiency. Various controltechniques have been developed in different fields to reduce vibration levels.Although vibration control has been studied for long time, it remains and indeedbecomes more challenging in many applications such as aerospace and aeronauticindustry. Furthermore, it is important to understand and analyze the vibrationperformance of the engineering system when we try to control the structural andmechanical vibration levels.
This book provides the background and techniques for modeling, analysis,design, and control of vibration in mechanical engineering systems. This book issuitable as a course textbook in a single-semester course for second-year or thirdyearundergraduate students, or for Master degree candidate in any branch ofengineering such as aeronautical and aerospace, aircraft design, and mechanicalengineering. The book can also serve as a valuable reference tool for practicingengineers with an interest in vibration problems.
This book is an outgrowth of the first author??s experience in teaching undergraduateand graduate courses in Basic of Mechanical Vibration, Active Noise and VibrationControl, for more than ten years.
The topics discussed in this book include free and forced vibration of the singledegree-of-freedom systems and multiple-degree-of-freedom systems. Vibrations ofcontinuous systems are also presented. This book is organized around the conceptsof vibration analysis of basic mechanical systems. Although the earlier chaptersprovide the basis for the later chapters, each chapter is written to be as selfcontainedas possible, with excerpts from earlier chapters provided as needed. Thisknowledge should be useful in the practice of vibration analysis or control,regardless of the application area or the branch of engineering.
Readers are expected to obtain skills ranging from the ability to performinsightful hand analysis to the ability to develop algorithms for numerical/computeranalysis. A large number of examples are presented throughout the book so thatthe subject can be better understood. To fully master the analytical techniques, it isessential that the students solve many of the homework problems which areprovided at the end of each chapter. In addition, many MATLAB programs arepresented for additional study. Furthermore, the MATLAB GUI (Graphical UserInterfaces) files were used to generate most of the results presented throughout thebook. The reader is welcome to use them freely.
Mao Qibo wishes to appreciate the financial support from the National NaturalScience Foundation of China (No.51975266, No.11464031), the Natural ScienceFoundation of Jiangxi, China (No. 20192BAB206024), the Six Talent PeaksProject of Jiangsu Province, China (No. 2017-KTHY-036) and the PublishingFoundation of Nanchang Hangkong University.
Moreover, we wish to thank the reviewers for their valuable comments andsuggestions. We appreciate Ms. Ying Zeduan from Macau University of Scienceand Technology for her excellent work in improving the readability of the book. Wegratefully acknowledge the editorial and production staff of Beihang UniversityPress including Ms. Dong Rui.
In the end, we dedicate this book to our families.
Mao Qibo and Li Yi
August 2019
NOTES TO STUDENTS
Is it not a delight to acquire knowledge and put it into practice from time totime?
—Confucius (Chinese great educator)
Although mechanical engineering is an exciting and challenging discipline, youmay be frightened during studying the mechanical vibration course. This book waswritten to try to prevent that. A good textbook and a good professor are anadvantage—but you are the one who does the learning. If you keep the followingideas in mind, you will do very well to study mechanical vibrations:
Ø The basic of mechanical vibration provides a foundation for other courses inthe mechanical engineering. For this reason, put in as much effort as youcan. Study the course regularly.
Ø Mechanical vibration is a problem-solving subject, so we should learnthrough practice. Solve as many problems as you can. The best way to learnmechanical vibration is to solve a lot of problems by hand or by usingcomputer software.
Ø MATLAB is a very useful software in signals analysis and other courses youmay be taking. A brief tutorial on MATLAB is given in Appendix B, whichhelp you to start learn this software. The best way to learn MATLAB is totry it once you know a few functions and commands.
Ø Attempt the review questions at the end of each chapter. They will help youdiscover some “tricks” not revealed in the class or in the textbook.
A short review of the mathematical formulas you may need is covered inAppendix A. Some terminology in English and in Chinese are listed in Appendix D.
Contents
Chapter 1 Introduction 1
1.1 Background 1
1.2 Study of Vibration 3
1.3 Basic Concepts of Vibration 6
1.3.1 Degrees of freedom 7
1.3.2 Newton??s laws 8
1.3.3 Vibration classification 8
1.4 Organization of the Book 9
Questions 10
Chapter 2 Free Vibration of Single-Degree-Of-Freedom(SDOF) Systems 11
2.1 The Basic Mechanical Components 11
2.2 Free Vibration of UndampedSystems 12
2.2.1 Modelling of undamped SDOF systems 12
2.2.2 Simple harmonic motion 15
2.2.3 MATLAB examples 15
2.2.4 Solution for undamped SDOF systems 18
2.3 Rotary Systems 20
2.4 Springs Connected in Series or in Parallel 22
2.5 Modelling Using Energy Method 23
2.6 Viscously Damped SDOF Systems 29
2.6.1 Case 1: Overdamped motion (ζ> 1) 31
2.6.2 Case 2: Underdamped motion (ζ< 1) 33
2.6.3 Case 3: Critically damped motion (ζ=1) 34
2.7 Evaluating Damping Ratio from Measurements (Logarithmic Decrement) 36
2.8 Summary: the Effects of Damping on an Unforced Mass-spring System 39
2.9 MATLAB Examples for Free Vibration of SDOF Systems 39
2.9.1 Free vibration for undamped SDOF systems 39
2.9.2 Free vibration for underdamped SDOF systems 40
2.9.3 Free vibration for critical damped SDOF systems 42
2.9.4 Free vibration for overdamped SDOF systems 43
2.9.5 The GUI program for free vibration of SDOF system 44
Questions 45
Chapter 3 Harmonic Excitation of SDOF Systems 50
3.1 Harmonic Excitation 50
3.2 Complex Analysis 53
3.3 Undamped SDOF System with Harmonic Excitation 54
3.3.1 Excitation frequency ≠ natural frequency 54
3.3.2 Excitation frequency=natural frequency (resonant condition) 56
3.3.3 Response ratio for undamped SDOF system 57
3.4 Damped SDOF System with Harmonic Excitation 59
3.4.1 Response for damped SDOF system with harmonic excitation 59
3.4.2 Dynamic magnification factor for damped SDOF system 60
3.4.3 Response ratio for β=1 63
3.5 Harmonic Base Excitation 65
3.5.1 Relative motion 66
3.5.2 Absolute motion 67
3.6 Transmissibility of Vibration 73
3.6.1 Motion transmissibility 74
3.6.2 Force transmissibility 75
3.7 Rotating Unbalance 77
3.8 MATLAB Examples for Forced SDOF System 81
3.8.1 Harmonic excitation of undamped SDOF systems 81
3.8.2 Rotating unbalance vibration of SDOF systems 83
Questions 85
Chapter 4 Vibration of SDOF Systems under General Excitation 92
4.1 The Impulse Response 92
4.2 The Principle of Superposition 95
4.3 Response of SDOF Systems under a General Periodic Force 97
4.4 System??s Response to General Excitation by Convolution 104
4.5 System??s Response to General Excitation by the Laplace Transform 108
4.6 The Transfer Function 112
4.7 Composite Function Excitation 113
Questions 116
Chapter 5 Vibration of Multiple-Degree-Of-Freedom (MDOF) Systems 119
5.1 Free Vibration of Structures with Two-Degree-of-Freedom 119
5.1.1 Equations of motion for free vibration of two-degree-of-freedom 120
5.1.2 Free vibration analysis 121
5.1.3 Free vibration responses 124
5.2 Modelling of a System with n Degrees of Freedom 125
5.2.1 Governing equations 125
5.2.2 Coordinate coupling 130
5.3 Influence Coefficient Method 133
5.3.1 The stiffness influence coefficients 134
5.3.2 Flexibility influence coefficients 136
5.4 The Lagrange??sEquation 137
5.4.1 Derivation of Lagrange??s equations in Cartesian coordinates 138
5.4.2 Extension to general coordinate systems 140
5.4.3 Application of Lagrange??sequation 140
5.5 Solving the Equations of Motion of Undamped Systems UsingMATLAB 144
5.6 Vibration of Undamped MDOF System 148
5.6.1 Free response of undamped MDOF systems 148
5.6.2 Harmonic excitation of undampedsystems 156
5.7 Vibration of Underdamped MDOF Systems 158
5.7.1 Free vibration of underdamped MDOF systems 158
5.7.2 Steady-state forced vibration response for MDOF systems 162
5.8 The Dynamic Vibration Absorber 166
Questions 169
Chapter 6 Vibration of Continuous Systems 175
6.1 Transverse Vibration of Cables 175
6.2 Transverse Vibration of Beams 181
6.2.1 The governing equation of motion for beams 181
6.2.2 The natural frequencies and mode shapes for beams 183
6.2.3 MATLAB examples 186
6.3 The Structural Modes for Plates 190
Questions 193
Appendix A Mathematical Background 195
A.1 Mathematical Relations 195
A.2 Ordinary Differential Equations (ODE) 195
Appendix B Basic of MATLAB 199
B.1 How to Read this MATLAB Tutorial 199
B.2 Making Matrix 200
B.2.1 The colon operator 201
B.2.2 Matrix manipulations 202
B.2.3 The end keyword 203
B.2.4 Transposing a matrix 203
B.2.5 Element-wise operations 204
B.3 Functions 205
B.4 Graphics and Plotting 205
B.5 Programming in MATLAB 208
B.5.1 The M-files 208
B.5.2 Repeating with “for” loops 208
B.5.3 “if” statements 209
B.5.4 Writing function subroutines 210
B.6 Saving and Loading 210
B.7 The Help Menu 211
B.8 Set the Display Format for Output 212
B.9 Closing Remarks and References 212
Appendix C Properties of Laplace Transform and Laplace Transform Pairs 214
Appendix D Technical Terms 216
References 228
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