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Mechanical Vibrations, Fifth Edition

Engineering Applications and Design
Engineering Vibrations

Mechanical Vibrations, Fifth Edition
Simgiresu S. Rao
1105 Pages

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Mechanical Vibrations, Fifth Edition

Preface

This book serves as an introduction to the subject of vibration engineering at the undergraduate level. Favorable reactions by professors and students to the fourth edition have encouraged me to prepare this fifth edition of the book. I have retained the style of the prior editions, presenting the theory, computational aspects, and applications of vibration in as simple a manner as possible, and emphasizing computer techniques of analysis. Expanded explanations of the fundamentals are given, emphasizing physical significance and interpretation that build upon previous experiences in undergraduate mechanics. Numerous examples and problems are used to illustrate principles and concepts.

TOC

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 Harmonic Motion 56
1.10.2 Complex-Number Representation of 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 Computation of Coefficients 72
1.12 Examples Using MATLAB 76
1.13 Vibration Literature 80

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 Second Law of Motion 129
2.2.2 Equation of Motion Using Other Methods 130
2.2.3 Equation of Motion of a Spring-Mass System 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 Systems and 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 Viscous Damping 166
2.6.5 Torsional Systems with Viscous Damping 168
2.7 Graphical Representation of Characteristic Roots and Corresponding Solutions 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 Locus Representations 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 Coulomb Damping 190
2.10 Free Vibration with Hysteretic Damping 192
2.11 Stability of Systems 198
2.12 Examples Using MATLAB 202

CHAPTER 3
Harmonically Excited Vibration 259
3.1 Introduction 261
3.2 Equation of Motion 261
3.3 Response of an Undamped System Under Harmonic Force 263
3.3.1 Total Response 267
3.3.2 Beating Phenomenon 267
3.4 Response of a Damped System Under Harmonic Force 271
3.4.1 Total Response 274
3.4.2 Quality Factor and Bandwidth 276
3.5 Response of a Damped System Under 278
3.6 Response of a Damped System Under the Harmonic 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 Rotating Unbalance 287
3.8 Forced Vibration with Coulomb Damping 293
3.9 Forced Vibration with Hysteresis Damping 298
3.10 Forced Motion with Other Types of Damping 300
3.11 Self-Excitation and Stability Analysis 301
3.11.1 Dynamic Stability Analysis 301
3.11.2 Dynamic Instability Caused by Fluid Flow 305
3.12 Transfer-Function Approach 313
3.13 Solutions Using Laplace Transforms 317
3.14 Frequency Transfer Functions 320
3.14.1 Relation Between the General Transfer function T(s) and the Frequency Transfer Function 322
3.14.2 Representation of Frequency-Response Characteristics 323
3.15 Examples Using MATLAB 326

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 Using Numerical Methods 431
4.10 Examples Using MATLAB 436

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 6
Multidegree-of-Freedom Systems 553
6.1 Introduction 555
6.2 Modeling of Continuous Systems as Multidegree-of- Freedom Systems 555
6.3 Using Newton s Second Law to Derive Equations of 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 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 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 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 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 Transformer Transducer 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 and Constant-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 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 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