Mechanical Vibrations in SI Units

Singiresu S. Rao

BOOK

Mechanical Vibrations in SI Units

Singiresu S. Rao

(2017)

Additional Information

Book Details

ISBN: 978-1-292-17861-5
Edition: 6
Language: English
Pages: 1152
Subjects

Abstract

For courses in vibration engineering.

Building Knowledge: Concepts of Vibration in Engineering

Retaining the style of previous editions, this Sixth SI Edition of Mechanical Vibrations effectively presents theory, computational aspects, and applications of vibration, introducing undergraduate engineering students to the subject of vibration engineering in as simple a manner as possible. Emphasizing computer techniques of analysis, Mechanical Vibrations thoroughly explains the fundamentals of vibration analysis, building on the understanding achieved by students in previous undergraduate mechanics courses. Related concepts are discussed, and real-life applications, examples, problems, and illustrations related to vibration analysis enhance comprehension of all concepts and material. In the Sixth SI Edition, several additions and revisions have been made—including new examples, problems, and illustrations—with the goal of making coverage of concepts both more comprehensive and easier to follow.

Section Title	Page	Action	Price
Front Cover	Front Cover
Equivalent Masses, Springs and Dampers	1
Title Page	5
Copyright Page	6
Contents	9
Preface	16
Acknowledgments	21
List of Symbols	23
Chapter 1 Fundamentals of Vibration	29
1.1 Preliminary Remarks	30
1.2 Brief History of the Study of Vibration	31
1.2.1 Origins of the Study of Vibration	31
1.2.2 From Galileo to Rayleigh	33
1.2.3 Recent Contributions	36
1.3 Importance of the Study of Vibration	38
1.3.1 Conversion of Vibrations to Sound by the Human Ear	40
1.4 Basic Concepts of Vibration	43
1.4.1 Vibration	43
1.4.2 Elementary Parts of Vibrating Systems	43
1.4.3 Number of Degrees of Freedom	44
1.4.4 Discrete and Continuous Systems	46
1.5 Classification of Vibration	46
1.5.1 Free and Forced Vibration	46
1.5.2 Undamped and Damped Vibration	47
1.5.3 Linear and Nonlinear Vibration	47
1.5.4 Deterministic and Random Vibration	47
1.6 Vibration Analysis Procedure	48
1.7 Spring Elements	52
1.7.1 Nonlinear Springs	53
1.7.2 Linearization of a Nonlinear Spring	55
1.7.3 Spring Constants of Elastic Elements	57
1.7.4 Combination of Springs	60
1.7.5 Spring Constant Associated with the Restoring Force due to Gravity	68
1.8 Mass or Inertia Elements	69
1.8.1 Combination of Masses	70
1.9 Damping Elements	74
1.9.1 Construction of Viscous Dampers	75
1.9.2 Linearization of a Nonlinear Damper	81
1.9.3 Combination of Dampers	81
1.10 Harmonic Motion	83
1.10.1 Vectorial Representation of Harmonic Motion	85
1.10.2 Complex-Number Representation of Harmonic Motion	86
1.10.3 Complex Algebra	87
1.10.4 Operations on Harmonic Functions	87
1.10.5 Definitions and Terminology	90
1.11 Harmonic Analysis	93
1.11.1 Fourier Series Expansion	93
1.11.2 Complex Fourier Series	95
1.11.3 Frequency Spectrum	96
1.11.4 Time- and Frequency-Domain Representations	97
1.11.5 Even and Odd Functions	98
1.11.6 Half-Range Expansions	100
1.11.7 Numerical Computation of Coefficients	101
1.12 Examples Using MATLAB	105
1.13 Vibration Literature	109
Chapter Summary	110
References	110
Review Questions	112
Problems	116
Design Projects	149
Chapter 2 Free Vibration of Single-Degree-of-Freedom Systems	153
2.1 \rIntroduction	155
2.2 Free Vibration of an Undamped Translational System	158
2.2.1 Equation of Motion Using Newton’s Second Law of Motion	158
2.2.2 Equation of Motion Using Other Methods	159
2.2.3 Equation of Motion of a Spring-Mass System in Vertical Position	161
2.2.4 Solution	162
2.2.5 Harmonic Motion	163
2.3 Free Vibration of an Undamped Torsional System	176
2.3.1 Equation of Motion	177
2.3.2 Solution	178
2.4 Response of First-Order Systems and Time Constant	181
2.5 Rayleigh’s Energy Method	183
2.6 Free Vibration with Viscous Damping	188
2.6.1 Equation of Motion	188
2.6.2 Solution	189
2.6.3 Logarithmic Decrement	198
2.6.4 Energy Dissipated in Viscous Damping	199
2.6.5 Torsional Systems with Viscous Damping	201
2.7 Graphical Representation of Characteristic Roots and Corresponding Solution	207
2.7.1 Roots of the Characteristic Equation	207
2.7.2 Graphical Representation of Roots and Corresponding Solutions	208
2.8 Parameter Variations and Root Locus Representations	209
2.8.1 Interpretations of ωn, ωd, ζ, and τ in the s-plane	209
2.8.2 Root Locus and Parameter Variations	212
2.9 Free Vibration with Coulomb Damping	218
2.9.1 Equation of Motion	219
2.9.2 Solution	220
2.9.3 Torsional Systems with Coulomb Damping	223
2.10 Free Vibration with Hysteretic Damping	225
2.11 Stability of Systems	231
2.12 Examples Using MATLAB	235
Chapter Summary	241
References	242
Review Questions	242
Problems	247
Design Projects	294
Chapter 3 Harmonically Excited Vibration	297
3.1 Introduction	299
3.2 Equation of Motion	299
3.3 Response of an Undamped System Under Harmonic Force	301
3.3.1 Total Response	305
3.3.2 Beating Phenomenon	305
3.4 Response of a Damped System Under Harmonic Force	309
3.4.1 Total Response	312
3.4.2 Quality Factor and Bandwidth	316
3.5 Response of a Damped System Under F(t) = F0eiwt	317
3.6 Response of a Damped System Under the Harmonic Motion of the Base	320
3.6.1 Force Transmitted	322
3.6.2 Relative Motion	323
3.7 Response of a Damped System Under Rotating Unbalance	326
3.8 Forced Vibration with Coulomb Damping	332
3.9 Forced Vibration with Hysteresis Damping	337
3.10 Forced Motion with Other Types of Damping	339
3.11 Self-Excitation and Stability Analysis	340
3.11.1 Dynamic Stability Analysis	340
3.11.2 Dynamic Instability Caused by Fluid Flow	344
3.12 Transfer-Function Approach	352
3.13 Solutions Using Laplace Transforms	356
3.14 Frequency Transfer Functions	359
3.14.1 Relation between the General Transfer Function T(s) and the Frequency Transfer Function T (iw)	361
3.14.2 Representation of Frequency-Response Characteristics	362
3.15 Examples Using MATLAB	365
Chapter Summary	371
References	371
Review Questions	372
Problems	375
Design Projects	402
Chapter 4 Vibration Under General Forcing Conditions	403
4.1 Introduction	404
4.2 Response Under a General Periodic Force	405
4.2.1 First-Order Systems	406
4.2.2 Second-Order Systems	412
4.3 Response Under a Periodic Force of Irregular Form	418
4.4 Response Under a Nonperiodic Force	420
4.5 Convolution Integral	421
4.5.1 Response to an Impulse	422
4.5.2 Response to a General Forcing Condition	425
4.5.3 Response to Base Excitation	426
4.6 Response Spectrum	434
4.6.1 Response Spectrum for Base Excitation	436
4.6.2 Earthquake Response Spectra	439
4.6.3 Design Under a Shock Environment	443
4.7 Laplace Transforms	446
4.7.1 Transient and Steady-State Responses	446
4.7.2 Response of First-Order Systems	447
4.7.3 Response of Second-Order Systems	449
4.7.4 Response to Step Force	454
4.7.5 Analysis of the Step Response	460
4.7.6 Description of Transient Response	461
4.8 Numerical Methods	467
4.8.1 Runge-Kutta Methods	469
4.9 Response to Irregular Forcing Conditions Using Numerical Methods	471
4.10 Examples Using MATLAB	476
Chapter Summary	480
References	480
Review Questions	481
Problems	484
Design Projects	506
Chapter 5 Two-Degree-of-Freedom Systems	509
5.1 Introduction	510
5.2 Equations of Motion for Forced Vibration	514
5.3 Free-Vibration Analysis of an Undamped System	516
5.4 Torsional System	525
5.5 Coordinate Coupling and Principal Coordinates	530
5.6 Forced-Vibration Analysis	536
5.7 Semidefinite Systems	539
5.8 Self-Excitation and Stability Analysis	542
5.9 Transfer-Function Approach	544
5.10 Solutions Using Laplace Transform	546
5.11 Solutions Using Frequency Transfer Functions	554
5.12 Examples Using MATLAB	557
Chapter Summary	564
References	565
Review Questions	565
Problems	568
Design Projects	594
Chapter 6 Multidegree-of-Freedom Systems	596
6.1 Introduction	598
6.2 Modeling of Continuous Systems as Multidegree-of-Freedom Systems	598
6.3 Using Newton’s Second Law to Derive Equations of Motion	600
6.4 Influence Coefficients	605
6.4.1 Stiffness Influence Coefficients	605
6.4.2 Flexibility Influence Coefficients	610
6.4.3 Inertia Influence Coefficients	615
6.5 Potential and Kinetic Energy Expressions in Matrix Form	617
6.6 Generalized Coordinates and Generalized Forces	619
6.7 Using Lagrange’s Equations to Derive Equations of Motion	620
6.8 Equations of Motion of Undamped Systems in Matrix Form	624
6.9 Eigenvalue Problem	626
6.10 Solution of the Eigenvalue Problem	628
6.10.1 Solution of the Characteristic (Polynomial) Equation	628
6.10.2 Orthogonality of Normal Modes	634
6.10.3 Repeated Eigenvalues	637
6.11 Expansion Theorem	639
6.12 Unrestrained Systems	639
6.13 Free Vibration of Undamped Systems	644
6.14 Forced Vibration of Undamped Systems Using Modal Analysis	646
6.15 Forced Vibration of Viscously Damped Systems	653
6.16 Self-Excitation and Stability Analysis	660
6.17 Examples Using MATLAB	662
Chapter Summary	670
References	670
Review Questions	671
Problems	675
Design Projects	696
Chapter 7 Determination of Natural Frequencies and Mode Shapes	699
7.1 Introduction	700
7.2 Dunkerley’s Formula	701
7.3 Rayleigh’s Method	703
7.3.1 Properties of Rayleigh’s Quotient	704
7.3.2 Computation of the Fundamental Natural Frequency	706
7.3.3 Fundamental Frequency of Beams and Shafts	708
7.4 Holzer’s Method	711
7.4.1 Torsional Systems	711
7.4.2 Spring-Mass Systems	714
7.5 Matrix Iteration Method	715
7.5.1 Convergence to the Highest Natural Frequency	717
7.5.2 Computation of Intermediate Natural Frequencies	718
7.6 Jacobi’s Method	723
7.7 Standard Eigenvalue Problem	725
7.7.1 Choleski Decomposition	726
7.7.2 Other Solution Methods	728
7.8 Examples Using MATLAB	728
Chapter Summary	731
References	731
Review Questions	733
Problems	735
Design Projects	744
Chapter 8 Continuous Systems	745
8.1 Introduction	746
8.2 Transverse Vibration of a String or Cable	747
8.2.1 Equation of Motion	747
8.2.2 Initial and Boundary Conditions	749
8.2.3 Free Vibration of a Uniform String	750
8.2.4 Free Vibration of a String with Both Ends Fixed	751
8.2.5 Traveling-Wave Solution	755
8.3 Longitudinal Vibration of a Bar or Rod	756
8.3.1 Equation of Motion and Solution	756
8.3.2 Orthogonality of Normal Functions	759
8.4 Torsional Vibration of a Shaft or Rod	764
8.5 Lateral Vibration of Beams	767
8.5.1 Equation of Motion	767
8.5.2 Initial Conditions	769
8.5.3 Free Vibration	769
8.5.4 Boundary Conditions	770
8.5.5 Orthogonality of Normal Functions	772
8.5.6 Forced Vibration	776
8.5.7 Effect of Axial Force	778
8.5.8 Effects of Rotary Inertia and Shear Deformation	780
8.5.9 Beams on Elastic Foundation	785
8.5.10 Other Effects	788
8.6 Vibration of Membranes	788
8.6.1 Equation of Motion	788
8.6.2 Initial and Boundary Conditions	790
8.7 Rayleigh’s Method	791
8.8 The Rayleigh-Ritz Method	794
8.9 Examples Using MATLAB	797
Chapter Summary	800
References	800
Review Questions	802
Problems	805
Design Project	818
Chapter 9 Vibration Control	819
9.1 Introduction	820
9.2 Vibration Nomograph and Vibration Criteria	821
9.3 Reduction of Vibration at the Source	825
9.4 Balancing of Rotating Machines	826
9.4.1 Single-Plane Balancing	826
9.4.2 Two-Plane Balancing	829
9.5 Whirling of Rotating Shafts	835
9.5.1 Equations of Motion	835
9.5.2 Critical Speeds	837
9.5.3 Response of the System	838
9.5.4 Stability Analysis	840
9.6 Balancing of Reciprocating Engines	842
9.6.1 Unbalanced Forces Due to Fluctuations in Gas Pressure	842
9.6.2 Unbalanced Forces Due to Inertia of the Moving Parts	843
9.6.3 Balancing of Reciprocating Engines	846
9.7 Control of Vibration	848
9.8 Control of Natural Frequencies	848
9.9 Introduction of Damping	849
9.10 Vibration Isolation	851
9.10.1 Vibration Isolation System with Rigid Foundation	854
9.10.2 Vibration Isolation System with Base Motion	864
9.10.3 Vibration Isolation System with Flexible Foundation	872
9.10.4 Vibration Isolation System with Partially Flexible Foundation	874
9.10.5 Shock Isolation	875
9.10.6 Active Vibration Control	878
9.11 Vibration Absorbers	883
9.11.1 Undamped Dynamic Vibration Absorber	884
9.11.2 Damped Dynamic Vibration Absorber	891
9.12 Examples Using MATLAB	895
Chapter Summary	903
References	903
Review Questions	905
Problems	907
Design Project	922
Chapter 10 Vibration Measurement and Applications	924
10.1 Introduction	925
10.2 Transducers	927
10.2.1 Variable-Resistance Transducers	927
10.2.2 Piezoelectric Transducers	930
10.2.3 Electrodynamic Transducers	931
10.2.4 Linear Variable Differential Transformer Transducer	932
10.3 Vibration Pickups	933
10.3.1 Vibrometer	935
10.3.2 Accelerometer	936
10.3.3 Velometer	940
10.3.4 Phase Distortion	942
10.4 Frequency-Measuring Instruments	944
10.5 Vibration Exciters	946
10.5.1 Mechanical Exciters	946
10.5.2 Electrodynamic Shaker	947
10.6 Signal Analysis	949
10.6.1 Spectrum Analyzers	950
10.6.2 Bandpass Filter	951
10.6.3 Constant-Percent Bandwidth and Constant-Bandwidth Analyzers	952
10.7 Dynamic Testing of Machines and Structures	954
10.7.1 Using Operational Deflection-Shape Measurements	954
10.7.2 Using Modal Testing	954
10.8 Experimental Modal Analysis	954
10.8.1 The Basic Idea	954
10.8.2 The Necessary Equipment	954
10.8.3 Digital Signal Processing	957
10.8.4 Analysis of Random Signals	959
10.8.5 Determination of Modal Data from Observed Peaks	961
10.8.6 Determination of Modal Data from Nyquist Plot	964
10.8.7 Measurement of Mode Shapes	966
10.9 Machine-Condition Monitoring and Diagnosis	969
10.9.1 Vibration Severity Criteria	969
10.9.2 Machine Maintenance Techniques	969
10.9.3 Machine-Condition Monitoring Techniques	970
10.9.4 Vibration Monitoring Techniques	972
10.9.5 Instrumentation Systems	978
10.9.6 Choice of Monitoring Parameter	978
10.10 Examples Using MATLAB	979
Chapter Summary	982
References	982
Review Questions	984
Problems	986
Design Projects	992
Chapter 11 Numerical Integration Methods in Vibration Analysis	993
11.1 Introduction	994
11.2 Finite Difference Method	995
11.3 Central Difference Method for Single-Degree-of-Freedom Systems	996
11.4 Runge-Kutta Method for Single-Degree-of-Freedom Systems	999
11.5 Central Difference Method for Multidegree-of-Freedom Systems	1001
11.6 Finite Difference Method for Continuous Systems	1005
11.6.1 Longitudinal Vibration of Bars	1005
11.6.2 Transverse Vibration of Beams	1009
11.7 Runge-Kutta Method for Multidegree-of-Freedom Systems	1014
11.8 Houbolt Method	1016
11.9 Wilson Method	1019
11.10 Newmark Method	1022
11.11 Examples Using MATLAB	1026
Chapter Summary	1032
References	1032
Review Questions	1033
Problems	1035
Chapter 12 Finite Element Method	1041
12.1 Introduction	1042
12.2 Equations of Motion of an Element	1043
12.3 Mass Matrix, Stiffness Matrix, and Force Vector	1045
12.3.1 Bar Element	1045
12.3.2 Torsion Element	1048
12.3.3 Beam Element	1049
12.4 Transformation of Element Matrices and Vectors	1052
12.5 Equations of Motion of the Complete System of Finite Elements	1055
12.6 Incorporation of Boundary Conditions	1057
12.7 Consistent- and Lumped-Mass Matrices	1066
12.7.1 Lumped-Mass Matrix for a Bar Element	1066
12.7.2 Lumped-Mass Matrix for a Beam Element	1066
12.7.3 Lumped-Mass Versus Consistent-Mass Matrices	1067
12.8 Examples Using MATLAB	1069
Chapter Summary	1073
References	1073
Review Questions	1074
Problems	1076
Design Projects	1088
Chapter 13 Nonlinear Vibration	13-1
13.1 Introduction	13-2
13.2 Examples of Nonlinear Vibration Problems	13-3
13.2.1 Simple Pendulum	13-3
13.2.2 Mechanical Chatter, Belt Friction System	13-5
13.2.3 Variable Mass System	13-5
13.3 Exact Methods	13-6
13.4 Approximate Analytical Methods	13-7
13.4.1 Basic Philosophy	13-8
13.4.2 Lindstedt’s Perturbation Method	13-10
13.4.3 Iterative Method	13-13
13.4.4 Ritz-Galerkin Method	13-17
13.5 Subharmonic and Superharmonic Oscillations	13-19
13.5.1 Subharmonic Oscillations	13-20
13.5.2 Superharmonic Oscillations	13-23
13.6 Systems with Time-Dependent Coefficients (Mathieu Equation)	13-24
13.7 Graphical Methods	13-29
13.7.1 Phase-Plane Representation	13-29
13.7.2 Phase Velocity	13-34
13.7.3 Method of Constructing Trajectories	13-34
13.7.4 Obtaining Time Solution from Phase-Plane Trajectories	13-36
13.8 Stability of Equilibrium States	13-37
13.8.1 Stability Analysis	13-37
13.8.2 Classification of Singular Points	13-40
13.9 Limit Cycles	13-41
13.10 Chaos	13-43
13.10.1 Functions with Stable Orbits	13-45
13.10.2 Functions with Unstable Orbits	13-45
13.10.3 Chaotic Behavior of Duffing’s Equation Without the Forcing Term	13-47
13.10.6 Chaotic Behavior of Duffing’s Equation with the Forcing Term	13-50
13.11 Numerical Methods	13-52
13.12 Examples Using MATLAB	13-53
Chapter Summary	13-62
References	13-62
Review Questions	13-64
Problems	13-67
Design Projects	13-75
Chapter 14 Random Vibration	14-1
14.1 Introduction	14-2
14.2 Random Variables and Random Processes	14-3
14.3 Probability Distribution	14-4
14.4 Mean Value and Standard Deviation	14-6
14.5 Joint Probability Distribution of Several Random Variables	14-7
14.6 Correlation Functions of a Random Process	14-9
14.7 Stationary Random Process	14-10
14.8 Gaussian Random Process	14-14
14.9 Fourier Analysis	14-16
14.9.1 Fourier Series	14-16
14.9.2 Fourier Integral	14-19
14.10 Power Spectral Density	14-23
14.11 Wide-Band and Narrow-Band Processes	14-25
14.12 Response of a Sngle-Degree-of-Freedom System	14-28
14.12.1 Impulse-Response Approach	14-28
14.12.2 Frequency-Response Approach	14-30
14.12.3 Characteristics of the Response Function	14-30
14.13 Response Due to Stationary Random Excitations	14-31
14.13.1 Impulse-Response Approach	14-32
14.13.2 Frequency-Response Approach	14-33
14.14 Response of a Multidegree-of-Freedom System	14-39
14.15 Examples Using MATLAB	14-46
Chapter Summary	14-49
References	14-49
Review Questions	14-50
Problems	14-53
Design Project	14-61
Appendix A Mathematical Relations and Material Properties	1092
Appendix B Deflection of Beams and Plates	1095
Appendix C Matrices	1097
Appendix D Laplace Transform	1104
Appendix E Units	1112
Appendix F Introduction to MATLAB	1116
Answers to Selected Problems	1126
Index	1135
Back Cover	Back Cover

Mechanical Vibrations in SI Units

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