Textbook of Mechanical Vibrations

Textbook of Mechanical Vibrations
اسم المؤلف
Mahesh Chandra Luintel
التاريخ
20 فبراير 2024
المشاهدات
121
التقييم
(لا توجد تقييمات)
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Textbook of Mechanical Vibrations
Mahesh Chandra Luintel
Contents
1 Basic Concepts of Vibration . 1
1.1 Introduction 1
1.1.1 Causes of Vibration 2
1.1.2 Effects of Vibration 3
1.2 Simple Harmonic Motion . 3
1.3 Vibration Analysis Procedure . 5
1.3.1 Mathematical Modeling . 5
1.3.2 Mathematical Solution 5
1.3.3 Physical Interpretation of Mathematical Solution . 6
1.4 Generalized Coordinates . 6
1.5 Degrees of Freedom . 7
1.6 Discrete and Continuous System . 8
1.7 Classification of Vibration 9
1.8 Review of Dynamics . 10
1.8.1 Kinematics . 10
1.8.2 Kinetics 13
1.8.3 Principle of Work and Energy . 15
1.8.4 Principle of Impulse and Momentum . 17
2 Modeling of Components of a Vibrating System . 53
2.1 Components of a Vibrating System 53
2.2 Inertia Elements and Kinetic Energy . 53
2.2.1 Kinetic Energy of a Discrete System Consisting
of Particles 54
2.2.2 Kinetic Energy of a Discrete System Consisting
of a Rigid Body . 54
2.2.3 Kinetic Energy of a Continuous System . 55
2.3 Stiffness Elements and Potential Energy 59
2.3.1 Potential Energy Stored by a Spring 59
2.3.2 Potential Energy or Strain Energy Stored
by a Continuous System 60
ixx Contents
2.3.3 Equivalent System and Equivalent Stiffness
for Different Combinations of Springs 63
2.3.4 Equivalent System and Equivalent Stiffness
for Continuous System with Negligible Weight . 65
2.4 Damper and Energy Dissipation . 69
2.4.1 Types of Damping . 69
2.4.2 Energy Dissipation Due to Damping 71
2.5 External Excitation 71
3 Derivation of Equation of Motion of a Vibrating System . 109
3.1 Classical Methods for Derivation of Equation of Motion 109
3.1.1 Newton’s Second Law of Motion . 109
3.1.2 Equivalent System Parameters Method 111
3.1.3 Principle of Conservation of Energy 113
3.2 Variational Formulation of Dynamic System 115
3.2.1 Independent Variable, Function and Functional . 115
3.2.2 Differentiation and Variation 116
3.2.3 Fundamental Lemma of Variational Calculus 119
3.3 Euler–Lagrange Equation . 120
3.4 Hamilton’s Principle . 122
3.5 Lagrange’s Equations for Conservative Discrete Systems 126
3.6 Lagrange’s Equations for Non-Conservative Discrete
Systems 126
4 Response of a Single Degree of Freedom System . 177
4.1 Un-damped Free Response of a SDOF System . 177
4.2 Damped Free Response of a SDOF System . 180
4.2.1 Response of an Over-Damped System 183
4.2.2 Response of a Critically Damped System 185
4.2.3 Response of an Under-Damped System . 186
4.3 Forced Harmonic Response of a SDOF System 190
4.4 Rotating Unbalance 198
4.5 Vibration Isolation and Transmissibility 201
4.6 Response of a System to an External Motion 205
4.7 Vibration Measuring Instruments 208
4.7.1 Seismometer 209
4.7.2 Accelerometer . 211
4.8 Response to Multi-Frequency and General Periodic
Excitations . 211
4.8.1 Response to Multi-Frequency Excitation 211
4.8.2 Response to a General Periodic Excitation . 212
4.9 Response to Transient Input Forces 213
4.9.1 Response Due to a Unit Impulse . 214
4.9.2 Response Due to a General Excitation 217
4.10 Solution Using the Method of Laplace Transform 219
4.11 Energy Dissipated in Viscous Damper 221Contents xi
4.12 Response of a System with Coulomb Damping 222
4.12.1 Free Response for a System with Coulomb Damping 222
4.12.2 Forced Response for a System with Coulomb
Damping . 225
4.13 Response of a System with Hysteretic Damping . 227
4.13.1 Free Response for a System with Hysteretic
Damping . 227
4.13.2 Forced Response for a System with Hysteretic
Damping . 229
5 Response of a Two Degree of Freedom System 339
5.1 Introduction 339
5.2 Free Response of an Undamped Two Degree of Freedom
System . 340
5.3 Free Response of a Damped Two Degree of Freedom System 343
5.4 Forced Harmonic Response of a Two Degree of Freedom
System . 345
5.4.1 Forced Harmonic Response of an Un-damped Two
Degree of Freedom System 345
5.4.2 Forced Harmonic Response of a Damped Two
Degree of Freedom System 347
5.5 Transfer Functions . 349
5.6 Vibration Absorber 350
5.7 Semi-definite System 355
5.8 Coordinate Coupling and Principal Coordinates 357
5.8.1 Equation of Motion Using x1 and x2 as Generalized
Coordinates . 358
5.8.2 Equation of Motion Using x and θ as Generalized
Coordinates . 359
6 Response of a Multi-Degree of Freedom System . 445
6.1 Introduction 445
6.2 Formulation of Equation of Motion in Matrix Form 445
6.3 Flexibility and Stiffness Matrices 447
6.3.1 Flexibility Influence Coefficients and Flexibility
Matrix . 447
6.3.2 Stiffness Influence Coefficients and Stiffness Matrix 448
6.3.3 Relationship Between Flexibility and Stiffness
Matrix . 449
6.3.4 Reciprocity Theorem . 449
6.4 Natural Frequencies and Mode Shapes of a MDOF System 451
6.5 Orthogonal Properties of the Eigen-Vectors . 452
6.6 Modal Analysis . 453
6.6.1 Modal Analysis for Un-damped Free Response
of a MDOF System 454xii Contents
6.6.2 Modal Analysis for Damped Free Response
of a MDOF System 456
6.6.3 Modal Analysis for Forced Response of a MDOF
System . 459
6.7 Review Questions . 517
7 Modeling and Response of Continuous System 559
7.1 Introduction 559
7.2 Lateral Vibration of a String 561
7.2.1 Derivation of Equation of Motion Using Newton’s
Second Law of Motion . 561
7.2.2 Derivation of Equation of Motion Using Hamilton’s
Principle 562
7.2.3 Free Response for Lateral Vibration of a String . 564
7.2.4 Forced Harmonic Response for Lateral Vibration
of a String 568
7.3 Longitudinal Vibration of a Bar . 570
7.3.1 Derivation of Equation of Motion Using Newton’s
Second Law of Motion . 570
7.3.2 Derivation of Equation of Motion Using Hamilton’s
Principle 571
7.4 Torsional Vibration of a Shaft . 573
7.4.1 Derivation of Equation of Motion Using Newton’s
Second Law of Motion . 573
7.4.2 Derivation of Equation of Motion Using Hamilton’s
Principle 574
7.5 Transverse Vibration of a Beam . 576
7.5.1 Derivation of Equation of Motion Using Newton’s
Second Law of Motion . 576
7.5.2 Derivation of Equation of Motion Using Hamilton’s
Principle 578
7.5.3 Free Response for Transverse Vibration of a Beam 580
7.5.4 Forced Harmonic Response for Lateral Vibration
of a Beam 584
7.6 Modal Analysis for a Continuous System . 587
7.6.1 Modal Analysis of a Continuous System Governed
by Wave Equation 587
7.6.2 Modal Analysis for Vibration Analysis of a Beam 590
8 Approximate Methods 667
8.1 Introduction 667
8.2 Rayleigh Method 668
8.2.1 Rayleigh Method for a Single Degree of Freedom
System . 668
8.2.2 Rayleigh Method for a Discrete Multi Degree
of Freedom System 669Contents xiii
8.2.3 Rayleigh Method for a Shaft or a Beam Carrying
a Number of Lumped Inertia Elements 672
8.2.4 Rayleigh Method for a Continuous System 673
8.3 Dunkerley’s Method . 677
8.4 Matrix Iteration Method 678
8.4.1 Matrix Iteration Using Flexibility Matrix 679
8.4.2 Determination of Higher Order Modes 679
8.4.3 Matrix Iteration Using Dynamic Matrix . 681
8.5 Stodola’s Method 682
8.6 Holzer’s Method 683
8.6.1 Holzer’s Method for a System Without a Branch . 683
8.6.2 Holzer’s Method for a Branched System 687
8.7 Myklestad-Prohl Method for Transverse Bending Vibration . 689
8.8 Rayleigh–Ritz Method . 694
8.9 Assumed Mode Method 696
8.10 Weighted Residual Method . 697

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