About the Author xiii
Preface xv
1 Introduction to Structural Vibrations 1
1.1 Terminology 1
1.2 Types of Vibration 5
1.3 Objectives of Vibration Analyses 9
1.3.1 Free Vibration Analysis 9
1.3.2 Forced Vibration Analysis 10
1.4 Global and Local Vibrations 14
1.5 Theoretical Approaches to Structural Vibrations 16
References 18
2 Analytical Solutions for Uniform Continuous Systems 19
2.1 Methods for Obtaining Equations of Motion of a Vibrating System 20
2.2 Vibration of a Stretched String 21
2.2.1 Equation of Motion 21
2.2.2 Free Vibration of a Uniform Clamped–Clamped String 22
2.3 Longitudinal Vibration of a Continuous Rod 25
2.3.1 Equation of Motion 25
2.3.2 Free Vibration of a Uniform Rod 28
2.4 Torsional Vibration of a Continuous Shaft 34
2.4.1 Equation of Motion 34
2.4.2 Free Vibration of a Uniform Shaft 36
2.5 Flexural Vibration of a Continuous Euler–Bernoulli Beam 41
2.5.1 Equation of Motion 41
2.5.2 Free Vibration of a Uniform Euler–Bernoulli Beam 43
2.5.3 Numerical Example 54
2.6 Vibration of Axial-Loaded Uniform Euler–Bernoulli Beam 60
2.6.1 Equation of Motion 60
2.6.2 Free Vibration of an Axial-Loaded Uniform Beam 62
2.6.3 Numerical Example 69
2.6.4 Critical Buckling Load of a Uniform Euler–Bernoulli Beam 72
2.7 Vibration of an Euler–Bernoulli Beam on the Elastic Foundation 82
2.7.1 Influence of Stiffness Ratio and Total Beam Length 86
2.7.2 Influence of Supporting Conditions of the Beam 87
2.8 Vibration of an Axial-Loaded Euler Beam on the Elastic Foundation 90
2.8.1 Equation of Motion 90
2.8.2 Free Vibration of a Uniform Beam 91
2.8.3 Numerical Example 93
2.9 Flexural Vibration of a Continuous Timoshenko Beam 96
2.9.1 Equation of Motion 96
2.9.2 Free Vibration of a Uniform Timoshenko Beam 98
2.9.3 Numerical Example 105
2.10 Vibrations of a Shear Beam and a Rotary Beam 107
2.10.1 Free Vibration of a Shear Beam 107
2.10.2 Free Vibration of a Rotary Beam 110
2.11 Vibration of an Axial-Loaded Timoshenko Beam 116
2.11.1 Equation of Motion 116
2.11.2 Free Vibration of an Axial-Loaded Uniform Timoshenko Beam 118
2.11.3 Numerical Example 124
2.12 Vibration of a Timoshenko Beam on the Elastic Foundation 126
2.12.1 Equation of Motion 126
2.12.2 Free Vibration of a Uniform Beam on the Elastic Foundation 128
2.12.3 Numerical Example 132
2.13 Vibration of an Axial-Loaded Timoshenko Beam on the Elastic Foundation 134
2.13.1 Equation of Motion 134
2.13.2 Free Vibration of a Uniform Timoshenko Beam 135
2.13.3 Numerical Example 139
2.14 Vibration of Membranes 142
2.14.1 Free Vibration of a Rectangular Membrane 142
2.14.2 Free Vibration of a Circular Membrane 148
2.15 Vibration of Flat Plates 157
2.15.1 Free Vibration of a Rectangular Plate 158
2.15.2 Free Vibration of a Circular Plate 162
References 171
3 Analytical Solutions for Non-Uniform Continuous Systems: Tapered Beams 173
3.1 Longitudinal Vibration of a Conical Rod 173
3.1.1 Determination of Natural Frequencies and Natural Mode Shapes 173
3.1.2 Determination of Normal Mode Shapes 180
3.1.3 Numerical Examples 182
3.2 Torsional Vibration of a Conical Shaft 188
3.2.1 Determination of Natural Frequencies and Natural Mode Shapes 188
3.2.2 Determination of Normal Mode Shapes 192
3.2.3 Numerical Example 194
3.3 Displacement Function for Free Bending Vibration of a Tapered Beam 200
3.4 Bending Vibration of a Single-Tapered Beam 204
3.4.1 Determination of Natural Frequencies and Natural Mode Shapes 204
3.4.2 Determination of Normal Mode Shapes 210
3.4.3 Finite Element Model of a Single-Tapered Beam 212
3.4.4 Numerical Example 213
3.5 Bending Vibration of a Double-Tapered Beam 217
3.5.1 Determination of Natural Frequencies and Natural Mode Shapes 217
3.5.2 Determination of Normal Mode Shapes 221
3.5.3 Finite Element Model of a Double-Tapered Beam 222
3.5.4 Numerical Example 224
3.6 Bending Vibration of a Nonlinearly Tapered Beam 226
3.6.1 Equation of Motion and Boundary Conditions 226
3.6.2 Natural Frequencies and Mode Shapes for Various Supporting Conditions 232
3.6.3 Finite Element Model of a Non-Uniform Beam 238
3.6.4 Numerical Example 239
References 243
4 Transfer Matrix Methods for Discrete and Continuous Systems 245
4.1 Torsional Vibrations of Multi-Degrees-of-Freedom Systems 245
4.1.1 Holzer Method for Torsional Vibrations 245
4.1.2 Transfer Matrix Method for Torsional Vibrations 257
4.2 Lumped-Mass Model Transfer Matrix Method for Flexural Vibrations 268
4.2.1 Transfer Matrices for a Station and a Field 269
4.2.2 Free Vibration of a Flexural Beam 272
4.2.3 Discretization of a Continuous Beam 279
4.2.4 Transfer Matrices for a Timoshenko Beam 279
4.2.5 Numerical Example 281
4.2.6 A Timoshenko Beam Carrying Multiple Various Concentrated Elements 291
4.2.7 Transfer Matrix for Axial-Loaded Euler Beam and Timoshenko Beam 300
4.3 Continuous-Mass Model Transfer Matrix Method for Flexural Vibrations 304
4.3.1 Flexural Vibration of an Euler–Bernoulli Beam 304
4.3.2 Flexural Vibration of a Timoshenko Beam with Axial Load 314
4.4 Flexural Vibrations of Beams with In-Span Rigid (Pinned) Supports 336
4.4.1 Transfer Matrix of a Station Located at an In-Span Rigid (Pinned) Support 336
4.4.2 Natural Frequencies and Mode Shapes of a Multi-Span Beam 340
4.4.3 Numerical Examples 348
References 353
5 Eigenproblem and Jacobi Method 355
5.1 Eigenproblem 355
5.2 Natural Frequencies, Natural Mode Shapes and Unit-Amplitude Mode Shapes 357
5.3 Determination of Normal Mode Shapes 364
5.3.1 Normal Mode Shapes Obtained From Natural Ones 364
5.3.2 Normal Mode Shapes Obtained From Unit-Amplitude Ones 365
5.4 Solution of Standard Eigenproblem with Standard Jacobi Method 367
5.4.1 Formulation Based on Forward Multiplication 368
5.4.2 Formulation Based on Backward Multiplication 371
5.4.3 Convergence of Iterations 372
5.5 Solution of Generalized Eigenproblem with Generalized Jacobi Method 378
5.5.1 The Standard Jacobi Method 378
5.5.2 The Generalized Jacobi Method 382
5.5.3 Formulation Based on Forward Multiplication 382
5.5.4 Determination of Elements of Rotation Matrix (a and g) 384
5.5.5 Convergence of Iterations 387
5.5.6 Formulation Based on Backward Multiplication 387
5.6 Solution of Semi-Definite System with Generalized Jacobi Method 398
5.7 Solution of Damped Eigenproblem 398
References 398
6 Vibration Analysis by Finite Element Method 399
6.1 Equation of Motion and Property Matrices 399
6.2 Longitudinal (Axial) Vibration of a Rod 400
6.3 Property Matrices of a Torsional Shaft 411
6.4 Flexural Vibration of an Euler–Bernoulli Beam 412
6.5 Shape Functions for a Three-Dimensional Timoshenko Beam Element 430
6.5.1 Assumptions for the Formulations 430
6.5.2 Shear Deformations Due to Translational Nodal Displacements V1 and V3 431
6.5.3 Shear Deformations Due to Rotational Nodal Displacements V2 and V4 435
6.5.4 Determination of Shape Functions Φyi(ξ) (i = 1 - 4) 437
6.5.5 Determination of Shape Functions Φxi(ξ) (i = 1 - 4) 440
6.5.6 Determination of Shape Functions φzi(ξ) (i = 1 - 4) 441
6.5.7 Determination of Shape Functions φxi(ξ) (i = 1 - 4) 443
6.5.8 Shape Functions for a 3D Beam Element 445
6.6 Property Matrices of a Three-Dimensional Timoshenko Beam Element 451
6.6.1 Stiffness Matrix of a 3D Timoshenko Beam Element 451
6.6.2 Mass Matrix of a 3D Timoshenko Beam Element 458
6.7 Transformation Matrix for a Two-Dimensional Beam Element 462
6.8 Transformations of Element Stiffness Matrix and Mass Matrix 464
6.9 Transformation Matrix for a Three-Dimensional Beam Element 465
6.10 Property Matrices of a Beam Element with Concentrated Elements 469
6.11 Property Matrices of Rigid–Pinned and Pinned–Rigid Beam Elements 472
6.11.1 Property Matrices of the R-P Beam Element 474
6.11.2 Property Matrices of the P-R Beam Element 476
6.12 Geometric Stiffness Matrix of a Beam Element Due to Axial Load 477
6.13 Stiffness Matrix of a Beam Element Due to Elastic Foundation 480
References 482
7 Analytical Methods and Finite Element Method for Free Vibration Analyses of Circularly Curved Beams 483
7.1 Analytical Solution for Out-of-Plane Vibration of a Curved Euler Beam 483
7.1.1 Differential Equations for Displacement Functions 484
7.1.2 Determination of Displacement Functions 485
7.1.3 Internal Forces and Moments 490
7.1.4 Equilibrium and Continuity Conditions 491
7.1.5 Determination of Natural Frequencies and Mode Shapes 493
7.1.6 Classical and Non-Classical Boundary Conditions 495
7.1.7 Numerical Examples 497
7.2 Analytical Solution for Out-of-Plane Vibration of a Curved Timoshenko Beam 503
7.2.1 Coupled Equations of Motion and Boundary Conditions 503
7.2.2 Uncoupled Equation of Motion for uy 507
7.2.3 The Relationships Between ψx, ψθ and uy 508
7.2.4 Determination of Displacement Functions Uy(θ), ψx(θ) and ψθ(θ) 509
7.2.5 Internal Forces and Moments 512
7.2.6 Classical Boundary Conditions 513
7.2.7 Equilibrium and Compatibility Conditions 515
7.2.8 Determination of Natural Frequencies and Mode Shapes 518
7.2.9 Numerical Examples 520
7.3 Analytical Solution for In-Plane Vibration of a Curved Euler Beam 521
7.3.1 Differential Equations for Displacement Functions 521
7.3.2 Determination of Displacement Functions 527
7.3.3 Internal Forces and Moments 529
7.3.4 Continuity and Equilibrium Conditions 530
7.3.5 Determination of Natural Frequencies and Mode Shapes 533
7.3.6 Classical Boundary Conditions 536
7.3.7 Mode Shapes Obtained From Finite Element Method and Analytical (Exact) Method 537
7.3.8 Numerical Examples 539
7.4 Analytical Solution for In-Plane Vibration of a Curved Timoshenko Beam 547
7.4.1 Differential Equations for Displacement Functions 547
7.4.2 Determination of Displacement Functions 552
7.4.3 Internal Forces and Moments 553
7.4.4 Equilibrium and Compatibility Conditions 554
7.4.5 Determination of Natural Frequencies and Mode Shapes 558
7.4.6 Classical and Non-Classical Boundary Conditions 560
7.4.7 Numerical Examples 562
7.5 Out-of-Plane Vibration of a Curved Beam by Finite Element Method with Curved Beam Elements 564
7.5.1 Displacement Functions and Shape Functions 565
7.5.2 Stiffness Matrix for Curved Beam Element 573
7.5.3 Mass Matrix for Curved Beam Element 575
7.5.4 Numerical Example 576
7.6 In-Plane Vibration of a Curved Beam by Finite Element Method with Curved Beam Elements 578
7.6.1 Displacement Functions 578
7.6.2 Element Stiffness Matrix 586
7.6.3 Element Mass Matrix 587
7.6.4 Boundary Conditions of the Curved and Straight Finite Element Methods 589
7.6.5 Numerical Examples 590
7.7 Finite Element Method with Straight Beam Elements for Out-of-Plane Vibration of a Curved Beam 595
7.7.1 Property Matrices of Straight Beam Element for Out-of-Plane Vibrations 596
7.7.2 Transformation Matrix for Out-of-Plane Straight Beam Element 599
7.8 Finite Element Method with Straight Beam Elements for In-Plane Vibration of a Curved Beam 601
7.8.1 Property Matrices of Straight Beam Element for In-Plane Vibrations 602
7.8.2 Transformation Matrix for the In-Plane Straight Beam Element 605
References 606
8 Solution for the Equations of Motion 609
8.1 Free Vibration Response of an SDOF System 609
8.2 Response of an Undamped SDOF System Due to Arbitrary Loading 612
8.3 Response of a Damped SDOF System Due to Arbitrary Loading 614
8.4 Numerical Method for the Duhamel Integral 615
8.4.1 General Summation Techniques 615
8.4.2 The Linear Loading Method 629
8.5 Exact Solution for the Duhamel Integral 633
8.6 Exact Solution for a Damped SDOF System Using the Classical Method 636
8.7 Exact Solution for an Undamped SDOF System Using the Classical Method 639
8.8 Approximate Solution for an SDOF Damped System by the Central Difference Method 642
8.9 Solution for the Equations of Motion of an MDOF System 645
8.9.1 Direct Integration Methods 645
8.9.2 The Mode Superposition Method 649
8.10 Determination of Forced Vibration Response Amplitudes 659
8.10.1 Total and Steady Response Amplitudes of an SDOF System 660
8.10.2 Determination of Steady Response Amplitudes of an MDOF System 662
8.11 Numerical Examples for Forced Vibration Response Amplitudes 668
8.11.1 Frequency-Response Curves of an SDOF System 668
8.11.2 Frequency-Response Curves of an MDOF System 670
References 675
Appendices 677
A.1 List of Integrals 677
A.2 Theory of Modified Half-Interval (or Bisection) Method 680
A.3 Determinations of Influence Coefficients 681
A.3.1 Determination of Influence Coefficients aiYM and aiψM 681
A.3.2 Determination of Influence Coefficients aiYQ and aiψQ 683
A.4 Exact Solution of a Cubic Equation 685
A.5 Solution of a Cubic Equation Associated with Its Complex Roots 686
A.6 Coefficients of Matrix [H] Defined by Equation (7.387) 687
A.7 Coefficients of Matrix [H] Defined by Equation (7.439) 689
A.8 Exact Solution for a Simply Supported Euler Arch 691
References 693
Index 695
