Analytical and numerical methods for vibration analyses

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''This book illustrates theories and associated mathematical expressions with numerical examples using various methods, leading to exact solutions, more accurate results, and more computationally efficient techniques. It presents the derivations of the equations of motion for all structure foundations using either the continuous model or the discrete model. It discusses applications for students taking courses including vibration mechanics, dynamics of structures, and finite element analyses of structures, the transfer matrix method, and Jacobi method''

Author(s): Wu, Jong-Shyong
Edition: 1
Publisher: John Wiley & Sons Inc
Year: 2013

Language: English
Pages: 672
Tags: Механика;Теория колебаний;

Content: 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 Systems19 
2.1 Methods for Obtaining Equations of Motion of a VibratingSystem 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 String22 
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 BernoulliBeam 41 
2.5.1 Equation of Motion 41 
2.5.2 Free Vibration of a Uniform Euler Bernoulli Beam43 
2.5.3 Numerical Example 54 
2.6 Vibration of Axial-Loaded Uniform Euler Bernoulli Beam60 
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 BernoulliBeam 72 
2.7 Vibration of an Euler Bernoulli Beam on the ElasticFoundation 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 ElasticFoundation 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 Beam118 
2.11.3 Numerical Example 124 
2.12 Vibration of a Timoshenko Beam on the Elastic Foundation126 
2.12.1 Equation of Motion 126 
2.12.2 Free Vibration of a Uniform Beam on the ElasticFoundation 128 
2.12.3 Numerical Example 132 
2.13 Vibration of an Axial-Loaded Timoshenko Beam on the ElasticFoundation 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 ModeShapes 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 ModeShapes 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 aTapered Beam 200 
3.4 Bending Vibration of a Single-Tapered Beam 204 
3.4.1 Determination of Natural Frequencies and Natural ModeShapes 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 ModeShapes 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 SupportingConditions 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 Systems245 
4.1 Torsional Vibrations of Multi-Degrees-of-Freedom Systems245 
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 FlexuralVibrations 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 ConcentratedElements 291 
4.2.7 Transfer Matrix for Axial-Loaded Euler Beam and TimoshenkoBeam 300 
4.3 Continuous-Mass Model Transfer Matrix Method for FlexuralVibrations 304 
4.3.1 Flexural Vibration of an Euler Bernoulli Beam304 
4.3.2 Flexural Vibration of a Timoshenko Beam with Axial Load314 
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 Beam340 
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-AmplitudeMode 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 Ones365 
5.4 Solution of Standard Eigenproblem with Standard JacobiMethod 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 JacobiMethod 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 JacobiMethod 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 BeamElement 430 
6.5.1 Assumptions for the Formulations 430 
6.5.2 Shear Deformations Due to Translational NodalDisplacements V1 and V3 431 
6.5.3 Shear Deformations Due to Rotational Nodal DisplacementsV2 and V4 435 
6.5.4 Determination of Shape Functions fyi djP (i 1 4) 437 
6.5.5 Determination of Shape Functions fxi djP (i 1 4) 440 
6.5.6 Determination of Shape Functions wzi djP (i 1 4) 441 
6.5.7 Determination of Shape Functions wxi djP (i 1 4) 443 
6.5.8 Shape Functions for a 3D Beam Element 445 
6.6 Property Matrices of a Three-Dimensional Timoshenko BeamElement 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 Element462 
6.8 Transformations of Element Stiffness Matrix and Mass Matrix464 
6.9 Transformation Matrix for a Three-Dimensional Beam Element465 
6.10 Property Matrices of a Beam Element with ConcentratedElements 469 
6.11 Property Matrices of Rigid Pinned andPinned 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 AxialLoad 477 
6.13 Stiffness Matrix of a Beam Element Due to ElasticFoundation 480 
References 482 
7 Analytical Methods and Finite Element Method for FreeVibration Analyses of Circularly Curved Beams 483 
7.1 Analytical Solution for Out-of-Plane Vibration of a CurvedEuler 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 Shapes493 
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 CurvedTimoshenko Beam 503 
7.2.1 Coupled Equations of Motion and Boundary Conditions503 
7.2.2 Uncoupled Equation of Motion for uy 507 
7.2.3 The Relationships Between cx, cu and uy 508 
7.2.4 Determination of Displacement Functions UyduP,CxduP and CuduP 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 Shapes518 
7.2.9 Numerical Examples 520 
7.3 Analytical Solution for In-Plane Vibration of a Curved EulerBeam 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 Shapes533 
7.3.6 Classical Boundary Conditions 536 
7.3.7 Mode Shapes Obtained From Finite Element Method andAnalytical (Exact) Method 537 
7.3.8 Numerical Examples 539 
7.4 Analytical Solution for In-Plane Vibration of a CurvedTimoshenko 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 Shapes558 
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 ElementMethod 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 Methodwith 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 FiniteElement Methods 589 
7.6.5 Numerical Examples 590 
7.7 Finite Element Method with Straight Beam Elements forOut-of-Plane Vibration of a Curved Beam 595 
7.7.1 Property Matrices of Straight Beam Element forOut-of-Plane Vibrations 596 
7.7.2 Transformation Matrix for Out-of-Plane Straight BeamElement 599 
7.8 Finite Element Method with Straight Beam Elements forIn-Plane Vibration of a Curved Beam 601 
7.8.1 Property Matrices of Straight Beam Element for In-PlaneVibrations 602 
7.8.2 Transformation Matrix for the In-Plane Straight BeamElement 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 Loading612 
8.3 Response of a Damped SDOF System Due to Arbitrary Loading614 
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 ClassicalMethod 636 
8.7 Exact Solution for an Undamped SDOF System Using theClassical Method 639 
8.8 Approximate Solution for an SDOF Damped System by theCentral Difference Method 642 
8.9 Solution for the Equations of Motion of an MDOF System645 
8.9.1 Direct Integration Methods 645 
8.9.2 The Mode Superposition Method 649 
8.10 Determination of Forced Vibration Response Amplitudes659 
8.10.1 Total and Steady Response Amplitudes of an SDOF System660 
8.10.2 Determination of Steady Response Amplitudes of an MDOFSystem 662 
8.11 Numerical Examples for Forced Vibration Response Amplitudes668 
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) Method680 
A.3 Determinations of Influence Coefficients 681 
A.3.1 Determination of Influence Coefficients aYM i and aCM i681 
A.3.2 Determination of Influence Coefficients aYQ i and aCQ i683 
A.4 Exact Solution of a Cubic Equation 685 
A.5 Solution of a Cubic Equation Associated with Its ComplexRoots 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

Analytical and numerical methods for vibration analyses

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