Analytical and numerical methods for vibration analyses

Content: About the Author xiii  <
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Preface xv <
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1 Introduction to Structural Vibrations 1 <
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1.1 Terminology 1 <
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1.2 Types of Vibration 5 <
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1.3 Objectives of Vibration Analyses 9 <
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1.3.1 Free Vibration Analysis 9 <
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1.3.2 Forced Vibration Analysis 10 <
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1.4 Global and Local Vibrations 14 <
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1.5 Theoretical Approaches to Structural Vibrations 16 <
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References 18 <
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2 Analytical Solutions for Uniform Continuous Systems19 <
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2.1 Methods for Obtaining Equations of Motion of a VibratingSystem 20 <
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2.2 Vibration of a Stretched String 21 <
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2.2.1 Equation of Motion 21 <
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2.2.2 Free Vibration of a Uniform Clamped   Clamped String22 <
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2.3 Longitudinal Vibration of a Continuous Rod 25 <
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2.3.1 Equation of Motion 25 <
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2.3.2 Free Vibration of a Uniform Rod 28 <
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2.4 Torsional Vibration of a Continuous Shaft 34 <
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2.4.1 Equation of Motion 34 <
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2.4.2 Free Vibration of a Uniform Shaft 36 <
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2.5 Flexural Vibration of a Continuous Euler   BernoulliBeam 41 <
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2.5.1 Equation of Motion 41 <
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2.5.2 Free Vibration of a Uniform Euler   Bernoulli Beam43 <
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2.5.3 Numerical Example 54 <
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2.6 Vibration of Axial-Loaded Uniform Euler   Bernoulli Beam60 <
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2.6.1 Equation of Motion 60 <
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2.6.2 Free Vibration of an Axial-Loaded Uniform Beam 62 <
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2.6.3 Numerical Example 69 <
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2.6.4 Critical Buckling Load of a Uniform Euler   BernoulliBeam 72 <
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2.7 Vibration of an Euler   Bernoulli Beam on the ElasticFoundation 82 <
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2.7.1 Influence of Stiffness Ratio and Total Beam Length 86 <
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2.7.2 Influence of Supporting Conditions of the Beam 87 <
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2.8 Vibration of an Axial-Loaded Euler Beam on the ElasticFoundation 90 <
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2.8.1 Equation of Motion 90 <
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2.8.2 Free Vibration of a Uniform Beam 91 <
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2.8.3 Numerical Example 93 <
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2.9 Flexural Vibration of a Continuous Timoshenko Beam 96 <
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2.9.1 Equation of Motion 96 <
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2.9.2 Free Vibration of a Uniform Timoshenko Beam 98 <
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2.9.3 Numerical Example 105 <
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2.10 Vibrations of a Shear Beam and a Rotary Beam 107 <
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2.10.1 Free Vibration of a Shear Beam 107 <
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2.10.2 Free Vibration of a Rotary Beam 110 <
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2.11 Vibration of an Axial-Loaded Timoshenko Beam 116 <
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2.11.1 Equation of Motion 116 <
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2.11.2 Free Vibration of an Axial-Loaded Uniform Timoshenko Beam118 <
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2.11.3 Numerical Example 124 <
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2.12 Vibration of a Timoshenko Beam on the Elastic Foundation126 <
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2.12.1 Equation of Motion 126 <
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2.12.2 Free Vibration of a Uniform Beam on the ElasticFoundation 128 <
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2.12.3 Numerical Example 132 <
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2.13 Vibration of an Axial-Loaded Timoshenko Beam on the ElasticFoundation 134 <
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2.13.1 Equation of Motion 134 <
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2.13.2 Free Vibration of a Uniform Timoshenko Beam 135 <
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2.13.3 Numerical Example 139 <
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2.14 Vibration of Membranes 142 <
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2.14.1 Free Vibration of a Rectangular Membrane 142 <
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2.14.2 Free Vibration of a Circular Membrane 148 <
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2.15 Vibration of Flat Plates 157 <
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2.15.1 Free Vibration of a Rectangular Plate 158 <
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2.15.2 Free Vibration of a Circular Plate 162 <
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References 171 <
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3 Analytical Solutions for Non-Uniform Continuous Systems:Tapered Beams 173 <
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3.1 Longitudinal Vibration of a Conical Rod 173 <
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3.1.1 Determination of Natural Frequencies and Natural ModeShapes 173 <
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3.1.2 Determination of Normal Mode Shapes 180 <
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3.1.3 Numerical Examples 182 <
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3.2 Torsional Vibration of a Conical Shaft 188 <
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3.2.1 Determination of Natural Frequencies and Natural ModeShapes 188 <
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3.2.2 Determination of Normal Mode Shapes 192 <
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3.2.3 Numerical Example 194 <
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3.3 Displacement Function for Free Bending Vibration of aTapered Beam 200 <
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3.4 Bending Vibration of a Single-Tapered Beam 204 <
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3.4.1 Determination of Natural Frequencies and Natural ModeShapes 204 <
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3.4.2 Determination of Normal Mode Shapes 210 <
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3.4.3 Finite Element Model of a Single-Tapered Beam 212 <
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3.4.4 Numerical Example 213 <
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3.5 Bending Vibration of a Double-Tapered Beam 217 <
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3.5.1 Determination of Natural Frequencies and Natural ModeShapes 217 <
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3.5.2 Determination of Normal Mode Shapes 221 <
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3.5.3 Finite Element Model of a Double-Tapered Beam 222 <
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3.5.4 Numerical Example 224 <
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3.6 Bending Vibration of a Nonlinearly Tapered Beam 226 <
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3.6.1 Equation of Motion and Boundary Conditions 226 <
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3.6.2 Natural Frequencies and Mode Shapes for Various SupportingConditions 232 <
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3.6.3 Finite Element Model of a Non-Uniform Beam 238 <
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3.6.4 Numerical Example 239 <
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References 243 <
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4 Transfer Matrix Methods for Discrete and Continuous Systems245 <
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4.1 Torsional Vibrations of Multi-Degrees-of-Freedom Systems245 <
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4.1.1 Holzer Method for Torsional Vibrations 245 <
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4.1.2 Transfer Matrix Method for Torsional Vibrations 257 <
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4.2 Lumped-Mass Model Transfer Matrix Method for FlexuralVibrations 268 <
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4.2.1 Transfer Matrices for a Station and a Field 269 <
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4.2.2 Free Vibration of a Flexural Beam 272 <
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4.2.3 Discretization of a Continuous Beam 279 <
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4.2.4 Transfer Matrices for a Timoshenko Beam 279 <
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4.2.5 Numerical Example 281 <
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4.2.6 A Timoshenko Beam Carrying Multiple Various ConcentratedElements 291 <
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4.2.7 Transfer Matrix for Axial-Loaded Euler Beam and TimoshenkoBeam 300 <
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4.3 Continuous-Mass Model Transfer Matrix Method for FlexuralVibrations 304 <
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4.3.1 Flexural Vibration of an Euler   Bernoulli Beam304 <
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4.3.2 Flexural Vibration of a Timoshenko Beam with Axial Load314 <
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4.4 Flexural Vibrations of Beams with In-Span Rigid (Pinned)Supports 336 <
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4.4.1 Transfer Matrix of a Station Located at an In-Span Rigid(Pinned) Support 336 <
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4.4.2 Natural Frequencies and Mode Shapes of a Multi-Span Beam340 <
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4.4.3 Numerical Examples 348 <
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References 353 <
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5 Eigenproblem and Jacobi Method 355 <
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5.1 Eigenproblem 355 <
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5.2 Natural Frequencies, Natural Mode Shapes and Unit-AmplitudeMode Shapes 357 <
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5.3 Determination of Normal Mode Shapes 364 <
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5.3.1 Normal Mode Shapes Obtained From Natural Ones 364 <
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5.3.2 Normal Mode Shapes Obtained From Unit-Amplitude Ones365 <
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5.4 Solution of Standard Eigenproblem with Standard JacobiMethod 367 <
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5.4.1 Formulation Based on Forward Multiplication 368 <
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5.4.2 Formulation Based on Backward Multiplication 371 <
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5.4.3 Convergence of Iterations 372 <
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5.5 Solution of Generalized Eigenproblem with Generalized JacobiMethod 378 <
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5.5.1 The Standard Jacobi Method 378 <
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5.5.2 The Generalized Jacobi Method 382 <
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5.5.3 Formulation Based on Forward Multiplication 382 <
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5.5.4 Determination of Elements of Rotation Matrix (a and g)384 <
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5.5.5 Convergence of Iterations 387 <
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5.5.6 Formulation Based on Backward Multiplication 387 <
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5.6 Solution of Semi-Definite System with Generalized JacobiMethod 398 <
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5.7 Solution of Damped Eigenproblem 398 <
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References 398 <
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6 Vibration Analysis by Finite Element Method 399 <
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6.1 Equation of Motion and Property Matrices 399 <
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6.2 Longitudinal (Axial) Vibration of a Rod 400 <
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6.3 Property Matrices of a Torsional Shaft 411 <
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6.4 Flexural Vibration of an Euler   Bernoulli Beam 412 <
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6.5 Shape Functions for a Three-Dimensional Timoshenko BeamElement 430 <
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6.5.1 Assumptions for the Formulations 430 <
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6.5.2 Shear Deformations Due to Translational NodalDisplacements V1 and V3 431 <
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6.5.3 Shear Deformations Due to Rotational Nodal DisplacementsV2 and V4 435 <
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6.5.4 Determination of Shape Functions fyi djP (i  1 4) 437 <
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6.5.5 Determination of Shape Functions fxi djP (i  1  4) 440 <
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6.5.6 Determination of Shape Functions wzi djP (i  1  4) 441 <
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6.5.7 Determination of Shape Functions wxi djP (i  1  4) 443 <
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6.5.8 Shape Functions for a 3D Beam Element 445 <
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6.6 Property Matrices of a Three-Dimensional Timoshenko BeamElement 451 <
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6.6.1 Stiffness Matrix of a 3D Timoshenko Beam Element 451 <
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6.6.2 Mass Matrix of a 3D Timoshenko Beam Element 458 <
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6.7 Transformation Matrix for a Two-Dimensional Beam Element462 <
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6.8 Transformations of Element Stiffness Matrix and Mass Matrix464 <
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6.9 Transformation Matrix for a Three-Dimensional Beam Element465 <
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6.10 Property Matrices of a Beam Element with ConcentratedElements 469 <
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6.11 Property Matrices of Rigid   Pinned andPinned   Rigid Beam Elements 472 <
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6.11.1 Property Matrices of the R-P Beam Element 474 <
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6.11.2 Property Matrices of the P-R Beam Element 476 <
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6.12 Geometric Stiffness Matrix of a Beam Element Due to AxialLoad 477 <
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6.13 Stiffness Matrix of a Beam Element Due to ElasticFoundation 480 <
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References 482 <
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7 Analytical Methods and Finite Element Method for FreeVibration Analyses of Circularly Curved Beams 483 <
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7.1 Analytical Solution for Out-of-Plane Vibration of a CurvedEuler Beam 483 <
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7.1.1 Differential Equations for Displacement Functions 484 <
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7.1.2 Determination of Displacement Functions 485 <
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7.1.3 Internal Forces and Moments 490 <
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7.1.4 Equilibrium and Continuity Conditions 491 <
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7.1.5 Determination of Natural Frequencies and Mode Shapes493 <
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7.1.6 Classical and Non-Classical Boundary Conditions 495 <
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7.1.7 Numerical Examples 497 <
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7.2 Analytical Solution for Out-of-Plane Vibration of a CurvedTimoshenko Beam 503 <
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7.2.1 Coupled Equations of Motion and Boundary Conditions503 <
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7.2.2 Uncoupled Equation of Motion for uy 507 <
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7.2.3 The Relationships Between cx, cu and uy 508 <
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7.2.4 Determination of Displacement Functions UyduP,CxduP and CuduP 509 <
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7.2.5 Internal Forces and Moments 512 <
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7.2.6 Classical Boundary Conditions 513 <
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7.2.7 Equilibrium and Compatibility Conditions 515 <
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7.2.8 Determination of Natural Frequencies and Mode Shapes518 <
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7.2.9 Numerical Examples 520 <
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7.3 Analytical Solution for In-Plane Vibration of a Curved EulerBeam 521 <
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7.3.1 Differential Equations for Displacement Functions 521 <
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7.3.2 Determination of Displacement Functions 527 <
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7.3.3 Internal Forces and Moments 529 <
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7.3.4 Continuity and Equilibrium Conditions 530 <
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7.3.5 Determination of Natural Frequencies and Mode Shapes533 <
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7.3.6 Classical Boundary Conditions 536 <
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7.3.7 Mode Shapes Obtained From Finite Element Method andAnalytical (Exact) Method 537 <
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7.3.8 Numerical Examples 539 <
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7.4 Analytical Solution for In-Plane Vibration of a CurvedTimoshenko Beam 547 <
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7.4.1 Differential Equations for Displacement Functions 547 <
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7.4.2 Determination of Displacement Functions 552 <
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7.4.3 Internal Forces and Moments 553 <
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7.4.4 Equilibrium and Compatibility Conditions 554 <
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7.4.5 Determination of Natural Frequencies and Mode Shapes558 <
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7.4.6 Classical and Non-Classical Boundary Conditions 560 <
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7.4.7 Numerical Examples 562 <
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7.5 Out-of-Plane Vibration of a Curved Beam by Finite ElementMethod with Curved Beam Elements 564 <
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7.5.1 Displacement Functions and Shape Functions 565 <
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7.5.2 Stiffness Matrix for Curved Beam Element 573 <
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7.5.3 Mass Matrix for Curved Beam Element 575 <
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7.5.4 Numerical Example 576 <
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7.6 In-Plane Vibration of a Curved Beam by Finite Element Methodwith Curved Beam Elements 578 <
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7.6.1 Displacement Functions 578 <
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7.6.2 Element Stiffness Matrix 586 <
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7.6.3 Element Mass Matrix 587 <
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7.6.4 Boundary Conditions of the Curved and Straight FiniteElement Methods 589 <
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7.6.5 Numerical Examples 590 <
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7.7 Finite Element Method with Straight Beam Elements forOut-of-Plane Vibration of a Curved Beam 595 <
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7.7.1 Property Matrices of Straight Beam Element forOut-of-Plane Vibrations 596 <
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7.7.2 Transformation Matrix for Out-of-Plane Straight BeamElement 599 <
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7.8 Finite Element Method with Straight Beam Elements forIn-Plane Vibration of a Curved Beam 601 <
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7.8.1 Property Matrices of Straight Beam Element for In-PlaneVibrations 602 <
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7.8.2 Transformation Matrix for the In-Plane Straight BeamElement 605 <
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References 606 <
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8 Solution for the Equations of Motion 609 <
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8.1 Free Vibration Response of an SDOF System 609 <
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8.2 Response of an Undamped SDOF System Due to Arbitrary Loading612 <
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8.3 Response of a Damped SDOF System Due to Arbitrary Loading614 <
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8.4 Numerical Method for the Duhamel Integral 615 <
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8.4.1 General Summation Techniques 615 <
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8.4.2 The Linear Loading Method 629 <
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8.5 Exact Solution for the Duhamel Integral 633 <
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8.6 Exact Solution for a Damped SDOF System Using the ClassicalMethod 636 <
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8.7 Exact Solution for an Undamped SDOF System Using theClassical Method 639 <
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8.8 Approximate Solution for an SDOF Damped System by theCentral Difference Method 642 <
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8.9 Solution for the Equations of Motion of an MDOF System645 <
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8.9.1 Direct Integration Methods 645 <
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8.9.2 The Mode Superposition Method 649 <
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8.10 Determination of Forced Vibration Response Amplitudes659 <
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8.10.1 Total and Steady Response Amplitudes of an SDOF System660 <
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8.10.2 Determination of Steady Response Amplitudes of an MDOFSystem 662 <
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8.11 Numerical Examples for Forced Vibration Response Amplitudes668 <
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8.11.1 Frequency-Response Curves of an SDOF System 668 <
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8.11.2 Frequency-Response Curves of an MDOF System 670 <
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References 675 <
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Appendices 677 <
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A.1 List of Integrals 677 <
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A.2 Theory of Modified Half-Interval (or Bisection) Method680 <
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A.3 Determinations of Influence Coefficients 681 <
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A.3.1 Determination of Influence Coefficients aYM i and aCM i681 <
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A.3.2 Determination of Influence Coefficients aYQ i and aCQ i683 <
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A.4 Exact Solution of a Cubic Equation 685 <
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A.5 Solution of a Cubic Equation Associated with Its ComplexRoots 686 <
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A.6 Coefficients of Matrix   H  Defined by Equation(7.387) 687 <
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A.7 Coefficients of Matrix   H  Defined by Equation(7.439) 689 <
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A.8 Exact Solution for a Simply Supported Euler Arch 691 <
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References 693 <
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Index 695