{"product_id":"analytical-and-numerical-methods-for-vibration-analyses-9781118632154","title":"Analytical and Numerical Methods for Vibration Analyses","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThis 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eAbout the Author xiii\u003c\/p\u003e \u003cp\u003ePreface xv\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction to Structural Vibrations 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Terminology 1\u003c\/p\u003e \u003cp\u003e1.2 Types of Vibration 5\u003c\/p\u003e \u003cp\u003e1.3 Objectives of Vibration Analyses 9\u003c\/p\u003e \u003cp\u003e1.3.1 Free Vibration Analysis 9\u003c\/p\u003e \u003cp\u003e1.3.2 Forced Vibration Analysis 10\u003c\/p\u003e \u003cp\u003e1.4 Global and Local Vibrations 14\u003c\/p\u003e \u003cp\u003e1.5 Theoretical Approaches to Structural Vibrations 16\u003c\/p\u003e \u003cp\u003eReferences 18\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Analytical Solutions for Uniform Continuous Systems 19\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Methods for Obtaining Equations of Motion of a Vibrating System 20\u003c\/p\u003e \u003cp\u003e2.2 Vibration of a Stretched String 21\u003c\/p\u003e \u003cp\u003e2.2.1 Equation of Motion 21\u003c\/p\u003e \u003cp\u003e2.2.2 Free Vibration of a Uniform Clamped–Clamped String 22\u003c\/p\u003e \u003cp\u003e2.3 Longitudinal Vibration of a Continuous Rod 25\u003c\/p\u003e \u003cp\u003e2.3.1 Equation of Motion 25\u003c\/p\u003e \u003cp\u003e2.3.2 Free Vibration of a Uniform Rod 28\u003c\/p\u003e \u003cp\u003e2.4 Torsional Vibration of a Continuous Shaft 34\u003c\/p\u003e \u003cp\u003e2.4.1 Equation of Motion 34\u003c\/p\u003e \u003cp\u003e2.4.2 Free Vibration of a Uniform Shaft 36\u003c\/p\u003e \u003cp\u003e2.5 Flexural Vibration of a Continuous Euler–Bernoulli Beam 41\u003c\/p\u003e \u003cp\u003e2.5.1 Equation of Motion 41\u003c\/p\u003e \u003cp\u003e2.5.2 Free Vibration of a Uniform Euler–Bernoulli Beam 43\u003c\/p\u003e \u003cp\u003e2.5.3 Numerical Example 54\u003c\/p\u003e \u003cp\u003e2.6 Vibration of Axial-Loaded Uniform Euler–Bernoulli Beam 60\u003c\/p\u003e \u003cp\u003e2.6.1 Equation of Motion 60\u003c\/p\u003e \u003cp\u003e2.6.2 Free Vibration of an Axial-Loaded Uniform Beam 62\u003c\/p\u003e \u003cp\u003e2.6.3 Numerical Example 69\u003c\/p\u003e \u003cp\u003e2.6.4 Critical Buckling Load of a Uniform Euler–Bernoulli Beam 72\u003c\/p\u003e \u003cp\u003e2.7 Vibration of an Euler–Bernoulli Beam on the Elastic Foundation 82\u003c\/p\u003e \u003cp\u003e2.7.1 Influence of Stiffness Ratio and Total Beam Length 86\u003c\/p\u003e \u003cp\u003e2.7.2 Influence of Supporting Conditions of the Beam 87\u003c\/p\u003e \u003cp\u003e2.8 Vibration of an Axial-Loaded Euler Beam on the Elastic Foundation 90\u003c\/p\u003e \u003cp\u003e2.8.1 Equation of Motion 90\u003c\/p\u003e \u003cp\u003e2.8.2 Free Vibration of a Uniform Beam 91\u003c\/p\u003e \u003cp\u003e2.8.3 Numerical Example 93\u003c\/p\u003e \u003cp\u003e2.9 Flexural Vibration of a Continuous Timoshenko Beam 96\u003c\/p\u003e \u003cp\u003e2.9.1 Equation of Motion 96\u003c\/p\u003e \u003cp\u003e2.9.2 Free Vibration of a Uniform Timoshenko Beam 98\u003c\/p\u003e \u003cp\u003e2.9.3 Numerical Example 105\u003c\/p\u003e \u003cp\u003e2.10 Vibrations of a Shear Beam and a Rotary Beam 107\u003c\/p\u003e \u003cp\u003e2.10.1 Free Vibration of a Shear Beam 107\u003c\/p\u003e \u003cp\u003e2.10.2 Free Vibration of a Rotary Beam 110\u003c\/p\u003e \u003cp\u003e2.11 Vibration of an Axial-Loaded Timoshenko Beam 116\u003c\/p\u003e \u003cp\u003e2.11.1 Equation of Motion 116\u003c\/p\u003e \u003cp\u003e2.11.2 Free Vibration of an Axial-Loaded Uniform Timoshenko Beam 118\u003c\/p\u003e \u003cp\u003e2.11.3 Numerical Example 124\u003c\/p\u003e \u003cp\u003e2.12 Vibration of a Timoshenko Beam on the Elastic Foundation 126\u003c\/p\u003e \u003cp\u003e2.12.1 Equation of Motion 126\u003c\/p\u003e \u003cp\u003e2.12.2 Free Vibration of a Uniform Beam on the Elastic Foundation 128\u003c\/p\u003e \u003cp\u003e2.12.3 Numerical Example 132\u003c\/p\u003e \u003cp\u003e2.13 Vibration of an Axial-Loaded Timoshenko Beam on the Elastic Foundation 134\u003c\/p\u003e \u003cp\u003e2.13.1 Equation of Motion 134\u003c\/p\u003e \u003cp\u003e2.13.2 Free Vibration of a Uniform Timoshenko Beam 135\u003c\/p\u003e \u003cp\u003e2.13.3 Numerical Example 139\u003c\/p\u003e \u003cp\u003e2.14 Vibration of Membranes 142\u003c\/p\u003e \u003cp\u003e2.14.1 Free Vibration of a Rectangular Membrane 142\u003c\/p\u003e \u003cp\u003e2.14.2 Free Vibration of a Circular Membrane 148\u003c\/p\u003e \u003cp\u003e2.15 Vibration of Flat Plates 157\u003c\/p\u003e \u003cp\u003e2.15.1 Free Vibration of a Rectangular Plate 158\u003c\/p\u003e \u003cp\u003e2.15.2 Free Vibration of a Circular Plate 162\u003c\/p\u003e \u003cp\u003eReferences 171\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Analytical Solutions for Non-Uniform Continuous Systems: Tapered Beams 173\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Longitudinal Vibration of a Conical Rod 173\u003c\/p\u003e \u003cp\u003e3.1.1 Determination of Natural Frequencies and Natural Mode Shapes 173\u003c\/p\u003e \u003cp\u003e3.1.2 Determination of Normal Mode Shapes 180\u003c\/p\u003e \u003cp\u003e3.1.3 Numerical Examples 182\u003c\/p\u003e \u003cp\u003e3.2 Torsional Vibration of a Conical Shaft 188\u003c\/p\u003e \u003cp\u003e3.2.1 Determination of Natural Frequencies and Natural Mode Shapes 188\u003c\/p\u003e \u003cp\u003e3.2.2 Determination of Normal Mode Shapes 192\u003c\/p\u003e \u003cp\u003e3.2.3 Numerical Example 194\u003c\/p\u003e \u003cp\u003e3.3 Displacement Function for Free Bending Vibration of a Tapered Beam 200\u003c\/p\u003e \u003cp\u003e3.4 Bending Vibration of a Single-Tapered Beam 204\u003c\/p\u003e \u003cp\u003e3.4.1 Determination of Natural Frequencies and Natural Mode Shapes 204\u003c\/p\u003e \u003cp\u003e3.4.2 Determination of Normal Mode Shapes 210\u003c\/p\u003e \u003cp\u003e3.4.3 Finite Element Model of a Single-Tapered Beam 212\u003c\/p\u003e \u003cp\u003e3.4.4 Numerical Example 213\u003c\/p\u003e \u003cp\u003e3.5 Bending Vibration of a Double-Tapered Beam 217\u003c\/p\u003e \u003cp\u003e3.5.1 Determination of Natural Frequencies and Natural Mode Shapes 217\u003c\/p\u003e \u003cp\u003e3.5.2 Determination of Normal Mode Shapes 221\u003c\/p\u003e \u003cp\u003e3.5.3 Finite Element Model of a Double-Tapered Beam 222\u003c\/p\u003e \u003cp\u003e3.5.4 Numerical Example 224\u003c\/p\u003e \u003cp\u003e3.6 Bending Vibration of a Nonlinearly Tapered Beam 226\u003c\/p\u003e \u003cp\u003e3.6.1 Equation of Motion and Boundary Conditions 226\u003c\/p\u003e \u003cp\u003e3.6.2 Natural Frequencies and Mode Shapes for Various Supporting Conditions 232\u003c\/p\u003e \u003cp\u003e3.6.3 Finite Element Model of a Non-Uniform Beam 238\u003c\/p\u003e \u003cp\u003e3.6.4 Numerical Example 239\u003c\/p\u003e \u003cp\u003eReferences 243\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Transfer Matrix Methods for Discrete and Continuous Systems 245\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Torsional Vibrations of Multi-Degrees-of-Freedom Systems 245\u003c\/p\u003e \u003cp\u003e4.1.1 Holzer Method for Torsional Vibrations 245\u003c\/p\u003e \u003cp\u003e4.1.2 Transfer Matrix Method for Torsional Vibrations 257\u003c\/p\u003e \u003cp\u003e4.2 Lumped-Mass Model Transfer Matrix Method for Flexural Vibrations 268\u003c\/p\u003e \u003cp\u003e4.2.1 Transfer Matrices for a Station and a Field 269\u003c\/p\u003e \u003cp\u003e4.2.2 Free Vibration of a Flexural Beam 272\u003c\/p\u003e \u003cp\u003e4.2.3 Discretization of a Continuous Beam 279\u003c\/p\u003e \u003cp\u003e4.2.4 Transfer Matrices for a Timoshenko Beam 279\u003c\/p\u003e \u003cp\u003e4.2.5 Numerical Example 281\u003c\/p\u003e \u003cp\u003e4.2.6 A Timoshenko Beam Carrying Multiple Various Concentrated Elements 291\u003c\/p\u003e \u003cp\u003e4.2.7 Transfer Matrix for Axial-Loaded Euler Beam and Timoshenko Beam 300\u003c\/p\u003e \u003cp\u003e4.3 Continuous-Mass Model Transfer Matrix Method for Flexural Vibrations 304\u003c\/p\u003e \u003cp\u003e4.3.1 Flexural Vibration of an Euler–Bernoulli Beam 304\u003c\/p\u003e \u003cp\u003e4.3.2 Flexural Vibration of a Timoshenko Beam with Axial Load 314\u003c\/p\u003e \u003cp\u003e4.4 Flexural Vibrations of Beams with In-Span Rigid (Pinned) Supports 336\u003c\/p\u003e \u003cp\u003e4.4.1 Transfer Matrix of a Station Located at an In-Span Rigid (Pinned) Support 336\u003c\/p\u003e \u003cp\u003e4.4.2 Natural Frequencies and Mode Shapes of a Multi-Span Beam 340\u003c\/p\u003e \u003cp\u003e4.4.3 Numerical Examples 348\u003c\/p\u003e \u003cp\u003eReferences 353\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Eigenproblem and Jacobi Method 355\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Eigenproblem 355\u003c\/p\u003e \u003cp\u003e5.2 Natural Frequencies, Natural Mode Shapes and Unit-Amplitude Mode Shapes 357\u003c\/p\u003e \u003cp\u003e5.3 Determination of Normal Mode Shapes 364\u003c\/p\u003e \u003cp\u003e5.3.1 Normal Mode Shapes Obtained From Natural Ones 364\u003c\/p\u003e \u003cp\u003e5.3.2 Normal Mode Shapes Obtained From Unit-Amplitude Ones 365\u003c\/p\u003e \u003cp\u003e5.4 Solution of Standard Eigenproblem with Standard Jacobi Method 367\u003c\/p\u003e \u003cp\u003e5.4.1 Formulation Based on Forward Multiplication 368\u003c\/p\u003e \u003cp\u003e5.4.2 Formulation Based on Backward Multiplication 371\u003c\/p\u003e \u003cp\u003e5.4.3 Convergence of Iterations 372\u003c\/p\u003e \u003cp\u003e5.5 Solution of Generalized Eigenproblem with Generalized Jacobi Method 378\u003c\/p\u003e \u003cp\u003e5.5.1 The Standard Jacobi Method 378\u003c\/p\u003e \u003cp\u003e5.5.2 The Generalized Jacobi Method 382\u003c\/p\u003e \u003cp\u003e5.5.3 Formulation Based on Forward Multiplication 382\u003c\/p\u003e \u003cp\u003e5.5.4 Determination of Elements of Rotation Matrix (a and g) 384\u003c\/p\u003e \u003cp\u003e5.5.5 Convergence of Iterations 387\u003c\/p\u003e \u003cp\u003e5.5.6 Formulation Based on Backward Multiplication 387\u003c\/p\u003e \u003cp\u003e5.6 Solution of Semi-Definite System with Generalized Jacobi Method 398\u003c\/p\u003e \u003cp\u003e5.7 Solution of Damped Eigenproblem 398\u003c\/p\u003e \u003cp\u003eReferences 398\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Vibration Analysis by Finite Element Method 399\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Equation of Motion and Property Matrices 399\u003c\/p\u003e \u003cp\u003e6.2 Longitudinal (Axial) Vibration of a Rod 400\u003c\/p\u003e \u003cp\u003e6.3 Property Matrices of a Torsional Shaft 411\u003c\/p\u003e \u003cp\u003e6.4 Flexural Vibration of an Euler–Bernoulli Beam 412\u003c\/p\u003e \u003cp\u003e6.5 Shape Functions for a Three-Dimensional Timoshenko Beam Element 430\u003c\/p\u003e \u003cp\u003e6.5.1 Assumptions for the Formulations 430\u003c\/p\u003e \u003cp\u003e6.5.2 Shear Deformations Due to Translational Nodal Displacements V\u003csub\u003e1\u003c\/sub\u003e and V\u003csub\u003e3\u003c\/sub\u003e 431\u003c\/p\u003e \u003cp\u003e6.5.3 Shear Deformations Due to Rotational Nodal Displacements V\u003csub\u003e2\u003c\/sub\u003e and V\u003csub\u003e4\u003c\/sub\u003e 435\u003c\/p\u003e \u003cp\u003e6.5.4 Determination of Shape Functions Φ\u003csub\u003eyi\u003c\/sub\u003e(ξ) (i = 1 - 4) 437\u003c\/p\u003e \u003cp\u003e6.5.5 Determination of Shape Functions Φ\u003csub\u003exi\u003c\/sub\u003e(ξ) (i = 1 - 4) 440\u003c\/p\u003e \u003cp\u003e6.5.6 Determination of Shape Functions φ\u003csub\u003ezi\u003c\/sub\u003e(ξ) (i = 1 - 4) 441\u003c\/p\u003e \u003cp\u003e6.5.7 Determination of Shape Functions φ\u003csub\u003exi\u003c\/sub\u003e(ξ) (i = 1 - 4) 443\u003c\/p\u003e \u003cp\u003e6.5.8 Shape Functions for a 3D Beam Element 445\u003c\/p\u003e \u003cp\u003e6.6 Property Matrices of a Three-Dimensional Timoshenko Beam Element 451\u003c\/p\u003e \u003cp\u003e6.6.1 Stiffness Matrix of a 3D Timoshenko Beam Element 451\u003c\/p\u003e \u003cp\u003e6.6.2 Mass Matrix of a 3D Timoshenko Beam Element 458\u003c\/p\u003e \u003cp\u003e6.7 Transformation Matrix for a Two-Dimensional Beam Element 462\u003c\/p\u003e \u003cp\u003e6.8 Transformations of Element Stiffness Matrix and Mass Matrix 464\u003c\/p\u003e \u003cp\u003e6.9 Transformation Matrix for a Three-Dimensional Beam Element 465\u003c\/p\u003e \u003cp\u003e6.10 Property Matrices of a Beam Element with Concentrated Elements 469\u003c\/p\u003e \u003cp\u003e6.11 Property Matrices of Rigid–Pinned and Pinned–Rigid Beam Elements 472\u003c\/p\u003e \u003cp\u003e6.11.1 Property Matrices of the R-P Beam Element 474\u003c\/p\u003e \u003cp\u003e6.11.2 Property Matrices of the P-R Beam Element 476\u003c\/p\u003e \u003cp\u003e6.12 Geometric Stiffness Matrix of a Beam Element Due to Axial Load 477\u003c\/p\u003e \u003cp\u003e6.13 Stiffness Matrix of a Beam Element Due to Elastic Foundation 480\u003c\/p\u003e \u003cp\u003eReferences 482\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Analytical Methods and Finite Element Method for Free Vibration Analyses of Circularly Curved Beams 483\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Analytical Solution for Out-of-Plane Vibration of a Curved Euler Beam 483\u003c\/p\u003e \u003cp\u003e7.1.1 Differential Equations for Displacement Functions 484\u003c\/p\u003e \u003cp\u003e7.1.2 Determination of Displacement Functions 485\u003c\/p\u003e \u003cp\u003e7.1.3 Internal Forces and Moments 490\u003c\/p\u003e \u003cp\u003e7.1.4 Equilibrium and Continuity Conditions 491\u003c\/p\u003e \u003cp\u003e7.1.5 Determination of Natural Frequencies and Mode Shapes 493\u003c\/p\u003e \u003cp\u003e7.1.6 Classical and Non-Classical Boundary Conditions 495\u003c\/p\u003e \u003cp\u003e7.1.7 Numerical Examples 497\u003c\/p\u003e \u003cp\u003e7.2 Analytical Solution for Out-of-Plane Vibration of a Curved Timoshenko Beam 503\u003c\/p\u003e \u003cp\u003e7.2.1 Coupled Equations of Motion and Boundary Conditions 503\u003c\/p\u003e \u003cp\u003e7.2.2 Uncoupled Equation of Motion for u\u003csub\u003ey\u003c\/sub\u003e 507\u003c\/p\u003e \u003cp\u003e7.2.3 The Relationships Between ψ\u003csub\u003ex\u003c\/sub\u003e, ψ\u003csub\u003eθ\u003c\/sub\u003e and u\u003csub\u003ey\u003c\/sub\u003e 508\u003c\/p\u003e \u003cp\u003e7.2.4 Determination of Displacement Functions U\u003csub\u003ey\u003c\/sub\u003e(θ), ψ\u003csub\u003ex\u003c\/sub\u003e(θ) and ψ\u003csub\u003eθ\u003c\/sub\u003e(θ) 509\u003c\/p\u003e \u003cp\u003e7.2.5 Internal Forces and Moments 512\u003c\/p\u003e \u003cp\u003e7.2.6 Classical Boundary Conditions 513\u003c\/p\u003e \u003cp\u003e7.2.7 Equilibrium and Compatibility Conditions 515\u003c\/p\u003e \u003cp\u003e7.2.8 Determination of Natural Frequencies and Mode Shapes 518\u003c\/p\u003e \u003cp\u003e7.2.9 Numerical Examples 520\u003c\/p\u003e \u003cp\u003e7.3 Analytical Solution for In-Plane Vibration of a Curved Euler Beam 521\u003c\/p\u003e \u003cp\u003e7.3.1 Differential Equations for Displacement Functions 521\u003c\/p\u003e \u003cp\u003e7.3.2 Determination of Displacement Functions 527\u003c\/p\u003e \u003cp\u003e7.3.3 Internal Forces and Moments 529\u003c\/p\u003e \u003cp\u003e7.3.4 Continuity and Equilibrium Conditions 530\u003c\/p\u003e \u003cp\u003e7.3.5 Determination of Natural Frequencies and Mode Shapes 533\u003c\/p\u003e \u003cp\u003e7.3.6 Classical Boundary Conditions 536\u003c\/p\u003e \u003cp\u003e7.3.7 Mode Shapes Obtained From Finite Element Method and Analytical (Exact) Method 537\u003c\/p\u003e \u003cp\u003e7.3.8 Numerical Examples 539\u003c\/p\u003e \u003cp\u003e7.4 Analytical Solution for In-Plane Vibration of a Curved Timoshenko Beam 547\u003c\/p\u003e \u003cp\u003e7.4.1 Differential Equations for Displacement Functions 547\u003c\/p\u003e \u003cp\u003e7.4.2 Determination of Displacement Functions 552\u003c\/p\u003e \u003cp\u003e7.4.3 Internal Forces and Moments 553\u003c\/p\u003e \u003cp\u003e7.4.4 Equilibrium and Compatibility Conditions 554\u003c\/p\u003e \u003cp\u003e7.4.5 Determination of Natural Frequencies and Mode Shapes 558\u003c\/p\u003e \u003cp\u003e7.4.6 Classical and Non-Classical Boundary Conditions 560\u003c\/p\u003e \u003cp\u003e7.4.7 Numerical Examples 562\u003c\/p\u003e \u003cp\u003e7.5 Out-of-Plane Vibration of a Curved Beam by Finite Element Method with Curved Beam Elements 564\u003c\/p\u003e \u003cp\u003e7.5.1 Displacement Functions and Shape Functions 565\u003c\/p\u003e \u003cp\u003e7.5.2 Stiffness Matrix for Curved Beam Element 573\u003c\/p\u003e \u003cp\u003e7.5.3 Mass Matrix for Curved Beam Element 575\u003c\/p\u003e \u003cp\u003e7.5.4 Numerical Example 576\u003c\/p\u003e \u003cp\u003e7.6 In-Plane Vibration of a Curved Beam by Finite Element Method with Curved Beam Elements 578\u003c\/p\u003e \u003cp\u003e7.6.1 Displacement Functions 578\u003c\/p\u003e \u003cp\u003e7.6.2 Element Stiffness Matrix 586\u003c\/p\u003e \u003cp\u003e7.6.3 Element Mass Matrix 587\u003c\/p\u003e \u003cp\u003e7.6.4 Boundary Conditions of the Curved and Straight Finite Element Methods 589\u003c\/p\u003e \u003cp\u003e7.6.5 Numerical Examples 590\u003c\/p\u003e \u003cp\u003e7.7 Finite Element Method with Straight Beam Elements for Out-of-Plane Vibration of a Curved Beam 595\u003c\/p\u003e \u003cp\u003e7.7.1 Property Matrices of Straight Beam Element for Out-of-Plane Vibrations 596\u003c\/p\u003e \u003cp\u003e7.7.2 Transformation Matrix for Out-of-Plane Straight Beam Element 599\u003c\/p\u003e \u003cp\u003e7.8 Finite Element Method with Straight Beam Elements for In-Plane Vibration of a Curved Beam 601\u003c\/p\u003e \u003cp\u003e7.8.1 Property Matrices of Straight Beam Element for In-Plane Vibrations 602\u003c\/p\u003e \u003cp\u003e7.8.2 Transformation Matrix for the In-Plane Straight Beam Element 605\u003c\/p\u003e \u003cp\u003eReferences 606\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Solution for the Equations of Motion 609\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Free Vibration Response of an SDOF System 609\u003c\/p\u003e \u003cp\u003e8.2 Response of an Undamped SDOF System Due to Arbitrary Loading 612\u003c\/p\u003e \u003cp\u003e8.3 Response of a Damped SDOF System Due to Arbitrary Loading 614\u003c\/p\u003e \u003cp\u003e8.4 Numerical Method for the Duhamel Integral 615\u003c\/p\u003e \u003cp\u003e8.4.1 General Summation Techniques 615\u003c\/p\u003e \u003cp\u003e8.4.2 The Linear Loading Method 629\u003c\/p\u003e \u003cp\u003e8.5 Exact Solution for the Duhamel Integral 633\u003c\/p\u003e \u003cp\u003e8.6 Exact Solution for a Damped SDOF System Using the Classical Method 636\u003c\/p\u003e \u003cp\u003e8.7 Exact Solution for an Undamped SDOF System Using the Classical Method 639\u003c\/p\u003e \u003cp\u003e8.8 Approximate Solution for an SDOF Damped System by the Central Difference Method 642\u003c\/p\u003e \u003cp\u003e8.9 Solution for the Equations of Motion of an MDOF System 645\u003c\/p\u003e \u003cp\u003e8.9.1 Direct Integration Methods 645\u003c\/p\u003e \u003cp\u003e8.9.2 The Mode Superposition Method 649\u003c\/p\u003e \u003cp\u003e8.10 Determination of Forced Vibration Response Amplitudes 659\u003c\/p\u003e \u003cp\u003e8.10.1 Total and Steady Response Amplitudes of an SDOF System 660\u003c\/p\u003e \u003cp\u003e8.10.2 Determination of Steady Response Amplitudes of an MDOF System 662\u003c\/p\u003e \u003cp\u003e8.11 Numerical Examples for Forced Vibration Response Amplitudes 668\u003c\/p\u003e \u003cp\u003e8.11.1 Frequency-Response Curves of an SDOF System 668\u003c\/p\u003e \u003cp\u003e8.11.2 Frequency-Response Curves of an MDOF System 670\u003c\/p\u003e \u003cp\u003eReferences 675\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendices 677\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 List of Integrals 677\u003c\/p\u003e \u003cp\u003eA.2 Theory of Modified Half-Interval (or Bisection) Method 680\u003c\/p\u003e \u003cp\u003eA.3 Determinations of Influence Coefficients 681\u003c\/p\u003e \u003cp\u003eA.3.1 Determination of Influence Coefficients a\u003csub\u003ei\u003c\/sub\u003e\u003csup\u003eYM \u003c\/sup\u003eand a\u003csub\u003ei\u003c\/sub\u003e\u003csup\u003eψM\u003c\/sup\u003e 681\u003c\/p\u003e \u003cp\u003eA.3.2 Determination of Influence Coefficients a\u003csub\u003ei\u003c\/sub\u003e\u003csup\u003eYQ\u003c\/sup\u003e and a\u003csub\u003ei\u003c\/sub\u003e\u003csup\u003eψQ\u003c\/sup\u003e 683\u003c\/p\u003e \u003cp\u003eA.4 Exact Solution of a Cubic Equation 685\u003c\/p\u003e \u003cp\u003eA.5 Solution of a Cubic Equation Associated with Its Complex Roots 686\u003c\/p\u003e \u003cp\u003eA.6 Coefficients of Matrix [H] Defined by Equation (7.387) 687\u003c\/p\u003e \u003cp\u003eA.7 Coefficients of Matrix [H] Defined by Equation (7.439) 689\u003c\/p\u003e \u003cp\u003eA.8 Exact Solution for a Simply Supported Euler Arch 691\u003c\/p\u003e \u003cp\u003eReferences 693\u003c\/p\u003e \u003cp\u003eIndex 695\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":52090800734551,"sku":"9781118632154","price":114.26,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781118632154.jpg?v=1762273509","url":"https:\/\/bookcurl.com\/products\/analytical-and-numerical-methods-for-vibration-analyses-9781118632154","provider":"Book Curl","version":"1.0","type":"link"}