Analytical and Numerical Methods for Vibration Analyses.
Title:
Analytical and Numerical Methods for Vibration Analyses.
Author:
Wu, Jong-Shyong.
ISBN:
9781118632345
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (726 pages)
Contents:
Analytical and Numerical Methods for Vibration Analyses -- Contents -- About the Author -- Preface -- 1 Introduction to Structural Vibrations -- 1.1 Terminology -- 1.2 Types of Vibration -- 1.3 Objectives of Vibration Analyses -- 1.3.1 Free Vibration Analysis -- 1.3.2 Forced Vibration Analysis -- 1.4 Global and Local Vibrations -- 1.5 Theoretical Approaches to Structural Vibrations -- References -- 2 Analytical Solutions for Uniform Continuous Systems -- 2.1 Methods for Obtaining Equations of Motion of a Vibrating System -- 2.2 Vibration of a Stretched String -- 2.2.1 Equation of Motion -- 2.2.2 Free Vibration of a Uniform Clamped-Clamped String -- 2.3 Longitudinal Vibration of a Continuous Rod -- 2.3.1 Equation of Motion -- 2.3.2 Free Vibration of a Uniform Rod -- 2.4 Torsional Vibration of a Continuous Shaft -- 2.4.1 Equation of Motion -- 2.4.2 Free Vibration of a Uniform Shaft -- 2.5 Flexural Vibration of a Continuous Euler-Bernoulli Beam -- 2.5.1 Equation of Motion -- 2.5.2 Free Vibration of a Uniform Euler-Bernoulli Beam -- 2.5.3 Numerical Example -- 2.6 Vibration of Axial-Loaded Uniform Euler-Bernoulli Beam -- 2.6.1 Equation of Motion -- 2.6.2 Free Vibration of an Axial-Loaded Uniform Beam -- 2.6.3 Numerical Example -- 2.6.4 Critical Buckling Load of a Uniform Euler-Bernoulli Beam -- 2.7 Vibration of an Euler-Bernoulli Beam on the Elastic Foundation -- 2.7.1 Influence of Stiffness Ratio and Total Beam Length -- 2.7.2 Influence of Supporting Conditions of the Beam -- 2.8 Vibration of an Axial-Loaded Euler Beam on the Elastic Foundation -- 2.8.1 Equation of Motion -- 2.8.2 Free Vibration of a Uniform Beam -- 2.8.3 Numerical Example -- 2.9 Flexural Vibration of a Continuous Timoshenko Beam -- 2.9.1 Equation of Motion -- 2.9.2 Free Vibration of a Uniform Timoshenko Beam -- 2.9.3 Numerical Example.
2.10 Vibrations of a Shear Beam and a Rotary Beam -- 2.10.1 Free Vibration of a Shear Beam -- 2.10.2 Free Vibration of a Rotary Beam -- 2.11 Vibration of an Axial-Loaded Timoshenko Beam -- 2.11.1 Equation of Motion -- 2.11.2 Free Vibration of an Axial-Loaded Uniform Timoshenko Beam -- 2.11.3 Numerical Example -- 2.12 Vibration of a Timoshenko Beam on the Elastic Foundation -- 2.12.1 Equation of Motion -- 2.12.2 Free Vibration of a Uniform Beam on the Elastic Foundation -- 2.12.3 Numerical Example -- 2.13 Vibration of an Axial-Loaded Timoshenko Beam on the Elastic Foundation -- 2.13.1 Equation of Motion -- 2.13.2 Free Vibration of a Uniform Timoshenko Beam -- 2.13.3 Numerical Example -- 2.14 Vibration of Membranes -- 2.14.1 Free Vibration of a Rectangular Membrane -- 2.14.2 Free Vibration of a Circular Membrane -- 2.15 Vibration of Flat Plates -- 2.15.1 Free Vibration of a Rectangular Plate -- 2.15.2 Free Vibration of a Circular Plate -- References -- 3 Analytical Solutions for Non-Uniform Continuous Systems: Tapered Beams -- 3.1 Longitudinal Vibration of a Conical Rod -- 3.1.1 Determination of Natural Frequencies and Natural Mode Shapes -- 3.1.2 Determination of Normal Mode Shapes -- 3.1.3 Numerical Examples -- 3.2 Torsional Vibration of a Conical Shaft -- 3.2.1 Determination of Natural Frequencies and Natural Mode Shapes -- 3.2.2 Determination of Normal Mode Shapes -- 3.2.3 Numerical Example -- 3.3 Displacement Function for Free Bending Vibration of a Tapered Beam -- 3.4 Bending Vibration of a Single-Tapered Beam -- 3.4.1 Determination of Natural Frequencies and Natural Mode Shapes -- 3.4.2 Determination of Normal Mode Shapes -- 3.4.3 Finite Element Model of a Single-Tapered Beam -- 3.4.4 Numerical Example -- 3.5 Bending Vibration of a Double-Tapered Beam -- 3.5.1 Determination of Natural Frequencies and Natural Mode Shapes.
3.5.2 Determination of Normal Mode Shapes -- 3.5.3 Finite Element Model of a Double-Tapered Beam -- 3.5.4 Numerical Example -- 3.6 Bending Vibration of a Nonlinearly Tapered Beam -- 3.6.1 Equation of Motion and Boundary Conditions -- 3.6.2 Natural Frequencies and Mode Shapes for Various Supporting Conditions -- 3.6.3 Finite Element Model of a Non-Uniform Beam -- 3.6.4 Numerical Example -- References -- 4 Transfer Matrix Methods for Discrete and Continuous Systems -- 4.1 Torsional Vibrations of Multi-Degrees-of-Freedom Systems -- 4.1.1 Holzer Method for Torsional Vibrations -- 4.1.2 Transfer Matrix Method for Torsional Vibrations -- 4.2 Lumped-Mass Model Transfer Matrix Method for Flexural Vibrations -- 4.2.1 Transfer Matrices for a Station and a Field -- 4.2.2 Free Vibration of a Flexural Beam -- 4.2.3 Discretization of a Continuous Beam -- 4.2.4 Transfer Matrices for a Timoshenko Beam -- 4.2.5 Numerical Example -- 4.2.6 A Timoshenko Beam Carrying Multiple Various Concentrated Elements -- 4.2.7 Transfer Matrix for Axial-Loaded Euler Beam and Timoshenko Beam -- 4.3 Continuous-Mass Model Transfer Matrix Method for Flexural Vibrations -- 4.3.1 Flexural Vibration of an Euler-Bernoulli Beam -- 4.3.2 Flexural Vibration of a Timoshenko Beam with Axial Load -- 4.4 Flexural Vibrations of Beams with In-Span Rigid (Pinned) Supports -- 4.4.1 Transfer Matrix of a Station Located at an In-Span Rigid (Pinned) Support -- 4.4.2 Natural Frequencies and Mode Shapes of a Multi-Span Beam -- 4.4.3 Numerical Examples -- References -- 5 Eigenproblem and Jacobi Method -- 5.1 Eigenproblem -- 5.2 Natural Frequencies, Natural Mode Shapes and Unit-Amplitude Mode Shapes -- 5.3 Determination of Normal Mode Shapes -- 5.3.1 Normal Mode Shapes Obtained From Natural Ones -- 5.3.2 Normal Mode Shapes Obtained From Unit-Amplitude Ones.
5.4 Solution of Standard Eigenproblem with Standard Jacobi Method -- 5.4.1 Formulation Based on Forward Multiplication -- 5.4.2 Formulation Based on Backward Multiplication -- 5.4.3 Convergence of Iterations -- 5.5 Solution of Generalized Eigenproblem with Generalized Jacobi Method -- 5.5.1 The Standard Jacobi Method -- 5.5.2 The Generalized Jacobi Method -- 5.5.3 Formulation Based on Forward Multiplication -- 5.5.4 Determination of Elements of Rotation Matrix (α and γ) -- 5.5.5 Convergence of Iterations -- 5.5.6 Formulation Based on Backward Multiplication -- 5.6 Solution of Semi-Definite System with Generalized Jacobi Method -- 5.7 Solution of Damped Eigenproblem -- References -- 6 Vibration Analysis by Finite Element Method -- 6.1 Equation of Motion and Property Matrices -- 6.2 Longitudinal (Axial) Vibration of a Rod -- 6.3 Property Matrices of a Torsional Shaft -- 6.4 Flexural Vibration of an Euler-Bernoulli Beam -- 6.5 Shape Functions for a Three-Dimensional Timoshenko Beam Element -- 6.5.1 Assumptions for the Formulations -- 6.5.2 Shear Deformations Due to Translational Nodal Displacements -- 6.5.3 Shear Deformations Due to Rotational Nodal Displacements V2 and V4 -- 6.5.4 Determination of Shape Functions Φyi(ξ) (i = 1 - 4) -- 6.5.5 Determination of Shape Functions Φxi(ξ) (i = 1 - 4) -- 6.5.6 Determination of Shape Functions Φzi(ξ) (i = 1 - 4) -- 6.5.7 Determination of Shape Functions φxi(ξ) (i = 1 - 4) -- 6.5.8 Shape Functions for a 3D Beam Element -- 6.6 Property Matrices of a Three-Dimensional Timoshenko Beam Element -- 6.6.1 Stiffness Matrix of a 3D Timoshenko Beam Element -- 6.6.2 Mass Matrix of a 3D Timoshenko Beam Element -- 6.7 Transformation Matrix for a Two-Dimensional Beam Element -- 6.8 Transformations of Element Stiffness Matrix and Mass Matrix -- 6.9 Transformation Matrix for a Three-Dimensional Beam Element.
6.10 Property Matrices of a Beam Element with Concentrated Elements -- 6.11 Property Matrices of Rigid-Pinned and Pinned-Rigid Beam Elements -- 6.11.1 Property Matrices of the R-P Beam Element -- 6.11.2 Property Matrices of the P-R Beam Element -- 6.12 Geometric Stiffness Matrix of a Beam Element Due to Axial Load -- 6.13 Stiffness Matrix of a Beam Element Due to Elastic Foundation -- References -- 7 Analytical Methods and Finite Element Method for Free Vibration Analyses of Circularly Curved Beams -- 7.1 Analytical Solution for Out-of-Plane Vibration of a Curved Euler Beam -- 7.1.1 Differential Equations for Displacement Functions -- 7.1.2 Determination of Displacement Functions -- 7.1.3 Internal Forces and Moments -- 7.1.4 Equilibrium and Continuity Conditions -- 7.1.5 Determination of Natural Frequencies and Mode Shapes -- 7.1.6 Classical and Non-Classical Boundary Conditions -- 7.1.7 Numerical Examples -- 7.2 Analytical Solution for Out-of-Plane Vibration of a Curved Timoshenko Beam -- 7.2.1 Coupled Equations of Motion and Boundary Conditions -- 7.2.2 Uncoupled Equation of Motion for uy -- 7.2.3 The Relationships Between ψx, ψθ and uy -- 7.2.4 Determination of Displacement Functions Uy(θ), ψx(θ) and ψθ(θ) -- 7.2.5 Internal Forces and Moments -- 7.2.6 Classical Boundary Conditions -- 7.2.7 Equilibrium and Compatibility Conditions -- 7.2.8 Determination of Natural Frequencies and Mode Shapes -- 7.2.9 Numerical Examples -- 7.3 Analytical Solution for In-Plane Vibration of a Curved Euler Beam -- 7.3.1 Differential Equations for Displacement Functions -- 7.3.2 Determination of Displacement Functions -- 7.3.3 Internal Forces and Moments -- 7.3.4 Continuity and Equilibrium Conditions -- 7.3.5 Determination of Natural Frequencies and Mode Shapes -- 7.3.6 Classical Boundary Conditions.
7.3.7 Mode Shapes Obtained From Finite Element Method and Analytical (Exact) Method.
Abstract:
Illustrates theories and associated mathematical expressions with numerical examples using various methods, leading to exact solutions, more accurate results, and more computationally efficient techniques This book presents the derivations of the equations of motion for all structure foundations using either the continuous model or the discrete model. This mathematical display is a strong feature of the book as it helps to explain in full detail how calculations are reached and interpreted. In addition to the simple 'uniform' and 'straight' beams, the book introduces solution techniques for the complicated 'non uniform' beams (including linear or non-linear tapered beams), and curved beams. Most of the beams are analyzed by taking account of the effects of shear deformation and rotary inertia of the beams themselves as well as the eccentricities and mass moments of inertia of the attachments. Demonstrates approaches which dramatically cut CPU times to a fraction of conventional FEM Presents "mode shapes" in addition to natural frequencies, which are critical for designers Gives detailed derivations for continuous and discrete model equations of motions Summarizes the analytical and numerical methods for the natural frequencies, mode shapes, and time histories of straight structures rods shafts Euler beams strings Timoshenko beams membranes/thin plates Conical rods and shafts Tapered beams Curved beams Has applications for students taking courses including vibration mechanics, dynamics of structures, and finite element analyses of structures, the transfer matrix method, and Jacobi method This book is ideal for graduate students in mechanical, civil, marine, aeronautical engineering courses as well as advanced undergraduates with a background in General Physics, Calculus, and Mechanics of Material. The book is also a handy reference for researchers
and professional engineers.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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