Vibration of Continuous Systems

1 Vibration of Continuous SystemsVibration of Continuous Systems . Singiresu S. Rao 2007 John Wiley & Sons, Inc. ISBN: 978-0-471-77171-5 Vibration of ContinuousSystemsSingiresu S. RaoProfessor and ChairmanDepartment of Mechanical and Aerospace EngineeringUniversity of MiamiCoral Gables, FloridaJOHN WILEY & SONS, book is printed on acid-free 2007 by John Wiley & Sons, Inc. All rights by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form orby any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except aspermitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriate per-copy fee tothe Copyright Clearance Center, Inc.

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4 Of Continuous Systems / Singiresu S. 978-0-471-77171-5 (cloth)ISBN-10 0-471-77171-6 (cloth)1. Vibration Textbooks. 2. Structural dynamics 71 dc222006008775 Printed in the United States of America10987654321To Lord Sri VenkateswaraContentsPreface xvSymbols xix1 Introduction: Basic Concepts of Vibration of Vibration and Developments in Mechanics andVibration of Vibration of ContinuousSystems and Continuous Systems Problems Analysis Functions Representation of HarmonicMotion Definitions and Terminology Periodic Functions and Fourier Series Nonperiodic Functions and FourierIntegrals Literature on Vibration of ContinuousSystems 29 References 29 Problems 312 Vibration of Discrete Systems .

5 Of a Single-Degree-of-FreedomSystem Free Vibration Forced Vibration under HarmonicForce Forced Vibration under GeneralForce of Multidegree-of-FreedomSystems eigenvalue problem Orthogonality of Modal Vectors Free Vibration Analysis of anUndamped System Using ModalAnalysis Forced Vibration Analysis of anUndamped System Using ModalAnalysis Forced Vibration Analysis of a Systemwith Proportional Damping Forced Vibration Analysis of a Systemwith General Viscous Damping Contributions 60 References 61 Problems 623 Derivation of Equations: s Second Law of Motion Alembert s Principle of Motion of a Bar in AxialVibration of Motion of a Beam in TransverseVibration of Motion of a Plate in TransverseVibration State of Stress Dynamic Equilibrium Equations Strain Displacement Relations Moment DisplacementRelations Equation of Motion in Terms ofDisplacement Initial and Boundary Conditions Contributions 80 References 80 Problems 814 Derivation of Equations.

6 Of a Single Variable of Variations Operator with Higher-OrderDerivatives with Several with Several of a Functional withConstraints Conditions Variational Methods in SolidMechanics Principle of Minimum PotentialEnergy Principle of Minimum ComplementaryEnergy Principle of Stationary ReissnerEnergy Hamilton s Principle Applications of Hamilton s Principle Equation of Motion for TorsionalVibration of a Shaft (FreeVibration) Transverse Vibration of a ThinBeam Recent Contributions 119 References 120 Problems 1205 Derivation of Equations: Integral of Integral Equations Classification Based on the NonlinearAppearance of (t) Classification Based on the Location ofUnknown Function (t)

7 Classification Based on the Limits ofIntegration Classification Based on the ProperNature of an Integral of Integral Equations Direct Method Derivation from the DifferentialEquation of Motion Formulation of the EigenvalueProblem One-Dimensional Systems General Continuous Systems Orthogonality ofEigenfunctions of Integral Equations Method of UndeterminedCoefficients Iterative Method Rayleigh Ritz Method Galerkin s Method Collocation Method Numerical Integration Method Contributions 147 References 148 Problems 1496 Solution Procedure: eigenvalue and ModalAnalysis problem of Homogeneous Equations:Separation-of-Variables Technique liouville problem Classification of sturm LiouvilleProblems Properties of Eigenvalues andEigenfunctions eigenvalue problem Self-Adjoint EigenvalueProblem Orthogonality ofEigenfunctions Expansion Theorem of NonhomogeneousEquations Response of Viscously DampedSystems Contributions 171 References 172 Problems 173 Contentsix7 Solution Procedure.

8 Integral Transforms Fourier Series Fourier Transforms Fourier Transform of Derivatives ofFunctions Finite Sine and Cosine FourierTransforms Vibration of a Finite String Vibration of a Finite String Vibration of a Beam Transforms Properties of LaplaceTransforms Partial Fraction Method Inverse Transformation Vibration of a String of FiniteLength Vibration of a Beam of FiniteLength Vibration of a Beam of FiniteLength Recent Contributions 201 References 202 Problems 2038 Transverse Vibration of of Motion Equilibrium Approach Variational Approach and Boundary Conditions Vibration of an Infinite String Traveling-Wave Solution Fourier Transform BasedSolution Laplace Transform BasedSolution Vibration of a String of FiniteLength Free Vibration of a String with BothEnds Fixed Vibration Contributions 231 References 232 Problems 2339 Longitudinal Vibration of of Motion Using SimpleTheory Using Newton s Second Law ofMotion Using Hamilton s Principle Vibration Solution and NaturalFrequencies Solution Using Separation ofVariables Orthogonality ofEigenfunctions Free Vibration Response due to InitialExcitation Vibration

9 Of a Bar Subjected to LongitudinalSupport Motion Theory Equation of Motion Natural Frequencies and ModeShapes s Theory Equation of Motion Natural Frequencies and ModeShapes Forced Vibration Using ModalAnalysis Contributions 267 References 268 Problems 26810 Torsional Vibration of Introduction Elementary Theory: Equation ofMotion Equilibrium Approach Variational Approach Free Vibration of Uniform Shafts Natural Frequencies of a Shaft withBoth Ends Fixed Natural Frequencies of a Shaft withBoth Ends Free Natural Frequencies of a Shaft Fixed atOne End and Attached to a TorsionalSpring at the Other Free Vibration Response due to InitialConditions: Modal Analysis Forced Vibration of a Uniform Shaft: ModalAnalysis Torsional Vibration of Noncircular Shafts:Saint-Venant s Theory Torsional Vibration of Noncircular Shafts,Including Axial Inertia Torsional Vibration of Noncircular Shafts.

10 Timoshenko Gere Theory Torsional Rigidity of NoncircularShafts Prandtl s Membrane Analogy Recent Contributions 313 References 314 Problems 31511 Transverse Vibration of Introduction Equation of Motion: Euler BernoulliTheory Free Vibration Equations Free Vibration Solution Frequencies and Mode Shapes of UniformBeams Beam Simply Supported at BothEnds Beam Fixed at Both Ends Beam Free at Both Ends Beam with One End Fixed and theOther Simply Supported Beam Fixed at One End and Free atthe Other Orthogonality of Normal Modes Free Vibration Response due to InitialConditions Forced Vibration Response of Beams under MovingLoads Transverse Vibration of Beams Subjected toAxial Force Derivation of Equations Free Vibration of a UniformBeam Vibration of a Rotating Beam Natural