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Contents12.4Flexural Vibrations at Right Angles to thePlane of a Ring40212.4.1 Classical Equations of Motion40212.4.2 Equations of Motion That IncludeEffects of Rotary Inertia and ShearDeformation40312.5Torsional Vibrations40612.6Extensional Vibrations40712.7Vibration of a Curved Beam with VariableCurvature40812.7.1 Thin Curved Beam40812.7.2 Curved Beam Analysis, Including theEffect of Shear Deformation41412.8Recent Contributions416References418Problems41913 Vibration of Membranes13.113.2420Introduction420Equation of Motion42013.2.1 Equilibrium Approach42013.2.2 Variational Approach42313.3Wave Solution42513.4Free Vibration of RectangularMembranes42613.4.1 Membrane with ClampedBoundaries42813.4.2 Mode Shapes43013.5Forced Vibration of RectangularMembranes43813.5.1 Modal Analysis Approach43813.5.2 Fourier Transform Approach44113.6Free Vibration of Circular Membranes44413.6.1 Equation of Motion44413.6.2 Membrane with a ClampedBoundary44613.6.3 Mode Shapes44713.7Forced Vibration of CircularMembranes44813.8Membranes with Irregular Shapes45213.9Partial Circular Membranes45313.10 Recent Contributions453References454Problems45514 Transverse Vibration of Plates14.114.2xi457Introduction457Equation of Motion: Classical PlateTheory45714.2.1 Equilibrium Approach45714.2.2 Variational Approach45814.3Boundary Conditions46514.4Free Vibration of Rectangular Plates47114.4.1 Solution for a Simply SupportedPlate47314.4.2 Solution for Plates with OtherBoundary Conditions47414.5Forced Vibration of Rectangular Plates47914.6Circular Plates48514.6.1 Equation of Motion48514.6.2 Transformation of Relations48614.6.3 Moment and Force Resultants48814.6.4 Boundary Conditions48914.7Free Vibration of Circular Plates49014.7.1 Solution for a Clamped Plate49214.7.2 Solution for a Plate with a FreeEdge49314.8Forced Vibration of Circular Plates49514.8.1 Harmonic Forcing Function49514.8.2 General Forcing Function49714.9Effects of Rotary Inertia and ShearDeformation49914.9.1 Equilibrium Approach49914.9.2 Variational Approach50514.9.3 Free Vibration Solution51114.9.4 Plate Simply Supported on All FourEdges51314.9.5 Circular Plates51514.9.6 Natural Frequencies of a ClampedCircular Plate52014.10 Plate on an Elastic Foundation52114.11 Transverse Vibration of Plates Subjected toIn-Plane Loads52314.11.1 Equation of Motion52314.11.2 Free Vibration52814.11.3 Solution for a Simply SupportedPlate52814.12 Vibration of Plates with VariableThickness52914.12.1 Rectangular Plates529

xiiContents14.12.2 Circular Plates53114.12.3 Free Vibration Solution14.13 Recent Contributions535References537Problems53915 Vibration of troduction and Shell Coordinates54115.1.1 Theory of Surfaces54115.1.2 Distance between Points in the MiddleSurface before Deformation54215.1.3 Distance between Points Anywhere inthe Thickness of a Shell beforeDeformation54715.1.4 Distance between Points Anywhere inthe Thickness of a Shell afterDeformation549Strain–Displacement Relations552Love’s Approximations556Stress–Strain Relations562Force and Moment Resultants563Strain Energy, Kinetic Energy, and Work Doneby External Forces57115.6.1 Strain Energy57115.6.2 Kinetic Energy57315.6.3 Work Done by External Forces573Equations of Motion from Hamilton’sPrinciple57515.7.1 Variation of Kinetic Energy57515.7.2 Variation of Strain Energy57615.7.3 Variation of Work Done by ExternalForces57715.7.4 Equations of Motion57715.7.5 Boundary Conditions579Circular Cylindrical Shells58215.8.1 Equations of Motion58315.8.2 Donnell–Mushtari–VlasovTheory58415.8.3 Natural Frequencies of VibrationAccording to DMV Theory58415.8.4 Natural Frequencies of TransverseVibration According to DMVTheory58615.8.5 Natural Frequencies of VibrationAccording to Love’s Theory587Equations of Motion of Conical and SphericalShells59115.9.1 Circular Conical Shells59115.9.2 Spherical Shells59115.10 Effect of Rotary Inertia and ShearDeformation59215.10.1 Displacement Components59215.10.2 Strain–Displacement Relations59315.10.3 Stress–Strain Relations59415.10.4 Force and Moment Resultants59415.10.5 Equations of Motion59515.10.6 Boundary Conditions59615.10.7 Vibration of Cylindrical Shells59715.10.8 Natural Frequencies of Vibration ofCylindrical Shells59815.10.9 Axisymmetric Modes60115.11 Recent Contributions603References604Problems60516 Elastic Wave tion607One-Dimensional Wave Equation607Traveling-Wave Solution60816.3.1 D’Alembert’s Solution60816.3.2 Two-Dimensional Problems61016.3.3 Harmonic Waves610Wave Motion in Strings61116.4.1 Free Vibration and HarmonicWaves61116.4.2 Solution in Terms of InitialConditions61316.4.3 Graphical Interpretation of theSolution614Reflection of Waves in One-DimensionalProblems61716.5.1 Reflection at a Fixed or RigidBoundary61716.5.2 Reflection at a Free Boundary618Reflection and Transmission of Waves at theInterface of Two Elastic Materials61916.6.1 Reflection at a Rigid Boundary62216.6.2 Reflection at a Free Boundary623Compressional and Shear Waves62316.7.1 Compressional or P Waves623

Contents16.7.2 Shear or S Waves625Flexural Waves in Beams628Wave Propagation in an Infinite ElasticMedium63116.9.1 Dilatational Waves63116.9.2 Distortional Waves63216.9.3 Independence of Dilatational andDistortional Waves63216.10 Rayleigh or Surface Waves63516.11 Recent Contributions643References644Problems64516.816.917 Approximate Analytical Rayleigh’s Quotient648Rayleigh’s Method650Rayleigh–Ritz Method661Assumed Modes Method670Weighted Residual Methods673Galerkin’s Method67364717.8Collocation Method68017.9Subdomain Method68417.10 Least Squares Method68617.11 Recent Contributions693References695Problems696A Basic Equations of lacement RelationsRotations702Stress–Strain Relations703Equations of Motion in Terms ofStresses704Equations of Motion in Terms ofDisplacements705B Laplace and Fourier TransformsIndex700713700707xiii

PrefaceThis book covers analytical methods of vibration analysis of continuous structuralsystems, including strings, bars, shafts, beams, circular rings and curved beams, membranes, plates, and shells. The propagation of elastic waves in structures and solidbodies is also introduced. The objectives of the book are (1) to make a methodical andcomprehensive presentation of the vibration of various types of structural elements,(2) to present the exact analytical and approximate analytical methods of analysis, and(3) to present the basic concepts in a simple manner with illustrative examples.Continuous structural elements and systems are encountered in many branchesof engineering, such as aerospace, architectural, chemical, civil, ocean, and mechanical engineering. The design of many structural and mechanical devices and systemsrequires an accurate prediction of their vibration and dynamic performance characteristics. The methods presented in the book can be used in these applications. The book isintended to serve as a textbook for a dual-level or first graduate-level course on vibrations or structural dynamics. More than enough material is included for a one-semestercourse. The chapters are made as independent and self-contained as possible so thata course can be taught by selecting appropriate chapters or through equivalent selfstudy. A successful vibration analysis of continuous structural elements and systemsrequires a knowledge of mechanics of materials, structural mechanics, ordinary and partial differential equations, matrix methods, variational calculus, and integral equations.Applications of these techniques are presented throughout. The selection, arrangement,and presentation of the material has been made based on the lecture notes for a coursetaught by the author. The contents of the book permit instructors to emphasize a variety of topics, such as basic mathematical approaches with simple applications, barsand beams, beams and plates, or plates and shells. The book will also be useful as areference book for practicing engineers, designers, and vibration analysts involved inthe dynamic analysis and design of continuous systems.Organization of the BookThe book is organized into 17 chapters and two appendixes. The basic concepts andterminology used in vibration analysis are introduced in Chapter 1. The importance,origin, and a brief history of vibration of continuous systems are presented. The difference between discrete and continuous systems, types of excitations, description ofharmonic functions, and basic definitions used in the theory of vibrations and representation of periodic functions in terms of Fourier series and the Fourier integralare discussed. Chapter 2 provides a brief review of the theory and techniques usedin the vibration analysis of discrete systems. Free and forced vibration of single- andmultidegree-of-freedom systems are outlined. The eigenvalue problem and its role inthe modal analysis used in the free and forced vibration analysis of discrete systemsare discussed.xv

xviPrefaceVarious methods of formulating vibration problems associated with continuoussystems are presented in Chapters 3, 4, and 5. The equilibrium approach is presentedin Chapter 3. Use of Newton’s second law of motion and D’Alembert’s principle isoutlined, with application to different types of continuous elements. Use of the variational approach in deriving equations of motion and associated boundary conditions isdescribed in Chapter 4. The basic concepts of calculus of variations and their applicationto extreme value problems are outlined. The variational methods of solid mechanics,including the principles of minimum potential energy, minimum complementary energy,stationary Reissner energy, and Hamilton’s principle, are presented. The use of Hamilton’s principle in the formulation of continuous systems is illustrated with torsionalvibration of a shaft and transverse vibration of a thin beam. The integral equationapproach for the formulation of vibration problems is presented in Chapter 5. A briefoutline of integral equations and their classification, and the derivation of integralequations, are given together with examples. The solution of integral equations usingiterative, Rayleigh–Ritz, Galerkin, collocation, and numerical integration methods isalso discussed in this chapter.The common solution procedure based on eigenvalue and modal analyses for thevibration analysis of continuous systems is outlined in Chapter 6. The orthogonality ofeigenfunctions and the role of the expansion theorem in modal analysis are discussed.The forced vibration response of viscously damped systems are also considered in thischapter. Chapter 7 covers the solution of problems of vibration of continuous systemsusing integral transform methods. Both Laplace and Fourier transform techniques areoutlined together with illustrative applications.The transverse vibration of strings is presented in Chapter 8. This problem findsapplication in guy wires, electric transmission lines, ropes and belts used in machinery,and the manufacture of thread. The governing equation is derived using equilibriumand variational approaches. The traveling-wave solution and separation of variablessolution are outlined. The free and forced vibration of strings are considered in thischapter. The longitudinal vibration of bars is the topic of Chapter 9. Equations ofmotion based on simple theory are derived using the equilibrium approach as well asHamilton’s principle. The natural frequencies of vibration are determined for bars withdifferent end conditions. Free vibration response due to initial excitation and forcedvibration of bars are both presented, as is response using modal analysis. Free and forcedvibration of bars using Rayleigh and Bishop theories are also outlined in Chapter 9.The torsional vibration of shafts plays an important role in mechanical transmissionof power in prime movers and other high-speed machinery. The torsional vibrationof uniform and nonuniform rods with both circular and noncircular cross sections isdescribed in Chapter 10. The equations of motion and free and forced vibration of shaftswith circular cross section are discussed using the elementary theory. The Saint-Venantand Timoshenko–Gere theories are considered in deriving the equations of motion ofshafts with noncircular cross sections. Methods of determining the torsional rigidity ofnoncircular shafts are presented using the Prandtl stress function and Prandtl membraneanalogy.Chapter 11 deals with the transverse vibration of beams. Starting with the equationof motion based on Euler–Bernoulli or thin beam theory, natural frequencies andmode shapes of beams with different boundary conditions are determined. The freevibration response due to initial conditions, forced vibration under fixed and moving

Prefacexviiloads, response under axial loading, rotating beams, continuous beams, and beams onan elastic foundation are presented using the Euler–Bernoulli theory. The effects ofrotary inertia (Rayleigh theory) and rotary inertia and shear deformation (Timoshenkotheory) on the transverse vibration of beams are also considered. Finally, coupled bending–torsional vibration of beams is discussed toward the end of Chapter 11. In-planeflexural and coupled twist-bending vibration of circular rings and curved beams isconsidered in Chapter 12. The equations of motion and free vibration solutions arepresented first using a simple theory. Then the effects of rotary inertia and shear deformation are considered. The vibration of rings is important in a study of the vibrationof ring-stiffened shells used in aerospace applications, gears, and stators of electricalmachines.The transverse vibration of membranes is the topic of Chapter 13. Membranesfind application in drums and microphone condensers. The equation of motion ofmembranes is derived using both the equilibrium and variational approaches. The freeand forced vibration of rectangular and circular membranes are both discussed in thischapter. Chapter 14 covers the transverse vibration of plates. The equation of motionand the free and forced vibration of both rectangular and circular plates are presented.The vibration of plates subjected to in-plane forces, plates on elastic foundation, andplates with variable thickness is also discussed. Finally, the effect of rotary inertiaand shear deformation on the vibration of plates is outlined according to Mindlin’stheory. The vibration of shells is the topic of Chapter 15. First the theory of surfacesis presented using shell coordinates. Then the strain–displacement relations accordingto Love’s approximations, stress–strain, and force and moment resultants are given.Then the equations of motion are derived from Hamilton’s principle. The equations ofmotion of circular cylindrical shells and their natural frequencies are considered usingDonnel–Mushtari–Vlasov and Love’s theories. Finally, the effect of rotary inertia andshear deformation on the vibration of shells is considered.Wave propagation in elastic solids is considered in Chapter 16. The onedimensional wave equation and the traveling-wave solution are presented. The wavemotion in strings and wave propagation in a semi-infinite medium, along with reflectionand transmission of waves at fixed and free boundaries, are discussed. The differencesbetween compressional or P waves and shear or S waves are discussed. The flexuralwaves in beams and the propagation of dilatational and distortional waves is consideredin an infinite elastic medium. Rayleigh or surface waves are also discussed. Finally,Chapter 17 is devoted to the approximate analytical methods useful for vibrationanalysis. The computational details of the Rayleigh, Rayleigh–Ritz, assumed modes,weighted residual, Galerkin, collocation, subdomain collocation, and least squares methods are presented along with numerical examples. Appendix A presents the basicequations of elasticity. Laplace and Fourier transform pairs associated with some simpleand commonly used functions are summarized in Appendix B.AcknowledgmentsI would like to thank the many graduate students who offered constructive suggestionswhen drafts of this book were used as class notes. I would also like to express mygratitude to my wife, Kamala, for her infinite patience, encouragement, and moralsupport in completing this book. She shares with me the fun and pain associated with

xviiiPrefacethe writing of the book. Finally, and most importantly, I would like to acknowledge themany intangible contributions made by my granddaughters, Siriveena and Samanthaka,that helped the completion of this work in a timely fashion.S. S. Raosrao@miami.eduMiami, November 2005

Symbolsaa, bAA, B, C, Dcc1 , c2[c], [cij ]CC1 , C2 , C3 , C4[d], [dij ]DEEAEIff f0f (t), F (t)f (x, t)f (x, y, t)F0F (m, n, t)F (s)F (ω)Fj (t)GGJhH (t)iII0Im , KmIp , Jradius of a circular membrane or platedimensions of a membrane or plate along the xand y directionscross-sectional area; area of a plate; amplitudeconstantsvelocity of wave propagation; damping constantdamping constants of dampersdamping matrixtorsional rigidityconstantsdamping matrixflexural rigidity of a plate or shell; domainYoung’s modulusaxial stiffness of a barbending stiffness of a beamlinear frequency (Hz)vector of forcesuniform load; amplitude of forceforceexternal force per unit lengthexternal transverse force per unit area on amembrane or plateconcentrated forceFourier transform of f (x, y, t)Laplace transform of f (t)Fourier transform of f (t) jconcentrated force at point Xshear modulustorsional rigiditythickness of a plate or shellHeavisidefunction 1area moment of inertia of cross section of abeam; functionalmass polar moment of inertia per unit lengthmodified Bessel functions of order m of the firstand second kindpolar moment of inertia of cross sectionxix

xxSymbolsJm , Ymkk 1 , k2[k], [kij ]lLL 1mm1 , m2[m], [mij ]MMx , My , MxynNi (t)Nx , Ny , NxyPQn (t)Qx , Qyr, θRsStT Tmaxu, v, wu(x, t)u0 (x), u̇0 (x)UUn (x)vVw(x, t)w(x, y, t)w0 (x), ẇ0 (x)w0 (x, y),ẇ0 (x, y)WW (m, n, t)Wi (x)Wmn (x, y)W0 (a), Ẇ0 (a)Bessel functions of order m of the first andsecond kindshear correction factor; spring stiffnessstiffnesses of springsstiffness matrixlength of a string, bar, sha

Both Ends Free 278 10.3.3 Natural Frequencies of a Shaft Fixed at One End and Attached to a Torsional Spring at the Other 279 10.4 Free Vibration Response due to Initial Conditions: Modal Analysis 289 10.5 Forced Vibration of a Uniform Shaft: Modal Analysis 292 10.6 Torsional V