Vibration of Continuous Systems

Singiresu S. Rao is a Professor in the Mechanical and Aerospace Engineering Department at the University of Miami. His main areas of research include structural dynamics, multi objective optimization and development of uncertainty models in engineering modeling, analysis, design and optimization. He is a Fellow of ASME and an Associate Fellow of the AIAA.

• Preface xvAcknowledgments xixAbout the Author xxi1 Introduction: Basic Concepts and Terminology 11.1 Concept of Vibration 11.2 Importance of Vibration 41.3 Origins and Developments in Mechanics and Vibration 51.4 History of Vibration of Continuous Systems 71.5 Discrete and Continuous Systems 121.6 Vibration Problems 151.7 Vibration Analysis 161.8 Excitations 171.9 Harmonic Functions 171.10 Periodic Functions and Fourier Series 241.11 Non periodic Functions and Fourier Integrals 251.12 Literature on Vibration of Continuous Systems 28References 29Problems 312 Vibration of Discrete Systems: Brief Review 332.1 Vibration of a Single-Degree-of-Freedom System 332.2 Vibration of Multi degree-of-Freedom Systems 432.3 Recent Contributions 60References 61Problems 623 Derivation of Equations: Equilibrium Approach 693.1 Introduction 693.2 Newton’s Second Law of Motion 693.3 D’Alembert’s Principle 703.4 Equation of Motion of a Bar in Axial Vibration 703.5 Equation of Motion of a Beam in Transverse Vibration 723.6 Equation of Motion of a Plate in Transverse Vibration 743.7 Additional Contributions 81References 81Problems 824 Derivation of Equations: Variational Approach 874.1 Introduction 874.2 Calculus of a Single Variable 874.3 Calculus of Variations 884.4 Variation Operator 914.5 Functional with Higher-Order Derivatives 934.6 Functional with Several Dependent Variables 954.7 Functional with Several Independent Variables 964.8 Extremization of a Functional with Constraints 984.9 Boundary Conditions 1024.10 Variational Methods in Solid Mechanics 1064.11 Applications of Hamilton’s Principle 1164.12 Recent Contributions 121Notes 121References 122Problems 1225 Derivation of Equations: Integral Equation Approach 1255.1 Introduction 1255.2 Classification of Integral Equations 1255.3 Derivation of Integral Equations 1275.4 General Formulation of the Eigenvalue Problem 1325.6 Recent Contributions 149References 150Problems 1516 Solution Procedure: Eigenvalue and Modal Analysis Approach 1536.1 Introduction 1536.2 General Problem 1536.3 Solution of Homogeneous Equations: Separation-of-Variables Technique 1556.4 Sturm–Liouville Problem 1566.5 General Eigenvalue Problem 1656.6 Solution of Nonhomogeneous Equations 1696.7 Forced Response of Viscously Damped Systems 1716.8 Recent Contributions 173References 174Problems 1757 Solution Procedure: Integral Transform Methods 1777.1 Introduction 1777.2 Fourier Transforms 1787.3 Free Vibration of a Finite String 1847.4 Forced Vibration of a Finite String 1867.5 Free Vibration of a Beam 1887.6 Laplace Transforms 1917.7 Free Vibration of a String of Finite Length 1977.8 Free Vibration of a Beam of Finite Length 2007.9 Forced Vibration of a Beam of Finite Length 2017.10 Recent Contributions 204References 205Problems 2068 Transverse Vibration of Strings 2098.1 Introduction 2098.2 Equation of Motion 2098.3 Initial and Boundary Conditions 2138.4 Free Vibration of an Infinite String 2158.5 Free Vibration of a String of Finite Length 2218.6 Forced Vibration 2318.7 Recent Contributions 235Note 236References 236Problems 2379 Longitudinal Vibration of Bars 2399.1 Introduction 2399.2 Equation of Motion Using Simple Theory 2399.3 Free Vibration Solution and Natural Frequencies 2419.4 Forced Vibration 2599.5 Response of a Bar Subjected toLongitudinal Support Motion 2629.6 Rayleigh Theory 2639.7 Bishop’s Theory 2659.8 Recent Contributions 272References 273Problems 27310 Torsional Vibration of Shafts 27710.1 Introduction 27710.2 Elementary Theory: Equation of Motion 27710.3 Free Vibration of Uniform Shafts 28210.4 Free Vibration Response due to Initial Conditions: Modal Analysis 29510.5 Forced Vibration of a Uniform Shaft: Modal Analysis 29810.6 Torsional Vibration of Noncircular Shafts: Saint-Venant’s Theory 30110.7 Torsional Vibration of Noncircular Shafts, Including Axial Inertia 30510.8 Torsional Vibration of Noncircular Shafts: The Timoshenko–Gere Theory 30610.9 Torsional Rigidity of Noncircular Shafts 30910.10 Prandtl’s Membrane Analogy 31410.11 Recent Contributions 319References 320Problems 32111 Transverse Vibration of Beams 32311.1 Introduction 32311.2 Equation of Motion: The Euler–Bernoulli Theory 32311.3 Free Vibration Equations 33111.4 Free Vibration Solution 33111.5 Frequencies and Mode Shapes of Uniform Beams 33211.6 Orthogonality of Normal Modes 34511.7 Free Vibration Response due to Initial Conditions 34711.8 Forced Vibration 35011.9 Response of Beams under Moving Loads 35611.10 Transverse Vibration of Beams Subjected to Axial Force 35811.11 Vibration of a Rotating Beam 36311.12 Natural Frequencies of Continuous Beams on Many Supports 36511.13 Beam on an Elastic Foundation 37011.14 Rayleigh’s Theory 37511.15 Timoshenko’s Theory 37711.16 Coupled Bending–Torsional Vibration of Beams 38611.17 Transform Methods: Free Vibration of an Infinite Beam 39111.18 Recent Contributions 393References 395Problems 39612 Vibration of Circular Rings and Curved Beams 39912.1 Introduction 39912.2 Equations of Motion of a Circular Ring 39912.3 In-Plane Flexural Vibrations of Rings 40412.4 Flexural Vibrations at Right Angles to the Plane of a Ring 40812.5 Torsional Vibrations 41312.6 Extensional Vibrations 41312.7 Vibration of a Curved Beam with Variable Curvature 41412.8 Recent Contributions 423References 424Problems 42513 Vibration of Membranes 42713.1 Introduction 42713.2 Equation of Motion 42713.3 Wave Solution 43213.4 Free Vibration of Rectangular Membranes 43313.5 Forced Vibration of Rectangular Membranes 44413.6 Free Vibration of Circular Membranes 45013.7 Forced Vibration of Circular Membranes 45413.8 Membranes with Irregular Shapes 45913.9 Partial Circular Membranes 45913.10 Recent Contributions 460Notes 461References 462Problems 46314 Transverse Vibration of Plates 46514.1 Introduction 46514.2 Equation of Motion: Classical Plate Theory 46514.3 Boundary Conditions 47314.4 Free Vibration of Rectangular Plates 47914.5 Forced Vibration of Rectangular Plates 48914.6 Circular Plates 49314.7 Free Vibration of Circular Plates 49814.8 Forced Vibration of Circular Plates 50314.9 Effects of Rotary Inertia and Shear Deformation 50714.10 Plate on an Elastic Foundation 52914.11 Transverse Vibration of Plates Subjected to In-Plane Loads 53114.12 Vibration of Plates with Variable Thickness 53714.13 Recent Contributions 543References 545Problems 54715 Vibration of Shells 54915.1 Introduction and Shell Coordinates 54915.2 Strain–Displacement Relations 56015.3 Love’s Approximations 56415.4 Stress–Strain Relations 57015.5 Force and Moment Resultants 57115.6 Strain Energy, Kinetic Energy, and Work Done by External Forces 57915.7 Equations of Motion from Hamilton’s Principle 58215.8 Circular Cylindrical Shells 59015.9 Equations of Motion of Conical and Spherical Shells 59915.10 Effect of Rotary Inertia and Shear Deformation 60015.11 Recent Contributions 611Notes 612References 612Problems 61416 Vibration of Composite Structures 61716.1 Introduction 61716.2 Characterization of a Unidirectional Lamina with Loading Parallel to the Fibers 61716.3 Different Types of Material Behavior 61916.4 Constitutive Equations or Stress–Strain Relations 62016.5 Coordinate Transformations for Stresses and Strains 62616.6 Lamina with Fibers Oriented at an Angle 63216.7 Composite Lamina in Plane Stress 63416.8 Laminated Composite Structures 64116.9 Vibration Analysis of Laminated Composite Plates 65916.10 Vibration Analysis of Laminated Composte Beams 66316.11 Recent Contributions 666References 667Problems 66817 Approximate Analytical Methods 67117.1 Introduction 67117.2 Rayleigh’s Quotient 67217.3 Rayleigh’s Method 67417.4 Rayleigh–Ritz Method 68517.5 Assumed Modes Method 69517.6 Weighted Residual Methods 69717.7 Galerkin’s Method 69817.8 Collocation Method 70417.9 Subdomain Method 70917.10 Least Squares Method 71117.11 Recent Contributions 718References 719Problems 72118 Numerical Methods: Finite Element Method 72518.1 Introduction 72518.2 Finite Element Procedure 72518.3 Element Matrices of Different Structural Problems 73918.4 Dynamic Response Using the Finite Element Method 75318.5 Additional and Recent Contributions 760Note 763References 763Problems 765A Basic Equations of Elasticity 769A.1 Stress 769A.2 Strain–Displacement Relations 769A.3 Rotations 771A.4 Stress–Strain Relations 772A.5 Equations of Motion in Terms of Stresses 774A.6 Equations of Motion in Terms of Displacements 774B Laplace and Fourier Transforms 777Index 783