Vibrations of Elastic Systems

1 Introduction.- 1.1 A Brief Historical Perspective.- 1.2 Importance of Vibrations.- 1.3 Analysis of Vibrating Systems.- 1.4 About the Book.- 2 Spring-Mass Systems.- 2.1 Introduction.- 2.2 Some Preliminaries.- 2.2.1 A Brief Review of Single Degree-of-Freedom Systems.- 2.2.2 General Solution: Harmonically Varying Forcing.- 2.2.3 Power Dissipated by a Viscous Damper.- 2.2.4 Structural Damping.- 2.3 Squeeze Film Air Damping.- 2.3.1 Introduction.- 2.3.2 Rectangular Plates.- 2.3.3 Circular Plates.- 2.3.4 Base Excitation with Squeeze Film Damping.- 2.3.5 Time-Varying Force Excitation of the Mass.- 2.4 Viscous Fluid Damping.- 2.4.1 Introduction.- 2.4.2 Single Degree-of-Freedom System in a Viscous Fluid.- 2.5 Electrostatic and van der Waals Attraction.- 2.5.1 Introduction.- 2.5.2 Single Degree-of-Freedom with Electrostatic Attraction.- 2.5.3 van der Waals Attraction and Atomic Force Microscopy.- 2.6 Energy Harvesters.- 2.6.1 Introduction.- 2.6.2 Piezoelectric Generator.- 2.6.3 Maximum Average Power of a Piezoelectric Generator.- 2.6.4 Permanent Magnet Generator.- 2.6.5 Maximum Average Power of a Permanent Magnet Generator.- 2.7 Two Degree-of-Freedom Systems.- 2.7.1 Introduction.- 2.7.2 Harmonic Excitation: Natural Frequencies and Frequency Response Functions.- 2.7.3 Enhanced Energy Harvester.- 2.7.4 MEMS Filters.- 2.7.5 Time-Domain Response.- 2.7.6 Design of an Atomic Force Microscope MotionScanner.- Appendix 2.1 Forces on a Submerged Vibrating Cylinder.- 3 Thin Beams: Part I .- 3.1 Introduction.- 3.2 Derivation of Governing Equation and Boundary Conditions.- 3.2.1 Contributions to the Total Energy.- 3.2.2 Governing Equation.- 3.2.3 Boundary Conditions.- 3.2.4 Non Dimensional Form of the Governing Equation and Boundary Conditions.- 3.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section and with Attachments.- 3.3.1 Introduction.- 3.3.2 Solution for Very General Boundary Conditions .- 3.3.3 General Solution in the Absence of an Axial Force and an Elastic Foundation.- 3.3.4 Numerical Results.- 3.3.5 Cantilever Beam as a Biosensor.- 3.4 Single Degree-of-Freedom Approximation of Beams with a Concentrated Mass.- 3.5 Beams with In-Span Spring-Mass Systems .- 3.5.1 Single Degree-of-Freedom System.- 3.5.2 Two Degree-of-Freedom System with Translation and Rotation.- 3.6 Effects of an Axial Force and an Elastic Foundation on the Natural Frequency.- 3.7 Beams with a Rigid Extended Mass.- 3.7.1 Introduction.- 3.7.2 Cantilever Beam with a Rigid Extended Mass.- 3.7.3 Beam with an In-span Rigid Extended Mass.- 3.8 Beams with Variable Cross Section.- 3.8.1 Introduction.- 3.8.2 Continuously Changing Cross Section.- 3.8.3 Linear Taper.- 3.8.4 Exponential Taper.- 3.8.5 Approximate Solution to Tapered Beams: Rayleigh-Ritz Method.- 3.8.6 Triangular Taper: Application to Atomic Force Microscopy.- 3.8.7 Constant Cross Section with aStep Change in Properties.- 3.8.8 Stepped Beam with an In-Span Rigid Support.- 3.9 Elastically Connected Beams.- 3.9.1 Introduction.- 3.9.2 Beams Connected by a Continuous Elastic Spring.- 3.9.3 Beams with Concentrated Masses Connected by an Elastic Spring.- 3.10 Forced Excitation.- 3.10.1 Boundary Conditions and the Generation of Orthogonal Functions.- 3.10.2 General Solution.- 3.10.3 Impulse Response.- 3.10.4 Time-Dependent Boundary Excitation.- 3.10.5 Forced Harmonic Oscillations.- 3.10.6 Harmonic Boundary Excitation.- 4 Thin Beams: Part II .- 4.1 Introduction.- 4.2 Damping.- 4.2.1 Generation of Governing Equation.- 4.2.2 General Solution.- 4.2.3 Illustration of the Effects of Various Types of Damping: Cantilever Beam.- 4.3 In-plane Forces and Electrostatic Attraction.- 4.3.1 Introduction.- 4.3.2 Beam Subjected to a Constant Axial Force.- 4.3.3 Beam Subject to In-plane Forces and Electrostatic Attraction.- 4.4 Piezoelectric Energy Harvesters.- 4.4.1 Governing Equations and Boundary Conditions.- 4.4.2 Power from the Harmonic Oscillations of a Base-Excited Cantilever Beam.- Appendix 4.1 Hydrodynamic Correction Function.- 5 Timoshenko Beams.- 5.1 Introduction.- 5.2 Derivation of the Governing Equations and Boundary Conditions.- 5.2.1 Introduction.- 5.2.2 Contributions to the Total Energy.- 5.2.3 Governing Equations.- 5.2.4 Boundary Conditions.- 5.2.5 Non Dimensional Form of the Governing Equationsand Boundary Conditions.- 5.2.6 Reduction of Timoshenko Equations to That of Euler-Bernoulli.- 5.3 Natural Frequencies and Mode Shapes of Beams with Constant Cross Section, Elastic Foundation, Axial Force and In-span Attachments.- 5.3.1 Introduction.- 5.3.2 Solution for Very General Boundary Conditions.- 5.3.3 Special Cases.- 5.3.4 Numerical Results.- 5.4 Natural Frequencies of Beams with Variable Cross Section.- 5.4.1 Beams with a Continuous Taper: Rayleigh-Ritz Method.- 5.4.2 Constant Cross Section with a Step Change in Properties.- 5.4.3 Numerical Results.- 5.5 Beams Connected by a Continuous Elastic Spring.- 5.6 Forced Excitation.- 5.6.1 Boundary Conditions and the Generation of Orthogonal Functions.- 5.6.2 General Solution.- 5.6.3 Impulse Response.- Appendix 5.1 Definitions of the Solution Functions fl and gl and Their Derivatives .- Appendix 5.2 Definitions of the Solution Functions fli and gli and Their Derivatives.- 6 Thin Plates.- 6.1 Introduction.- 6.2 Derivation of Governing Equation and Boundary Conditions: Rectangular Plates.- 6.2.1 Introduction.- 6.2.2 Contributions to the Total Energy.- 6.2.3 Governing Equation.- 6.2.4 Boundary Conditions.- 6.2.5 Non Dimensional Form of the Governing Equation and Boundary Conditions.- 6.3 Governing Equations and Boundary Conditions: Circular Plates.- 6.4 Natural Frequencies and Mode Shapes of Circular Plates for Very General Boundary Conditions.- 6.4.1 Introduction.- 6.4.2 Natural Frequencies and Mode Shapes of Annular and Solid Circular Plates.- 6.4.3Numerical Results.- 6.5 Natural Frequencies and Mode Shapes of Rectangular and Square Plates: Rayleigh-Ritz Method.- 6.5.1 Introduction.- 6.5.2 Natural Frequencies and Mode Shapes of Rectangular and Square Plates.- 6.5.3 Numerical Results.- 6.5.4 Comparison with Thin Beams.- 6.6 Forced Excitation of Circular Plates.- 6.6.1 General Solution to the Forced Excitation of Circular Plates.- 6.6.2 Impulse Response of a Solid Circular Plate.- 6.7 Circular Plate with Concentrated Mass Revisited.- 6.8 Extensional Vibrations of Plates.- 6.8.1 Introduction.- 6.8.2 Contributions to the Total Energy.- 6.8.3 Governing Equations and Boundary Conditions.- 6.8.4 Natural Frequencies and Mode Shapes of a Circular Plate.- 6.8.5 Numerical Results.- Appendix 6.1 Elements of Matrices in Eq. (6.100).- 7 Cylindrical Shells and Carbon Nanotube Approximations .- 7.1 Introduction.- 7.2 Derivation of Governing Equations and Boundary Conditions: Flügge's Theory.- 7.2.1 Introduction.- 7.2.2 Contributions to the Total Energy.- 7.2.3 Governing Equations.- 7.2.4 Boundary Conditions.- 7.2.5 Boundary Conditions and the Generation of Orthogonal Functions.- 7.3 Derivation of Governing Equations and Boundary Conditions: Donnell's Theory.- 7.3.1 Introduction.- 7.3.2 Contributions to the Total Energy.- 7.3.3 Governing Equations 7.3.4 Boundary Conditions.- 7.4 Natural Frequencies of Clamped and Cantilever Shells: Single-Wall Carbon Nanotube Approximations.- 7.4.1 Rayleigh-Ritz Solution.- 7.4.2 Numerical Results.- 7.5Natural Frequencies of Hinged Shells: Double-Wall Carbon Nanotube Approximation.- Appendix A Strain Energy in Linear Elastic Bodies .- Appendix B Variational Calculus: Generation of Governing Equations, Boundary Conditions, and Orthogonal Functions.- B.1 Variational Calculus.- B.1.1 System with One Dependent Variable.- B.1.2 A Special Case for Systems with One Dependent Variable.- B.1.3 Systems with N Dependent Variables.- B.1.4 A Special Case for Systems with N Dependent Variables.- B.2 Orthogonal Functions.- B.2.1 Systems with One Dependent Variable.- B.2.2 Systems with N Dependent Variables.- B.3 Application of Results to Specific Elastic Systems.- Appendix C Laplace Transforms and the Solutions to Ordinary Differential Equations.- C.1 Definition of the Laplace Transform .- C.2 Solution to Second-Order Equation.- C.3 Solution to Fourth-Order Equation.- C.4 Table of Laplace Transform Pairs.