Applications of differential equations in engineering and mechanics /
Saved in:
Main Author: | |
---|---|
Corporate Author: | |
Format: | Electronic eBook |
Language: | English |
Published: |
Boca Raton, FL :
CRC Press, Taylor & Francis Group,
[2019]
|
Subjects: | |
Online Access: | Connect to this title online (unlimited simultaneous users allowed; 325 uses per year) |
Table of Contents:
- Machine generated contents note: 1.1. Introduction
- 1.2. Beam Bending
- 1.2.1. Euler-Bernoulli Beam
- 1.2.2. Simply-Supported Beam
- 1.2.3. Cantilever Beam
- 1.2.4. Cable Load
- 1.2.5. Green's Function for Simply-Supported Beams
- 1.3. Beam Vibrations
- 1.3.1. Simply-Supported Beams
- 1.3.2. Orthogonality of the Eigenfunctions
- 1.3.3. Cantilever Beam with Suddenly Removed Point Force
- 1.3.4. Cantilever Beam with a Tip Lump Mass
- 1.3.5. Simply-Supported Beam Subject to an Impulse
- 1.3.6. Seismograph as Vibrations of Rigid Beam
- 1.4. Rocket/Missile Launch Pad as Beam
- 1.5. Beam on Elastic Foundation
- 1.5.1. Formulation
- 1.5.2. Boundary Conditions
- 1.5.3. Infinite Beam under Concentrated Load
- 1.6. Euler's Column Buckling
- 1.7. Vibrations of Beams under Axial Compression
- 1.7.1. Free Vibrations of Cantilever Beams under Axial Compression
- 1.7.2. Orthogonal Approximation
- 1.7.3. Rigorous Approach
- 1.8. Timoshenko Beam Theory
- 1.8.1. Variational Formulation
- 1.8.2. Static Solution for Timoshenko Beam
- 1.8.3. Free Vibrations of Timoshenko Beams
- 1.8.4. Free Vibrations of Simply-Supported Timoshenko Beams
- 1.8.5. Free Vibrations of Cantilever Timoshenko Beams
- 1.8.6. Free Vibrations of Fixed End Timoshenko Beams
- 1.9. Summary and Further Reading
- 1.10. Problems
- 2.1. Introduction
- 2.2. Kirchhoff Plate Theory
- 2.2.1. Equilibrium Equations
- 2.2.2. Forces and Moments
- 2.2.3. Governing Equations
- 2.2.4. Edge Conditions
- 2.3. Simply-Supported Plates
- 2.3.1. Navier's Solution
- 2.3.2. Levy's Solution
- 2.4. Clamped Rectangular Plates
- 2.4.1. Galerkin Method
- 2.4.2. Approximation for Clamped Plates
- 2.5. Deflection of Circular Plates
- 2.5.1. Clamped Plate with Uniform Load
- 2.5.2. Clamped Plate with Patch Load
- 2.5.3. Plates under Central Point Force
- 2.6. Buckling of Plates
- 2.7. Bending of Anisotropic Plates
- 2.8. Plate on Elastic Foundation
- 2.9. Plate Vibrations
- 2.9.1. Free Vibrations
- 2.9.2. Forced Vibrations
- 2.9.3. Approximation by Rayleigh Quotient
- 2.9.4. Strain Energy of Plates
- 2.9.5. Rayleigh-Ritz Method
- 2.10. Vibrations of Circular Plates
- 2.11. Hertz Problem of Circular Plate under Point Load
- 2.11.1. Series Solution
- 2.11.2. Variational Principle
- 2.11.3. Rayleigh-Ritz Method
- 2.11.4. General Solution in Kelvin Functions
- 2.11.5. Matching of Boundary Condition
- 2.11.6. Wyman's Solution
- 2.12. Summary and Further Reading
- 2.13. Problems
- 3.1. Introduction
- 3.2. Stresses, Forces, and Moments in Shells
- 3.3. Membrane Theory for Axisymmetric Shells
- 3.3.1. Dome under Concentrated Apex Load
- 3.3.2. Truncated Dome under Ring Load
- 3.3.3. Compatibility at Ring Foundation
- 3.4. Shell of Revolution under Uniform Load
- 3.4.1. Spherical Shell with Opening
- 3.4.2. Spherical Fluid Container
- 3.4.3. Conical Shells
- 3.5. Membrane Theory for Cylindrical Shells
- 3.5.1. Governing Equations
- 3.5.2. General Solutions for Axisymmetric Case
- 3.5.3. Simply-Supported Tube
- 3.5.4. Circular Tube under Dead Load
- 3.5.5. Membrane Theory versus Beam Theory
- 3.5.6. Pipe Subject to Edge Load
- 3.5.7. Simply-Supported Cylindrical Shell Roof
- 3.6. Bending Theory of Cylindrical Shells
- 3.6.1. Governing Equation for Axisymmetric Cylindrical Shells
- 3.6.2. Deformation Kinematics
- 3.6.3. Shell Bending Theory versus Beam on Elastic Foundation
- 3.6.4. General Solutions
- 3.7. Circular Pipe
- 3.7.1. Semi-Infinite Pipe Subject to End Force
- 3.7.2. Decay of Edge Disturbance
- 3.7.3. Infinite Pipes under Ring Load
- 3.7.4. Effective Length
- 3.8. Buckling of Cylindrical Shell under Axial Load
- 3.9. Bending Theory for Shell of Revolution
- 3.9.1. Force and Moment Equilibrium
- 3.9.2. Hooke's Law
- 3.9.3. Change of Curvature
- 3.9.4. Reissner Formulation
- 3.10. Spherical Shell of Constant Thickness
- 3.10.1. Solution in Terms of Hypergeometric Functions
- 3.10.2. Superposition for Various Boundary Conditions
- 3.11. Thin Spherical Shell
- 3.11.1. Geckeler-Staerman Approximation
- 3.11.2. Hetenyi Approximation
- 3.12. Symmetrical Bending of Thin Shallow Spherical Shell
- 3.12.1. Reissner Formulation
- 3.12.2. Governing Equations for Negligible Self-Weight
- 3.12.3. Solution in Kelvin Functions
- 3.13. Bending of Cylindrical Shell
- 3.13.1. Governing Equations
- 3.13.2. Vlasov's Stress Function
- 3.13.3. Cylindrical Roof Shells
- 3.13.4. Particular Solution
- 3.13.5. Homogeneous Solution
- 3.13.6. General Solution
- 3.13.7. Vertical Load on Shell Surface
- 3.14. Summary and Further Reading
- 3.15. Problems
- 4.1. Introduction
- 4.2. Static Deflection versus Natural Vibration
- 4.3. Single-Story Building
- 4.4. Damped and Undamped Responses
- 4.4.1. Undamped Responses
- 4.4.2. Damped-Free Responses
- 4.4.3. Damping Ratio by Hammer Test
- 4.4.4. Damped Forced Responses
- 4.5. Duhamel Integral for General Ground Motions
- 4.5.1. Formulation of Equation of Motion
- 4.5.2. Duhamel Integral
- 4.6. Response Spectrum
- 4.6.1. Pseudo-Response Spectrum
- 4.6.2. Nonlinear Response Spectrum
- 4.7. Multiple-Story Buildings
- 4.8. Modal Analysis
- 4.8.1. Free Vibrations
- 4.8.2. Decoupling of the Undamped Dynamic System
- 4.8.3. Decoupling of the Damped Dynamic System
- 4.8.4. Rayleigh Damping
- 4.8.5. Caughey and Liu-Gorman Proportional Damping
- 4.8.6. Rayleigh Quotient Technique
- 4.8.7. Response Spectrum Method for MDOF System
- 4.9. Summary and Further Reading
- 4.10. Problems
- 5.1. Introduction
- 5.2. Vibrations of Hanging Chains
- 5.3. Catenary
- 5.4. Inverted Catenary and Arch
- 5.5. Stone Arches
- 5.5.1. Formulation of Stone Arches
- 5.5.2. Inglis Solution
- 5.6. Cable Suspension Bridge
- 5.7. Cable-Stay Bridge
- 5.8. Vibrations of Cable Suspension Bridge
- 5.8.1. Governing Equations for Flexible Deck
- 5.8.2. Symmetric Modes
- 5.8.3. Anti-Symmetric Modes
- 5.8.4. Suspension Bridge with Stiffened Truss
- 5.9. Summary and Further Reading
- 5.10. Problems
- 6.1. Introduction
- 6.1.1. Column Buckling
- 6.1.2. Plate Buckling
- 6.1.3. Shell Buckling
- 6.2. Lagrangian or Green's Strain
- 6.3. Euler-Bernoulli Beam
- 6.3.1. Strain Energy Function
- 6.3.2. Hamiltonian Principle
- 6.3.3. Calculus of Variations
- 6.3.4. Applied Force versus Applied Displacement
- 6.4. Static Buckling Theory of Beam
- 6.5. Linear Dynamic Stability of Static States
- 6.5.1. Perturbation Method
- 6.5.2. Stability of Straight State
- 6.5.3. Stability of Buckled States
- 6.6. Nonlinear Dynamic Stability
- 6.6.1. Undamped Motions
- 6.6.2. Damped Motions
- 6.7. Multi-Time Perturbation and Stability
- 6.8. Governing Equations of Crooked Beams
- 6.8.1. Lagrangian Strain for Crooked Beams
- 6.8.2. Variational Principle for Crooked Beams
- 6.9. Snap-Through Buckling of Elastic Arches
- 6.9.1. Static Solution under Pressure
- 6.9.2. Linear Dynamic Stability
- 6.9.3. Transitions of Snap-Through Buckling
- 6.9.4. Linear Dynamic Stability for Unsymmetric State
- 6.10. Summary and Further Reading
- 6.11. Problems
- 7.1. Introduction
- 7.2. Error Function
- 7.2.1. Definition
- 7.2.2. Relation to Normal Distribution
- 7.2.3. Complementary Error Function
- 7.2.4. Some Results of Error Function
- 7.3. Diffusion of Pollutants in River
- 7.4. Ogata and Banks Solution
- 7.5. Solution for Decaying Pollutants
- 7.6. Dispersion of Decaying Substances
- 7.7. Taylor's Point Source Solution
- 7.7.1. Taylor's Approach
- 7.7.2. Taylor's Solution by Dimensional Analysis
- 7.8. Decaying Pollutant in Flowing Fluid
- 7.8.1. Point Source Solution
- 7.8.2. Continuous Source Solution
- 7.9. Diffusion in Higher Dimension
- 7.9.1. Two-Dimensional Point Source Solution
- 7.9.2. Three-Dimensional Point Source Solution
- 7.9.3. Two-Dimensional Line Source
- 7.10. Summary and Further Reading
- 7.11. Problems
- 8.1. Introduction
- 8.2. Coriolis Force Due to Rotation
- 8.2.1. Coriolis Force for High Altitude
- 8.2.2. Coriolis Force for All Altitudes
- 8.3. Hydrodynamic Equations for Geophysical Flows
- 8.3.1. Continuity Condition
- 8.3.2. Momentum Equations
- 8.3.3. Mass Conservation
- 8.3.4. Constitutive Law
- 8.3.5. Energy Equation
- 8.3.6. Equation of State
- 8.4. System of Equations for Geophysical Flows
- 8.4.1. Consideration of Scales
- 8.4.2. Governing Equations
- 8.4.3. Rossby, Ekman and Reynolds Numbers
- 8.5. Storm Surges
- 8.5.1. Storm Surges by Inverse Barometer Effect
- 8.5.2. Storm Surges with Moving Disturbance
- 8.5.3. Wind-Induced Storm Surges
- 8.5.4. Current Profile
- 8.6. Ekman Transport
- 8.6.1. Ekman Transport with No Internal Currents
- 8.6.2. Ekman Transport with Internal Currents
- 8.7. Geostrophic Flows
- 8.7.1. Taylor-Proudman Theorem
- 8.7.2. Homogeneous Geostrophic Flows
- 8.8. 2-D Shallow Water Equations
- 8.9. Vorticity and Tornado Dynamics
- 8.9.1. Helmholtz Vorticity Equation
- 8.9.2. Conservation of Angular Momentum
- 8.9.3. Vorticity in Tornadoes
- 8.9.4. Potential Vortex Model
- 8.9.5. Rankine Vortex Model
- 8.9.6. Burgers-Rott Vortex Model
- 8.9.7. Oseen-Lamb Vortex Model
- Contents note continued: 8.9.8. Sullivan Vortex Model
- 8.10. Summary and Further Reading
- 8.17. Problems
- 9.1. Introduction
- 9.2. Nonlinear Transport and Shocks
- 9.3. Dispersive Waves
- 9.4. Shock Waves in Traffic Flow
- 9.5. KdV Equation
- 9.5.1. Formulation of KdV
- 9.5.2. Scale Invariance
- 9.5.3. Physical Interpretation of KdV
- 9.5.4. Dispersion versus Nonlinearity
- 9.5.5. Soliton Solution
- 9.6. Hirota's Direct Method
- 9.6.1. Bilinear Form of KdV Equation
- 9.6.2. One-Soliton Solution
- 9.6.3. Two-Soliton Solution
- 9.6.4. N-Soliton Solution
- 9.6.5. Hirota's D-Operator
- 9.7. KdV Equation and Other Nonlinear Equations
- 9.7.1. KdV Equation and mKdV Equation
- 9.7.2. KdV Equation and Boussinesq Equation
- 9.7.3. KdV Equation and Nonlinear Schrodinger Equation
- 9.7.4. KdV Equation and First Painleve Equation
- 9.7.5. mKdV Equation and Second Painleve Equation
- 9.8. Conservation Laws of KdV
- 9.9. Nonlinear SchrOdinger Equation
- 9.9.1. mKdV Equation and NLSE
- 9.9.2. Bright Soliton
- 9.9.3. Dark Soliton
- 9.9.4. Rogue Waves in Oceans
- 9.10. Other Nonlinear Wave Equations
- 9.11. Summary and Further Reading
- 9.12. Problems
- 10.1. Introduction
- 10.2. Microscopic Maxwell Equations
- 10.2.1. Gauss Law for Electric Field
- 10.2.2. Gauss Law for Magnetism
- 10.2.3. Maxwell-Faraday Law
- 10.2.4. Ampere Circuital Law (with Maxwell Correction)
- 10.2.5. Dual Symmetry of Electromagnetic Waves in Vacuum Space
- 10.3. Integral versus Differential Forms
- 10.4. Macroscopic Maxwell Equations
- 10.5. Constitutive Relation and Ohm's Law
- 10.6. Electromagnetic Waves in Vacuum
- 10.7. Maxwell Equations in Gauss Unit
- 10.8. Boundary Conditions
- 10.9. Maxwell's Vector and Scalar Potentials
- 10.10. Gauge Freedom
- 10.10.1. Coulomb Gauge
- 10.10.2. Lorenz Gauge
- 10.10.3. Aharonov-Bohm Effect (Physical Meaning of Wave Potentials)
- 10.11. Solutions of Maxwell Equations: Jefimenko's Equations
- 10.11.1. Gradient Identity of Jefimenko
- 10.11.2. Curl Identity of Jefimenko
- 10.12. Electromagnetic Waves in Materials
- 10.13. Mathematical Theory for Lorenz Gauge
- 10.13.1. Hertz Vector for Electric Field
- 10.13.2. Gauge Invariance of Hertz Vector
- 10.13.3. Hertz Vector for Magnetic Polarization
- 10.13.4. Debye Potential Function for Transverse Magnetic Waves
- 10.13.5. Debye Potential Function for Transverse Electric Waves
- 10.14. Duality and Symmetry
- 10.15. Mathematical Theory for Coulomb Gauge
- 10.15.1. Scalar and Vector Potentials
- 10.15.2. Transverse Waves or Radiation Gauge
- 10.15.3. General Solution for Poisson Equation
- 10.15.4. Single-and Double-Layer Potentials
- 10.16. Kirchhoff Integral Formula for Waves
- 10.17. Summary and Further Reading
- 10.18. Problems
- 11.1. Introduction
- 11.2. Black Body Radiation and Quantized Energy
- 11.3. Schrodinger Equation
- 11.3.1. One-Dimensional Schrodinger Equation
- 11.3.2. Three-Dimensional Schrodinger Equation
- 11.3.3. Wave Functions of Particles
- 11.3.4. Expectation Values
- 11.3.5. Stationary State of Energy E
- 11.4. Operators and Expectation Values
- 11.5. Classical Mechanics versus Quantum Mechanics
- 11.6. Hydrogen-Like Atom Model
- 11.6.1. Schrodinger Equation in Polar Form
- 11.6.2. Separation of Variables
- 11.6.3. Constraints Imposed by Wavefunctions
- 11.6.4. Laguerre and Associated Laguerre Polynomials
- 11.6.5. Orthogonality of Associated Laguerre Polynomials
- 11.6.6. Admissible Form of the Wavefunctions
- 11.7. Electron Spins
- 11.8. Schrodinger Equation for General Atoms
- 11.9. Radiative Transitions from Atoms
- 11.10. Summary and Further Reading
- 11.11. Problems
- 12.1. Introduction
- 12.2. Equation of Motion for a Rigid Mass
- 12.3. Mass under Gravitational Pull
- 12.4. Orbital Equations for an Artificial Satellite
- 12.5. Orbital Equations in Polar Form
- 12.6. Kepler's 1st Law
- 12.7. First Escape Velocity (Orbital Speed)
- 12.8. Second Escape Velocity (from Earth)
- 12.9. Third Escape Velocity (from Our Solar System)
- 12.10. Travel to the Moon
- 12.11. Kepler's Second Law
- 12.12. Kepler's Third Law (Newton's Law)
- 12.13. Energy in an Elliptic Orbit
- 12.14. Interplanetary Travel
- 12.14.1. Hohmann Transfer Orbit
- 12.14.2. Launching Time Window
- 12.15. Striking Speed of Meteors on Earth
- 12.16. Precession of the Perihelion of Mercury
- 12.16.1. Schwarzschild Metric for Curved Space-Time
- 12.16.2. Energy Term Due to Relativity
- 12.16.3. Contribution to Perihelion Precession
- 12.17. Motion near the Earth's Surface
- 12.18. Rocket and Missile Problem
- 12.19. Dynamic of Atmospheric Re-Entry
- 12.19.1. Formulation
- 12.19.2. Yaroshevsky Solution
- 12.20. Restricted Problem of Three Bodies
- 12.20.1. Formulation of the Three-Body Problem
- 12.20.2. Triangular Lagrangian Points
- 12.20.3. Three Collinear Lagrangian Points
- 12.20.4. Approximate Solution to Lagrange's Quintic Equation
- 12.21. Summary and Further Reading
- 12.22. Problems
- 13.1. Introduction
- 13.2. Papkovitch-Neuber Potentials for Axisymmetric Elasticity
- 13.3. Mixed Boundary Value Problems as Potential Problems
- 13.4. Formulation of Dual Integral Equations
- 13.5. Penny-Shaped Crack Problem
- 13.5.1. Reduction of Dual Integral Equations to Abel Integral
- 13.5.2. Displacement Field Due to Uniform Pressure
- 13.5.3. Energy Change Due to Crack Presence
- 13.6. Papkovitch-Neuber Potentials for Plane Elasticity
- 13.7. Formulation of Dual Integral Equations
- 13.8. Griffith Crack Problem
- 13.8.1. Reduction of Dual Integral Equations to Abel Integral
- 13.8.2. Solutions
- 13.9. Fracture Dynamics in Wave Equations
- 13.10. Reduction of Wave to Harmonic Problem by Galilean Transform
- 13.11. Mode I Asymptotic Field at Moving Crack Tip
- 13.11.1. Eigenvalue Problem
- 13.11.2. Asymptotic Fields
- 13.12. Mode II Asymptotic Field at Moving Crack Tip
- 13.12.1. Eigenvalue Problem
- 13.12.2. Asymptotic Fields
- 13.13. Mode III Asymptotic Field at Moving Crack Tip
- 13.13.1. Eigenvalue Problem
- 13.13.2. Asymptotic Fields
- 13.14. Asymptotic Field of Transient Crack Growth
- 13.15. Crack Growth with Intersonic Speed
- 13.15.1. Formulation
- 13.15.2. Mode I
- 13.15.3. Mode II
- 13.15.4. Asymptotic Field for Mode II Crack
- 13.16. Summary and Further Reading
- 13.17. Problems
- References.