Applications of differential equations in engineering and mechanics /

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Bibliographic Details
Main Author:	Chau, K. T. (Author)
Corporate Author:	ProQuest (Firm)
Format:	Electronic eBook
Language:	English
Published:	Boca Raton, FL : CRC Press, Taylor & Francis Group, [2019]
Subjects:	Engineering mathematics. Differential equations.
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.

Applications of differential equations in engineering and mechanics /

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