The Resource Mathematical Models for Suspension Bridges : Nonlinear Structural Instability
Mathematical Models for Suspension Bridges : Nonlinear Structural Instability
 Summary
 This work provides a detailed and uptotheminute survey of the various stability problems that can affect suspension bridges. In order to deduce some experimental data and rules on the behavior of suspension bridges, a number of historical events are first described, in the course of which several questions concerning their stability naturally arise. The book then surveys conventional mathematical models for suspension bridges and suggests new nonlinear alternatives, which can potentially supply answers to some stability questions. New explanations are also provided, based on the nonlinear structural behavior of bridges. All the models and responses presented in the book employ the theory of differential equations and dynamical systems in the broader sense, demonstrating that methods from nonlinear analysis can allow us to determine the thresholds of instability
 Language
 eng
 Extent
 1 online resource (274 pages)
 Contents

 "Preface"  "Acknowledgements"  "About this book"  "Brief History of Suspension Bridges"  "OneDimensional Models"  "A FishBone Beam Model"  "Models with Interacting Oscillators"  "Plate Models"  "Conclusions"  "Contents"  "Notations"  "1 Brief History of Suspension Bridges"  "1.1 First Suspension Bridges"  "1.2 Collapses Due to an External Resonance"  "1.3 Collapses Due to Unexpected Oscillations"  "1.4 The Tacoma Narrows Bridge Collapse"  "1.5 Some Bridges That Did Not Collapse"  "1.6 Some Doubts and Questions"  "1.7 Partial Explanations of the Tacoma Narrows Bridge Collapse"  "1.7.1 Structural Failure"  "1.7.2 External Resonance"  "1.7.3 Vortices"  "1.7.4 Flutter"  "1.7.5 Parametric Resonance"  "1.7.6 Partial Conclusions: Aerodynamic Effects"  "1.8 Nonlinear Behavior of Suspension Bridges"  "1.9 Bibliographical Notes"  "2 One Dimensional Models "  "2.1 From Navier to Melan"  "2.2 Linear and Quasilinear Beam Equations"  "2.3 Deflection of Cables Under Vertical Loads"  "2.4 Suspension Bridges Modeled by Beams and Cables"  "2.5 The Melan Equation"  "2.5.1 How to Compute the Additional Tension of the Cables"  "2.5.1.1 First Approximation"  "2.5.1.2 Second Approximation"  "2.5.1.3 Third Approximation"  "2.5.1.4 Parabolic Shape"  "2.5.1.5 Simplest Symmetric Beam Shape"  "2.5.1.6 Asymmetric Beams"  "2.5.2 Existence and Uniqueness Results"  "2.5.3 Numerical Implementations with a Stable Fixed Point"  "2.5.4 Numerics with an Unstable Fixed Pointfor an Actual Bridge"  "2.6 Selfexcited Oscillations in Semilinear Beam Equations"  "2.6.1 A Model with Superlinear Springs"  "2.6.2 Unbounded Beams and Selfexcited Oscillations"  "2.6.3 Hinged Beams Subject to Nonlinear Elastic Forces"  "2.7 The Birth of Aerodynamics"  "2.7.1 From Melan Until the Wake of Tacoma"
 "2.7.2 More Recent Models and the Sin of Mathematics"  "2.8 McKenna and the Awakening of Nonlinearity"  "2.8.1 Beam Suspended by Possibly Slackening Hangers"  "2.8.2 A CableBeam System with Possibly Slackening Hangers"  "2.8.3 Stretching Energy in a Compressed Beam"  "2.9 Bibliographical Notes"  "3 A FishBone Beam Model"  "3.1 A Beam Showing Torsional Oscillations"  "3.2 Parametric Resonance in a Linearised Model"  "3.3 A Nonlinear Version"  "3.3.1 Well Posedness"  "3.3.2 Dropping the Trigonometric Functions"  "3.3.3 Choosing the Nonlinearity"  "3.4 Finite Dimensional Torsional Stability"  "3.4.1 Why Can We Neglect High Torsional Modes?"  "3.4.2 Stability of the Low Modes"  "3.4.3 The Approximated 1Mode System"  "3.4.4 The Approximated 2Modes System"  "3.5 The Flutter Energy"  "3.6 Which Residual Mode Captures the Energy of the Dominant Mode?"  "3.6.1 Stability for Low Energy"  "3.6.2 Numerical Computation of the Flutter Energy"  "3.6.3 More General Nonlinearities"  "3.6.4 Mechanical Interpretation and Structural Remedies"  "3.7 The Role of Aerodynamic Forces"  "3.7.1 Numerical Results"  "3.7.2 The Pattern Creating Oscillations in Suspension Bridges"  "3.8 Brief History of the Hill and the Mathieu Equations"  "3.9 Bibliographical Notes"  "4 Models with Interacting Oscillators"  "4.1 Coupled Oscillators Modeling the Cross Section of a Bridge"  "4.2 Energy Transfer and PoincarÃ© Maps"  "4.3 A Link Between the PoincarÃ© Maps and the Hill Equations"  "4.4 Interactions Between Multiple Cross Sections"  "4.5 Computation of the Flutter Energy"  "4.6 Damped and Forced Systems"  "4.7 How to Construct the PoincarÃ© Maps"  "4.8 Bibliographical Notes"  "5 Plate Models"  "5.1 Linear or Nonlinear Models?"  "5.2 The Elastic Bending Energy of a Plate"
 "5.2.1 The Plate as a Model for Suspension Bridges"  "5.2.2 A Linear Model for Small Deformations"  "5.2.3 A Nonlinear Model for Large Deformations"  "5.3 The Linear Equation with No Stretching Term"  "5.3.1 Variational Setting: Existence and Uniqueness"  "5.3.2 Vertical and Torsional Modes"  "5.3.3 Quantitative Analysis of the Oscillating Modes"  "5.4 The Action of Cables and Hangers: Semilinear Equations"  "5.5 Torsional Instability and Flutter Energy"  "5.5.1 Finite Dimensional Approximation of the Solution"  "5.5.2 A Theoretical Characterisation of Torsional Stability"  "5.5.3 Sufficient Conditions for the Torsional Stability"  "5.5.4 Numerical Computation of the Flutter Energy"  "5.6 The Quasilinear von KÃ¡rmÃ¡n Plate System"  "5.6.1 The Equations and the Boundary Conditions"  "5.6.2 Uniqueness and Multiplicity of the Equilibrium Positions"  "5.7 Alternative Plate Models with Stretching Energy"  "5.7.1 Should the Stretching Energy Be Included in the Model?"  "5.7.2 The Equation with a Linearised Stretching Term"  "5.7.3 The Surface Increment Quasilinear Equation"  "5.7.4 A Nonlocal Quasilinear Equation"  "5.8 Bibliographical Notes"  "6 Conclusions"  "6.1 Flutter Energy in Nonlinear Models"  "6.2 Answers to the Main Questions"  "References"  "Index"  "Author Index"  "Bridges Index"
 Isbn
 9783319154343
 Label
 Mathematical Models for Suspension Bridges : Nonlinear Structural Instability
 Title
 Mathematical Models for Suspension Bridges
 Title remainder
 Nonlinear Structural Instability
 Language
 eng
 Summary
 This work provides a detailed and uptotheminute survey of the various stability problems that can affect suspension bridges. In order to deduce some experimental data and rules on the behavior of suspension bridges, a number of historical events are first described, in the course of which several questions concerning their stability naturally arise. The book then surveys conventional mathematical models for suspension bridges and suggests new nonlinear alternatives, which can potentially supply answers to some stability questions. New explanations are also provided, based on the nonlinear structural behavior of bridges. All the models and responses presented in the book employ the theory of differential equations and dynamical systems in the broader sense, demonstrating that methods from nonlinear analysis can allow us to determine the thresholds of instability
 Cataloging source
 MiAaPQ
 Literary form
 non fiction
 Nature of contents
 dictionaries
 Series statement
 MS&A
 Series volume
 v.15
 Label
 Mathematical Models for Suspension Bridges : Nonlinear Structural Instability
 Link
 http://libproxy.rpi.edu/login?url=https://ebookcentral.proquest.com/lib/rpi/detail.action?docID=2120602
 Carrier category
 online resource
 Carrier category code
 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code
 txt
 Content type MARC source
 rdacontent
 Contents

 "Preface"  "Acknowledgements"  "About this book"  "Brief History of Suspension Bridges"  "OneDimensional Models"  "A FishBone Beam Model"  "Models with Interacting Oscillators"  "Plate Models"  "Conclusions"  "Contents"  "Notations"  "1 Brief History of Suspension Bridges"  "1.1 First Suspension Bridges"  "1.2 Collapses Due to an External Resonance"  "1.3 Collapses Due to Unexpected Oscillations"  "1.4 The Tacoma Narrows Bridge Collapse"  "1.5 Some Bridges That Did Not Collapse"  "1.6 Some Doubts and Questions"  "1.7 Partial Explanations of the Tacoma Narrows Bridge Collapse"  "1.7.1 Structural Failure"  "1.7.2 External Resonance"  "1.7.3 Vortices"  "1.7.4 Flutter"  "1.7.5 Parametric Resonance"  "1.7.6 Partial Conclusions: Aerodynamic Effects"  "1.8 Nonlinear Behavior of Suspension Bridges"  "1.9 Bibliographical Notes"  "2 One Dimensional Models "  "2.1 From Navier to Melan"  "2.2 Linear and Quasilinear Beam Equations"  "2.3 Deflection of Cables Under Vertical Loads"  "2.4 Suspension Bridges Modeled by Beams and Cables"  "2.5 The Melan Equation"  "2.5.1 How to Compute the Additional Tension of the Cables"  "2.5.1.1 First Approximation"  "2.5.1.2 Second Approximation"  "2.5.1.3 Third Approximation"  "2.5.1.4 Parabolic Shape"  "2.5.1.5 Simplest Symmetric Beam Shape"  "2.5.1.6 Asymmetric Beams"  "2.5.2 Existence and Uniqueness Results"  "2.5.3 Numerical Implementations with a Stable Fixed Point"  "2.5.4 Numerics with an Unstable Fixed Pointfor an Actual Bridge"  "2.6 Selfexcited Oscillations in Semilinear Beam Equations"  "2.6.1 A Model with Superlinear Springs"  "2.6.2 Unbounded Beams and Selfexcited Oscillations"  "2.6.3 Hinged Beams Subject to Nonlinear Elastic Forces"  "2.7 The Birth of Aerodynamics"  "2.7.1 From Melan Until the Wake of Tacoma"
 "2.7.2 More Recent Models and the Sin of Mathematics"  "2.8 McKenna and the Awakening of Nonlinearity"  "2.8.1 Beam Suspended by Possibly Slackening Hangers"  "2.8.2 A CableBeam System with Possibly Slackening Hangers"  "2.8.3 Stretching Energy in a Compressed Beam"  "2.9 Bibliographical Notes"  "3 A FishBone Beam Model"  "3.1 A Beam Showing Torsional Oscillations"  "3.2 Parametric Resonance in a Linearised Model"  "3.3 A Nonlinear Version"  "3.3.1 Well Posedness"  "3.3.2 Dropping the Trigonometric Functions"  "3.3.3 Choosing the Nonlinearity"  "3.4 Finite Dimensional Torsional Stability"  "3.4.1 Why Can We Neglect High Torsional Modes?"  "3.4.2 Stability of the Low Modes"  "3.4.3 The Approximated 1Mode System"  "3.4.4 The Approximated 2Modes System"  "3.5 The Flutter Energy"  "3.6 Which Residual Mode Captures the Energy of the Dominant Mode?"  "3.6.1 Stability for Low Energy"  "3.6.2 Numerical Computation of the Flutter Energy"  "3.6.3 More General Nonlinearities"  "3.6.4 Mechanical Interpretation and Structural Remedies"  "3.7 The Role of Aerodynamic Forces"  "3.7.1 Numerical Results"  "3.7.2 The Pattern Creating Oscillations in Suspension Bridges"  "3.8 Brief History of the Hill and the Mathieu Equations"  "3.9 Bibliographical Notes"  "4 Models with Interacting Oscillators"  "4.1 Coupled Oscillators Modeling the Cross Section of a Bridge"  "4.2 Energy Transfer and PoincarÃ© Maps"  "4.3 A Link Between the PoincarÃ© Maps and the Hill Equations"  "4.4 Interactions Between Multiple Cross Sections"  "4.5 Computation of the Flutter Energy"  "4.6 Damped and Forced Systems"  "4.7 How to Construct the PoincarÃ© Maps"  "4.8 Bibliographical Notes"  "5 Plate Models"  "5.1 Linear or Nonlinear Models?"  "5.2 The Elastic Bending Energy of a Plate"
 "5.2.1 The Plate as a Model for Suspension Bridges"  "5.2.2 A Linear Model for Small Deformations"  "5.2.3 A Nonlinear Model for Large Deformations"  "5.3 The Linear Equation with No Stretching Term"  "5.3.1 Variational Setting: Existence and Uniqueness"  "5.3.2 Vertical and Torsional Modes"  "5.3.3 Quantitative Analysis of the Oscillating Modes"  "5.4 The Action of Cables and Hangers: Semilinear Equations"  "5.5 Torsional Instability and Flutter Energy"  "5.5.1 Finite Dimensional Approximation of the Solution"  "5.5.2 A Theoretical Characterisation of Torsional Stability"  "5.5.3 Sufficient Conditions for the Torsional Stability"  "5.5.4 Numerical Computation of the Flutter Energy"  "5.6 The Quasilinear von KÃ¡rmÃ¡n Plate System"  "5.6.1 The Equations and the Boundary Conditions"  "5.6.2 Uniqueness and Multiplicity of the Equilibrium Positions"  "5.7 Alternative Plate Models with Stretching Energy"  "5.7.1 Should the Stretching Energy Be Included in the Model?"  "5.7.2 The Equation with a Linearised Stretching Term"  "5.7.3 The Surface Increment Quasilinear Equation"  "5.7.4 A Nonlocal Quasilinear Equation"  "5.8 Bibliographical Notes"  "6 Conclusions"  "6.1 Flutter Energy in Nonlinear Models"  "6.2 Answers to the Main Questions"  "References"  "Index"  "Author Index"  "Bridges Index"
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