Introduction to Aeroelasticity: With Case-Studies

Preface
Acknowledgements
General Information
Author’s Claim
Structured Approach for Learning and Solving Problems
A. Problem-Solving Paradigms
To Solve any Problem
“Forget Me Not’s” Paradigm
To Facilitate Obtaining Solutions
B. Problem-Solving Schemes
Scheme 1
Scheme 2
Contents
About the Author
Part I Foundation of Aeroelasticity
1 Introduction and Overview
1.1 Basic Introduction
1.1.1 Physical Answer Related to Observations
1.2 Historical Background of Aeroelasticity [1–4]
1.3 Scientific Aspects in Aeroelasticity
1.4 Definition of Aeroelasticity
1.5 Aeroelastic Problems in Engineering
1.6 Extended Concept: Hydro-elasticity, Aeroservoelasticity and Envaeroelastomechanics
1.7 Extended Concept: Envaeroelastomechanics —Enviromental Forces Schematics
1.8 Trend of Modern Aircrafts Development
1.9 Examples of Aeroelastic Problems and Their System (Block Diagram) Representation
1.9.1 Two Major Categories of Aeroelastic Problems
1.9.2 Adverse Interactions: Flutter and Divergence
1.9.3 Basics of Aeroelasticity and Flutter Analysis
1.10 Some Illustrative Examples in Figures
1.11 Influence of Aeroelastic Phenomena on Aircraft Design
1.11.1 Dynamic Loads Problem
1.12 Some Modern Examples of Aeroelastic Testing and Experimental Studies in Aeroelasticity
1.13 Concluding Remarks
Appendices
Appendix 1: Studies in Aeroelasticity—For Creative and Proactive Class Discussions
Appendix 2: Problems and Issues for Creative and Proactive Class Discussions in Aeroelasticity
References
2 Fundamental Concepts from Theory of Elasticity
2.1 Equilibrium and Compatibility Equations for Elastically Deformable Structures
2.1.1 Equilibrium Equation
2.1.2 Equilibrium Equation and Internal Stresses
2.1.3 Compatibility Equation and Equation of State
2.1.4 Conditions for Solving the Equations
2.2 Thermodynamic Behavior of Elastic (Deformable Bodies) Under Dynamic and Thermal Loading
2.3 Concepts from Strength of Materials
2.4 Fundamentals of Elasticity
2.5 Elastic Properties of Structures
2.6 Strain Energy in Terms of Influence Coefficients
2.7 Deformation Under Distributed Forces and Influence Functions
2.8 Properties of Influence Functions
2.9 The Simplified Elastic Airplane
2.10 Deformations of Airplane Wings
2.11 Energy Methods in Deflection Calculations
2.11.1 Deflections Determination Using the Principle of Minimum Potential Energy
2.11.2 The Principle of Minimum Potential Energy Applied to Continuous Systems; Rayleigh–Ritz Method
2.11.3 Deflection by Castigliano’s Theorem
2.12 Deformations of Slender Unswept Wings
2.12.1 Bending and Shearing Deformation
2.12.2 Influence Functions and Coefficients
2.12.3 Torsional Deformation and Influence Function
2.12.4 Elastic Axis
2.13 Example
2.14 Case Study: Vibration Analysis of a Cantilevered Beam with Spring Loading at the Tip as a Generic Elastic Structure
2.14.1 Introduction
2.14.2 Detailed Vibration Analysis of Beam with Hinged Spring
2.14.3 Numerical Method-Finite Element Approach
2.14.4 Results and Discussions
2.14.5 Conclusions
2.15 Concluding Remarks
Appendix
References
3 Conservation Principles in Fluid Mechanics and Potential Flow Aerodynamics
3.1 Potential Flow Fluid Dynamics; Conservation Principles
3.1.1 General Assumptions
3.1.2 Thermodynamic Laws
3.1.3 Thermodynamic Properties
3.1.4 Conservation Principle
3.2 Elaboration of Conservation and Compatibility Principle
3.2.1 Continuity Equation
3.2.2 Gauss Theorem
3.3 Dynamics of Fluid Flow
3.3.1 Equation of Motion in Mechanics and Euler Equation in Fluid Dynamics
3.3.2 The Derivation of the Equation of Motion Using Differential Approach
3.3.3 The Derivation of the Equation of Motion Using Integral Approach
3.3.4 Other Forms of the Euler Equation for Fluid Motion
3.3.5 Law of Conservation of Thermodynamic Energy (For Adiabatic Fluid)
3.3.6 The Equation of Motion in a Non-inertial Coordinate System
3.3.7 Bernoulli’s Equation
3.4 Momentum and Moment of Momentum Equations
3.4.1 Momentum Theorem
3.4.2 Angular Momentum or Moment of Momentum
3.5 Energy Equations
3.5.1 The Derivation of Energy Equation Using Differential Method [1]
3.5.2 The Derivation of Energy Equation in Integral Form
3.6 Some Geometric and Kinematic Properties of the Velocity Field
3.6.1 Gauss’ Divergence Theorem
3.6.2 Stokes’ Theorem on Rotation
3.7 Vortex Theorems for the Ideal Fluid
3.7.1 First Vortex Theorem
3.7.2 Second Vortex Theorem
3.7.3 Third Vortex Theorem
3.8 Irrotational Flow and Velocity Potential
3.8.1 The Bernoulli Equation for Irrotational Flow (Kelvin’s Equation)
3.8.2 The Partial Differential Equation for Φ
3.9 Problem Examples
3.9.1 Example 1: Transient Flow During Valve Opening
3.9.2 Example 2: Oscillating Fluid in a U-tube with Unequal Initial Column Height
3.9.3 Example 3: Fluid Flow Through Converging Pipe
3.9.4 Example 4: Water Flow from an Orifice
Appendix: One-Dimensional Fluid Dynamics
Relation between Velocity and Cross-Sectional Area in One-Dimensional Gas Dynamics [1, 10]
The Continuity Equation
Energy Equation Considerations
Flow at Constant Area
References
4 Concepts of Typical Section
4.1 Typical Section—General
4.2 Typical Section—Definition
4.3 Unswept Uniform Wing Beam Model Aeroelastic Equations—Differential Equations of Motion
4.3.1 General: Wing Beam Under the Action of a Dynamic Transverse Load Along the Half-Wingspan
4.3.2 Differential Equation of Free Vibration of a Slender Beam
4.3.3 Differential Equations of a Wing as a Slender Beam Under the Action of Torsional Load
4.4 Aeroelastic Equations for a Simple Typical Wing Section Without Control Surface
4.5 Aeroelastic Equations for a Simple Typical Wing Section with Control Surface
4.5.1 General
4.5.2 Flutter Equation
4.5.3 Dynamic Response Equation
4.5.4 Divergence Equation
4.5.5 Static Response Equation
4.5.6 Some Important Notes
4.5.7 Linearity
References
5 Static Aeroelasticity-Typical Section, One-Dimensional Model and Lifting Surface
5.1 Typical Wing Section
5.2 Torsional Divergence
5.2.1 General
5.2.2 Physical Meaning of Torsional Divergence
5.2.3 Typical Section with Control Surface
5.3 Aileron Reversal
5.3.1 General
5.3.2 Comparison with Rigid Wing
5.4 One-Dimensional Aeroelastic Wing Model
5.4.1 Modeling of High-Aspect Ratio Wing as a Slender Beam (Beam-Rod) [8]
5.4.2 Static Aeroelasticity Using Eigenvalue and Eigenfunction Approach
5.5 Galerkin Method
5.6 Rolling of a Straight Wing
5.6.1 General
5.6.2 Integral Equation of Equilibrium
5.6.3 Derivation of the Equilibrium Equation
5.6.4 The Determination of Cαα
5.7 Aerodynamic Forces
5.8 Aeroelastic Equilibrium Equation and Lumped Mass Method
5.9 Control Surface Reversal and Rolling Effectiveness
5.10 Two-Dimensional Aeroelastic Model of Lifting Surface
5.10.1 Two-Dimensional Structure—Integral Representation
5.10.2 Two-Dimensional Aerodynamic Surfaces—Integral Representation
5.10.3 Solution by Lumped Mass Method
5.11 Closing Remarks
References
6 Flutter Stability of a Typical Section-An Elementary Discussion
6.1 Steady Aerodynamic Model
6.2 Conditions for Critical Instabilities
6.3 Physical Explanation of Flutter Mechanism
6.3.1 Phase Differences in Flutter Vibration Modes
6.3.2 Work Done by Aerodynamic Forces
6.4 Example of Typical Section with Steady Aerodynamic Model
6.5 Low-Frequency Refinement of Aerodynamic Model
6.6 Example of Typical Section with Low-Frequency Aerodynamic Model
6.7 Closing Remarks
Appendix: Verification/Proof oF Equation 6.35
References
7 Introduction to Unsteady Aerodynamics
7.1 Basic Fluid Dynamic Equation
7.2 Review of Fluid Dynamics
7.2.1 Conservation of Mass
7.2.2 Conservation of Momentum
7.2.3 Irrotational Flow, Kelvin’s Theorem and Bernoulli’s Equation
7.3 Differential Equations Based on Velocity Potential
7.3.1 Kelvin’s Theorem and Velocity Potential
7.3.2 Derivation of Single Equation for Velocity Potential
7.4 Small-Perturbation Theory
7.4.1 Differential Equation
7.4.2 Boundary Conditions
7.5 Subsonic Flow
7.5.1 Derivation of the Integral Equation by Transform Methods and Solution by Collocation
7.5.2 An Alternative Determination of the Kernel Function Using Green’s Theorem
7.5.3 Incompressible, Three-Dimensional Flow
7.5.4 Incompressible, Two-Dimensional Flow
7.6 Aerodynamic Lift and Moment for a Harmonically Oscillating Airfoil
7.7 Oscillatory Aerodynamic Derivatives
7.8 Aerodynamic Damping and Stiffness
7.9 Unsteady Aerodynamics of Thin Wing in Subsonic Flow, Derivation of Lift and Moment, Theodorsen Function,
7.9.1 Two-Dimensional, Constant-Density Flow
7.10 Concluding Remarks
Appendix 1: Some Complementary Information Regarding Bessel and Hankel Functions Associated with Theodorsen Function C(k)
Appendix 2: Some Excerpts from the Method of Asymptotic (Inner and Outer) Expansion for Two-Dimensional Unsteady Aerodynamics
Expansion Procedure for the Equations of Motion
Appendix 3: Applications of Two-Dimensional Unsteady Aerodynamics
Appendix 4: Transonic Small-Disturbance Flow
Introduction
Small-Perturbation Flow Equations
Appendix 5: Thin Airfoils in Supersonic Flow
Thin Airfoils in Supersonic Flow
Appendix 6: Three-Dimensional Thin Wings in Steady Supersonic Flow
Introduction
Non-Lifting Wings
Lifting Wings of Simple Planform
References
8 Unsteady Aerodynamics of Oscillating Objects with a Case Study
8.1 Introduction
8.2 Formulation of Unsteady Flow Problem
8.2.1 Irrotational Flow
8.2.2 Basic Flow Equation
8.2.3 Boundary Conditions
8.2.4 Unsteady Boundary Condition
8.3 Introduction to Acceleration Potential
8.4 Aerodynamic Quantities Required
8.5 Methods of Solution for Harmonic Motions
8.5.1 Boundary Value Problem
8.5.2 Solution by Superposition of Elementary Solutions
8.6 Classical Methods
8.6.1 Velocity Potential Method [2]
8.6.2 Integral Equation Method for Acceleration Potential [5]
8.7 Methods of Solution for Harmonic Motions
8.7.1 Integral Equation for Acceleration Potential
8.8 Standard Solution Methods of Integral Equation
8.8.1 Kernel Function Method
8.8.2 Doublet Lattice Method
8.8.3 Relative Merits of Both Methods
8.9 Aerodynamic Loads Due to Oscillatory Translation
8.10 Aerodynamic Loads Due to Oscillatory Pitching
8.11 Physical Interpretation of Unsteady Aerodynamic Forces
8.12 Constant-Density Inviscid Flow Foundation for Unsteady Aerodynamic Applications
8.12.1 Green’s Theorem
8.12.2 Kinetic Energy
8.13 Case Study—Simple Method to Calculathe the Oscillating Lift on a Circular Cylinder in Potential Flow [10]
8.13.1 Introduction
8.13.2 Analytical Model
8.13.3 The Motion of the Vortex
8.13.4 Calculation of Vortex Strength
8.13.5 Frequency of Vortex Shedding
8.13.6 Calculation of Vortex Trajectory
8.13.7 Result, Discussions and Concluding Remarks
8.14 List of Symbol
Appendix 1: Acceleration or Pressure Potential
Appendix 2: Velocity Potential and Stream Function
References
9 Flutter Calculation Methods
9.1 Review of Theoretical Foundation for Flutter Stability for Binary-Bending-Torsion Flutter of Typical Section
9.1.1 Quasi-Steady,
9.1.2 GT Done-Type Analysis
9.1.3 Preliminary Discussion—K-Method-Type Analysis,,
9.2 K-Method
9.2.1 Reduction of Eigenvalue Problem
9.2.2 Equivalent Modal Structural Damping
9.2.3 Agard Method [4, 13]
9.2.4 Low-Frequency Refinement [4]
9.3 Other Flutter Calculation Methods—The Expanding Domain of Aeroelasticity
9.3.1 Introduction to the P-K Method of Solution of the Flutter Solution—An Approximate True Damping Solution of the Flutter Equation by Determinant Iteration
9.3.2 Non-Iterative P-K (NIPK) Method—A New Non-Iterative P-K Match Point Flutter Solution Method
9.3.3 An Analysis of the Flutter and Damping Characteristics Using the P-K Method of Flutter Solution [15]
9.3.4 A Method for Efficient Flutter Analysis of Systems with Uncertain Modeling Parameters [16]
9.3.5 Solution of the Uncertain Flutter Eigenvalue Problem Using µ-p Analysis [18]
9.3.6 Modified P-K Method for Flutter Solution with Damping Iteration [20]
9.3.7 H Flutter Analysis Method—A Direct Harmonic Interpolation Method [22]
9.4 Concluding Remarks
Appendix 1: Discussions on K-Method, Following 9.1.3
Remarks 1: Rechecking Flutter Stability Equation Consistency
Remarks 2: Further Check and Validation
Remarks 3: Flutter Stability Analysis Similar to Done’s Approach
Appendix 2: Solution of Eigenvalue Problem Example of 9.2.3
Appendix 3: Another Approach on Example Problem 9.2.3
Notes
References
10 Dynamic Aeroelasticity of Typical Section with a Case Study
10.1 Dynamic Aeroelasticity of Typical Section
10.1.1 Sinusoidal Motion
10.1.2 Periodic Motion
10.1.3 Arbitrary Motion
10.2 Parametric Study of Aeroelastic Stability and Flutter Characteristics of Aircraft Wings as a Case Study
10.2.1 Introduction
10.2.2 Typical Section Representation of 3D Wing for Aeroelastic Analysis
10.2.3 Case Studies
10.2.4 Solution of Problems Addressed in Case Studies
10.2.5 Case Study: Boeing 747-Like Wing
10.2.6 Case Study 3: Determination of the Onset of Flutter for Typical Wing Section Using K-Method
10.2.7 Case Study 4: Parametric Study of Typical Section Subject to Changes in Its Sectional Characteristics
10.2.8 Discussion and Analysis
10.2.9 Remarks on Case Studies Addressed
10.3 Concluding Remarks
References
Part II Advanced Topics
11 Unsteady Aerodynamics with Case Studies
11.1 Introduction
11.2 Formulation of Unsteady Flow Problem
11.2.1 Irrotational Flow
11.2.2 Basic Flow Equation
11.2.3 Boundary Condition
11.2.4 Unsteady Boundary Condition
11.3 Introduction to Acceleration Potential
11.4 Case Study I: Calculation of Nonlinear, Unsteady Lifting Potential Flow Problems
11.4.1 Introduction
11.4.2 Formulation of the Problem
11.4.3 Dynamical Condition Governing the Wake
11.4.4 The Kutta–Joukowski Condition and the Generation of the Wake
11.4.5 Pressure, Forces and Moment
11.4.6 Method of Solution
11.4.7 Step-by-Step Procedure
11.4.8 Numerical Method
11.4.9 Systematic Expansion of the Velocity Influence Coefficient
11.4.10 Application
11.5 Case Study II: Calculation of 3D Unsteady Subsonic Flow with Separation Bubble Using Singularity Method [2]
11.5.1 Introduction
11.5.2 Problem Formulation
11.5.3 Governing Equation
11.5.4 Boundary Conditions
11.5.5 Systems of Integral Equations
11.5.6 Induced Velocity at the Attached Flow Region
11.5.7 Induced Velocity at the Separated Flow Region
11.5.8 Induced Pressure at the Attached Flow Region
11.5.9 Induced Pressure at the Separated Flow Region
11.6 Case Study III: A Preliminary Study on Buffeting Problem Utilizing Dynamic Response Approach
11.6.1 Introduction
11.6.2 Problem Formulation
11.6.3 Unsteady Airloads
11.6.4 Discussions on the Computational Results
11.6.5 Concluding Remarks for Case-Study III
11.7 Case Study IV: Unified Aerodynamic-Acoustic Formulation for Aero-Acoustic Structure Coupling
11.7.1 Introduction
11.7.2 Governing Equation for Acousto-Aeroelastic Problem
11.7.3 Review of Linearized Unsteady Aerodynamics and Acoustics Differential Equations
11.7.4 Boundary Element Formulation of the Solution of the Linearized Unsteady Aerodynamics Equations
11.7.5 Acoustic-Aerodynamic Analogy
11.7.6 Case Study and Numerical Results
11.7.7 Concluding Remarks for Case-Study IV
Appendix: For Case Study II—Calculation of 3D Unsteady Subsonic Flow with Separation Bubble Using Singularity Method
The Singular Part of Kernel Functions
Integration Along the Chordwise Direction
References
12 Introduction to Aeroservoelasticity with Case Studies
12.1 Introduction
12.2 Mathematical Modeling of a Simple Aeroelastic System with a Control Surface
12.3 Incorporation of the Gust Terms
12.4 Application of a Control System
12.5 Determination of Closed-Loop System Stability
12.6 Aeroelastic Analysis of an Aircraft with Standby Actuator [8]
12.6.1 Introduction
12.6.2 A State Space Form for an Aeroservoelastic System of an Aircraft with Stand-By Actuator
12.6.3 A Simple Approximation for the Unsteady Aerodynamics in the Time Domain
12.6.4 Linear State Space Approach to the Aeroservoelastic System of an Aircraft with Stand-By Actuator
12.6.5 Results and Discussions
12.7 The Application of Artificial Neural Networks on Flutter Suppression System [9]
12.7.1 Introduction
12.7.2 The Basic Concept of Artificial Neural Network (ANN)
12.7.3 Aeroservoelastic System
12.7.4 Neural Adaptive Control
12.7.5 Indirect Adaptive Control for Flutter Suppression
12.7.6 Results and Discussion
12.7.7 Conclusions and Further Works
12.8 Design and Optimization of an Aeroservoelastic Wind Tunnel Model
12.8.1 Model Specification
12.9 Aeroservoelastic Modeling and Analysis of a Highly Flexible Flutter Demonstrator [35]
12.9.1 Finite Element Model
12.9.2 Equations of Motion
12.9.3 Aerodynamics
12.9.4 Model Integration
12.9.5 Model Analysis
12.10 Concluding Remarks
References
13 Introduction to Aircraft Loads
13.1 Introduction
13.2 Basic Elements of Analysis for Aircraft Loads
13.3 Loads Analysis in Overall Aircraft Predesign
13.3.1 Multi-fidelity Loads Process
13.4 Relevance of Aircraft Structural Loads Analysis from Predesign to Loads Flight Testing
13.4.1 Loads and Fatigue
13.4.2 The Determination of Design Loads
13.5 Load and Certification of Aircraft
13.5.1 Structural Design Criteria (SDC)
13.6 Free-Body Diagrams for Loads Analysis
13.6.1 V–n Diagram
13.6.2 V–n Diagram Without Gust Effect
13.6.3 Gust V–n Diagram
13.7 Loads Analysis Models
13.7.1 Equations of Motion
13.7.2 Aerodynamics
13.7.3 Vortex Lattice Method
13.7.4 Aerodynamic Loads from Database
13.7.5 Finite Element Model of the Structural
13.7.6 Structure and Aerodynamics Interaction
13.8 Load Recovery
13.9 Concluding Remarks
References
14 Aeroelastic Experiments-Ground Vibration and Flight Flutter Test Case Studies
14.1 Introduction
14.2 Ground Vibration Testing
14.3 Modal Analysis
14.4 Structural Dynamic Testing Equipments
14.4.1 Sample Hardware for Structural Dynamic Modal Testing
14.4.2 Some Softwares That Can Be Utilized for Structural Dynamics Modal Testing Analysis
14.5 Wind Tunnel Flutter Model Testing
14.6 Aircraft Ground Vibration Test of N-219 Prototype for Certification—A Case Study
14.6.1 Objective
14.6.2 Scope
14.6.3 Applicable Requirements
14.6.4 Description of Test Article
14.6.5 Description of Test Set-Up
14.7 Flight Flutter Test
14.8 Aircraft Prototype Flight Flutter Testing—A Case Study [8]
14.8.1 Introduction
14.8.2 General Description of Aircraft
14.8.3 Objectives of Flight Flutter Tests
14.8.4 N250 Prototype 1 Flight Flutter Testing Development
14.8.5 Excitation System
14.8.6 Excitation Methods Used for N250 Prototype 1
14.8.7 Modes of Operation for the N250 Prototype 1
14.8.8 Instrumentation
14.8.9 Flight Test Procedure
14.8.10 Aircraft Configurations and Test Points
14.8.11 Analysis Methods of Flight Flutter Test Data
14.8.12 Analysis of Measured Vibration Signals
14.9 Conclusions
14.10 Concluding Remarks
References
15 Piezoaeroelastic Wing Section for Energy Harvester
15.1 Introduction and Brief Review
15.2 Mechanics of Piezoelectric-Patched Cantilevered Beam as Energy Harvester
15.3 Synthesis of Baseline Solution Procedure
15.4 Fundamental Solution Procedure—Decoupled Linear Approach
15.5 Principal Piezoaeroelastic Stability Equation
15.5.1 Eigenvalue Problem in Frequency Domain
15.5.2 Time-Integration Problem in the Time Domain/State Space
15.6 Results
15.6.1 Baseline Aeroelastic Stability Results
15.6.2 Decoupled Linear Equations Approach for the Binary Aeroelasticity Based Piezoaeroelastic System for Solving the System Output Voltage
15.7 Discussions and Concluding Remarks
References
16 Introduction and Selected Case Studies in Hydroelasticity
16.1 Review and Introduction to Hydroelasticity of Ships
16.2 Definition and Scope of Hydroelasticity
16.3 Some Important Phenomena
16.3.1 Springing and Whipping
16.3.2 Research Techniques
16.4 Global Hydroelastic Response of LNG Ships
16.5 Numerical Boundary Element Computation of Submerged Body-Surface Interaction—A Case Study
16.5.1 Basic Problem Analyzed
16.5.2 Green Identity Formulation
16.5.3 Solution of the Nonlinear. Free Surface Condition
16.5.4 Computational Detail
16.5.5 Two-Dimensional Results
16.5.6 Three-Dimensional Case
16.5.7 Closing Remarks
16.6 Hydroelastic Equation of Motion, Dynamic Response and Stability
16.7 Inviscid FSI Coupling Model
16.8 Experimental Analysis of Hydroelastic Response of Flexible Hydrofoils
16.9 Hydro Structure Interaction Models
16.10 Influence of Waves in the Hydroelasticity of Ships
16.11 Hydroelastic Modeling of a Bulk Carrier in Regular Waves
16.12 Design Applications for Hydroelasticity
16.13 Service Factor Assessment
16.14 Concluding Remarks—Progress and Future Developments
References
17 Introduction to Envaeroelasticity with Vibro-acoustics as a Case Study
17.1 Aerospace Vibro-acoustics as a Case Study on Envaeroelasticity
17.2 Vibro-Acoustic Analysis of the Acoustic-Structure Interaction of Flexible Structure Due to Acoustic Excitation
17.2.1 Introductory Remarks
17.2.2 Problem Formulation
17.2.3 Discretization and Treatment of Helmholtz Integral Equation for the Acoustic Field Following Conventional BEM Formulation
17.2.4 Acoustic-Structure Coupling
17.2.5 Finite Element Numerical Simulation of a Rectangular Plate Under Pressure
17.2.6 Flexible Structure Subject to Harmonic External Forces in Acoustic Medium
17.3 Flexible Structure Subject to Acoustic Excitation in a Confined Medium
17.3.1 Normal Mode Analysis
17.3.2 Numerical Results for Acoustic Boundary Element Validation
17.4 Closing Remarks
Appendix 1: Discretization of Helmholtz Integral Equation
Appendix 2: BEM-FEM Acoustic-Aeroelastic Coupling (AAC)
Appendix 3: Further Treatment for Acoustic-Aeroelastic Coupling; Acoustic-Aerodynamic Analogy
References
18 Introduction and Case Studies in Aeroelasticity of Bridges and Tall Structures
18.1 Aeroelastic Problems in Bridges and Tall Structures
18.1.1 Bridges
18.1.2 Vortex Shedding
18.1.3 Tall Buildings
18.1.4 Tall Structure Shaping Strategies for Aerodynamic Wind Excitation Response Modifications
18.2 A Semi-analytical Approach Based of Wind Tunnel Tests on Rigid Models to Account for Across-Wind Aeroelastic Response of Square Tall Buildings
18.3 Computer Modeling Example of Aeroelastic Analysis of Bridge Girder Section
18.4 Aeroelastic Effects and Phenomena
18.4.1 Dynamic Behavior
18.4.2 Aeroelastic Instability
18.4.3 Flow Pattern
18.4.4 Galloping
18.5 Torsional Divergence or Quasi-static Divergence
18.6 Flutter and Forced Oscillation
18.6.1 Flutter
18.6.2 Free Oscillation Solution Procedure
18.6.3 Forced Oscillation Procedure
18.6.4 The Aeroelastic Stability Problem of Long-Span Cable-Stayed Bridges Under an Approaching Crosswind Flow
18.7 Aeroelastic Equilibrium of the Bridge
18.7.1 Aeroelastic Equilibrium of the Bridge: A Continuous Model
18.8 Closing Remarks
References
Part III Case Studies on Application Examples
19 Aeroelastic Optimization of Tapered Wing Structure
19.1 Introduction
19.2 General Approach
19.3 Minimum Weight Optimization
19.4 Aeroelastic Constraint
19.5 Sensitivity of Aeroelastic Constraint
19.6 Optimization Procedure
19.7 Example Problem
19.8 Concluding Remarks
References
20 Acoustic Effects on Binary Aeroelastic Model
20.1 Introduction
20.2 Computational Method
20.2.1 Binary Aeroelastic Model
20.2.2 Structural-Acoustic Coupling
20.2.3 Flutter Solution
20.3 Results and Discussion
20.4 Conclusion
References
21 Application of a Multipole Secondary Source for Propeller Active Noise Control
21.1 Introduction
21.2 Secondary Source Strength Using Direct Approach
21.3 Secondary Source Strength Using Optimized Approach
21.4 Applications
21.4.1 The Case of Simple-Multiple Frequency Noise Reduction of Aircraft Air-Conditioning Blower
21.4.2 Noise Reduction of Cessna 150 Propeller
21.4.3 Noise Reduction of Broadband White Noise
21.5 Implementation
21.5.1 Preparation of Multipole Secondary Sources
21.6 Conclusion
References
22 Kinematic and Unsteady Aerodynamic Study of Bi- and Quad-Wing Ornithopter
22.1 Introduction
22.2 Theoretical Development of the Generic Aerodynamics of Flapping Wings
22.3 Results
22.3.1 Results for Bi-Wing
22.3.2 Analysis and Results for Quad-Wing
22.4 Comprehensive Assessment of Modeling Result
22.5 Conclusions
References
23 Analysis and Computational Study of the Aerodynamics, Aeroelasticity and Flight Dynamics of Flapping-Wing Ornithopter Using Linear Approximation
23.1 Introduction
23.2 Theoretical Development of the Aerodynamic, Aeroelastic and Flight Dynamic Modeling of Flapping Wings
23.2.1 Aerodynamic Model
23.3 Synthesis of Aeroelastic Approach
23.4 Typical Section Representation of Flapping Wing for Aeroelastic Analysis
23.5 Aeroelastic Analysis of Flapping-Wing Ornithopter Represented as Typical Section with Low-Frequency Aerodynamics
23.6 Computational Results
23.6.1 Theodorsen Unsteady Aerodynamic Aeroelastic Analysis of Flapping-Wing Ornithopter Model Represented as Typical Section
23.6.2 Computational Results
23.6.3 Incorporation of Quasi-Steady Aerodynamics Flexibility in a Heuristic Model for Aerodynamic Performance Estimation
23.7 Flight Dynamics Considerations
23.7.1 Flight Dynamic Model
23.7.2 Formulation of Overall Force and Moment
23.8 Results and Analysis
23.8.1 Parametric Study
23.9 Concluding Remarks
References
24 BEM–FEM Coupling for Acoustic Effects on Aeroelastic Stability of Structures
24.1 Introduction
24.2 Discretization of the Helmholtz Integral Equation for the Acoustic Field
24.3 BEM–FEM Acoustic-Aeroelastic Coupling (AAC)
24.4 Further Treatment for AAC; Acoustic-Aerodynamic Analogy
24.5 Acoustic Modified Flutter Formulation (Stability Problem Using k-Method)
24.6 Numerical Results
24.6.1 Acoustic Boundary Element Simulation
24.6.2 Coupled BEM–FEM Numerical Simulation
24.6.3 Flutter Calculation for Coupled Unsteady Aerodynamic and Acoustic Excitations
24.7 AAC (Acoustic-Aeroelastic Coupling) Parametric Study
24.8 Concluding Remarks
References
25 Active Vibration Suppression of a Generic Smart Composite Structure
25.1 Introduction
25.2 Formulation of Generic Problems
25.3 Solution Scheme
25.3.1 Equation of Motion of the Euler–Bernoulli Beam Using Hamilton’s Principle
25.3.2 Finite Element Approach
25.3.3 Results and Discussions of the Baseline Problem
25.4 The Utilization of Piezoelectric Sensors and Actuators
25.4.1 Actuator
25.4.2 Sensor
25.5 Equation of Motion of Euler–Bernoulli Beam with Piezoelectric Patches
25.6 Solution of the Free Vibration of Beam with Piezoelectric Patches Using Finite Element Method
25.7 Control and Control Performance
25.7.1 System Response
25.7.2 Modal Order Reduction
25.7.3 State-Space Representation
25.7.4 Control Strategy Formulation
25.8 Results and Discussions
25.9 Conclusions
Appendix 1: Comparative Study of the Baseline Results with Other Works
Appendix 2: The Spill–Over Effect on the Vibration Control of Structures
Appendix 3: Uncertainty Analysis of the Current Model
References
26 Transonic Flow Computation of Slender Body of Revolution Using Transonic Small Disturbance and Navier–Stokes Equations
26.1 Introduction
26.2 Governing Equation for Transonic Flow About Slender Bodies of Revolution
26.3 Transonic Small Disturbance Approach with Boundary Condition Derived from Transonic Small Perturbation Integral Equation (1st TSD Method)
26.3.1 Boundary Conditions
26.3.2 Grid Generation and Solution in the Vicinity of the Surface
26.4 Finite Difference Computation
26.4.1 Disturbance Velocity Along x Direction on the Surface
26.4.2 Discretization of the Governing Equation for the Interior Points
26.5 TSD Equation for Axisymmetric Body (2nd TSD Method)
26.5.1 Entropy Correction
26.6 The Navier–Stokes Equations for Axisymmetric Body
26.7 Results
26.8 MBB Bodies of Revolution
26.9 Conclusions
References
27 Computational Modeling, Simulation and Tailoring of Non-penetrating and Impact Resilient Generic Structure
27.1 Introduction
27.2 Philosophical Approach in the Modeling and Simulation of Non-penetrating Impact
27.3 Method of Approach in the Modeling and Simulation of Non-penetrating Impact
27.4 Cross-Validation of Analytical and Numerical Simulation
27.5 Parametric Study of Plates Under Impact by Finite Element Simulation for Structural Tailoring of Non-penetrating Case
27.5.1 von Mises Stress Evaluation
27.5.2 Selection of Laminated Metal Composites for the Present Study
27.6 Analysis of Surface Impact Using Hertz Elastic Contact Impact Theory
27.6.1 Surface Stresses and Distribution of Displacement Induced by a Spherical Indentation Ball
27.6.2 Indentation Response of Materials
27.7 Analytical and Computational Results
27.7.1 Collapse Load Applied at Midspan of Beam: L1 = L/2
27.7.2 Static and Dynamic Analysis of Isotropic Flat Plate Bending
27.7.3 Methodological Scheme
27.7.4 Theoretical Foundation and Generic Analysis
27.7.5 Static and Dynamic Analysis
27.8 Finite Element Impact Simulation of Flat Plate Subject to Impact for Exploring Resilient Structure
27.9 Case Studies Simulation Rationale
27.9.1 Simulation Studies
27.9.2 Simulation Studies on Hypothetical Metal Laminate (HML) by Assumed Directional Properties
27.10 Concluding Remarks
Appendix
References
Appendix A MATLAB Program for Chap. 9 Flutter Calculation Method
Appendix B SI Units and Conversion Tables