Theory of Adaptive Fiber Composites: From Piezoelectric Material Behavior to Dynamics of Rotating Structures Book

1 Introduction ; 1.1 Adaptive Structural Systems ; 1.2 Objective and Scope ; 1.3 Outline and Overview ; 2 Helicopter Applications ; 2.1 Noise and Vibration ; 2.1.1 Generation ; 2.1.2 Areas of Relevance ; 2.2 Main Rotor ; 2.2.1 Rotational Sources ; 2.2.2 Impulsive Sources ; 2.2.3 Broadband Sources ; 2.3 Passive Concepts ; 2.3.1 External Devices ; 2.3.2 Aeroelastic Conformability; 2.4 Active and Adaptive Concepts ; 2.4.1 Pitch Control at the Blade Root ; 2.4.2 Discrete Flap Actuation ; 2.4.3 Integral Blade Actuation ; 2.5 Adaptive Beam Aspects ; 2.5.1 Beam Actuation Concepts ; 2.5.2 Adaptive System Concepts ; 2.5.3 Development Status; 3 Fundament a1 Considerations; 3.1 Mathematical Preliminaries; 3.1.1 Euclidean Vectors; 3.1.2 Tensor Representation ; 3.1.3 Matrix Representation ; 3.2 Deformable Structures . Mechanical Fields; 3.2.1 Loads ; 3.2.2 Stresses; 3.2.3 Mechanical Equilibrium; 3.2.4 Strains; 3.2.5 Transformations ; 3.3 Dielectric Domains - Electrostatic Fields; 3.3.1 Electric Charge; 3.3.2 Electric Flux Density; 3.3.3 Electrostatic Equilibrium; 3.3.4 Electric Field Strengths; 3.4 Principle of Viual Work; 3.4.1 General Principle of Virtual Work; 3.4.2 Principle of Virtual Displacements ; 3.4.3 Principle of Virtual Loads ; 3.4.4 Principle of Virtual Electric Potential; 3.4.5 D'Alembert's Principle in the Lagrangian Version; 3.4.6 Summation of Virtual Work Contributions ; 3.5 Other Variational Principles ; 3.5.1 Extended Dirichlet's Principle of Minimum Potential Energy ; 3.5.2 Extended General Hamilton's Principle; 4 Piezoelectric Materials; 4.1 Piezoelectric Effect ; 4.1.1 Historical Development; 4.1.2 Crystal Structures; 4.2 Constitutive Formulation ; 4.2.1 Mechanical Fields; 4.2.2 Electrostatic Fields; 4.2.3 Electromechanical Coupling ; 4.2.4 Spatial Rotation; 4.2.5 Analogy of Electrically and Thermally Induced Deformations; 4.3 Constitutive Examination ; 4.3.1 Constitutive Relation ; 4.3.2 Converse Piezoelectric Effect; 4.3.3 Direct Piezoelectric Effect; 4.4 Constitutive Reduction; 4.4.1 Unidirectional Electrostatic Fields; 4.4.2 Planar Mechanical Fields; 4.4.3 Planar Rotation ; 4.4.4 Negated Electric Field Strength; 4.5 Actuator and Sensor Conditions; 4.5.1 Actuator Application with Voltage and Current Source ; 4.5.2 Sensor Application with Voltage and Current Measurement; 5 Piezoelectric Composites; 5.1 Classification of General Composites ; 5.1.1 Topology of the Inclusion Phase; 5.1.2 Laminated Composites and Laminated Fiber Composites; 5.2 Conception of Piezoelectric Composites ; 5.2.1 Interdigitated Electrodes and Piezoelectric Fibers ; 5.2.2 Electroding Implications ; 5.2.3 Development Status ; 5.2.4 Representative Volume Element and Fiber Geometry ; 5.2.5 Modeling Preliminaries; 5.3 Micro-Electromechanics with Equivalent Inclusions; 5.3.1 Mean Fields and Concentration Matrices; 5.3.2 Elementary Rules of Mixture ; 5.3.3 Equivalence of Inclusion and Inhomogenity’5.3.4 Non-Dilute Concentrations ; 5.4 Micro-Electromechanics with Sequential Stacking ; 5.4.1 Stacking of Constituents with Uniform Fields; 5.4.2 Normal Mode Stacking Coefficients; 5.4.3 Shear Mode Stacking Coefficients; 5.4.4 Stacking Sequences; 5.4.5 Non-Homogeneous Electrostatic Fields; 5.4.6 Stacking Sequences for Non-Homogeneous Electrostatic Fields; 5.5 Validation of the Micro-Electromechanics; 5.5.1 Experiments and Finite Element Models; 5.5.2 Dielectric, Piezoelectric, and Mechanical Properties ; 6 Adaptive Laminated Composite Shells; 6.1 Macro-Electromechanics; 6.1.1 Lamination Theory; 6.1.2 Laminates with Groups of Electrically Paralleled Laminae ; 6.2 Kinematics and Equilibrium; 6.2.1 General Thin Shell Kinematics; 6.2.2 Cylindrical Thin Shell Kinematics; 6.2.3 Cylindrical Thin Shell Equilibrium ; 6.3 Constitutive Reduction ; 6.3.1 Negligence of Strain and Stress Components ; 6.3.2 Potential Energy Considerations; 7 Adaptive Thin-Walled Beams; 7.1 General Beam Kinematics; 7.1.1 Positions and Displacements; 7.1.2 Rotations; 7.1.3 Simplifications; 7.1.4 Strains; 7.2 Thin-Walled Beam Kinematics ; 7.2.1 Differential Geometry; 7.2.2 Cartesian and Curvilinear Positions and Displacements ; 7.2.3 Strains of Wall and Beam ; 7.2.4 Electric Field Strength; 7.3 Torsional Out-of-Plane Warping for Thin Walls; 7.3.1 General Formulation ; 7.3.2 Non-Branched Open and Closed Cross-Sections ; 7.3.3 General Cross-Sections with Open Branches and Closed Cells ; 7.3.4 Exemplary Configurations ; 7.3.5 Consistency Contemplations ; 7.4 Rotating Beams; 7.4.1 Rotor Kinematics; 7.4.2 Transformation Properties ; 8 Virtual Work Statements; 8.1 Internal Virtual Work; 8.1.1 Internal Loads of Beam and Wall; 8.1.2 Constitutive Relation ; 8.1.3 Constitutive Coefficients ; 8.1.4 Partially Prescribed Electric Potential ; 8.2 External Virtual Work; 8.2.1 Applied Load Contributions ; 8.2.2 ; 8.2.3 Equilibrium and Boundary Conditions ; 8.3 Second-Order Theory ; 8.3.1 Additional Internal Load Contributions; .8.3.2 Reformulation; 9 Solution Variants ; 9.1 Statics of the Non-Rotating Structure; 9.1.1 Configuration Restrictions; 9.1.2 Extension, Torsion, and Warping Solution; 9.1.3 Shear and Bending Solution; 9.2 Dynamics of the Rotating Structure; 9.2.1 Virtual WorkRoundup; 9.2.2 Finite Element Formulation; 9.2.3 Solution; 10 Demonstration and Validation ; 10.1 Beam Configurations; 10.1.1 Actuation and Sensing Schemes; 10.1.2 Set-Up of Walls ; 10.1.3 Set-Up of Cross-Sections; 10.1.4 Constitutive Coefficients; 10.2 Elementary Examinations ; 10.2.1 Beam Geometry Influences on the Actuation Schemes; 10.2.2 Beam Property Adaptation ; 10.2.3 Wall Geometry Optimization; 10.3 Validation and Evaluation; 10.3.1 Reference Configurations; 10.3.2 Reference Calculations; 10.3.3 Static Behavior; 10.3.4 Free Vibrations; 10.3.5 Forced Vibrations ; Conclusion ; 11.1 Summary ; 11.2 Perspective; Material Properties; B Helicopter Rotor Properties ; References ; Index