Theory of Adaptive Fiber Composites (eBook)

Contents 6
List of Figures 11
List of Tables 13
List of Symbols 14
Indices 14
Greek Symbols 14
Latin Symbols 15
Introduction 18
Adaptive Structural Systems 18
Objective and Scope 19
Outline and Overview 20
Helicopter Applications 21
Noise and Vibration 21
Generation 21
Main Rotor 22
Tail Rotor 22
Engine and Drivetrain 23
Areas of Relevance 23
Noise in the Distance 23
Vibrations of the Structure 23
Noise and Vibrations Inside the Cabin 23
Main Rotor 24
Rotational Sources 24
Impulsive Sources 25
Blade Vortex Interaction 25
High Speed Flow Conditions 25
Retreating Blade Stall 26
Broadband Sources 26
Passive Concepts 26
External Devices 26
Absorbers 26
Dampers 27
Aeroelastic Conformability 27
Elastomechanic Modifications 28
Aerodynamic Modifications 28
Active and Adaptive Concepts 29
Pitch Control at the Blade Root 29
Higher Harmonic Control 29
Individual Blade Control 29
Discrete Flap Actuation 30
Integral Blade Actuation 30
Adaptive Beam Aspects 31
Beam Actuation Concepts 32
Adaptive System Concepts 33
Development Status 33
Fundamental Considerations 35
Mathematical Preliminaries 35
Euclidean Vectors 35
Vectorial Products 36
Tensor Representation 36
Tensorial Products 36
Theorems 37
Matrix Representation 37
Substitution of Vectorial Products 38
Deformable Structures-Mechanical Fields 38
Loads 39
Stresses 39
Mechanical Equilibrium 40
Strains 41
Transformations 42
Dielectric Domains-Electrostatic Fields 44
Electric Charge 45
Electric Flux Density 45
Electrostatic Equilibrium 45
Electric Field Strengths 46
Principle of Virtual Work 47
General Principle of Virtual Work 47
Principle of Virtual Displacements 48
Principle of Virtual Loads 49
Principle of Virtual Electric Potential 50
D'Alembert's Principle in the Lagrangian Version 51
Summation of Virtual Work Contributions 53
Other Variational Principles 54
Extended Dirichlet's Principle of Minimum Potential Energy 54
Extended General Hamilton's Principle 55
Piezoelectric Materials 57
Piezoelectric Effect 57
Historical Development 57
Crystal Structures 58
Polar Crystals and Polar-Neutral Crystals 58
Ferroelectric Polar Crystals 59
Monocrystalline Examples 60
Polycrystalline Characteristics 60
Semicrystalline Characteristics 61
Constitutive Formulation 61
Mechanical Fields 62
Electrostatic Fields 63
Electromechanical Coupling 64
Spatial Rotation 65
Analogy of Electrically and Thermally Induced Deformations 65
Constitutive Examination 66
Constitutive Relation 66
Partial Coupling 67
Converse Piezoelectric Effect 68
Normal Mode Actuation 68
Shear Mode Actuation 68
Direct Piezoelectric Effect 69
External Electric Influences 69
Normal Mode Sensing 70
Shear Mode Sensing 70
Complications 72
Constitutive Reduction 72
Unidirectional Electrostatic Fields 73
Omission of Shear Associated Electrostatic Fields 73
Unidirectional Electric Field Strength 74
Unidirectional Electric Flux Density 75
Summary of Unidirectional Electrostatic Fields 75
Transition between Unidirectional Electrostatic Fields 76
Planar Mechanical Fields 77
Planar Stress Transverse to Polarization 78
Planar Stress in Plane with Polarization 78
Planar Rotation 79
Negated Electric Field Strength 80
Electroelastic Energy Density 80
Actuator and Sensor Conditions 81
Actuator Application with Voltage and Current Source 81
Voltage Source 81
Current Source 82
Sensor Application with Voltage and Current Measurement 82
Strain Sensor-Voltage Measurement 82
Strain Rate Sensor-Current Measurement 83
Piezoelectric Composites 84
Classification of General Composites 84
Topology of the Inclusion Phase 84
Laminated Composites and Laminated Fiber Composites 85
Conception of Piezoelectric Composites 85
Interdigitated Electrodes and Piezoelectric Fibers 86
Electroding Implications 87
Areal Electrodes 87
Interdigitated Electrodes 87
Development Status 88
Representative Volume Element and Fiber Geometry 89
Fiber Volume Fraction 90
Fiber Cross-Sectional Shape 91
Modeling Preliminaries 92
Micro-Electromechanics with Equivalent Inclusions 92
Mean Fields and Concentration Matrices 93
Elementary Rules of Mixture 94
Equivalence of Inclusion and Inhomogenity 94
Non-Dilute Concentrations 96
Mori-Tanaka Approach 96
Other Approaches 96
Micro-Electromechanics with Sequential Stacking 97
Stacking of Constituents with Uniform Fields 97
Normal Mode Stacking Coefficients 98
Shear Mode Stacking Coefficients 101
Stacking Sequences 102
Non-Homogeneous Electrostatic Fields 104
Stacking in Fiber Direction 104
Determination of the Stacking Coefficients 105
Discussion of the Stacking Coefficients 106
Stacking Sequences for Non-Homogeneous Electrostatic Fields 107
Validation of the Micro-Electromechanics 108
Experiments and Finite Element Models 109
Dielectric, Piezoelectric, and Mechanical Properties 110
Dielectric Properties 110
Piezoelectric Properties 111
Mechanical Properties 112
Adaptive Laminated Composite Shells 114
Macro-Electromechanics 114
Lamination Theory 114
Laminates with Groups of Electrically Paralleled Lami-nae 116
Kinematics and Equilibrium 118
General Thin Shell Kinematics 118
Cylindrical Thin Shell Kinematics 119
Cylindrical Thin Shell Equilibrium 121
Constitutive Reduction 122
Negligence of Strain and Stress Components 122
Membrane Response 122
Reduced Shell Response 123
Potential Energy Considerations 124
Constrained Variational Problem of Several Independent Variables 125
Actual Problem 126
Implications of the Potential Energy Minimization 127
Example 128
Adaptive Thin-Walled Beams 129
General Beam Kinematics 129
Positions and Displacements 129
Rotations 130
Simplifications 131
Strains 133
Thin-Walled Beam Kinematics 134
Differential Geometry 134
Cartesian and Curvilinear Positions and Displacements 135
Strains of Wall and Beam 137
Shell Strain Comprehension 138
Beam Strain Comprehension 138
Electric Field Strength 139
Torsional Out-of-Plane Warping for Thin Walls 140
General Formulation 140
Non-Branched Open and Closed Cross-Sections 142
Open Cross-Sections 143
Closed Cross-Sections 143
General Cross-Sections with Open Branches and Closed Cells 144
Open Branches 144
Closed Cells 145
Junctions 145
Exemplary Configurations 146
Double Cell Cross-Section 146
Combined Cross-Section 147
Consistency Contemplations 148
Resulting Simplifications 149
Rotating Beams 150
Rotor Kinematics 150
Transformation Properties 151
Virtual Work Statements 153
Internal Virtual Work 153
Internal Loads of Beam and Wall 154
Constitutive Relation 154
Constitutive Coefficients 155
Mechanical Coefficients on the Principal Diagonal 156
Off-Diagonal Mechanical Coefficients without Laminae Level Coupling 156
Off-Diagonal Mechanical Coefficients with Laminae Level Coupling 157
Electromechanical Coefficients 159
Open Cross-Section Peculiarity 160
Partially Prescribed Electric Potential 160
External Virtual Work 161
Applied Load Contributions 162
Inertia Load Contributions 162
Equilibrium and Boundary Conditions 164
Second-Order Theory 165
Additional Internal Load Contributions 166
Known Initial Internal Loads 167
Reformulation 167
Solution Variants 169
Statics of the Non-Rotating Structure 169
Configuration Restrictions 169
Extension, Torsion, and Warping Solution 170
Twisting Angle 171
Lengthwise Displacement 172
Shear and Bending Solution 173
Inclination Angles 173
Transverse Displacements 173
Dynamics of the Rotating Structure 174
Virtual Work Roundup 174
Finite Element Formulation 175
Discretization 175
Element Matrices 177
System Assembly 178
Simplifications 179
Solution 179
Steady-State Solution 180
Homogeneous Solution 180
Particular Solution 181
Solution Assembly 182
Demonstration and Validation 183
Beam Configurations 183
Actuation and Sensing Schemes 183
Wall Strain Modes 183
Wall Electroding Sectors 184
Beam Schemes 184
Example Configuration Considerations 185
Example Configuration Schemes 185
Set-Up of Walls 186
Arrangement of Fibers 187
Arrangement of Layers 187
Set-Up of Cross-Sections 188
Rectangular Single-Cell Cross-Section 189
Convex Double-Cell Cross-Section 190
Constitutive Coefficients 190
General Procedure 190
Specific Illustration 191
Elementary Examinations 192
Beam Geometry Influences on the Actuation Schemes 192
Solution in Terms of Characteristic Ratios 192
Geometry Influence Discussion 193
Beam Property Adaptation 194
Cross-Sectional Aspect Ratio 194
Beam Aspect Ratio 195
Relative Shear Stiffness 195
Deviation Contemplations 196
Wall Geometry Optimization 196
Satisfaction of the Stiffness and Geometry Constraints 197
Compilation of the Objective Function 198
Comparison of the Different Schemes 199
Influence of the Fiber Volume Fraction 200
Validation and Evaluation 201
Reference Configurations 201
Rectangular Single-Cell Cross-Section 201
Convex Double-Cell Cross-Section 202
Reference Calculations 203
Analytic Approach 203
Beam Finite Elements 204
Shell Finite Elements 204
Static Behavior 204
Beam Extension due to Centrifugal Forces 205
Beam Torsion due to Piezoelectric Coupling 206
Free Vibrations 207
Influence of the Rotation 208
Influence of the Modeling Approach 209
Influence of the Cross-Section 211
Forced Vibrations 211
Conclusion 213
Summary 213
Perspective 214
Material Properties 216
Helicopter Rotor Properties 218
References 219
Index 229