Theory of Adaptive Fiber Composites

From Piezoelectric Material Behavior to Dynamics of Rotating Structures
1. Auflage 2009
ISBN: 978-90-481-2435-0
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 219 Seiten

Adaptive structural systems in conjunction with multifunctional materials facilitate technical solutions with a wide spectrum of applications and a high degree of integration. By virtue of combining the actuation and sensing capabilities of piezoelectric materials with the advantages of fiber composites, the anisotropic constitutive properties may be tailored according to requirements and the failure behavior can be improved. Such adaptive fiber composites are very well-suited for the task of noise and vibration reduction. In this respect the helicopter rotor system represents a very interesting and widely perceptible field of application. The occurring oscillations can be reduced with aid of aerodynamic couplings via fast manipulation of the angle of attack, being induced by twist actuation of the rotor blade. On the one hand the sensing properties may be used to determine the current state of deformation, while on the other hand the actuation properties may be used to attain the required state of deformation. The implementation of such concepts requires comprehensive knowledge of the theoretical context, which shall be illuminated in the work at hand from the examination of the material behavior to the simulation of the rotating structure.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Contents;6
2;List of Figures;11
3;List of Tables;13
4;List of Symbols;14
4.1;Indices;14
4.2;Greek Symbols;14
4.3;Latin Symbols;15
5;Introduction;18
5.1;Adaptive Structural Systems;18
5.2;Objective and Scope;19
5.3;Outline and Overview;20
6;Helicopter Applications;21
6.1;Noise and Vibration;21
6.1.1;Generation;21
6.1.1.1;Main Rotor;22
6.1.1.2;Tail Rotor;22
6.1.1.3;Engine and Drivetrain;23
6.1.2;Areas of Relevance;23
6.1.2.1;Noise in the Distance;23
6.1.2.2;Vibrations of the Structure;23
6.1.2.3;Noise and Vibrations Inside the Cabin;23
6.2;Main Rotor;24
6.2.1;Rotational Sources;24
6.2.2;Impulsive Sources;25
6.2.2.1;Blade Vortex Interaction;25
6.2.2.2;High Speed Flow Conditions;25
6.2.2.3;Retreating Blade Stall;26
6.2.3;Broadband Sources;26
6.3;Passive Concepts;26
6.3.1;External Devices;26
6.3.1.1;Absorbers;26
6.3.1.2;Dampers;27
6.3.2;Aeroelastic Conformability;27
6.3.2.1;Elastomechanic Modifications;28
6.3.2.2;Aerodynamic Modifications;28
6.4;Active and Adaptive Concepts;29
6.4.1;Pitch Control at the Blade Root;29
6.4.1.1;Higher Harmonic Control;29
6.4.1.2;Individual Blade Control;29
6.4.2;Discrete Flap Actuation;30
6.4.3;Integral Blade Actuation;30
6.5;Adaptive Beam Aspects;31
6.5.1;Beam Actuation Concepts;32
6.5.2;Adaptive System Concepts;33
6.5.3;Development Status;33
7;Fundamental Considerations;35
7.1;Mathematical Preliminaries;35
7.1.1;Euclidean Vectors;35
7.1.1.1;Vectorial Products;36
7.1.2;Tensor Representation;36
7.1.2.1;Tensorial Products;36
7.1.2.2;Theorems;37
7.1.3;Matrix Representation;37
7.1.3.1;Substitution of Vectorial Products;38
7.2;Deformable Structures-Mechanical Fields;38
7.2.1;Loads;39
7.2.2;Stresses;39
7.2.3;Mechanical Equilibrium;40
7.2.4;Strains;41
7.2.5;Transformations;42
7.3;Dielectric Domains-Electrostatic Fields;44
7.3.1;Electric Charge;45
7.3.2;Electric Flux Density;45
7.3.3;Electrostatic Equilibrium;45
7.3.4;Electric Field Strengths;46
7.4;Principle of Virtual Work;47
7.4.1;General Principle of Virtual Work;47
7.4.2;Principle of Virtual Displacements;48
7.4.3;Principle of Virtual Loads;49
7.4.4;Principle of Virtual Electric Potential;50
7.4.5;D'Alembert's Principle in the Lagrangian Version;51
7.4.6;Summation of Virtual Work Contributions;53
7.5;Other Variational Principles;54
7.5.1;Extended Dirichlet's Principle of Minimum Potential Energy;54
7.5.2;Extended General Hamilton's Principle;55
8;Piezoelectric Materials;57
8.1;Piezoelectric Effect;57
8.1.1;Historical Development;57
8.1.2;Crystal Structures;58
8.1.2.1;Polar Crystals and Polar-Neutral Crystals;58
8.1.2.2;Ferroelectric Polar Crystals;59
8.1.2.3;Monocrystalline Examples;60
8.1.2.4;Polycrystalline Characteristics;60
8.1.2.5;Semicrystalline Characteristics;61
8.2;Constitutive Formulation;61
8.2.1;Mechanical Fields;62
8.2.2;Electrostatic Fields;63
8.2.3;Electromechanical Coupling;64
8.2.4;Spatial Rotation;65
8.2.5;Analogy of Electrically and Thermally Induced Deformations;65
8.3;Constitutive Examination;66
8.3.1;Constitutive Relation;66
8.3.1.1;Partial Coupling;67
8.3.2;Converse Piezoelectric Effect;68
8.3.2.1;Normal Mode Actuation;68
8.3.2.2;Shear Mode Actuation;68
8.3.3;Direct Piezoelectric Effect;69
8.3.3.1;External Electric Influences;69
8.3.3.2;Normal Mode Sensing;70
8.3.3.3;Shear Mode Sensing;70
8.3.3.4;Complications;72
8.4;Constitutive Reduction;72
8.4.1;Unidirectional Electrostatic Fields;73
8.4.1.1;Omission of Shear Associated Electrostatic Fields;73
8.4.1.2;Unidirectional Electric Field Strength;74
8.4.1.3;Unidirectional Electric Flux Density;75
8.4.1.4;Summary of Unidirectional Electrostatic Fields;75
8.4.1.5;Transition between Unidirectional Electrostatic Fields;76
8.4.2;Planar Mechanical Fields;77
8.4.2.1;Planar Stress Transverse to Polarization;78
8.4.2.2;Planar Stress in Plane with Polarization;78
8.4.3;Planar Rotation;79
8.4.4;Negated Electric Field Strength;80
8.4.4.1;Electroelastic Energy Density;80
8.5;Actuator and Sensor Conditions;81
8.5.1;Actuator Application with Voltage and Current Source;81
8.5.1.1;Voltage Source;81
8.5.1.2;Current Source;82
8.5.2;Sensor Application with Voltage and Current Measurement;82
8.5.2.1;Strain Sensor-Voltage Measurement;82
8.5.2.2;Strain Rate Sensor-Current Measurement;83
9;Piezoelectric Composites;84
9.1;Classification of General Composites;84
9.1.1;Topology of the Inclusion Phase;84
9.1.2;Laminated Composites and Laminated Fiber Composites;85
9.2;Conception of Piezoelectric Composites;85
9.2.1;Interdigitated Electrodes and Piezoelectric Fibers;86
9.2.2;Electroding Implications;87
9.2.2.1;Areal Electrodes;87
9.2.2.2;Interdigitated Electrodes;87
9.2.3;Development Status;88
9.2.4;Representative Volume Element and Fiber Geometry;89
9.2.4.1;Fiber Volume Fraction;90
9.2.4.2;Fiber Cross-Sectional Shape;91
9.2.5;Modeling Preliminaries;92
9.3;Micro-Electromechanics with Equivalent Inclusions;92
9.3.1;Mean Fields and Concentration Matrices;93
9.3.2;Elementary Rules of Mixture;94
9.3.3;Equivalence of Inclusion and Inhomogenity;94
9.3.4;Non-Dilute Concentrations;96
9.3.4.1;Mori-Tanaka Approach;96
9.3.4.2;Other Approaches;96
9.4;Micro-Electromechanics with Sequential Stacking;97
9.4.1;Stacking of Constituents with Uniform Fields;97
9.4.2;Normal Mode Stacking Coefficients;98
9.4.3;Shear Mode Stacking Coefficients;101
9.4.4;Stacking Sequences;102
9.4.5;Non-Homogeneous Electrostatic Fields;104
9.4.5.1;Stacking in Fiber Direction;104
9.4.5.2;Determination of the Stacking Coefficients;105
9.4.5.3;Discussion of the Stacking Coefficients;106
9.4.6;Stacking Sequences for Non-Homogeneous Electrostatic Fields;107
9.5;Validation of the Micro-Electromechanics;108
9.5.1;Experiments and Finite Element Models;109
9.5.2;Dielectric, Piezoelectric, and Mechanical Properties;110
9.5.2.1;Dielectric Properties;110
9.5.2.2;Piezoelectric Properties;111
9.5.2.3;Mechanical Properties;112
10;Adaptive Laminated Composite Shells;114
10.1;Macro-Electromechanics;114
10.1.1;Lamination Theory;114
10.1.2;Laminates with Groups of Electrically Paralleled Lami-nae;116
10.2;Kinematics and Equilibrium;118
10.2.1;General Thin Shell Kinematics;118
10.2.2;Cylindrical Thin Shell Kinematics;119
10.2.3;Cylindrical Thin Shell Equilibrium;121
10.3;Constitutive Reduction;122
10.3.1;Negligence of Strain and Stress Components;122
10.3.1.1;Membrane Response;122
10.3.1.2;Reduced Shell Response;123
10.3.2;Potential Energy Considerations;124
10.3.2.1;Constrained Variational Problem of Several Independent Variables;125
10.3.2.2;Actual Problem;126
10.3.2.3;Implications of the Potential Energy Minimization;127
10.3.2.4;Example;128
11;Adaptive Thin-Walled Beams;129
11.1;General Beam Kinematics;129
11.1.1;Positions and Displacements;129
11.1.2;Rotations;130
11.1.3;Simplifications;131
11.1.4;Strains;133
11.2;Thin-Walled Beam Kinematics;134
11.2.1;Differential Geometry;134
11.2.2;Cartesian and Curvilinear Positions and Displacements;135
11.2.3;Strains of Wall and Beam;137
11.2.3.1;Shell Strain Comprehension;138
11.2.3.2;Beam Strain Comprehension;138
11.2.4;Electric Field Strength;139
11.3;Torsional Out-of-Plane Warping for Thin Walls;140
11.3.1;General Formulation;140
11.3.2;Non-Branched Open and Closed Cross-Sections;142
11.3.2.1;Open Cross-Sections;143
11.3.2.2;Closed Cross-Sections;143
11.3.3;General Cross-Sections with Open Branches and Closed Cells;144
11.3.3.1;Open Branches;144
11.3.3.2;Closed Cells;145
11.3.3.3;Junctions;145
11.3.4;Exemplary Configurations;146
11.3.4.1;Double Cell Cross-Section;146
11.3.4.2;Combined Cross-Section;147
11.3.5;Consistency Contemplations;148
11.3.5.1;Resulting Simplifications;149
11.4;Rotating Beams;150
11.4.1;Rotor Kinematics;150
11.4.2;Transformation Properties;151
12;Virtual Work Statements;153
12.1;Internal Virtual Work;153
12.1.1;Internal Loads of Beam and Wall;154
12.1.2;Constitutive Relation;154
12.1.3;Constitutive Coefficients;155
12.1.3.1;Mechanical Coefficients on the Principal Diagonal;156
12.1.3.2;Off-Diagonal Mechanical Coefficients without Laminae Level Coupling;156
12.1.3.3;Off-Diagonal Mechanical Coefficients with Laminae Level Coupling;157
12.1.3.4;Electromechanical Coefficients;159
12.1.3.5;Open Cross-Section Peculiarity;160
12.1.4;Partially Prescribed Electric Potential;160
12.2;External Virtual Work;161
12.2.1;Applied Load Contributions;162
12.2.2;Inertia Load Contributions;162
12.2.3;Equilibrium and Boundary Conditions;164
12.3;Second-Order Theory;165
12.3.1;Additional Internal Load Contributions;166
12.3.1.1;Known Initial Internal Loads;167
12.3.2;Reformulation;167
13;Solution Variants;169
13.1;Statics of the Non-Rotating Structure;169
13.1.1;Configuration Restrictions;169
13.1.2;Extension, Torsion, and Warping Solution;170
13.1.2.1;Twisting Angle;171
13.1.2.2;Lengthwise Displacement;172
13.1.3;Shear and Bending Solution;173
13.1.3.1;Inclination Angles;173
13.1.3.2;Transverse Displacements;173
13.2;Dynamics of the Rotating Structure;174
13.2.1;Virtual Work Roundup;174
13.2.2;Finite Element Formulation;175
13.2.2.1;Discretization;175
13.2.2.2;Element Matrices;177
13.2.2.3;System Assembly;178
13.2.2.4;Simplifications;179
13.2.3;Solution;179
13.2.3.1;Steady-State Solution;180
13.2.3.2;Homogeneous Solution;180
13.2.3.3;Particular Solution;181
13.2.3.4;Solution Assembly;182
14;Demonstration and Validation;183
14.1;Beam Configurations;183
14.1.1;Actuation and Sensing Schemes;183
14.1.1.1;Wall Strain Modes;183
14.1.1.2;Wall Electroding Sectors;184
14.1.1.3;Beam Schemes;184
14.1.1.4;Example Configuration Considerations;185
14.1.1.5;Example Configuration Schemes;185
14.1.2;Set-Up of Walls;186
14.1.2.1;Arrangement of Fibers;187
14.1.2.2;Arrangement of Layers;187
14.1.3;Set-Up of Cross-Sections;188
14.1.3.1;Rectangular Single-Cell Cross-Section;189
14.1.3.2;Convex Double-Cell Cross-Section;190
14.1.4;Constitutive Coefficients;190
14.1.4.1;General Procedure;190
14.1.4.2;Specific Illustration;191
14.2;Elementary Examinations;192
14.2.1;Beam Geometry Influences on the Actuation Schemes;192
14.2.1.1;Solution in Terms of Characteristic Ratios;192
14.2.1.2;Geometry Influence Discussion;193
14.2.2;Beam Property Adaptation;194
14.2.2.1;Cross-Sectional Aspect Ratio;194
14.2.2.2;Beam Aspect Ratio;195
14.2.2.3;Relative Shear Stiffness;195
14.2.2.4;Deviation Contemplations;196
14.2.3;Wall Geometry Optimization;196
14.2.3.1;Satisfaction of the Stiffness and Geometry Constraints;197
14.2.3.2;Compilation of the Objective Function;198
14.2.3.3;Comparison of the Different Schemes;199
14.2.3.4;Influence of the Fiber Volume Fraction;200
14.3;Validation and Evaluation;201
14.3.1;Reference Configurations;201
14.3.1.1;Rectangular Single-Cell Cross-Section;201
14.3.1.2;Convex Double-Cell Cross-Section;202
14.3.2;Reference Calculations;203
14.3.2.1;Analytic Approach;203
14.3.2.2;Beam Finite Elements;204
14.3.2.3;Shell Finite Elements;204
14.3.3;Static Behavior;204
14.3.3.1;Beam Extension due to Centrifugal Forces;205
14.3.3.2;Beam Torsion due to Piezoelectric Coupling;206
14.3.4;Free Vibrations;207
14.3.4.1;Influence of the Rotation;208
14.3.4.2;Influence of the Modeling Approach;209
14.3.4.3;Influence of the Cross-Section;211
14.3.5;Forced Vibrations;211
15;Conclusion;213
15.1;Summary;213
15.2;Perspective;214
16;Material Properties;216
17;Helicopter Rotor Properties;218
18;References;219
19;Index;229