Solid Mechanics and Its Applications

From Piezoelectric Material Behavior to Dynamics of Rotating Structures

By: T. H. Brockmann

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Hardcover | 21 July 2009

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Hardcover
240 Pages

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24.77 x 17.15 x 1.91

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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.

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Paperback

Published: 29th November 2011

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Introduction	p. 1
Adaptive Structural Systems	p. 1
Objective and Scope	p. 2
Outline and Overview	p. 3
Helicopter Applications	p. 5
Noise and Vibration	p. 5
Generation	p. 5
Areas of Relevance	p. 7
Main Rotor	p. 8
Rotational Sources	p. 8
Impulsive Sources	p. 9
Broadband Sources	p. 10
Passive Concepts	p. 10
External Devices	p. 10
Aeroelastic Conformability	p. 11
Active and Adaptive Concepts	p. 13
Pitch Control at the Blade Root	p. 13
Discrete Flap Actuation	p. 14
Integral Blade Actuation	p. 14
Adaptive Beam Aspects	p. 15
Beam Actuation Concepts	p. 16
Adaptive System Concepts	p. 17
Development Status	p. 17
Fundamental Considerations	p. 19
Mathematical Preliminaries	p. 19
Euclidean Vectors	p. 19
Tensor Representation	p. 20
Matrix Representation	p. 21
Deformable Structures-Mechanical Fields	p. 22
Loads	p. 23
Stresses	p. 23
Mechanical Equilibrium	p. 24
Strains	p. 25
Transformations	p. 26
Dielectric Domains-Electrostatic Fields	p. 28
Electric Charge	p. 29
Electric Flux Density	p. 29
Electrostatic Equilibrium	p. 29
Electric Field Strengths	p. 30
Principle of Virtual Work	p. 31
General Principle of Virtual Work	p. 31
Principle of Virtual Displacements	p. 32
Principle of Virtual Loads	p. 33
Principle of Virtual Electric Potential	p. 34
D'Alembert's Principle in the Lagrangian Version	p. 35
Summation of Virtual Work Contributions	p. 37
Other Variational Principles	p. 38
Extended Dirichlet's Principle of Minimum Potential Energy	p. 38
Extended General Hamilton's Principle	p. 39
Piezoelectric Materials	p. 41
Piezoelectric Effect	p. 41
Historical Development	p. 41
Crystal Structures	p. 42
Constitutive Formulation	p. 45
Mechanical Fields	p. 46
Electrostatic Fields	p. 47
Electromechanical Coupling	p. 48
Spatial Rotation	p. 49
Analogy of Electrically and Thermally Induced Deformations	p. 49
Constitutive Examination	p. 50
Constitutive Relation	p. 50
Converse Piezoelectric Effect	p. 52
Direct Piezoelectric Effect	p. 53
Constitutive Reduction	p. 56
Unidirectional Electrostatic Fields	p. 57
Planar Mechanical Fields	p. 61
Planar Rotation	p. 63
Negated Electric Field Strength	p. 64
Actuator and Sensor Conditions	p. 65
Actuator Application with Voltage and Current Source	p. 65
Sensor Application with Voltage and Current Measurement	p. 66
Piezoelectric Composites	p. 69
Classification of General Composites	p. 69
Topology of the Inclusion Phase	p. 69
Laminated Composites and Laminated Fiber Composites	p. 70
Conception of Piezoelectric Composites	p. 70
Interdigitated Electrodes and Piezoelectric Fibers	p. 71
Electroding Implications	p. 72
Development Status	p. 73
Representative Volume Element and Fiber Geometry	p. 74
Modeling Preliminaries	p. 77
Micro-Electromechanics with Equivalent Inclusions	p. 77
Mean Fields and Concentration Matrices	p. 78
Elementary Rules of Mixture	p. 79
Equivalence of Inclusion and Inhomogeneity	p. 79
Non-Dilute Concentrations	p. 81
Micro-Electromechanics with Sequential Stacking	p. 82
Stacking of Constituents with Uniform Fields	p. 82
Normal Mode Stacking Coefficients	p. 83
Shear Mode Stacking Coefficients	p. 86
Stacking Sequences	p. 87
Non-Homogeneous Electrostatic Fields	p. 89
Stacking Sequences for Non-Homogeneous Electrostatic Fields	p. 92
Validation of the Micro-Electromechanics	p. 93
Experiments and Finite Element Models	p. 94
Dielectric, Piezoelectric, and Mechanical Properties	p. 95
Adaptive Laminated Composite Shells	p. 99
Macro-Electromechanics	p. 99
Lamination Theory	p. 99
Laminates with Groups of Electrically Paralleled Laminae	p. 101
Kinematics and Equilibrium	p. 103
General Thin Shell Kinematics	p. 103
Cylindrical Thin Shell Kinematics	p. 104
Cylindrical Thin Shell Equilibrium	p. 106
Constitutive Reduction	p. 107
Negligence of Strain and Stress Components	p. 107
Potential Energy Considerations	p. 109
Adaptive Thin-Walled Beams	p. 115
General Beam Kinematics	p. 115
Positions and Displacements	p. 115
Rotations	p. 116
Simplifications	p. 117
Strains	p. 119
Thin-Walled Beam Kinematics	p. 120
Differential Geometry	p. 120
Cartesian and Curvilinear Positions and Displacements	p. 121
Strains of Wall and Beam	p. 123
Electric Field Strength	p. 125
Torsional Out-of-Plane Warping for Thin Walls	p. 126
General Formulation	p. 126
Non-Branched Open and Closed Cross-Sections	p. 128
General Cross-Sections with Open Branches and Closed Cells	p. 130
Exemplary Configurations	p. 132
Consistency Contemplations	p. 134
Rotating Beams	p. 136
Rotor Kinematics	p. 136
Transformation Properties	p. 137
Virtual Work Statements	p. 139
Internal Virtual Work	p. 139
Internal Loads of Beam and Wall	p. 140
Constitutive Relation	p. 140
Constitutive Coefficients	p. 141
Partially Prescribed Electric Potential	p. 146
External Virtual Work	p. 147
Applied Load Contributions	p. 148
Inertia Load Contributions	p. 148
Equilibrium and Boundary Contributions	p. 150
Second-Order Theory	p. 151
Additional Internal Load Contributions	p. 152
Reformulation	p. 153
Solution Variants	p. 155
Statics of the Non-Rotating Structure	p. 155
Configuration Restrictions	p. 155
Extension, Torsion, and Warping Solution	p. 156
Shear and Bending Solution	p. 159
Dynamics of the Rotating Structure	p. 160
Virtual Work Roundup	p. 160
Finite Element Formulation	p. 161
Solution	p. 165
Demonstration and Validation	p. 169
Beam Configurations	p. 169
Actuation and Sensing Schemes	p. 169
Set-Up of Walls	p. 172
Set-Up of Cross-Sections	p. 174
Constitutive Coefficients	p. 176
Elementary Examinations	p. 178
Beam Geometry Influences on the Actuation Schemes	p. 178
Beam Property Adaptation	p. 180
Wall Geometry Optimization	p. 182
Validation and Evaluation	p. 187
Reference Configurations	p. 187
Reference Calculations	p. 189
Static Behavior	p. 190
Free Vibrations	p. 193
Forced Vibrations	p. 197
Conclusion	p. 199
Summary	p. 199
Perspective	p. 200
Material Properties	p. 203
Helicopter Rotor Properties	p. 205
References	p. 207
Index	p. 217
Table of Contents provided by Ingram. All Rights Reserved.

Solid Mechanics and Its Applications

From Piezoelectric Material Behavior to Dynamics of Rotating Structures

At a Glance

Hardcover

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