Vibrations of Linear Piezostructures

Vibrations of Linear Piezostructures
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A thorough guide to the fundamental development of linear piezoelectricity for vibrations  Vibrations of Linear Piezostructures  is an introductory text that offers a concise examination of the general theory of vibrations of linear piezostructures. This important book brings together in one comprehensive volume the most current information on the theory for modeling and analysis of piezostructures. The authors explore the fundamental principles of piezostructures, review the relevant mathematics, continuum mechanics and elasticity, and continuum electrodynamics as they are applied to electromechanical piezostructures, and include the work that pertains to linear constitutive laws of piezoelectricity.  The book addresses modeling of linear piezostructures via Newton’s approach and Variational Methods. In addition, the authors explore the weak and strong forms of the equations of motion, Galerkin approximation methods for the weak form, Fourier or modal methods, and finite element methods. This important book:  Covers the fundamental developments to vibrational theory for linear piezostructures Provides an introduction to continuum mechanics, elasticity, electrodynamics, variational calculus, and applied mathematics Offers in-depth coverage of Newton’s formulation of the equations of motion of vibrations of piezo-structures Discusses the variational methods for generation of equations of motion of piezo-structures Written for students, professionals, and researchers in the field,  Vibrations of Linear Piezostructures  is an up-to-date volume to the fundamental development of linear piezoelectricity for vibrations from initial development to fully modeled systems using various methods.

Оглавление

Andrew J. Kurdila. Vibrations of Linear Piezostructures

Table of Contents

List of Tables

List of Illustrations

Guide

Pages

Wiley‐ASME Press Series

Vibrations of Linear Piezostructures

Foreword

Preface

Acknowledgments

List of Symbols

1 Introduction. 1.1 The Piezoelectric Effect

1.1.1 Ferroelectric Piezoelectrics

1.1.2 One Dimensional Direct and Converse Piezoelectric Effect

1.2 Applications

1.2.1 Energy Applications

1.2.2 Sensors

1.2.3 Actuators or Motors

1.3 Outline of the Book

2 Mathematical Background

2.1 Vectors, Bases, and Frames

2.2 Tensors

2.3 Symmetry, Crystals, and Tensor Invariance

2.3.1 Geometry of Crystals

2.3.2 Symmetry of Tensors

2.4 Problems

3 Review of Continuum Mechanics

3.1 Stress

3.1.1 The Stress Tensor

3.1.2 Cauchy's Formula

3.1.3 The Equations of Equilibrium

3.2 Displacement and Strain

3.3 Strain Energy

3.4 Constitutive Laws for Linear Elastic Materials

3.4.1 Triclinic Materials

3.4.2 Monoclinic Materials

3.4.3 Orthotropic Materials

3.4.4 Transversely Isotropic Materials

3.5 The Initial‐Boundary Value Problem of Linear Elasticity

3.6 Problems

4 Review of Continuum Electrodynamics

4.1 Charge and Current

4.2 The Electric and Magnetic Fields

4.2.1 The Definition of the Static Electric Field

4.2.2 The Definition of the Static Magnetic Field

4.3 Maxwell's Equations

4.3.1 Polarization and Electric Displacement

4.3.2 Magnetization and Magnetic Field Intensity

4.3.3 Maxwell's Equations in Gaussian Units

4.3.4 Scalar and Vector Potentials

4.4 Problems

5 Linear Piezoelectricity

5.1 Constitutive Laws of Linear Piezoelectricity

5.2 The Initial‐Value Boundary Problem of Linear Piezoelectricity. 5.2.1 Piezoelectricity and Maxwell's Equations

5.2.2 The Initial‐Boundary Value Problem

5.3 Thermodynamics of Constitutive Laws

5.4 Symmetry of Constitutive Laws for Linear Piezoelectricity

5.4.1 Monoclinic Crystals

5.4.2 Monoclinic Crystals

5.4.3 Trigonal Crystals

5.4.4 Hexagonal Crystals

5.5 Problems

6 Newton's Method for Piezoelectric Systems

6.1 An Axial Actuator Model

6.2 An Axial, Linear Potential, Actuator Model

6.3 A Linear Potential, Beam Actuator

6.4 Composite Plate Bending

6.5 Problems

7 Variational Methods

7.1 A Review of Variational Calculus

7.2 Hamilton's Principle

7.2.1 Uniaxial Rod

7.2.2 Bernoulli–Euler Beam

7.3 Hamilton's Principle for Piezoelectricity

7.3.1 Uniaxial Rod

7.3.2 Bernoulli–Euler Beam

7.4 Bernoulli–Euler Beam with a Shunt Circuit

7.5 Relationship to other Variational Principles

7.6 Lagrangian Densities

7.7 Problems

8 Approximations

8.1 Classical, Strong, and Weak Formulations

8.1.1 Classical Solutions

8.1.2 Strong and Weak Solutions

8.2 Modeling Damping and Dissipation

8.3 Galerkin Approximations

8.3.1 Modal or Eigenfunction Approximations

8.3.2 Finite Element Approximations

8.4 Problems

Supplementary Material. S.1 A Review of Vibrations

S.1.1 SDOF Systems

Homogeneous Solutions

Particular Solutions and FRFs

Total Solutions

Generalizations of SDOF Systems

S.1.2 Distributed Parameter Systems

Homogeneous Solutions: Separation of Variables

Eigenvalue Problems in Hilbert Spaces

Particular Solutions and FRFs

S.1.3 MDOF Equations of Motion. Generalized Algebraic Eigenvalue Problems

S.2 Tensor Analysis

S.3 Distributional and Weak Derivatives

Bibliography

Index

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