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PREFACE
INTRODUCTION TO SMART MATERIAL SYSTEMS
TYPES OF SMART MATERIALS
A visual representation of the concept of physical domains and the coupling between them is shown in Figure 1.2. The three vertices of the triangle represent the physical domains, and the connections between the vertices are materials studied in this book that exhibit coupling behavior.
HISTORICAL OVERVIEW OF PIEZOELECTRIC MATERIALS, SHAPE MEMORY ALLOYS, AND ELECTROACTIVE POLYMERS
At the same time that piezoelectric devices were advancing, fundamental research was being done on shape memory alloys and electroactive polymers. Work continued on the development of improved piezoelectric materials in parallel with the fundamental development of shape memory materials and electroactive polymers.
RECENT APPLICATIONS OF SMART MATERIALS AND SMART MATERIAL SYSTEMS
Applications of shape memory alloys have included some of the same engineering systems as those of piezoelectric materials. The use of piezoelectric materials and shape memory alloys for control of a rotating aircraft has also been studied in depth (Figure 1.10).
ADDITIONAL TYPES OF SMART MATERIALS
More important for the development of the textbook is the fact that limiting the scope of the treatment allows us to introduce topics within a general framework based on the analysis of constitutive properties and fundamental thermodynamic principles. Another consequence of the pedagogical approach is the inclusion of numerous homework assignments at the end of each chapter.
SMART MATERIAL PROPERTIES
The smart materials discussed in this book generally fall in the middle of the range of density and modulus values. In these applications, we see that materials that are in the upper right part of the diagram are most desirable.
ORGANIZATION OF THE BOOK
Chapters 4 and 5 present a detailed analysis of piezoelectric materials, starting with the discussion of their constitutive behavior in Chapter 4, which naturally leads to the development of what is called the transducer model of material behavior. A large part of the chapter is devoted to the use of piezoelectric materials as active-passive dampers using electronic shunts and switching state regulation.
SUGGESTED COURSE OUTLINES
An understanding of these topics leads to an understanding of how to analyze power requirements for systems containing piezoelectric materials, shape memory alloys or electroactive polymers. The second semester would place a strong emphasis on the nonlinear constitutive properties of shape memory materials, electrostrictive materials, and nonlinear electroactive polymers.
UNITS, EXAMPLES, AND NOMENCLATURE
The main purpose of the examples is to illustrate the analysis and calculations discussed in the text. A secondary purpose of the examples is to provide the reader with a sense of the numbers involved in common engineering analyzes of smart materials and systems.
MODELING MECHANICAL AND ELECTRICAL
FUNDAMENTAL RELATIONSHIPS IN MECHANICS AND ELECTROSTATICS
- Mechanics of Materials
- Linear Mechanical Constitutive Relationships
- Electrostatics
- Electronic Constitutive Properties of Conducting and Insulating Materials
The direction of the electric field from charge 1 (the negative charge) to the test point is . The electric field is the sum of the electric field due to the individual charges.
WORK AND ENERGY METHODS
- Mechanical Work
- Electrical Work
Another such function is the potential energy of a system, defined as the negative of the energy function U and denoted As shown, the force can be interpreted as the negative of the slope of the potential energy function at the specified elongation.
BASIC MECHANICAL AND ELECTRICAL ELEMENTS
- Axially Loaded Bars
- Bending Beams
- Capacitors
- Summary
The total energy stored in the bar is the product of the strain energy and the volume, AL,. The exact expression for the stored energy will be a function of the loading and the boundary conditions. The total energy stored is the product of equation (2.154) and the volume of the material, Atd,.
ENERGY-BASED MODELING METHODS
- Variational Motion
The need to define a set of small changes in the state variables consistent with the constraints of the problem. A variational change in the state variable is a differential change that conforms to the geometric constraints of the problem. The differential motions in Figure 2.17a and both are valid, but we see that only the differential motion in Figure 2.17 conforms to the constraints of the problem.
VARIATIONAL PRINCIPLE OF SYSTEMS IN STATIC EQUILIBRIUM The definitions of variational motion and its relationship to work enables the devel-
- Generalized State Variables
Finally, the derivation of the variational principle somewhat obscures the way in which the principle is used. For the same reason, the choice of coordinate countryθ could also form a complete and independent set. After expressing displacement in terms of the generalized coordinates, the total potential energy is written as the function.
VARIATIONAL PRINCIPLE OF DYNAMIC SYSTEMS
Now we can interpret the momentum term as simply a force due to the motion of the particle. The terms evaluated at t1 and t2 are equal to zero according to our definition of the variation displacement, and the term in the integral can be rewritten. Substitute the variation of the Lagrangian in equation (2.197) and combine it with the expressions for the generalized mechanical forces and the electrical work yields.
CHAPTER SUMMARY
A set of fixed charges is located in free space as shown in Figure 2.21. a) Calculate the electric field at the origin of the coordinate system. A representative charge density plot is shown in Figure 2.22. a) Calculate the function for the electrical displacement. The dielectric material in the capacitor has a relative dielectric constant of 850 and a surface area of 10 mm2. b) Calculate the charge stored in the capacitor with the given properties.
MATHEMATICAL REPRESENTATIONS
ALGEBRAIC EQUATIONS FOR SYSTEMS IN STATIC EQUILIBRIUM The equations that govern the response of a system in static equilibrium are a set
System outputs are a function of generalized state variables, and the observation matrix defines this functional relationship. In the most general form, the relationship between the observed outcomes, y, and the generalized state variables is In most cases in this book we assume that results can be written as a linear combination of generalized states; therefore, the results can be written in matrix form as
SECOND-ORDER MODELS OF DYNAMIC SYSTEMS
If the equations are linear and there are no terms due to the first derivative of the generalized states, the equations of motion can be written. The ratio of stiffness to mass is denoted k. and is called the unbeaten natural frequency of the system. The solution for the homogeneous undamped multiple-degree-of-freedom (MDOF) case is obtained by assuming a solution of the form
FIRST-ORDER MODELS OF DYNAMIC SYSTEMS
- Transformation of Second-Order Models to First-Order Form Second-order models of dynamic systems can be transformed to first-order form
- Output Equations for State Variable Models
Multiplying equation (3.37) by the inverse of the mass matrix and solving for ˙z2 produces the expression. 3.38) Equation (3.38) is one of two first-order equations required to model the second-order system. It is rare that all internal states of the system can be directly observed at the output of the system; therefore, we must define a second set of equations that express the internal states that can be measured at the output. If the observed outputs can be written as a linear combination of states and inputs, we can write the output expression as
INPUT–OUTPUT MODELS AND FREQUENCY RESPONSE
- Frequency Response
An input-output representation of a dynamic LTI system leads to the concept of frequency response. The output amplitude is scaled by the magnitude of the estimated transfer function ats= jω. Solution Describing the frequency response requires that we obtain an expression for the magnitude and phase of the transfer function.
IMPEDANCE AND ADMITTANCE MODELS
- System Impedance Models and Terminal Constraints
Solution As can be seen from equation (3.104), the entry is the ratio between the flux and the force. Another interpretation of the coupling coefficient is related to the change in impedance from a zero force to a zero flux constraint at location k. Determine the relationship between the output voltage and the input voltage with this constraint on terminal 2.
CHAPTER SUMMARY
Determine the expressions for the static response of the system and analyze the response as α→0 and α→. The equations of motion for the mass-spring system are 2 0. a) Calculate the eigenvalues and eigenvectors of the system. The equations of motion for the mass-spring system are a) Calculate the eigenvalues and eigenvectors of the system.
PIEZOELECTRIC MATERIALS
ELECTROMECHANICAL COUPLING IN PIEZOELECTRIC DEVICES: ONE-DIMENSIONAL MODEL
- Direct Piezoelectric Effect
- Converse Effect
The slope of the curve, called the piezoelectric strain coefficient (Figure 4.2), is indicated by the variable. Consider the application of a constant potential across the electrodes of the piezoelectric material, as shown in Figure 4.3. Dipole rotation will occur and an electrical displacement will be measured at the electrodes of the material.
PHYSICAL BASIS FOR ELECTROMECHANICAL COUPLING IN PIEZOELECTRIC MATERIALS
- Manufacturing of Piezoelectric Materials
- Effect of Mechanical and Electrical Boundary Conditions
- Interpretation of the Piezoelectric Coupling Coefficient
As discussed in Section 4.1, the piezoelectric effect is strongly related to the existence of electric dipoles in the crystal structure of ceramics. Electromechanical coupling in piezoelectric devices results in the fact that material properties are also a function of mechanical and electrical boundary conditions. Note that the piezoelectric strain coefficient is independent of mechanical or electrical boundary conditions.
CONSTITUTIVE EQUATIONS FOR LINEAR PIEZOELECTRIC MATERIAL
- Compact Notation for Piezoelectric Constitutive Equations Equations (4.34) and (4.35) represent the full set of constitutive relationships for a
The components of stress and strain that are normal to the surfaces of the cube are. The compact form of the constitutive relations is based on the fact that the stress and strain tensors are symmetric; therefore,. Visualizing the expression in this way, we see that we can write the compact form of the constitutive equations as a matrix expression, .
COMMON OPERATING MODES OF A PIEZOELECTRIC TRANSDUCER
- Transducer Equations for a 33 Piezoelectric Device
- Piezoelectric Stack Actuator
- Piezoelectric Stack Actuating a Linear Elastic Load
For the case shown in figure 4.14, us=uand f = −klu, where is the stiffness of the load. At the opposite extreme we see that the strength is almost equal to the trapped strength of the stack. The maximum work efficiency of the device is equal to the product of the trapped force and the free displacement.
DYNAMIC FORCE AND MOTION SENSING
- Extensional 31 Piezoelectric Devices
- Bending 31 Piezoelectric Devices
- Transducer Equations for a Piezoelectric Bimorph
- Piezoelectric Bimorphs Including Substrate Effects
The reason is that the free displacement in direction 1 is amplified by the geometry of the transducer. Note that the free stress is approximately three times the maximum stress in the bimorph. Increasing the ratio of length to thickness will produce a trade-off in trapped force and free deflection.
TRANSDUCER COMPARISON
- Energy Comparisons
Equation (4.166) is the same as the expression for the energy density of the material in the 33 mode of operation. Considering the reduction in strain coefficient, we see that the energy density of the piezoelectric bimorph can be only 10 to 20% of the energy density of the stack actuator. The energy density for other electric fields can be obtained by multiplying the value given in the table by the square of the applied electric field in MV/m.
ELECTROSTRICTIVE MATERIALS
- One-Dimensional Analysis
Electrostrictive materials are those in which electromechanical coupling is represented by the quadratic relationship between strain and electric field. In the case where the applied electric field is only in one direction, the constitutive relations are The linear relationship between stress and field produces a mechanical response that will change polarity when the polarity of the electric field changes.
