piezo book

Passive Vibration Attenuation

Viscoelastic Damping, Shunt Piezoelectric Patches, and Periodic Structures

Mohammad Tawfik

Piezoelectric Materials and Structures Piezoelectric Structures: A part of The Smart Structure Family

Passive Vibration Attenuation 2

Contents

1. Piezoelectric Materials and Structures ............................................................................................... 4

1.1. Piezoelectric Structures: A part of The Smart Structure Family .................................................. 4

1.2. Classification of Piezoelectric Structures ..................................................................................... 5

1.2.1. Structures with Surface-Bonded Piezoelectric Patches ........................................................ 5

1.2.2. Structures with Embedded Piezoelectric Laminas ................................................................ 6

1.2.3. Structures with Piezoelectric Fibres ...................................................................................... 6

1.3. Applications of Piezoelectric Structures in Control ..................................................................... 8

1.3.1. Piezoelectric Sensor/Actuator Modeling .............................................................................. 8

1.3.2. Self-Sensing Piezoelectric Actuators ..................................................................................... 9

1.3.3. Passively Shunted Piezoelectrics ......................................................................................... 10

1.4. Modelling of Piezoelectric Structures ........................................................................................ 15

1.4.1. The Electromechanical coupling of Piezoelectric Material ................................................. 15

1.4.2. Simplified 1-D model ........................................................................................................... 15

1.4.3. A Bar with Piezoelectric Patches ......................................................................................... 17

1.5. Finite Element Modelling of Plates with Piezoelectric Actuators .............................................. 22

1.5.1. Displacement Function ....................................................................................................... 24

1.5.2. Strain-Displacement Relation ............................................................................................. 26

1.5.3. Constitutive Relations of Piezoelectric Lamina ................................................................... 27

1.5.4. Stiffness and Mass Matrices of The Element ...................................................................... 28

1.6. Performance Characteristics of a Plate with Shunted Piezoelectric Patches ............................ 30

1.6.1. Overview ............................................................................................................................. 30

1.6.2. Experimental Setup ............................................................................................................. 30

1.6.3. Synthetic Inductor ............................................................................................................... 31

1.6.4. Performance Characteristics ............................................................................................... 32

1.7. Appendices ................................................................................................................................. 44

1.7.1. References and Bibliography ..................................................................................... 44

1.7.2. Constitutive model for 1-3 composites............................................................................... 52

1.7.3. Constitutive model for Active Fibre Composites ................................................................ 57



This book presents an introduction of different techniques used in passive vibration attenuation. The

aim of the book is to give the reader the most important tools needed to understand more details of

the different subjects that may be found in other literature. The author prepared the book based on

lecture notes prepared for a graduate course taught in Cairo University, thus, the book is a step by

step approach to the subjects discussed supported by simple computer based examples that

demonstrate the different topics. It is intended that a reader can read through the book and learn

without the extra support of an instructor or other literature.



1. Piezoelectric Materials and Structures

1.1. Piezoelectric Structures: A part of The Smart Structure Family

Piezoelectric materials belong to a family of engineering materials that is characterized by having the capability of transforming electric energy into strain energy and vice versa. The piezoelectric materials were first reported in the late 19th century, and all the research that was performed on it was regarding their capability of generating electric charges on their surface when mechanical loads are applied on them, that is known as the forward piezoelectric action or the sensing action.

The piezoelectric materials also exhibit what is known as the reverse action, that is, when the piezoelectric material is subjected to an electric field, they undergo mechanical deformations. This is also known as the actuator action.

Due to those two characteristics of the piezoelectric materials, they have been the centre of attention of different researches that were concerned with the sensing and the control of motion of structures. The piezoelectric materials were embedded into the structures and connected with monitoring device to detect motions in those structures. Those sensors were very useful especially in detecting damages in structures or indicating excessive vibration in different locations. On the other hand, the actuators were placed on structures to impose controlled deformations as those utilized by aircraft to modify the aerodynamic shape of the airfoils or for vibration control of different structural elements. A new type of piezoelectric materials was also introduced by embedding piezoelectric fibres in a matrix material to enhance the control or sensing characteristics

The applications and theory of piezoelectric materials have been described in many review articles 1-

5. Wada et al.1, presented one of the earliest review articles. In that article, a classification of adaptive structures is presented dividing these structures into 5 groups as shown in Figure 1.1. These groups include sensory structures incorporating sensors to monitor the dynamics or the health of structures, adaptive structures with attached or embedded actuator elements that influence the dynamics or the shape of the structure; controlled structures involving both sensors and actuators together with a controller; active structures with control elements acting as structural elements; and finally, intelligent structures which are active structures with learning elements.

Figure 1.1. Adaptive structures framework as suggested by Wada et al.1

Crawley2 presented an overview of the general trends in the applications of intelligent structures and classified the requirements of an intelligent structure into four main categories; actuators,

sensors, control methodologies, and controller hardware. Rao and Sunar3 focused their review on

the application of piezoelectric sensors and actuators to structure control. Park and Baz4 reviewed

Piezoelectric Materials and Structures Classification of Piezoelectric Structures


the state of the art of the applications and development of active constrained layer damping (ACLD) technique. In their paper, a broad variety of applications and configurations of ACLD are shown together with a variety of analysis methods.

In a very comprehensive review, Benjeddou5 presented the different methods and the number of papers published in the area of vibration suppression using hybrid active-passive techniques (Figure 1.2). He classified the available literature according to two criteria; the modelling technique, and type of structural elements used. With the aid of sketches and tables, he was able to present a clear picture of the accomplishments, trends, and gaps in the development of active-passive control techniques.

Figure 1.2. Number of papers published on hybrid active-passive damping treatments of structural elements. (Benjeddou

5)

1.2. Classification of Piezoelectric Structures

1.2.1. Structures with Surface-Bonded Piezoelectric Patches

This type of structures is the most common one among all piezoelectric structures. A patch of piezoelectric material is usually bonded to the surface of the structure usually for the purpose of sensing motion or controlling motion. When the base structure vibrates, the bonded piezoelectric patch will move simultaneously producing electric charges on the surface. Those charges are collected by a conductive layer, usually of silver, and then allowed to pass through an electric conductor to a measuring device. This sequence is the sensing sequence.

When electric potential is applied to the surface of the piezoelectric material it undergoes strain. As it is bonded to the surface of a structure, it will simultaneously strain causing the whole structure to move. This sequence is what is known as the actuating sequence.

In both cases, the bonding material, usually epoxy, should withstand the sheer stresses that are generated between the piezoelectric patch and the surface of the structure. Figure 1.3 presents a sketch for a typical piezoelectric sensor-actuator-controller configuration.

0

2

4

6

8

10

12

14

Num

ber o

f Pap

ers

1993 1994 1995 1996 1997 1998

Year of Publication

Beams

Plates

Shells



Figure 1.3. Non-Collocated sensor and actuator.

1.2.2. Structures with Embedded Piezoelectric Laminas

In many applications, piezoelectric patches/laminas are embedded under the surface of the structure. This usually is needed in structures where the applications are sensitive to the outer surface shape like in aircraft. In this case, the piezoelectric material has less bending authority since it becomes nearer to the neutral surface, nevertheless, the sheer stresses that were concentrated on one surface are distributed on two. This definitely reduces the requirements on the bonding material.

1.2.3. Structures with Piezoelectric Fibres

Piezoelectric materials have the highest coupling factor between the strain/stress in one direction and the electric potential/charge on surfaces in the same direction. In the previously mentioned configurations, the coupling is between stress/strain in the plain of the structure and the electric charges/potential on the surfaces parallel to it. It was suggested to embed piezoelectric fibers in the direction parallel to the application of the loads, 1-3 composites, or parallel to the direction of the strain, MFC and AFC.

The main penalties that are imposed by using piezoelectric sensors and actuators is that they are relatively heavy and that the control action they offer is always equal in the two planar directions which restricts the control applications. The active fibre composites concept was introduced to minimize or eliminate both the above-mentioned back draws of the piezoelectric sensors and actuators.



1-3 Piezocomposites

Figure 1.4. A sketch for 1-3 composites

The modelling of the 1-3 piezocomposites drew much of the research attention due to their apparent efficiency as sensors and actuators especially in the sound applications59,60. The formulation of the constitutive equations of the piezoelectric fibre composites in general has imposed a challenge on the researchers in the mechanics of materials field. Models have been developed using three-dimensional finite element analysis were proposed61 and gave accurate results compared to analytical models. Other models were proposed to calculate the effective material properties such as the method of cells proposed by Aboudi62 which is an extension to the original63 and modified64 method of cells.

Smith and Auld65 presented a formulation for the constitutive equations of the 1-3 composites that are suited for the thickness mode oscillations. Their model presented the composite material parameters in terms of the volume fraction and the material properties of the constituent piezoelectric ceramic and matrix polymer that is more or less a formulation similar to the conventional composite material constitutive equations (See Appendix A).

Avellaneda and Swart66,67 studied the effect of the Poisson's ratio of the piezocomposite material on its performance as a hydrophone. In the course of their study, they introduced the hydrostatic electromechanical coupling coefficient and the hydrostatic figure of merit with a great emphasis on the effect of the polymer matrix Poisson's ration. They showed that the reduction of the matrix Poisson's ratio greatly affects the performance and sensitivity of the overall hydrostatic sensor.

Shields et al.68 developed a model for the use of the active piezoelectric-damping composites (APDC), which is based on the use of 1-3 composites. They applied their model for the attenuation of acoustic transmission through a thin plate into an acoustic cavity using active control methods. The results obtained from their finite element model were validated with an experiment that verified the accuracy of the model. They concluded that the use of hard matrix material for the APDC results in higher sound level attenuation. Another important result was the ability to use APDC in the attenuation of low frequency vibrations.

Piezoelectric Materials and Structures Applications of Piezoelectric Structures in Control


Piezoelectric Fibre Composites

Figure 1.5. a sketch of active fiber composites with interdigitated electrodes

Recently, the attention was drawn toward applying the active fibre composites in the planar direction (Error! Reference source not found.). This configuration allows the control of bending and in-plane vibration and torsion (due to the non-orthotropic piezoelectric effect) simultaneously.

Bent69 and Bent and Hagood70,71 introduced a constitutive model for active fibre composites (See Appendix B for more details about the constitutive equations), and applied it with the interdigitated electrodes71 which was introduced earlier by Hagood et al.72. Applying the interdigitated electrodes to piezoelectric fibre composites allowed the use of the higher electromechanical coupling coefficient d33 which in turn provided higher control authority in the plane of actuation.

The piezoelectric fibre composites have not yet been introduced to many applications, though, McGowan et al.73 have introduced the concept of using the active composite for the twist control of rotor blades, and Goddu et al.74 applied it to the control of sound radiation from a cylindrical shell. Bent and Pizzochero75 studied the different factors affecting the manufacturing and performance of the active fiber composites. They demonstrated their effectiveness with applications to helicopter rotor blade harmonic control, tail buffet load alleviation, and torpedo silencing.

1.3. Applications of Piezoelectric Structures in Control

1.3.1. Piezoelectric Sensor/Actuator Modeling

In a review paper, dedicated to piezoelectric sensors and actuators, Chee et al.6 presented a classification of the different mathematical models that simulate the dynamics of these control elements. Linear as well as non-linear piezoelectric constitutive equations have been discussed. Emphasis has also been placed on PZT ceramics, PVDF layers, piezoelectric rod 1-3 composites, piezoelectric fibre composites, and inter-digitated electrode piezocomposites.

Crawley and de Luis7 presented an analytical model for the piezoelectric sensors and actuators that are either surface-bonded or embedded in the structure. The model is limited to Euler-Bernaulli beams and ignored the variation of strain in the piezoelectric material by assuming relatively thin piezoelectric layers (Figure 1.6). It was concluded that the use of segmented actuators is more effective than the use of continuous ones.

Hagood et al.8 presented a derivation of the equations of motion of an arbitrary elastic structure with piezoelectric elements coupled with passive electronics. They used a very important concept when applying their equations, that is; the electro-dynamics of the piezoelectric material are ignored when the material is used as an actuator and the effect of the piezoelectric material on the structure



is ignored when it is used as a sensor. They developed a state-space model and applied it to beams using the Rayleigh-Ritz formulation. Their results were verified experimentally. The concepts developed in that paper are limited only to the case of thin actuators and sensors. But in the case of thick piezoceramic patches attached on the surface of thin plates, ignoring the effect of the piezoceramic actuator/sensor on the dynamics of the plate would certainly produce inadequate results.

Figure 1.6. Strain distribution in a beam with piezoelectric material: (a) surface attached (b) embedded7.

Koshigoe and Murdock9 introduced a formulation for the sensor/actuator associated with plate dynamics together with a shunted active/passive circuit. They then solved the equations of motion in the modal coordinates presenting a simplified analytical formulation for plates with piezoelectric elements. Their model is verified experimentally on a plate using an accelerometer as a sensor and surface bonded PZT patches as actuators.

In a most recent study, Vel and Batra10 presented an analytical method for the analysis of laminated

plates with segmented actuators and sensors. The Eshelby-Stroh formulation is used for the case of plain-strain problem. The inter-laminar stresses for different boundary conditions are presented.

1.3.2. Self-Sensing Piezoelectric Actuators

The concept of self-sensing piezoelectric actuators is based on the simple use of one piezoelectric element as sensor and actuator simultaneously instead of two separate elements. That concept achieves two important goals; first, the reduction of the weight of the piezoelectric elements involved in the structure. Second, it achieves a truly collocated sensor/actuator arrangement which is preferred in control applications as it ensures the stability of the control system (see Figure 1.7, Figure 1.8, and Figure 1.9).

Figure 1.7. Non-Collocated sensor and actuator.

Figure 1.8. Collocated sensor and actuator.



Figure 1.9. Self-sensing piezoelectric actuator.

Dosch et al.11 introduced a formulation for the self-sensing piezoactuator as a special case of collocated sensor/actuators pair. They suggested implementing a complementary circuit (Figure 1.10) to the piezoelectric sensor-actuator to enable the measurement of the sensor potential separately. They presented two different configurations for measuring the strain and the rate of strain. They verified the accuracy of their model with an experiment on the suppression of the vibration of a cantilever beam.

(a)

(b)

Figure 1.10. A sketch of the circuit suggested by Dosch and Inman11 to measure the (a) rate of change of

piezoelectric voltage and (b) voltage.

Anderson et al.12 used similar models for the analysis of the behaviour of a self-sensing piezoactuator. They converted the equations into state-space model and applied the model to a cantilevered beam. They concluded experimentally validated the predictions of the models and

demonstrated the effectiveness of the self-sensing piezoelectric actuators. Vipperman and Clark13 extended their analysis toward the implementation of an adaptive controller. They used a hybrid analogue and digital compensator and implemented the model on a cantilever beam. Their results were verified experimentally.

Dongi et al.14 implemented the concept of self-sensing piezoactuators to the suppression of panel flutter. They used the principle of virtual work to derive a finite element model which is based on the von Karman non-linear strain-displacement relation for a plate. They used different control strategies to ensure high robustness properties.

1.3.3. Passively Shunted Piezoelectrics

The concept of passive shunting is a simple one. As the piezoelectric material can be viewed as a transformer of energy, from mechanical to electric energy and vice versa, a part of the electric energy generated by that transformer could be allowed to flow in a circuit that is connected to the electrodes of the piezoelectric patch. The dissipation characteristics of the shunt circuit would, naturally, be determined by the electric components involved.



The most widely used shunt circuit is that consisting of an inductance and a resistance. That circuit when connected to the piezoelectric patch, acting like a capacitance, would create and RLC circuit which has dynamic characteristics analogous to mass-spring-damper system. If the resonance frequency of the circuit is tuned to some frequency value, the circuit will draw a large value of current from the attached piezoelectric patch at that frequency, that current will be dissipated in the resistance in the form of heat energy; thus, the electromechanical system loses some of its energy through that dissipation process.

The concept of using the piezoelectric material as a member element of an electric circuit that has

dynamically designed characteristics was introduced as early as 1922 by Cady15 for the radio applications. In a review article about shunted piezoelectric elements, Lesieutre16 presented a classification of the shunted circuits into inductive, resistive, capacitive, and switched circuits. He emphasized that the inductive circuits which include an inductor and a resistance in parallel with the piezo-capacitor (Figure 1.11) are the most widely used circuits in damping as they are analogous to the mechanical vibration absorber.

Inductive Resistive Capacitive Switched

Figure 1.11. Configurations of the different shunt circuit.

Hagood and von Flotow17 presented a quantitative analysis of piezo-shunting with passive networks.

They introduced a non-dimensional model that indicates that the damping effect of shunted circuit resembles that of viscoelastic materials (Figure 1.12, Figure 1.13, and Figure 1.14). They applied their model to a cantilever beam and verified the accuracy of the model experimentally. A drawback of the model stems from the fact that the piezoelectric patch with the shunt circuit is assumed to damp vibration even if it was placed symmetrically on a vibration node, thus contradicting the basic properties of the piezoelectric patches as integral elements (Figure 1.15).

Figure 1.12. Mechanical (physical) model of the piezoelectric patch with shunted circuit.

Figure 1.13. Analogous spring-mass-damper model as suggested by Hagood and von Flotow17

.



Figure 1.14. Analogous electrical model as presented by Hagood and von Flotow17

.

Figure 1.15. Sketch to illustrate the dissipation argument.

Different studies18-20 investigated the use of passively shunted piezoelectric patches for vibration

damping using the technique introduced earlier by Hagood and von Flotow17. Law et al.21 presented a new method for analyzing the damping behaviour of resistor-shunted piezoelectric material. Their model is based on the energy conversion rather than the mechanical approach that describes the behaviour of the material as a change in the stiffness (Figure 1.16). Two equivalent models are proposed including: an electrical model (resistance, capacitance, electric sources), and a mechanical model (force, spring, damper). A two-degree of freedom experiment was set up to test the accuracy of the model, and the experimental results were in good agreement with the predictions of the model.

Figure 1.16. The piezoelectric material is used as an energy converter.

Tsai and Wang22,23 applied the concept of using active and passive control to simultaneously damp the vibration of a beam using piezoelectric materials as shown in Figure 1.17. The objective of their study was to answer four questions namely; 1- Do the passive elements always complement the active actions? 2- If the active and passive elements do not always complement each other, should they be separated? 3- Should the active and passive control parameters be selected simultaneously or sequentially? 4- How should the bandwidth of the active passive piezoelectric network (APPN) affect the design? Tsai and Wang presented an analytical formulation for the problem and the control low derivation which is then discretized using the Galerkin method. They concluded that the passive shunt not only provided passive damping but also enhanced the active control authority around the tuned frequency.



Figure 1.17. A sketch of hybrid control for a cantilever beam.

The extension of using shunt circuits for damping multiple vibration modes was also investigated.

Hollkamp24 presented an extension to the analysis of single mode damping formulation to cover

multiple-mode damping by introducing extra circuits in parallel to the initial shunt circuit. He showed that the attempt to damp more than one mode resulted in less damping for each mode than when damping each separately. Nevertheless, the damping of the multiple modes proved to be effective.

Wu25 also investigated the damping of multiple modes using a different configuration of shunt circuits in which sets of resistance and inductance or capacitance and inductance connected in parallel are connected together in series (Figure 1.18). These circuits were designed to provide infinite impedance (anti-resonance) at the design frequencies.

(a) (b)

Figure 1.18. Circuit configurations as suggested by (a) Hollkamp24

and (b) Wu25

.

Recently, different attempts for broadband vibration attenuation were introduced using “Negative-

Capacitance” shunt circuits26-28. The realization and application of the circuit in vibration damping

was also introduced by different patents29-31. The method has proven effective in damping out vibrations over a broadband of frequencies.

As a more practical application of the shunt circuit damping, McGowan32 utilized shunt circuit in damping out the aeroelastic response of a wing below flutter speed. She developed the structural model based on the typical section technique and the aerodynamic model based on Theodorsen’s method. She concluded that the passive control methodology is effective for controlling the flutter of lightly damped structures. Also experimental and analytical study was performed to investigate the effect of using passive shunt circuits for the control of flow induced vibration of turbomachine

blades33. The study concluded the effectiveness of that technique in the attenuation of blade vibration.

Zhank et al.34 presented another application for shunted piezoelectric material by applying it to damping the acoustic reflections from a rigid surface. They used a one-dimensional model to investigate the effectiveness of the model and they concluded that it is a promising application. They also proposed the use of negative capacitance for the same application.



Warkentin and Hagood35 attempted the enhancement of the shunt circuit sensitivity by introducing

non-linear electric elements. They investigated the use of diode and variable resistance elements in the shunt circuits and applied their model to develop a one-dimensional electromechanical circuit model. They concluded that the nonlinear shunt networks had a potential for providing significant advantages over conventional piezoelectric shunts for structural damping.

Davis and Lesieutre36 used a modal strain energy approach to predict the damping generated by

shunted resistance. They introduced a variable that measures the contribution of the circuit to the energy dissipation. This variable depends on the strain induced in the piezoelectric material. Then, they applied the finite element method to determine the effective strain energy. Finally, they presented their results in terms of the conventional loss factor and confirmed their results experimentally.

Saravanos37 presented an analytical solution of the problem of plate vibration with embedded

piezoelectric elements shunted to resistance circuit (Figure 1.19). The study used the Ritz method to solve the resulting coupled electromechanical equations. The paper presents a very good starting point for further development of analytical or numerical methods for the analysis of plates with

shunted piezoelectric elements. Saravanos and Christoforou38 developed a model to investigate the response of a plate under low-velocity impact. The analysis presented is more rigorous than the previously introduced methods as it includes explicitly the circuit dynamics into the equations, thus,

avoiding the problem introduced ealier by Hagood and von Flotow17.

Figure 1.19. Shunted piezoelectric material with composite structures.

Park and Inman39 compared the results of shunting the piezoelectric elements with an R-L circuit connected either in parallel or in Series. They developed an analytical model to predict the behaviour of a beam with a shunted circuit. The predictions of the model are verified experimentally. They noted that the amount of energy dissipated in the series shunting case is directly dependent on the shunting resistance, while in the parallel case, the energy dissipated depends on the inductance and capacitance as well.

Recently, Caruso40 presented a comparative theoretical and experimental study of different shunt circuits. He incorporated the structural damping in his analysis which did increase the complexity of the analysis. However, he modelled the piezoshunted system using the traditional approach as a viscoelastic material.

Piezoelectric Materials and Structures Modelling of Piezoelectric Structures


1.4. Modelling of Piezoelectric Structures

1.4.1. The Electromechanical coupling of Piezoelectric Material

The behaviour of the piezoelectric material, as mentioned before, is characterized by the coupling between the mechanical and the electric states. The constitutive relations of piezoelectric material are presented in many publications76-77. In general, piezoelectric material have 6 components of

mechanical stresses and strains 1, 2, 3, 4, 5, 6, 1, 2,3,4,5,6, respectively, where the components with subscripts 1 through 3 are the normal components while the ones with subscripts 4 through 6 are the sheer components. In addition, each surface of the piezoelectric material have its electric field E and its electric displacement D. E and D are in the direction of the surface. The constitutive relation of the piezoelectric material may be written as:

E

Sd

dDE

Where the components are:

3

2

1

D

D

D

D ,

3

2

1

E

E

E

E ,

6

5

4

3

2

1

,

6

5

4

3

2

1

000

00000

00000

333131

15

15

ddd

d

d

d

E

E

E

EEE

EEE

EEE

E

s

s

s

sss

sss

sss

S

66

44

44

332313

231112

131211

00000

00000

00000

000

000

000

3

1

1

00

00

00

‘E’ is the electric field (Volt/m), ‘s’ (small s) is the compliance; 1/stiffness (m2/N), ‘D’ is the electric displacement, charge per unit area (Coulomb/m2), is the electric permittivity (Farade/m) or (Coulomb/mV), dij is called the electromechanical coupling factor (m/Volt).

1.4.2. Simplified 1-D model

Let’s focus our attention on the case of one dimensional case. The stresses and strains will be taken

as the ones in the ‘1’ direction, while the electric field will be that in the ‘3’ direction. We may then

reduce all the matrices and vectors into scalar quantities.



Recall that the electric displacement is the charge per unit area

A

QD

And that the rate of change of the charge is the current

As

IIdt

AD

1

Where ‘s’ is the Laplace parameter. Also, the electric field is the electric potential difference per unit

length

t

VE

Substituting in the constitutive relations, we get

Vt

sAsAdI

Vt

ds

33131

311111

Introducing the electric capacitance, we get

CsVsAdI 131

Which can also be presented as the electrical admittance (reciprocal of the impedance)

YVsAdI 131

Now, if you focus on the case of open circuit (no current or constant electric displacement), the

equation above may be written as

131

Y

sAdV

Which may be used into the strain equation to get

1

2

311111

tY

Asds

Or

1111

1133

2

31111 1

Ds

s

ds



Which indicates that the effective structure compliance D

s11 will be less (higher stiffness). While for

the case of short circuit (zero impedance or constant electric field) Ess 11 .

On the other hand, when no mechanical strain is applied on the structure, we get the electric

relations as

VYVs

dYI S

1133

2

311

Indicating that the effective admittance is less (higher impedance)

1.4.3. A Bar with Piezoelectric Patches

Now, let us consider the case of a bar with piezoelectric patches attached to both upper and lower

surfaces. In the case when the problem is static, we may have the piezoelectric patch in either a

state of open circuit or open circuit. This produces the simple relations for the bar displacement

differential equation

With the boundary conditions at any side will be

Where the subscripts ‘s’ stands for structure and ‘p’ stands for piezoelectric patches. The modulus of

elasticity of the piezoelectric patches will be

in the case of open circuit and

in

the case of short circuit. Also is a given value for the displacement and ‘P’ is a given value for t he

end load.

If an electric potential is applied on the patch, the problem may be described by the same

differential equation, however the boundary conditions at the end of the piezoelectric patch will be

However, for the bar with the shunted piezoelectric patch, the equations may be found from the

Hamilton’s principle. First we need to rewrite the constitutive relations such that the stress and the

electric voltage are the primary variables.



Now, we get:

Writing down the relation for the total potential energy of the structure with the piezoelectric patch,

we get:

Substituting with the constitutive relations, we get:

Expanding and rearranging the terms,

Applying the variation principles to obtain the first variation



As for the kinetic energy,

Applying the first variation,

The external work exerted on the structure is through the circuit that is shunted to the piezoelectric

patch, thus, we may write:

Finally, applying the Hamilton principle which states that

Applying for each term, we get:



Finally, we may sum up the three terms to get:



Separating the equation above into two terms each multiplied by the variation of one of the

variables, we get the space equation

Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric Actuators


Subject to the boundary conditions

As for the electric equation

Now, we have obtained two coupled partial differential equations in the bar deflection and the

electric displacement as the primary variables. It can be shown, that in the case of harmonic

vibration and absence of electric displacement, when the excitation frequency becomes equal to

that of the electric circuit natural frequency, the mechanical displacement amplitude will essentially

become zero; this is analogous to the problem of the vibration absorber.

Similar derivation for the equation of motion of a beam with piezoelectric patches can be performed

and a similar conclusion will be obtained for the vibration absorber analogy.

1.5. Finite Element Modelling of Plates with Piezoelectric Actuators

It has to be noted that the previously presented literature presented a wide variety of methods to analyze structures with bonded piezoelectric elements. The different methods were applicable in special cases but lacked the generality that can be introduced by numerical methods. However, those analytical approaches paved the way for the development of numerical methods that could be of more practical use. In the following, an introduction is presented to the finite element models used for modelling piezoelectric sensors and actuators for different applications.

Benjeddou41 presented a comprehensive survey of the available literature on the finite element modelling of structures with piezoelectric elements. In that survey, he showed the trend of increasing interest in the field of structural control with piezoelectric elements (Figure 1.20). The common assumptions that are used in the piezoelectric modelling, as pointed out in that paper, were; linear variation of electric potential through thickness, poling direction along the thickness and only longitudinal stress or strain could be induced by monolithic piezoelectric materials, and only the transverse components electric field and displacement are retained.



Figure 1.20. Number of published papers involving finite element modeling of piezoelectric structures. (Benjeddou41

)

Tzou and Tseng42 developed a finite element model for the sensors and actuators attached to the

surface of plates and shells. The finite element they considered was for a thin piezoelectric solid with internal degrees of freedom (DOF). Then, Hamilton principle is used to formulate the dynamics

problem in the finite element form, and Guyan43 reduction to condense the DOF’s associated with

electrical potential. The time response of the system was calculated using the Wilson- method44.

Their results were obtained using two different control lows; constant gain velocity feedback and constant amplitude velocity feedback, and the effect of the feedback gain was illustrated.

Hwang and Park45 introduced a model for the plate elements with attached piezoelectric sensors and

actuators. They used the classical plate theory and the Hamilton principle to develop their model. They introduced four-node quadrilateral non-conforming element. In their paper, they investigated the effect of different piezoelectric sensor/actuator configurations on the vibration control.

Zhou et al.46 extended the finite element model to cover nonlinear regimes using the von Karman non-linear strain-displacement relation and the principle of virtual work. The effects of aerodynamic and thermal loading were added as well. The controller was designed using the LQR method. The equations of motion were transformed to the modal coordinates then cast into a state-space model. They concluded that the piezoelectric-based controller is effective in suppressing the panel flutter.

Later, Oh et al.47 presented a formulation for the post-buckling vibration of plates. Their model was developed using the layer-wise plate theory. In their study, they investigated the phenomena of snapthrough.

Liu et al.48 developed a finite element model for the control of laminated composite plates containing integrated piezoelectric sensors and actuators, rather than attached piezoelectric patches. They built their model using the classical laminated composite plate theory and the principle of virtual displacement, then derived the equations for a four-node non-conforming element. With the use of negative velocity feedback control scheme, they investigated the vibration suppression of a beam and a plate with different piezoelectric embedding configurations.

Several attempts were made to develop finite element models that have higher accuracy by increasing the polynomial order of the elements or by using higher order mechanical modelling to accurately describe the mechanical behaviour of the structure. Further, higher order electrical models were used to accurately describe the non-linear electric field in the piezoelectric material.

Bhattacharya et al.49 developed a finite element model based on the Raleigh-Ritz principle to represent the dynamic behaviour of a laminated plate with piezoelectric layers. They used an eight-node isoparametric quadrilateral element with both structural and electrical degrees of freedom. They applied the first order shear deformation theory. In their results, they presented different configurations of piezoelectric stacking, boundary conditions, and electric voltage application.



Hamdi et al.50 presented a finite element formulation for a beam element with piezoelectric laminas

using the Argyris’ natural mode method51 for the first time. The method is characterized by being free of shear locking problem. They applied the model for the shape control of a beam. They concluded that the proposed formulation is effective in reducing the computational effort with high

accuracy results. Zhou et al.52 presented another development in the finite element models by

introducing a higher order potential field that should accurately describe the field in the

piezoelectric elements. While Peng et al.53 introduced the third order shear theory to their finite

element model to increase the modelling accuracy.

Kim and Moon54,55 presented, for the first time, a finite element formulation for piezoelectric plate

elements with passively shunted circuit elements that incorporated the electric circuit dynamics. They used the Hamilton’s principle to derive the non-linear finite element model. The electric

degrees of freedom of an element were presented as one per node54 or one per element55. They

applied their model for the prediction of plate behaviour subjected to aerodynamic loading (panel-flutter). Their model was based on the von Karman non-linear strain-displacement relations. They compared the results obtained from an active control model using LQR method with those obtained from a passive RL circuit. They concluded that, the suppression using the passive control in not more than that obtained using active control. However, the need of controller, power supplies, and amplifiers for the active control case would reduce its efficiency compared to the passive elements that only require the addition of a resistance and an inductance.

Saravanos56,57 presented a formulation for the finite element problem of a composite shell with

piezoelectric laminas. He proposed the “Mixed Piezoelectric Shell Theory” 56 (MPST) that utilizes the first order shear theory for the displacement and the discrete-layer approximation for the electric potential. He used the Love assumption for shallow shells (radius is much larger than thickness). The model he developed was for an eight-node curvilinear shell element. The model is applied to different cases of composite layouts and geometric boundary conditions and concluded that the model is accurate in predicting the dynamics of the shells. Further; he included a passively shunted

circuit to damp out the vibration of the shells57. Meanwhile, Chen et al.

58 presented a similar finite element formulation but for thin shell elements which presents a special case of the formulation presented by Saravanos.

Later Tawfik and Baz127 presented an experimental and finite element study of the vibration of plates with piezoelectric patches shunted with LR circuits. The study introduced, for the first time, a spectral finite element model for the plate vibration and emphasised the effectiveness of the shunted piezoelectric patches in damping the vibration as well as localization effects when using several ones. On the other hand, Tawfik128 presented the spectral finite element model and compared it to the performance of different other models for plate vibration and confirmed, numerically, that 4 and 9-node C1 elements were adequate for the modelling of the problem.

In the following subsections, the derivation procedure of the finite element model with any number of nodes and shunted piezoelectric patches will be presented.

1.5.1. Displacement Function

The numerical construction of the propagation surfaces, which will be introduced later, requires high order elements [97]. Thus, a 16-node element is considered (Figure 1.21), with 4 DOF per node which provides a full 7th order interpolation function.



Figure 1.21. Sketch of the 16-node element.

The transverse displacement w(x,y), at any location x and y inside the plate element, is expressed by

(1)

where wH is a 64 element row vector and {a} is the vector of unknown coefficients. For the plate

element under consideration, the bending degrees of freedom associated with each node are

64

2

1

2

,,

,

,

a

a

a

H

H

H

H

yx

w

y

wx

ww

yx

y

x

w

w

w

w

(2)

where Hw,i is the partial derivative of Hw with respect to i. Substituting the nodal coordinates

into equation (13), the nodal bending displacement vector {wb} is obtained as follows,

(3)

where

3/2,3/

0,0

0,0

0,0

0,0

][&

,,

,,

,

,

16

2

1

2

1

1

1

baH

H

H

H

H

T

yx

w

yx

w

y

wx

w

w

w

yx

yx

y

x

w

w

w

w

w

bb

(4)

From equation (14), we can obtain

(5)

Substituting equation (16) into equation (12) gives

aHyxw w),(

aTw bb

bb wTa1



(6)

where [Nw] is the shape function for bending given by

(7)

Similarly, the electric displacement associated with the piezoelectric patch could be written in the form

(8)

where DH is a 16 element row vector with its terms resulting from the multiplication of two 3rd

order polynomials in both x and y-directions and {b} is the vector of unknown coefficients.

Substituting the nodal coordinates into equation (19), we obtain the nodal electric displacement vector {wD} in terms of {b} and following the same procedure as for the mechanical degrees of freedom, we get,

(9)

where [ND] is the shape function for electric displacement given by

(10)

1.5.2. Strain-Displacement Relation

Consider the classical plate theory, for the strain vector {} can be written in terms of the lateral deflections as follows

z

xy

y

x

(11)

where z is the vertical distance from the neutral plane and { } is the curvature vector which can be written as,

(12)

where

bwbbw wNwTHyxw 1

),(

1 bww THN

bHyxD D),(

DDDDD wNwTHyxD 1

),(

1 DDD THN

}{

22

2

2

2

2

aC

yx

w

y

wx

w

b



(13)

Substituting equation (17) into equation (23), gives

(14)

where

(15)

Thus, the strain-nodal displacement relationship can be written as

(16)

1.5.3. Constitutive Relations of Piezoelectric Lamina

The general form of the constitutive equation of the piezoelectric patch are written as follows

(17)

where, are the stress in the x-direction, stress in the y-direction, and the planar shear

stress respectively; are the corresponding mechanical strains; D is the electric

displacement (Culomb/m2), is the electric field (Volt/m), piezoelectric material constant

relating the stress to the electric field, is the material dielectric constant at constant stress

(Farad/m), and is the mechanical stress-strain constitutive matrix at constant electric field.

is given by,

where E is the Young’s modulus of elasticity at constant electric field, and is the Poisson’s ratio.

Equation (28) can be rearranged as follows

xy

yy

xx

w

w

w

b

H

H

H

C

,

,

,

2

}{}{1

bbbbb wBwTC

1 bbb TCB

bb wBzz }{

E

e

eQ

D

xy

y

x

T

E

xy

y

x

xyyx ,,

xyyx ,,

E e

EQ

EQ

1200

011

011

22

22

E

EE

EE

QE



D

e

eeeQ

E

xy

y

x

T

TE

xy

y

x

(18)

or

(19)

and

(20)

where .

1.5.4. Stiffness and Mass Matrices of The Element

The principal of virtual work states that

(21)

where is the total energy of the system, U is the strain energy, T is the kinetic energy, W is the

external work done, and (.) denotes the first variation.

The Potential Energy

The variation of the mechanical and electrical potential energies is given by

(22)

where V is the volume of the structure. Substituting equation (30) and (31) into equation (33) gives,

(23)

Substituting from equations (20) and (27), we get,

(24)

The terms of the expansion of equation (35) can be recast as follows

DeQ

xy

y

x

D

xy

y

x

DeE

xy

y

x

T

1

0 WTU

VV

TdVEDdVU

V

T

V

DTdVDzeDdVDezQzU

V

DDbb

TT

DD

V

DDbb

DT

bb

dVwNwBzewN

dVwNewBzQwBzU



,

,

,

and ;

where [kb] is bending stiffness matrix, [kbD] is bending displacement-electric displacement coupling matrix, and [kD] is the electric stiffness matrix.

The Kinetic Energy

The variation of the kinetic energy T of the plate/piezo patch element is given by,

(25)

where is the density/equivalent density and h is the thickness of the element. The above equation can be rewritten in terms of nodal displacements as follows

(26)

where [mb] is the element bending mass matrix.

The external work

The variation of the external work done exerted by the shunt circuit is given by

A

dAqDLW (27)

where A is the element area, L is the shunted inductance, and q is the charge flowing in the circuit. But, as the charge is the integral of the electric displacement over the element area; then equation (38) reduces to,

AA

dADLDdAW (28)

Substituting from equation (20), gives

A

DD

A

T

D

T

D dAwLNdANwW (29)

bb

T

b

V

bb

DT

bb wkwdVwBQwBz 2

DbD

T

b

V

DD

T

bb wkwdVwNewBz

b

T

bD

T

DbDb

T

D

V

bb

TT

DD wkwwkwdVwBzewN

DD

T

D

V

DD

T

DD wkwdVwNwN

A

dAt

whwT

2

2

bb

T

b

A

bw

T

w

T

b

A

wmwdAwNNwhdAt

whw

2

2

Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric Patches


which can be recast in the following form,

DD

T

D wmwW (30)

where [mD] is the element electric mass matrix.

Finally, the element equation of motion with no external forces can be written as

0

0

0

0

D

b

DDb

bDb

D

b

D

b

w

w

kk

kk

w

w

m

m

(31)

1.6. Performance Characteristics of a Plate with Shunted Piezoelectric

Patches

1.6.1. Overview

In sections 1.4 and 1.5, the foundations required to handle the problem of plates with shunted piezoelectric patches were laid. Different finite element models were developed to handle the different aspects of the problems.

This section presents experimental performance characteristics of a plate with shunted piezoelectric networks. The experiments aim at monitoring the modal parameters of the plate using scanning laser vibrometer (Polytec PI –V2000, Auburn, MA).

The modal parameters considered are the natural frequencies and mode shapes. These experimental parameters are used to validate the predictions of the finite element model presented in section 1.5.

The experiments aim also at monitoring the frequency response of the plate when it is controlled first with only two shunted piezoelectric patches which are arranged in a non-periodic manner. Then the frequency response is monitored when the plate is provided with nine shunted piezoelectric patches organized in periodic manner over the plate surface.

The obtained results are compared with those recorded when patches are not shunted. Such comparisons are essential to quantify, in general, the passive damping imported to the plate due to the shunting. For the case of the periodic arrangement, the experiments aim at demonstrating the localization effects when the patches are non-uniformly shunted. Finally, the propagation surfaces of a plate with partial coverage with a piezoelectric patch are going to be presented as a natural expansion of the models developed earlier.

1.6.2. Experimental Setup

An experiment was set up and conducted on a square plate clamped from all sides. The aluminium

(6061 alloy) plate has the following properties: modulus of elasticity (E) 71 GPa, Poisson’s ratio ()

0.3, density () 2700 kg/m3, length 0.507 m, and thickness 1 mm. Symmetric piezoelectric square patches (model T110-H4E-602 Piezo Systems Inc.) were bonded on two positions of the plate. The

piezoceramic properties are: modulus of elasticity (E) 68 GPa, Poisson’s ratio () 0.3, density () 7800 kg/m3, length 0.073 m, thickness 0.27 mm, dielectric constant ( ) 2.37*10-8 Farad/m, and piezoelectric coefficient (d) -320*10-12 m/V.



The plate is excited using an electro-mechanical speaker (model TS-W26C, 350W Woofer, Pioneer, Japan) (Figure 1.22) driven by a power amplifier (model PA7E, Wilcoxon Research), and the resulting response is measured with an accelerometer (model 357C10, PCB, Depew, NY). The excitation function and the accelerometer output signal are processed using spectrum analyzer (model SR780, SRS, Sunnyvale, CA) (Figure 1.23).

Figure 1.22. A picture of the speaker used to excite the plate.

Figure 1.23. A picture of the Spectrum analyzer.

1.6.3. Synthetic Inductor

The values of inductance required to create resonating shunt circuit for the damping purposes are always higher than those available commercially. Thus, synthetic inductors are used instead. Several versions of these synthetic inductors are used in the various resonating circuits employed in structural damping24,39. The version used in this study is sketched in Figures 5.3 and 5.4 as presented by Chen125. This configuration was selected after proving to be more stable in maintaining the inductance value it is tuned to when compared to another design suggested in literature39.

Figure 1.24. A schematic of the synthetic inductor circuit.



Figure 1.25. Shunting network used in present study.

To tune and measure the performance characteristics of the circuit, it was connected to a capacitance, to present the piezo-patch, and the frequency response of the circuit was measured using the spectrum analyzer (Figure 1.26).

Figure 1.26. Connection sketch for the circuit performance analysis.

For the experiment purpose, the synthetic inductor circuit was realized using a 1458 dual amplifier IC

with R1=R3=R4=10 k, C=10 nF, and R2=50 k potentiometer, while the resistance connected in

series with the inductor was a 10 k potentiometer.

1.6.4. Performance Characteristics

Numerical vs. Analytical Prediction

A case study for the verification of the prediction of the finite element model was considered for a plate with different boundary conditions. These conditions include clamped from all sides (CCCC), cantilever (CFFF), clamped from two opposite sides and free from the other two (CFCF), and simply supported from all sides (SSSS). The plate aspect ratio is 1 and Poisson’s ratio is 0.3. The model

predictions of the frequency parameter, , where L is the plate length for a square plate,

for different modes for the four different boundary conditions using a 7x7 uniform mesh are presented in Table 1.6.1. The predictions are compared with the analytical predictions presented by Leissa124 and the results obtained from a finite element model using traditional polynomial interpolation functions. (Bogner-Fox-Schmidt (BFS) C1 conforming element125)

The presented results demonstrate the high accuracy of the developed finite element model. A maximum relative error of 2.82% was obtained for mode (1,1) for the case of CCCC plate.

Table 1.6.1. Comparison of numerical and analytical results for the frequency parameter of the four different test cases (Poisson’s ratio = 0.3)

Mode # Analytical

Spectral BFS

Frequency % Error Frequency % Error

SSSS

1,1 19.75 19.85 0.53 19.33 -2.12

1,2 49.32 49.37 0.11 49.11 -0.42

2,2 78.99 78.89 -0.13 77.84 -1.46

3,1 98.74 98.74 0.00 101.35 2.65

PDL /



3,2 128.31 128.00 -0.24 127.48 -0.65

4,1 167.81 168.23 0.25 169.53 1.03

3,3 177.63 176.32 -0.74 174.23 -1.91

CCCC

1,1 35.11 35.79 1.93 33.70 -4.02

1,2 72.93 72.88 -0.07 70.27 -3.65

2,2 107.52 106.84 -0.63 104.75 -2.58

3,1 131.65 131.65 0.00 132.18 0.40

3,2 164.36 162.17 -1.34 159.87 -2.73

4,1 210.33 210.80 0.22 211.07 0.35

3,3 219.32 215.25 -1.86 213.68 -2.57

CFCF

1 22.17 21.68 -2.20 20.11 -9.27

2 43.60 42.84 -1.74 44.67 2.45

3 120.10 117.81 -1.91 120.16 0.05

4 136.90 136.62 -0.21 N/A

5 149.30 145.50 -2.55 146.28 -2.02

CFFF

1 3.49 3.40 -2.70 3.40 -2.70

2 8.55 8.36 -2.23 8.88 3.88

3 21.44 21.94 2.34 21.16 -1.31

4 27.46 27.17 -1.07 29.00 5.59

5 31.17 30.56 -1.95 31.87 2.24

Experimental Results with Two Piezo-Patches

Different experiments were conducted on the plate setup described in section 1.6.2. Two piezoelectric patches were bonded to the plate as shown in Figure 1.27. All the results showed very high effectiveness of the proposed damping circuit in reducing the amplitude of vibration of the targeted frequency.

Figure 1.27. A sketch of the plate with dimensions.

For the purpose of comparison of the numerical and experimental models, the numerical model is modified to accommodate the effect of the flexible boundary conditions. Also, the material damping ratio was tuned for each mode for the purpose of matching the experimental results. The experiments were conducted by exciting the plate using the speaker and a sine sweep function generated by the analyzer. The damping was applied by attaching the central PZT patch to the synthetic inductor (Figure 1.24) in series with a resistance.



Contour plots of the different modes of vibration of the plate together with picture generated by the laser vibrometer for the shape of the plate as being exited at a frequency equal to that of the natural frequency are presented in Figures 5.7 through 5.14.

Numerical

Experimental

Figure 1.28 A contour plot of mode (1,1) and picture of the same mode .

Numerical

Experimental


Numerical

Experimental




Numerical

Experimental


Numerical

Experimental


Numerical

Experimental




Numerical

Experimental


Numerical

Experimental


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Numer. Closed Circuit



(b)

Figure 1.36. Comparison of (a) numerical and (b) experimental results for damped and undamped cases around mode (3,1).

(a)

(b)

Figure 1.37. Comparison of (a) numerical and (b) experimental results for damped and undamped cases around mode (3,3).

The modes targeted for damping were the (3,1) mode at 111 Hz and the (3,3) mode at 195 Hz. Figures 5.15 and 5.16 present a comparison between the experimental results obtained with the accelerometer placed at the centre of the plate with those predicted by the developed finite element model. Reduction in the vibration amplitude of 7 dB was obtained at mode (3,1) and 12 dB at mode (3,3). The numerical model predicted 6 dB at mode (3,1) and 8 dB at mode (3,3).The obtained results indicate close agreement between the numerical prediction and the experimental results for modes (3,1) and (3,3) for open circuit cases.

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Experimental Results with Nine Periodic Piezo-Patches and Speaker Excitation

Another set of experiments were conducted with all the nine piezoelectric patches attached to the plate. Figure 1.38 presents a schematic drawing of the plate with all the piezoelectric patches attached to it and numbered 1 to 9. Four other locations of interest are marked on Figure 1.38 and given numbers 1 to 4.

Figure 1.38. A sketch presenting the location and numbering of the piezoelectric patches and the four points of interest.

The plate was excited with the speaker as in the previous set of experiments and measurements were made with an accelerometer placed at point #3. Figure 1.39 presents the response to sine sweep excitation with and without all the circuits connected to the patches. The circuits were tuned to maximize the damping of mode (3,3) (192 Hz). It is obvious from the displayed results that when all the circuits were connected, damping is obtained over a broad band.

Note the approximately 20dB attenuation obtained for the frequency-band of 140-220 Hz. Also, more than 5 dB reduction at 112 Hz, 4 dB at 276 Hz and another attenuation band in the range 310-350 Hz. The broadband attenuation characteristics of the results are very promising for further study.

Figure 1.39. Response of the plate with the all the patches attached.

To investigate the effect of introducing disorder in the system on localizing the vibration, the disconnection of each of the circuits was investigated. The resulting response is compared to the case when all the circuits are connected to see how much the vibration increases or decreases at a certain mode by disconnecting that patch. Figures 5.19 through 5.27 show the effect of disconnecting each of the patches’ circuits one at a time.

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All Off All On



Figure 1.40. The resulting response at point #3 when patch #1 is switched off compared to when all the patches are turned on.




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Figure 1.48 The resulting response at point #3 when patch #9 is switched off compared to when all the patches are turned on.

Investigating the disorder effects demonstrated in the previous figures, it was decided to select three patches to show the localization effect at the sensor location and other three patches to localize the vibration away from the sensor location. In the first case, patches numbers 1, 2, and 4 were disconnected from their corresponding circuits (Note that those patches are the farthest from location #3 (see Figure 1.38)). Figure 1.49 presents the response of the accelerometer at point #3 when patches 1,2, and 4 are disconnected. It can be seen very clearly how the response amplitude dropped in two bands each due to this intentional electronic aperiodicity.

Figure 1.49. Localizing the vibration away from the sensor location by disconnecting patches 1,2&4.

The second case was set up to investigate the ability to localize the vibration at point #3 where the accelerometer is placed. Patches 6, 8, and 9 where disconnected to localize the vibration at the point where the accelerometer is located. Figure 1.50 presents the results when the patches nearest to the accelerometer where disconnected from the corresponding circuits.

Figure 1.50. Localizing the vibration at the sensor location by disconnecting patches 6,8,&9.

The results presented in Figure 1.49 and Figure 1.50 clearly indicates the ability of the configuration to localize the vibration energy. Further, the localization is controlled by selection of the circuits to be turned on or off.

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Experimental Results with Nine Periodic Piezo-Patches and Point Excitation

In the previous set of experiments, the results of the numerical model could not be compared to those of the experimental model as the acoustic response load distribution resulting from the excitation speaker was not known accurately. Thus, another set of experiments were conducted using point excitation. A piezoelectric stack (model AE0203D08, Tokin Inc., Union City, CA) with a 12 gm proof mass (Figure 1.51) was placed at point #4 and an accelerometer was placed at the same point to measure its vibration as an input to the system. The output measurements were taken using another accelerometer at point #2. The measurements from point #4 were then used as a boundary condition for the spectral element model to ensure that both models have matching inputs.

Figure 1.51. The piezoelectric stack with the proof mass attached to the plate.

Figure 1.52 presents the numerical and experimental results with one patch connected to its corresponding circuit; which is the nearest to the measuring accelerometer (patch #3). It is obvious that the effect in reducing the vibration at that particular point is dramatic (~25 dB).

Figure 1.53 presents the numerical and the experimental results when all the circuits were turned on and tuned to damp mode (3,3). It is seen that the resulting damping characteristics are, again, those of a broad band, but of smaller bandwidth. The ability of the numerical model to predict the response of the plate could be noticed by comparing the results of Figure 1.53.

(a)

(b)

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Figure 1.52. Comparison of numerical and experimental results with only patch #3 connected to the shunt circuit.

(a)

(b)

Figure 1.53. Comparison of (a) experimental and (b) numerical results for both; numerical nad experimental results. (All circuits connected)

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)

Numer. Open Circuit


Piezoelectric Materials and Structures Appendices


1.7. Appendices

1.7.1. References and Bibliography

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VA, American Institute of Aeronautics and Astronautics, 1999, p. 2681-2694.

124. Leissa,A.,“VibrationofPlates,”2nd edition, Acoustical Society of America, 1993.

125. Chen,W., “Passive andActive Filters: Theory and Implimentation,” 1st edition, John

Wiley and Sons, 1986.

126. Zienkiewicz, O. C. and Taylor, R. L., “The Finite ElementMethod,” 4th

ed, Vol. 2,

McGraw-Hill Book Company, London, 1991.



127. Tawfik, M. and Baz,A. , "Experimental and Spectral Finite Element Study of Plates with

Shunted Piezoelectric Patches" International Journal of Acoustics and Vibration, Vol. 9,

No. 2, pp87-97, 2004.

128. Tawfik, M. ,“ASpectralFiniteElementModelforThinPlateVibration,”International

Congress on Sound and Vibration (ICSV14), Cairns, Australia, 9-12 July 2007.



1.7.2. Constitutive model for 1-3 composites

(The following derivation is a detailed one based on that presented by Smith and Auld65)

For the polymer

3113

2112

1111

6446

5445

4444

3112121123

3122111122

3122121111

PP

PP

PP

PP

PP

PP

PPPP

PPPP

PPPP

ED

ED

ED

ScT

ScT

ScT

ScScScT

ScScScT

ScScScT

For the piezo

3333332311313

2114152

1115151

6666

1155555

2154444

3333332321313

3313232221212

3313132121111

CSCCCC

CSCC

CSCC

CEC

CCEC

CCEC

CCECECEC

CCECECEC

CCECECEC

ESeSeSeD

ESeD

ESeD

ScT

EeScT

EeScT

EeScScScT

EeScScScT

EeScScScT

Assumption #1: Strain and electric field are independent of (x,y) (1,2)

Assumption #2: Large thin electroded plate

1- Symmetry in x-y plane: c11=c22, c31=c32, S1=S2

2- E1=E2=0

3- S4=S5= S6=0

For the polymer

3113

3111123

2312112111

2

PP

PPP

PPPP

ED

ScScT

TScSccT



For the piezo

3333331313

3333331313

2331313112111

2

2

CSCCC

CCECEC

CCCECEEC

ESeSeD

EeScScT

TEeScSccT

Assumption #3: Polymer and ceramic have same strain and electric field in the oscillation direction (z)

333 SSS CC

333 EEE CC

Giving

3333331313

3333331313

2331313112111

3113

3111123

2312112111

2

2

2

ESeSeD

EeScScT

TEeScSccT

ED

ScScT

TScSccT

SCC

ECEC

CECEEC

P

PP

PPP

Assumption #4: Lateral stresses are equal

Assumption #5: The resultant lateral strain is zero (laterally clamped)

01 111

111

CP

CP

vSSvS

TTT

Where v is the ceramic volume fraction. From The above assumption

01 11

3313123131121111211

CP

ECEEP

vSSv

EeScScSccScc

Which can be written as

01 11

1211

3313121311

1211

1211

CP

EE

ECP

EE

vSSv

cc

EeSccvvSS

cc

ccv

Adding

1211

3313121311

1211

1211 1EE

EPP

EE cc

EeSccvSvS

cc

ccv



Solving, gives

12111211

331313131

1 EE

EP

ccvccv

EeSccvS

12111211

331312131

11

EE

EC

ccvccv

EeSccvS

The strain stress relations can be written as

1

1211

13313131

1

1211

3121

C

EE

E

P

Scc

EeScT

Scc

ScT

From which

01

01

1211

3121

1211

13313131

11

cc

ScTv

cc

EeScTv

SvvS

EE

E

PC

Giving

1211

312

1211

1331313

12111211

12111211

1 11 cc

Scv

cc

EeScv

ccvccv

ccccT

EE

E

EE

EE

3

12111211

121131

312111211

121112121113

1

1

1

1

Eccvccv

ccve

Sccvccv

cccvccvcT

EE

EE

EEE

Which can be written in the form

3313131 EeScT

Where

12111211

12113131

12111211

12111212111313

1

1

1

EE

EE

EEE

ccvccv

ccvee

ccvccv

cccvccvcc

For the 3-direction:



33333312111211

33131213

313

333333

12111211

33131213313

3113

33312111211

33131213

313

112

112

12

ESeccvccv

EeSccveD

EeScccvccv

EeSccvcT

ED

Scccvccv

EeSccvcT

S

EE

EC

E

EE

EEC

P

EE

EP

Collecting terms of S3 and E3, gives

3

12111211

2

3133

312111211

131331333

33312111211

3131

312111211

131331333

3113

312111211

3112

33312111211

1213313

1

12

1

12

1

12

1

12

1

2

1

2

Eccvccv

ve

Sccvccv

ccveeD

Eeccvccv

evc

Sccvccv

ccvccT

ED

Eccvccv

vec

Scccvccv

ccvcT

EE

S

EE

EC

EE

E

EE

EEEC

P

EE

EE

EP

Assumption #6: The lateral periodicity is sufficiently fine (averages are acceptable)

333

333

1

1

CP

CP

vDDvD

vTTvT

3

12111211

3112

33312111211

121312

33312111211

3131

312111211

121331333

333

1

21

1

21

1

12

1

12

1

Eccvccv

vecv

Scccvccv

ccvcv

Eeccvccv

evcv

Sccvccv

ccvccvT

TvvTT

EE

EE

E

EE

E

EE

EEE

PC



311312111211

2

3133

312111211

121331333

333

11

12

1

12

1

EvEccvccv

vev

Sccvccv

ccveevD

DvvDD

EE

S

EE

E

PC

312111211

2

311133

312111211

121331333

33312111211

311213

312111211

2

1213

33333

1

121

1

12

1

12

1

121

Eccvccv

vvevv

Sccvccv

ccveevD

Eveccvccv

evvcc

Sccvccv

ccvvcvvcT

EE

S

EE

E

EE

E

EE

EE

Which can be written as

3333333

3333333

ESeD

EeScT

S

E

Where

12111211

2

31113333

3312111211

31121333

12111211

2

1213

333333

1

121

1

12

1

121

EE

SS

EE

E

EE

EEE

ccvccv

vvevv

veccvccv

evvcce

ccvccv

ccvvcvvcc



1.7.3. Constitutive model for Active Fibre Composites

(The following derivation is a detailed one based on that presented by Bent69,70,71)

According to the assumptions introduced by Bent, the behavior of the piezoelectric fiber composites can be described in each of the three directions separately.

For the Piezo, we have

xCS

zC

yC

xC

xC

xC

zCE

yCE

xCE

zC

xC

zCE

yCE

xCE

yC

xC

zCE

yCE

xCE

xC

ESeSeSeD

EeScScScT

EeScScScT

EeScScScT

33313133

31111213

31121113

33131333

For the polymer, we have

xP

xP

zP

yP

xP

zP

zP

yP

xP

yP

zP

yP

xP

xP

ED

ScScScT

ScScScT

ScScScT

11

111212

121112

121211

For Case B, recognize the variables that are equal in both phases as independent variables. The

independent variable would be xzyx ESTS ,,, .

zP

xP

yP

yP

zC

Ez

C

E

E

xC

E

E

yC

Ey

C

Sc

cS

c

cT

cS

Ec

eS

c

cS

c

cT

cS

11

12

11

12

11

11

31

11

12

11

13

11

1

1



For the Piezo, we can write

x

z

y

x

E

SE

E

EE

EE

EE

E

EE

E

EE

E

E

E

EEEE

EE

E

EE

E

E

EE

E

EEEE

E

E

E

EEE

xC

zC

yC

xC

E

S

T

S

c

ec

c

ecec

c

e

c

ecec

c

ecec

c

cc

c

c

c

cccc

c

e

c

c

cc

cc

ecec

c

cccc

c

c

c

ccc

D

T

S

T

11

2

313311

11

31123311

11

31

11

31133111

11

31113112

11

2

12

2

11

11

12

11

13121113

11

31

11

12

1111

13

11

33113113

11

12131113

11

13

11

2

131133

1

For the polymer, we can write

x

z

y

x

xP

zP

yP

xP

E

S

T

S

c

cc

c

c

c

ccc

c

c

cc

c

c

ccc

c

c

c

cc

D

T

S

T

11

11

2

12

2

11

11

12

11

2

121211

11

12

1111

12

11

2

121211

11

12

11

2

12

2

11

000

0

01

0

For the dependent variables, the values are obtained using weighted addition

xP

zP

yP

xP

p

y

xC

zC

yC

xC

C

y

x

z

y

x

D

T

S

T

v

D

T

S

T

v

D

T

S

T

Substituting,

x

z

y

x

p

y

E

SE

E

EE

EE

EE

E

EE

E

EE

E

E

E

EEEE

EE

E

EE

E

E

EE

E

EEEE

E

E

E

EEE

C

y

x

z

y

x

E

S

T

S

c

cc

c

c

c

ccc

c

c

cc

c

c

ccc

c

c

c

cc

v

c

ec

c

ecec

c

e

c

ecec

c

ecec

c

cc

c

c

c

cccc

c

e

c

c

cc

cc

ecec

c

cccc

c

c

c

ccc

v

D

T

S

T

11

11

2

12

2

11

11

12

11

2

121211

11

12

1111

12

11

2

121211

11

12

11

2

12

2

11

11

2

313311

11

31123311

11

31

11

31133111

11

31113112

11

2

12

2

11

11

12

11

13121113

11

31

11

12

1111

13

11

33113113

11

12131113

11

13

11

2

131133

000

0

01

0

1



Which we need to be cast in the following form,

x

z

y

x

effeffeffeff

effeffeffeff

effeffeffeff

effeffeffeff

x

z

y

x

E

S

S

S

eee

eccc

eccc

eccc

D

T

T

T

33313233

31111213

32122223

33132333

Solving for the stress in the y-direction,

zyx

p

yxEzE

E

yExE

EC

yy Sc

cT

cS

c

cvE

c

eS

c

cT

cS

c

cvS

11

12

1111

12

11

31

11

12

1111

13 11

xE

C

yyE

C

y

y

p

y

z

p

yzE

EC

yx

p

yxE

EC

yy Ec

evT

c

vT

c

vS

c

cvS

c

cvS

c

cvS

c

cvS

11

31

111111

12

11

12

11

12

11

13

xEp

y

C

y

C

y

zEp

y

C

y

Ep

y

EC

y

yEp

y

C

y

E

xEp

y

C

y

Ep

y

EC

y

y

Ecvcv

cev

Scvcv

ccvccvS

cvcv

ccS

cvcv

ccvccvT

1111

1131

1111

12111112

1111

1111

1111

12111113

1111

12111113

32Ep

y

C

y

Ep

y

EC

yeff

cvcv

ccvccvc

1111

111122

Ep

y

C

y

Eeff

cvcv

ccc

1111

12111112

12Ep

y

C

y

Ep

y

EC

yeff

cvcv

ccvccvc

1111

113123

Ep

y

C

y

C

yeff

cvcv

ceve

For the stress in the x-direction

xE

EEC

yy

p

yE

EC

y

z

p

yE

EEEEC

yx

p

yE

EEEC

yx

Ec

ececvT

c

cv

c

cv

Sc

cccv

c

ccccvS

c

ccv

c

cccvT

11

33113113

11

12

11

13

11

2

121211

11

12131113

11

2

12

2

11

11

2

131133

Simplifying



xE

EEC

y

yE

Ep

y

EC

y

zE

Ep

y

EEEEC

y

xE

Ep

y

EEEC

y

x

Ec

ececvT

cc

ccvccv

Scc

ccccvcccccv

Scc

cccvccccvT

11

33113113

1111

12111113

1111

2

121211111213111311

1111

2

12

2

1111

2

13113311

Substituting with the value of the stress in the y-direction,

xE

EEC

y

xEp

y

C

y

C

y

zEp

y

C

y

Ep

y

EC

y

yEp

y

C

y

E

xEp

y

C

y

Ep

y

EC

y

E

Ep

y

EC

y

zE

Ep

y

EEEEC

y

xE

Ep

y

EEEC

y

x

Ec

ececv

Ecvcv

cev

Scvcv

ccvccv

Scvcv

ccS

cvcv

ccvccv

cc

ccvccv

Scc

ccccvcccccv

Scc

cccvccccvT

11

33113113

1111

1131

1111

12111112

1111

1111

1111

12111113

1111

12111113

1111

2

121211111213111311

1111

2

12

2

1111

2

13113311

Extracting the effective coefficients,

1111

12111113

1111

12111113

1111

2

12

2

1111

2

1311331133

Ep

y

C

y

Ep

y

EC

y

E

Ep

y

EC

y

E

Ep

y

EEEC

yeff

cvcv

ccvccv

cc

ccvccv

cc

cccvccccvc

1211111312111113

2

12

2

1111

2

131133111111

11111111

33

1

ccvccvccvccv

cccvccccvcvcv

cvcvccc

Ep

y

EC

y

Ep

y

EC

y

Ep

y

EEEC

y

Ep

y

C

y

Ep

y

C

y

E

eff



2

121112111113

11111111

11121113

2

1113

11111111

2

12

2

112

11

2

11

2

1211113

1111

11111111

2

131111

2

131111111111

33

33

1

1

1

1

ccvccccvvcvcvcc

ccccvvccvcvcvcc

ccvvccvv

cccvvccvv

cvcvcc

cccvvccvcvcvcc

vc

c

Ep

y

EEp

y

C

yEp

y

C

y

E

EEp

y

C

y

EC

yEp

y

C

y

E

Ep

y

p

y

Ep

y

p

y

EC

y

p

y

EC

y

p

y

Ep

y

C

y

E

EEp

y

C

y

EC

yEp

y

C

y

E

C

y

E

eff

11111111

121111131211

11111111

121111131113

11111111

2

12

2

11111111

11111111

1111

2

1311

33

33

Ep

y

C

y

E

Ep

y

EC

y

Ep

y

Ep

y

C

y

E

Ep

y

EC

y

EC

y

Ep

y

C

y

E

Ep

y

C

y

Ep

y

Ep

y

C

y

E

Ep

y

C

y

EC

y

C

y

E

eff

cvcvcc

ccvccvccv

cvcvcc

ccvccvccv

cvcvcc

cccvcvcv

cvcvcc

cvcvccv

vc

c

11111111

111311131311

11111111

121111131113

11111111

121112111211

11111111

121111131211

11111111

1111

2

1111

33

33

Ep

y

C

y

E

EEp

y

EC

y

EC

y

Ep

y

C

y

E

Ep

y

EC

y

EC

y

Ep

y

C

y

E

Ep

y

C

y

Ep

y

Ep

y

C

y

E

Ep

y

EC

y

Ep

y

Ep

y

C

y

E

Ep

y

C

y

Ep

yC

y

E

eff

cvcvcc

ccvccvccv

cvcvcc

ccvccvccv

cvcvcc

ccvccvccv

cvcvcc

ccvccvccv

cvcvcc

cvcvccvvc

c

Finally,

1111

2

1213

113333Ep

y

C

y

EC

y

p

yp

y

C

y

Eeff

cvcv

ccvvcvvcc

For the second term,

1111

12111112

1111

12111113

1111

2

12121111121311131131

Ep

y

C

y

Ep

y

EC

y

E

Ep

y

EC

y

E

Ep

y

EEEEC

yeff

cvcv

ccvccv

cc

ccvccv

cc

ccccvcccccvc

1211111212111113

2

1212111112131113111111

11111111

31

1

ccvccvccvccv

ccccvcccccvcvcv

cvcvccc

Ep

y

EC

y

Ep

y

EC

y

Ep

y

EEEEC

y

Ep

y

C

y

Ep

y

C

y

E

eff



121111131211121111131112

1111

2

12111111121111

11111213111111111311

11111111

31

1

ccvccvccvccvccvccv

cvcvccvcvcvcccv

cvcvcccvcvcvcccv

cvcvccc

Ep

y

EC

y

Ep

y

Ep

y

EC

y

EC

y

Ep

y

C

y

Ep

y

Ep

y

C

y

Ep

y

Ep

y

C

y

EEC

y

Ep

y

C

y

EEC

y

Ep

y

C

y

E

eff

2

12111112111311

111211122

111312

2

12

2

1111

2

1211

1211

2

1111111211

111112132

111213

11

2

11132

111113

11111111

31

1

cccvvccccvv

ccccvvcccvv

ccvvcccvv

cccvvccccvv

ccccvvcccvv

cccvvcccvv

cvcvccc

EEp

y

p

y

EEC

y

p

y

EEC

y

p

y

EEC

y

C

y

Ep

y

p

y

EC

y

p

y

Ep

y

p

y

EC

y

p

y

EEEC

y

p

y

EEC

y

C

y

EEC

y

p

y

EEC

y

C

y

Ep

y

C

y

E

eff

2

121213121312121111

11111111121111111113

11111111

31

1

cccccccccvv

cccvcvcvcccvcvcv

cvcvccc

EEEEEC

y

p

y

EEp

y

C

y

p

y

EEp

y

C

y

EC

y

Ep

y

C

y

E

eff

Finally,

1111

12131212

121331Ep

y

C

y

EEC

y

p

yp

y

EC

y

eff

cvcv

ccccvvcvcvc

For the electromechanical coupling coefficient,

1111

1131

1111

12111113

11

33113113

33Ep

y

C

y

C

y

E

Ep

y

EC

y

E

EEC

yeff

cvcv

cev

cc

ccvccv

c

ececve

111111

121111133133113113111133

Ep

y

C

y

E

Ep

y

EC

y

C

y

EEC

y

Ep

y

C

yeff

cvcvc

ccvccvevececvcvcve

1211111331

11113311

11113113

111111

33

1

ccvccvev

cvcvvec

cvcvvec

cvcvce

Ep

y

EC

y

C

y

Ep

y

C

y

C

y

E

Ep

y

C

y

C

y

E

Ep

y

C

y

E

eff

311211311113

331111331111

311113311113

111111

33

1

eccvveccvv

eccvveccvv

eccvveccvv

cvcvce

EC

y

p

y

EC

y

C

y

EEp

y

C

y

EC

y

C

y

EEp

y

C

y

EC

y

C

y

Ep

y

C

y

E

eff

1111

121331

3333Ep

y

C

y

Ep

y

C

yC

y

eff

cvcv

ccevveve

For the stress in the z-direction,



xE

EEC

yz

p

yE

EEC

y

y

p

yE

EC

yx

p

yE

EEEEC

yz

Ec

ececvS

c

ccv

c

ccv

Tc

cv

c

cvS

c

cccv

c

ccccvT

11

31113112

11

2

12

2

11

11

2

12

2

11

11

12

11

12

11

2

121211

11

13121113

xE

EE

C

yz

p

yE

EEC

y

xEp

y

C

y

C

y

zEp

y

C

y

Ep

y

EC

y

yEp

y

C

y

E

xEp

y

C

y

Ep

y

EC

y

p

yE

EC

y

x

p

yE

EEEEC

yz

Ec

ececvS

c

ccv

c

ccv

Ecvcv

cevS

cvcv

ccvccv

Scvcv

ccS

cvcv

ccvccv

c

cv

c

cv

Sc

cccv

c

ccccvT

11

31113112

11

2

12

2

11

11

2

12

2

11

1111

1131

1111

12111112

1111

1111

1111

12111113

11

12

11

12

11

2

121211

11

13121113

From which,

1111

12111112

11

12

11

12

11

2

12

2

11

11

2

12

2

1111

Ep

y

C

y

Ep

y

EC

yp

yE

EC

y

p

yE

EEC

y

eff

cvcv

ccvccv

c

cv

c

cv

c

ccv

c

ccvc

1111

12111112

1111

12111112

1111

2

121111

2

12

111111

Ep

y

C

y

Ep

y

EC

y

E

Ep

y

EC

y

E

Ep

y

EC

yp

y

EC

y

eff

cvcv

ccvccv

cc

ccvccv

cc

ccvccvcvcvc

11111111

1211111212111112

11111111

1111

2

121111

2

12

111111

Ep

y

C

y

E

Ep

y

EC

y

Ep

y

EC

y

Ep

y

C

y

E

Ep

y

C

y

Ep

y

EC

yp

y

EC

y

eff

cvcvcc

ccvccvccvccv

cvcvcc

cvcvccvccvcvcvc

11111111

2

12

2

1111121112111211122

11

2

12

11111111

2

12

2

1111

2

12111111

2

122

11

2

12

111111

Ep

y

C

y

E

Ep

y

p

y

EEp

y

C

y

EEp

y

C

y

EC

y

C

y

Ep

y

C

y

E

Ep

y

p

y

Ep

y

C

y

EEp

y

C

y

EC

y

C

yp

y

EC

y

eff

cvcvcc

ccvvccccvvccccvvccvv

cvcvcc

ccvvcccvvcccvvccvvcvcvc

1111

2

1212

111111Ep

y

C

y

Ep

y

C

yp

y

EC

y

eff

cvcv

ccvvcvcvc

The electromechanical coupling coefficient,

1111

1131

11

12

11

12

11

3111311231

Ep

y

C

y

C

yp

yE

EC

yE

EEC

y

eff

cvcv

cev

c

cv

c

cv

c

ececve



11111111

121111121131

11111111

1111113112

3131Ep

y

C

y

E

Ep

y

EC

y

C

y

Ep

y

C

y

E

Ep

y

C

y

EC

yC

y

eff

cvcvcc

ccvccvcev

cvcvcc

cvcvcecveve

11111111

3112111131

2

11123111111231

2

1112

3131Ep

y

C

y

E

Ep

y

C

y

EC

y

C

y

EEp

y

C

y

EC

y

C

yC

y

eff

cvcvcc

ecccvveccvvecccvveccvveve

1111

311212

3131Ep

y

C

y

Ep

y

C

yC

y

eff

cvcv

eccvveve

For the electric displacement equation,

x

p

yE

SEC

y

zE

EEC

yyE

C

yxE

EEC

yx

Evc

ecv

Sc

ececvT

c

evS

c

ececvD

1111

2

313311

11

31123311

11

31

11

31133111

xEp

y

C

y

C

y

zEp

y

C

y

Ep

y

EC

y

yEp

y

C

y

E

xEp

y

C

y

Ep

y

EC

y

C

y

x

p

y

SC

yz

EC

yx

EC

y

Ecvcv

cevS

cvcv

ccvccv

Scvcv

ccS

cvcv

ccvccv

c

ev

Evc

ecvS

c

ececvS

c

ececvD

1111

1131

1111

12111112

1111

1111

1111

12111113

11

31

11

11

2

313311

11

31123311

11

31133111

Giving,

1111

1131

11

3111

11

2

31331133

Ep

y

C

y

C

y

E

C

y

p

yE

SEC

y

effS

cvcv

cev

c

evv

c

ecv

111111

1131311111

2

31

113333Ep

y

C

y

C

y

C

y

Ep

y

C

y

C

yp

y

SC

y

effS

cvcvc

cevevcvcvevvv

111111

1131311111

2

31

113333Ep

y

C

y

E

C

y

C

y

Ep

y

C

y

C

yp

y

SC

y

effS

cvcvc

cevevcvcvevvv

1111

2

31

113333Ep

y

C

y

p

y

C

yp

y

SC

y

effS

cvcv

evvvv

Summarizing, the effective values can be given by:

1111

2

1213

113333Ep

y

C

y

EC

y

p

yp

y

C

y

Eeff

cvcv

ccvvcvvcc

1111

12111113

32Ep

y

C

y

Ep

y

EC

yeff

cvcv

ccvccvc



1111

113123

Ep

y

C

y

C

yeff

cvcv

ceve

Similarly, for Case A, the independent variable become xzyx ETSS ,,, , and the effective material

properties can be written as

1111

2

1213

113333Ep

z

C

z

EC

z

p

zp

z

C

z

Eeff

cvcv

ccvvcvvcc

1111

1211111331

Ep

z

C

z

Ep

z

EC

zeff

cvcv

ccvccvc

1111

1211111212

Ep

z

C

z

Ep

z

EC

zeff

cvcv

ccvccvc

1111

111111

Ep

z

C

z

Eeff

cvcv

ccc

1111

2

1212

111122Ep

z

C

z

Ep

z

C

zp

z

EC

z

eff

cvcv

ccvvcvcvc

1111

12131212121331

Ep

z

C

z

EEC

z

p

zp

z

EC

z

eff

cvcv

ccccvvcvcvc

1111

12111112

12Ep

y

C

y

Ep

y

EC

yeff

cvcv

ccvccvc

1111

111122

Ep

y

C

y

Eeff

cvcv

ccc

1111

2

1212

111111Ep

y

C

y

Ep

y

C

yp

y

EC

y

eff

cvcv

ccvvcvcvc

1111

12131212

121331Ep

y

C

y

EEC

y

p

yp

y

EC

y

eff

cvcv

ccccvvcvcvc

1111

311212

3131Ep

y

C

y

Ep

y

C

yC

y

eff

cvcv

eccvveve

1111

121331

3333Ep

y

C

y

Ep

y

C

yC

y

eff

cvcv

ccevveve

1111

2

31

113333Ep

y

C

y

p

y

C

yp

y

SC

y

effS

cvcv

evvvv



1111

113131

Ep

z

C

z

C

zeff

cvcv

ceve

1111

311212

3132Ep

z

C

z

Ep

z

C

zC

z

eff

cvcv

eccvveve

1111

1213313333

Ep

z

C

z

Ep

z

C

zC

z

eff

cvcv

ccevveve

111111

2

3111

113333Ep

z

C

z

Ep

z

C

zp

z

SC

z

effS

cvcvc

ecvvvv

For Case C, the independent material properties become xzyx DSST ,,, . For the Piezo, we have

xC

ES

xC

ES

S

zC

yC

ES

ES

xC

Dce

e

Tce

SSce

ceeS

33332

33

33

33332

33

33

33332

33

13333133

x

C

ES

E

xC

ESz

Cy

C

ES

EE

xC

Dce

c

Tce

eSS

ce

ceceE

33332

33

33

33332

33

33

33332

33

13333331

For the polymer, we have

11

1111

12

11

12 1

xP

xP

xP

zP

yP

xP

DE

Tc

Sc

cS

c

cS

For case C, the contribution of the polymer material in the mechanical properties is ignored, giving,

xpp

yxCC

y

zC

yC

xC

x

z

y

x

EvEv

T

T

S

E

T

T

S

0

0

0

Following similar procedure as outlined above, the effective material properties can be found to be,

3333

2

311111

SpC

p

Eeff

vv

evcc

Appendices


3333

2

311212

SpC

p

Eeff

vv

evcc

3333

33311313

SpC

p

Eeff

vv

eevcc

3333

2

312222

SpC

p

Eeff

vv

evcc

3333

33312323

SpC

p

Eeff

vv

eevcc

3333

2

333333

SpC

C

Eeff

vv

evcc

3333

313331

SpC

eff

vv

ee

3333

313332

SpC

eff

vv

ee

3333

333333

SpC

eff

vv

ee

3333

333333

SpC

S

eff

vv

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