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

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

Passive Vibration Attenuation 2


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. These
1
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.

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

Crawley‎ presented an overview of the general trends in the applications of intelligent structures
2

and classified the requirements of an intelligent structure into four main categories; actuators,
sensors, control methodologies, and controller hardware. Rao and Sunar‎ focused their review on
3
‎
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 ‎ .2). He classified the available literature according to two criteria; the modelling technique,
1
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.

14

12

10
Number of Papers

8

Beams
6
Plates
Shells
4

2

0
1993 1994 1995 1996 1997 1998

Ye ar of Publication

Figure ‎ .2. Number of papers published on hybrid active-passive damping treatments of structural elements.
1
‎
5
(Benjeddou )

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 ‎ .3 presents a
1
sketch for a typical piezoelectric sensor-actuator-controller configuration.



Figure ‎ .3. Non-Collocated sensor and actuator.
1

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 ‎ .4. A sketch for 1-3 composites
1

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 ‎ .5. a sketch of active fiber composites with interdigitated electrodes
1

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.‎ presented a
6

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 ‎ .6). It was concluded that the use of segmented actuators is more
1
effective than the use of continuous ones.

Hagood et al.‎ presented a derivation of the equations of motion of an arbitrary elastic structure
8

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.

‎
7
Figure ‎ .6. Strain distribution in a beam with piezoelectric material: (a) surface attached (b) embedded .
1

‎
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 Batra‎ 0 presented an analytical method for the analysis of laminated
1

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 ‎ .7,
1
Figure ‎ .8, and Figure ‎ .9).
1 1

Figure ‎ .7. Non-Collocated sensor and actuator.
1 Figure ‎ .8. Collocated sensor and actuator.
1



Figure ‎ .9. Self-sensing piezoelectric actuator.
1

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 ‎ .10) to the piezoelectric sensor-actuator to enable the measurement of the sensor potential
1
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 ‎ .10. A sketch of the circuit suggested by Dosch and Inman‎11 to measure the (a) rate of change of
1
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 Clark‎ 3
1

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.‎ 4 implemented the concept of self-sensing piezoactuators to the suppression of panel
1

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 Cady‎ 5 for the radio
1

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 ‎ .11) are the most widely used circuits in damping as they are analogous to
1
the mechanical vibration absorber.

Inductive Resistive Capacitive Switched

Figure ‎ .11. Configurations of the different shunt circuit.
1

Hagood and von Flotow‎ 7 presented a quantitative analysis of piezo-shunting with passive networks.
1

They introduced a non-dimensional model that indicates that the damping effect of shunted circuit
resembles that of viscoelastic materials (Figure ‎ .12, Figure ‎ .13, and Figure ‎ .14). They applied their
1 1 1
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 ‎ .15).
1

Figure ‎ .12. Mechanical (physical) model of the piezoelectric patch with shunted circuit.
1

‎7
1
Figure ‎ .13. Analogous spring-mass-damper model as suggested by Hagood and von Flotow .
1



‎7
1
Figure ‎ .14. Analogous electrical model as presented by Hagood and von Flotow .
1

Figure ‎ .15. Sketch to illustrate the dissipation argument.
1

Different studies‎ 8-‎ 0 investigated the use of passively shunted piezoelectric patches for vibration
1 2
‎
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 ‎ .16). Two equivalent models are
1
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 ‎ .16. The piezoelectric material is used as an energy converter.
1

Tsai and Wang‎22,‎ 3 applied the concept of using active and passive control to simultaneously damp
2

the vibration of a beam using piezoelectric materials as shown in Figure ‎ .17. The objective of their
1
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 ‎ .17. A sketch of hybrid control for a cantilever beam.
1

The extension of using shunt circuits for damping multiple vibration modes was also investigated.
Hollkamp‎ 4 presented an extension to the analysis of single mode damping formulation to cover
2

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.
Wu‎ 5 also investigated the damping of multiple modes using a different configuration of shunt
2

circuits in which sets of resistance and inductance or capacitance and inductance connected in
parallel are connected together in series (Figure ‎ .18). These circuits were designed to provide
1
infinite impedance (anti-resonance) at the design frequencies.

(a) (b)
‎4
2 ‎5
2
Figure ‎ .18. Circuit configurations as suggested by (a) Hollkamp and (b) Wu .
1

Recently, different attempts for broadband vibration attenuation were introduced using “Negative-
Capacitance” shunt circuits‎ 6-‎28. The realization and application of the circuit in vibration damping
2

was also introduced by different patents‎ 9-‎ 1. The method has proven effective in damping out
2 3

vibrations over a broadband of frequencies.

As a more practical application of the shunt circuit damping, McGowan‎ 2 utilized shunt circuit in
3

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
blades‎ 3. The study concluded the effectiveness of that technique in the attenuation of blade
3

vibration.

Zhank et al.‎ 4 presented another application for shunted piezoelectric material by applying it to
3

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 Lesieutre‎ 6 used a modal strain energy approach to predict the damping generated by
3

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.

Saravanos‎ 7 presented an analytical solution of the problem of plate vibration with embedded
3

piezoelectric elements shunted to resistance circuit (Figure ‎ .19). The study used the Ritz method to
1
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 Flotow‎ 7. 1

Figure ‎ .19. Shunted piezoelectric material with composite structures.
1

Park and Inman‎ 9 compared the results of shunting the piezoelectric elements with an R-L circuit
3

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, Caruso‎ 0 presented a comparative theoretical and experimental study of different shunt
4

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:

D  d  E 
   
   d S E   

Where the components are:

 1   1 
   
 D1   E1   2  2
     
   
 
D   D2  , E  E2  ,    3,    3
D  E   4   4 
 3  3  5   5 
   
 6 
   6 
 

 s11
E E
s12 E
s13 0 0 0
 E E E 
0 0 0 0 d15 0  s12 s11 s23 0 0 0
0 s E E E
0
0 0
s s 0 0
d  0 0 d15  S E   13 23 33
E 
d 31 d 31 d 33 0 0 0 0 0 0 s44 0 0
  0 0 0 0 E
s 44 0
 
0 s66 
E
 0 0 0 0 

1 0 0
 
   0 1
0
0 0  
 3 

‘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

Q
D
A

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

1 I
D
A  Idt  As
Where ‘s’ is the Laplace parameter. Also, the electric field is the electric potential difference per unit
length

V
E
t

Substituting in the constitutive relations, we get

d 31
1  s11 1  V
t
A 33 s
I  Ad 31s 1  V
t

Introducing the electric capacitance, we get

I  Ad 31s 1  CsV

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

I  Ad 31s 1  YV

Now, if you focus on the case of open circuit (no current or constant electric displacement), the
equation above may be written as

Ad 31s
V  1
Y

Which may be used into the strain equation to get

2
Asd 31
1  s11 1  1
tY

Or

 d31 
2
1  s111 
  1  s11D 1
 33 s11 




D
Which indicates that the effective structure compliance s11 will be less (higher stiffness). While for
the case of short circuit (zero impedance or constant electric field)   s11  s  .
E

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

 d2 
1  31 V  Y SV
I  Y 
 33 s11 

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.

Benjeddou‎ 1 presented a comprehensive survey of the available literature on the finite element
4

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 ‎ .20). The
1
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.



‎1
4
Figure ‎ .20. Number of published papers involving finite element modeling of piezoelectric structures. (Benjeddou )
1

Tzou and Tseng‎ 2 developed a finite element model for the sensors and actuators attached to the
4

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 Guyan‎ 3 reduction to condense the DOF’s associated with
4

electrical potential. The time response of the system was calculated using the Wilson- method‎ 4. 4

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.‎ 6 extended the finite element model to cover nonlinear regimes using the von Karman
4

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.‎ 7 presented a formulation for the post-buckling vibration of plates. Their model was
4

developed using the layer-wise plate theory. In their study, they investigated the phenomena of
snapthrough.

Liu et al.‎ 8 developed a finite element model for the control of laminated composite plates
4

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.‎ 9 developed a finite element model based on the Raleigh-Ritz principle to
4

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 method‎ 1 for the first time. The method is characterized by being
5

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.‎ 2 presented another development in the finite element models by
5

introducing a higher order potential field that should accurately describe the field in the
piezoelectric elements. While Peng et al.‎ 3 introduced the third order shear theory to their finite
5

element model to increase the modelling accuracy.

Kim and Moon‎ 4,‎55 presented, for the first time, a finite element formulation for piezoelectric plate
5

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.

Saravanos‎ 6,‎ 7 presented a formulation for the finite element problem of a composite shell with
5 5
‎
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 shells‎ 7. Meanwhile, Chen et al.‎ 8 presented a similar finite
5 5

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 [‎ 7]. Thus, a 16-node element is considered (Figure ‎ .21), with 4 DOF per node
9 1
which provides a full 7th order interpolation function.



Figure ‎ .21. Sketch of the 16-node element.
1
The transverse displacement w(x,y), at any location x and y inside the plate element, is expressed by

w( x, y)  H w a (1)

where H w  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

 w 
 w   H   a 
 x   w   1 
 w   H w, x   a2 
    H   (2)
 y   w, y    
  2 w   H w, x , y  a64 
    
 xy 

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,

 wb    Tb  a  (3)

 w1 
 w1 
   H w 0,0 
 x   H w, x 0,0 
 w1   
 y   H w, y 0,0 
where wb    w  [Tb ]    (4)
 H w, x , y 0,0  
2 &
 1 

 xy    
    
  2 w16   H w, x , y a / 3,2b / 3
 
 
 xy 

From equation (14), we can obtain

a  Tb 1wb  (5)

Substituting equation (16) into equation (12) gives



w( x, y)   H w  Tb   wb    N w  wb 
1
(6)

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

 N w    H w  Tb 1 (7)

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

D( x, y)  H D b (8)

where H D  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,

D( x, y)   H D  TD   wD    N D  wD 
1
(9)

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

 N D    H D TD 1 (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

x 
     y   z 
  (11)
 
 xy 

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

 2w 
  2 
 2 x
  w 
     2    Cb  { a } (12)
 y2 
  w
 2 xy 
 

where



 Hw 
 , xx 
Cb    H w, yy  (13)
2 H 
 w, xy 

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

    Cb  Tb 1{wb }   Bb { wb } (14)

where

 Bb    Cb Tb 1 (15)

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

   z{ }  z Bb  wb  (16)

1.5.3. Constitutive Relations of Piezoelectric Lamina

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

 x   x 
   E  
 y  Q   e   y 
  T   (17)
 xy   e
    xy 

D
  E 
 

where,  x , y , xy are the stress in the x-direction, stress in the y-direction, and the planar shear
stress respectively;  x ,  y ,  xy are the corresponding mechanical strains; D is the electric
displacement (Culomb/m2), E is the electric field (Volt/m), e piezoelectric material constant
relating the stress to the electric field,  is the material dielectric constant at constant stress
(Farad/m), and Q  is the mechanical stress-strain constitutive matrix at constant electric field.
E

QE is given by,

 E E 
1   2 12
0 
 E 
QE 
E
0 
1   12 
2

 E 
 0 0
21   
 

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



 x   x 
   E
 y
  

Q   ee
T  
  e   y 
  (18)
 xy     e    xy 
T
 
E
  D
 

 x   x 
  D 
or   y   Q    y    eD (19)
   
 xy   xy 

 x 
 
E    e   y   D
T
and (20)
 
 xy 

1
where   .

1.5.4. Stiffness and Mass Matrices of The Element

The principal of virtual work states that

   U  T  W   0 (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

U      dV   D EdV
T
(22)
V V

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

 
U  z T QD z    eD dV  D   eT z   D dV
    (23)
V V

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

U   z B w  Q z B  w    e N  w dV
T D
b b b b D D
V
(24)
   N w    e z B  w     N  w dV
T T
D D b b D D
V

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



 z  B w  Q  B w dV  w   k w ,
2 T D T
b b b b b b b
V

  z B w   e N  wD dV  wb T  kbD  wD ,
T
b b D
V

    N
V
D wD T  eT z Bb  wb dV  wD T  k Db  wb   wD T  kbD T  wb  ,

and
   N
V
D wD T  N D  wD dV  wD T  k D  wD ;

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,

 2w 
A



T  w  h 2  dA
t 

(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

 
 2w 
dA  h wb T N w T N w wb  dA  wb T mb wb 

A
w  h


2 
t  A
  (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

W    DLqdA
 (27)
A

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,

W    DdA LDdA
 (28)
A A

Substituting from equation (20), gives

W  wD T  N D T dA N D LwD dA
 (29)
A A


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

which can be recast in the following form,

W  wD T mD wD 
 (30)

where [mD] is the element electric mass matrix.

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

mb  0   wb   k b 
 kbD   wb  0
   
 0 mD  wD  k Db  k D  wD  0
(31)
      

1.6. Performance Characteristics of a Plate with Shunted Piezoelectric
Patches
1.6.1. Overview

In sections ‎ .4 and ‎ .5, the foundations required to handle the problem of plates with shunted
1 1
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 ‎ .5.
1

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 ‎ .22) driven by a power amplifier (model PA7E, Wilcoxon Research), and the resulting
1
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 ‎ .23).
1

Figure ‎ .22. A picture of the speaker used to excite the plate.
1

Figure ‎ .23. A picture of the Spectrum analyzer.
1

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 ‎ .24. A schematic of the synthetic inductor circuit.
1



Figure ‎ .25. Shunting network used in present study.
1

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 ‎ .26).
1

Figure ‎ .26. Connection sketch for the circuit performance analysis.
1

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, L  / DP , 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 ‎ .6.1. The predictions are compared with the analytical predictions presented by
1
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 ‎ .6.1. Comparison of numerical and analytical results for the frequency parameter of the four different test cases
1
(Poisson’s ratio = 0.3)
Spectral BFS
Mode # Analytical
Frequency % Error Frequency % Error
1,1 19.75 19.85 0.53 19.33 -2.12
1,2 49.32 49.37 0.11 49.11 -0.42
SSSS
2,2 78.99 78.89 -0.13 77.84 -1.46
3,1 98.74 98.74 0.00 101.35 2.65



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
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
CCCC 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
1 22.17 21.68 -2.20 20.11 -9.27
2 43.60 42.84 -1.74 44.67 2.45
CFCF 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
1 3.49 3.40 -2.70 3.40 -2.70
2 8.55 8.36 -2.23 8.88 3.88
CFFF 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 ‎ .6.2. Two
1
piezoelectric patches were bonded to the plate as shown in Figure ‎ .27. All the results showed very
1
high effectiveness of the proposed damping circuit in reducing the amplitude of vibration of the
targeted frequency.

Figure ‎ .27. A sketch of the plate with dimensions.
1

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 ‎ .24) in series with a resistance.
1



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


1


1




1

Experimental
Numerical

1


1




1


1

0

-5

-10
Amplitude (dB)

-15

-20 Numer. Open Circuit
Numer. Closed Circuit

-25

-30
80 85 90 95 100 105 110 115 120 125 130
Frequency (Hz)

(a)



0

-5

-10

Amplitude (dB)
-15

-20

Exp. Open Circuit

Exp. Closed Circuit
-25

-30
80 85 90 95 100 105 110 115 120 125 130
Frequency (Hz)

(b)

Figure ‎ .36. Comparison of (a) numerical and (b) experimental results for damped and undamped cases around mode
1
(3,1).

0

-5

-10
Amplitude (dB)

-15

-20

Numer. Open Circuit

-25 Numer. Closed Circuit

-30
170 175 180 185 190 195 200 205 210 215 220
Frequency (Hz)

(a)
0

-5

-10
Amplitude (dB)

-15

-20

Exp. Open Circuit
-25 Exp. Closed Circuit

-30
170 175 180 185 190 195 200 205 210 215 220
Frequency (Hz)

(b)

Figure ‎ .37. Comparison of (a) numerical and (b) experimental results for damped and undamped cases around mode
1
(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.



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 ‎ .38 presents a schematic drawing of the plate with all the piezoelectric patches
1
attached to it and numbered 1 to 9. Four other locations of interest are marked on Figure ‎ .38 and
1
given numbers 1 to 4.

Figure ‎ .38. A sketch presenting the location and numbering of the piezoelectric patches and the four points of interest.
1

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 ‎ .39 presents the response to sine
1
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.
0

-10

-20
Amplitude (dB)

-30

-40

-50

-60

All Off All On
-70
100 150 200 250 300 350 400 450 500
Frequency (Hz)

Figure ‎ .39. Response of the plate with the all the patches attached.
1

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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