shape and vibration control of smart laminated plates_phd thesis

7/27/2019 Shape and Vibration Control of Smart Laminated Plates_phd Thesis

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SHAPE AND VIBRATION CONTROLOF

SMART LAMINATED PLATES

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Amitesh Punhani, M.S.

* * * * *

The Ohio State University2008

Dissertation Committee:

Professor Gregory N. Washington, Adviser Approved by

Professor Vadim Utkin

Professor Rajendra Singh______________________________

Professor Daniel Mendelsohn AdviserMechanical Engineering Graduate Program


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ABSTRACT

Active structures flexible enough to be molded in desired shapes and coupled with the

ability to be controlled have been pursued for many high precision applications.

Membrane-thin but extremely large (>10 m) optical mirrors and reflectors for combined

RF-Optical applications are one of the important high precision applications where shape

and vibration control of a structure is highly desirable. In this application the precision

demands of the optical surface mitigate the combined benefit of the large aperture.

Polyvinylidene Fluoride or PVDF is a semi-crystalline piezoelectric polymer with strong

orthotropic in-plane properties. This material is suitable for making large reflectors due to

its availability in thin sheets and almost linear and nonhysteretic behavior at low to

moderate operating voltages. Its low cost and ease of manufacture also make it suitable

for shape and vibration control of large reflecting structures.

This research focuses on a three-layer laminated actuator with two layers of PVDF film

bonded with a layer of epoxy. The electrodes are applied externally on the top PVDF film

in a given pattern such that the applied electric field will yield the desired shape of the

laminate. The bottom layer of the bimorph is the reflecting surface and acts as the

ground. The actuator itself acts as the RF and optical surface and therefore requires no

secondary surface.


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Research has been performed by Sumali et. al. and Massad et. al. for quasi-static

deflection of a PVDF bimorph with both simply supported and corner supported

boundary conditions under an applied electric field. This methodology produced

excellent results under ideal conditions with no disturbances. Due to lack of any kind of

feedback, the methodology was an open loop technique lacking the ability to acclimatize

under inclement real world conditions.

This research takes a step further and removes this demerit by dynamic modeling of a

three layer PVDF laminated plate with simply supported boundary conditions and

developing a Closed Loop Control Methodology which is capable of rejecting external

disturbances. This will not only help in controlling the shape, but also will allow the

structure to maintain it under inclement environment. The orthotropic properties of the

laminate / actuator are also incorporated into the model, reducing the error due to

unmodeled dynamics. Using the developed model and closed loop control methodology,

the laminate can be precisely and accurately shaped to function as a satellite antenna,

optical reflector or a solar reflector. Typically, these are the type of applications where a

shape change is difficult once the system is installed.


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Dedicated to my Father my Idol,for guiding me in the right direction

throughout my life. I would have neverreached where I am without his support and guidance


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ACKNOWLEDGMENTS

I would like to extend my sincere appreciation to my adviser, Dr. Gregory Washington

for his support, technical guidance, encouragement and patience throughout the course of

this research. I would also like to thank Dr. Vadim Utkin, Dr. Daniel Mendelsohn and Dr.

Rajendra Singh for being a part my Doctoral Committee and giving invaluable inputs on

my dissertation.

I would like to thank Leon, Vijay, LeAnn, Farzad and other Intelligent Structures and

Systems Laboratory members for their help during my graduate school.

Special thanks to my wife for her patience and support during the more difficult times.

Finally, I would like to thank my family in India for their continuous support throughout

my graduate school at The Ohio State University.


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VITA

June 01, 1976. Born Shahjahanpur, U.P., India

July 1998 B.E., Mechanical Engineering

Birla Institute of Technology, Mesra, India

July 1998-July 1999Senior Officer, Tata Steel, India

August 2001 .. M.S., Mechanical Engineering

The Ohio State University

1999-June 2006.. Graduate Research Associate

Department of Mechanical Engineering

The Ohio State University

July 2006 Present.Engineer, ICM Projects Team

Rockwell Automation Inc.,

FIELDS OF STUDY

Major Field: Mechanical Engineering

Major Areas of Specialization: Dynamics Systems, Smart Materials, Vibration and

Control Systems.


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TABLE OF CONTENTS

ABSTRACT........................................................................................................................ iiDEDICATION................................................................................................................... ivACKNOWLEDGMENTS .................................................................................................. vVITA.................................................................................................................................. viLIST OF FIGURES ........................................................................................................... ix

CHAPTER 1 INTRODUCTION....................................................................................... 1

1.1 Motivation................................................................................................................. 11.2 Background............................................................................................................... 51.3 Smart Materials......................................................................................................... 7

1.3.1 Piezoelectricity................................................................................................... 91.4 Research Objectives................................................................................................ 13

1.4.1 Conceptual Development................................................................................. 141.4.2 Design and Modeling ....................................................................................... 141.4.3 Control Methodology Development and Implementation ................................ 15

1.5 Thesis Organization ................................................................................................ 15

CHAPTER 2 LAMINATED PLATE: EQUATIONS OF MOTION.............................. 17

2.1 Piezoelectric Effect ................................................................................................. 262.2 Energy Formulation ................................................................................................ 36

2.2.1 Kinetic Energy: ................................................................................................ 362.2.2 Strain Energy: .................................................................................................. 372.2.3 Potential Energy: ............................................................................................. 38

2.3 Equations of Motion ............................................................................................... 39

CHAPTER 3 LAMINATED PLATE: DYNAMIC MODELING................................... 45

3.1 PVDF Laminate (Bimorph) .................................................................................... 453.2 Dynamic Equations................................................................................................. 463.3 Input Function......................................................................................................... 493.4 Boundary Conditions .............................................................................................. 53

3.4.1 Simply Supported............................................................................................. 543.4.2 Corner Supported: ........................................................................................... 68


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CHAPTER 4 CONTROL METHODOLOGY ................................................................ 76

4.1 Open Loop Control ................................................................................................. 764.2 Closed Loop (Feedback) Control............................................................................ 83

4.2.1 State Space Modeling....................................................................................... 85

4.2.2 Controller Design ............................................................................................ 884.2.2 Reference Tracking .......................................................................................... 904.2.3 Empirical Observer.......................................................................................... 97

4.3 Application to a Simply Supported Laminated Plate............................................ 1004.3.1 State Space Modeling..................................................................................... 101

CHAPTER 5 CONCLUSIONS AND FUTURE WORK.............................................. 104

5.1 Research Summary and Contributions.................................................................. 1045.2 Future Work:......................................................................................................... 109

BIBLIOGRAPHY........................................................................................................... 111


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LIST OF FIGURES

Figure 1.1: Offset Cassegrain Reflector Antenna............................................................... 2

Figure 1.2: Three stages of a piezoelectric material (a) Unpolarized, (b) DuringPolarization, and (c) After Polarization. ................................................................... 11

Figure 1.3: PVDF Chemical Formula [23] .................................................................... 12

Figure 2.1: An Element showing the coordinate system .................................................. 18

Figure 2.2: Deformation of a cross-section due to bending.............................................. 31

Figure 3.1: A three layer laminate with PVDF and epoxy ............................................... 46

Figure 3.2: PVDF Bimorph with rectangular patch of electrode for input signal ............ 50

Figure 3.3: Three layer laminate with an electrode pattern. ............................................. 52

Figure 3.4: PVDF Bimorph with electrode grids on the top and bottom layer................. 53

Figure 3.5: Constant electric field of 200+ Volts applied to the whole laminate............. 66

Figure 3.6: Opposite polarity electric fields of 200 Volts applied to each half of the

plate........................................................................................................................... 66

Figure 3.7: Alternate polarity electric fields of 200 Volts applied to each quarter. ...... 67

Figure 3.8: One quarter of the laminate is excited by 100+ Volts, others by 200 Volts.

................................................................................................................................... 67

Figure 4.1: Open Loop Control System............................................................................ 77

Figure 4.2: Beam with four patch actuators...................................................................... 77


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Figure 4.3: Actual shape calculated using the Open Loop Control, exactly matches the

Desired or Reference Shape...................................................................................... 82

Figure 4.4: Block diagram for a typical feedback control system with sensor and control.

................................................................................................................................... 83

Figure 4.5: Block diagram representing the State Space model of the beam................... 87

Figure 4.6: Closed Loop Control System ......................................................................... 89

Figure 4.7: Reference tracking with input directly proportional to the error.................... 91

Figure 4.8: Simulink Model for Reference Tracking of Beam Shape Coefficients.......... 93

Figure 4.9: Reference Tracking response Steady State Error. ....................................... 94

Figure 4.10: Simulink Model with Integral Control for Reference Tracking and

Disturbance Rejection............................................................................................... 95

Figure 4.11: Reference Tracking Disturbance Rejection and No Steady State Error.... 96

Figure 4.12: Block diagram showing the application of feedback control to the system. 97

Figure 4.13: Schematic Outline of Experimental Setup ................................................. 100

Figure 5.1: a) Alternated Electric field of 200 V applied to each of the four quadrants.

b) One quadrant of the plate is excited by +100V, others by 200 V .................... 107

Figure 5.2: Quasi-static deflection of a corner supported plate, excited by +200V of

Electric Field........................................................................................................... 108


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CHAPTER 1

INTRODUCTION

The primary objective of this chapter is to provide an introduction to this research. It

starts by talking about the motivation behind pursuing this area of research, where

different practical applications are discussed. This is followed by the literature review,

where work done by different people in this area is discussed and the novel aspects of this

research are explained. As the research is based on smart material, the next section gives

a brief introduction to different smart materials and their properties. PVDF, the

piezoelectric material used in the research is discussed in detail. The section for research

objectives breaks down the research into three major sections and briefly discusses each

one of them. Finally, the organization of this dissertation is discussed by stating the area

of work explained in each subsequent chapter.

1.1 Motivation

Active structures flexible enough to be molded in desired shapes and coupled with the

ability to be controlled have been pursued for many high precision applications.

Membrane-thin but extremely large (>10m) optical mirrors and reflectors for combined

RF-Optical applications are one of the important high precision applications where shape


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and vibration control of a structure is highly desirable. In this application the reflection

demands of the optical surface mitigate the combined benefit of the large aperture.

Figure 1.1: Offset Cassegrain Reflector Antenna

Much work has been going on under Dr. G. Washington at the Intelligent Structures and

Systems Laboratory (ISSL) in the Department of Mechanical Engineering at The Ohio

State University since 1999 to change the shape of a sub-reflector (Figure 1.1) of a

satellite antenna to get the desired radiation pattern. Fukashi Andoh et.al.[1] tried

controlling the shape of the sub-reflector by attaching actuators on the back of the


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structure. This method had limitations because only a finite number of actuators can be

attached, so not all the shapes were practically possible. Bruce Isler [2] also worked with

Dr. Washington to implement uniform damping control on a distributed system by using

specially etched PVDF films. Mark Angelino [3] worked at ISSL for the design and

construction of a piezoelectric point actuated aperture antenna.

Bailey and Hubbard [4] were one of the first to use the piezoelectric actuators as one of

the layers of the structure. Crawley and Luis [5] showed the use of smart actuators by

surface binding and embedding it into the structure. Extending Crawleys work, Main,

Garcia and Howard [6] determined the optimal actuators location and layer thickness for

beam and plate control. Wang, B. and Rogers [7] also gave predictions for piezoelectric

patches embedded in anisotropic plates. Ha, Keilers and Chang [8] used the finite

element method to discuss the possibility of plate shape control by distributed

piezoelectric patches. Wang, X. et. al. [9] analytically investigated the static models of

piezoelectric patches attached to beams and plates. But all the work has similar

limitations, as the number of locations of surface or embedded actuators actually

determined the achievable shapes of the structure. The main advantage of applying

external actuators on the structure is that the structural dynamics are not needed

beforehand. This is also a demerit, as we cannot incorporate the system dynamics to

derive the actuator signals, hence accuracy in the system response is more difficult to

achieve.

Polyvinylidene Fluoride or PVDF is a synthetic polymer with strong piezoelectric

properties. It is a semi-crystalline polymer with unidirectional alignment of molecule


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chains and has shown strong orthotropic in-plane properties parallel and perpendicular to

the orientation of the molecular chain [10]. This material is really suitable for making

large reflectors due to its availability in thin sheets, 9 m to 800 m thick. Additionally,

its almost linear and non-hysteretic behavior at low to moderate operating voltages and

its relative low cost and ease of manufacturability make it a suitable material for shape

and vibration control of large reflecting structures.

This research focuses on a three-layer laminated actuator with two layers of PVDF film

bonded with a layer of epoxy. This differs from the previous research of Lee, Main,

Hubbard and others because the actuator itself will act as the RF and optical surface and

will thus require no secondary surface. This means that the orthotropic properties of the

laminate cant be neglected as in previous studies. The electrodes are applied externally

on the PVDF film in a given pattern such that the applied electric field will yield the

desired shape of the laminate. As this research assumes the orthotropic properties of the

PVDF film in developing the model, the results are more accurate and realistic thereby

increasing the practicality of implementation. Recently research has been conducted for

the quasi-static shape control of the laminate [11, 12 and 13]. The primary aim of this

research is to include the dynamic case thus controlling the vibration and shape of the

laminated plate. Controlling vibration will help in negating external disturbances hence

increasing system performance. This phenomenon is really useful for making large mirror

-like reflectors to be used in satellite antennas where unknown disturbances (like micro

meteorite impact and thermal cycling gradients) may negate the validity of the quasi-

static analysis.


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1.2 Background

The polymer Polyvinylidene Fluoride (PVDF or PVF2) was first discovered by Kawai in

1969 [14]. Being a polymer it embodied many of the general characteristics of polymers

like low mass, high flexibility, high toughness, and relatively low production cost. These

characteristics were very different than the more conventional piezoceramic materials and

hence, PVDF had extensive application capabilities primarily as a sensor. PVDF is

available in thin sheets, which are easy to cut into different shapes. This enabled

distributed sensors and actuators to become some of the more popular applications. The

most comprehensive research employing PVDF as distributed sensors and actuators was

conducted by Lee [15]. He used composite laminate theory coupled with piezoelectric

constitutive relationships to develop the equations of motion of a multilayer laminate.

The PVDF films were considered to be isotropic, so the Youngs modulus of the material

was the same in x and y directions. The input was applied by the electrodes on the surface

of the structure. Sensor equations were also derived for the same electrodes. Lee and

Moon [16] also studied the dynamic model of laminated plates for torsion and bending

sensors and actuators. They used different laminae skew angles and electrode patterns to

create both bending and torsion in the structure under applied electric field. This theory

has major applications in control of large continuous structures. Yang and Ngoi [17]

extended the work by Lee and Moon by incorporating the orthotropic properties of the

PVDF film in the sensor and actuator equations.


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Hubbard and Burke [18] developed an open loop control to get the desired shape of a

beam with patch actuators. They proved that the number of actuators required should be

atleast equal to or more than the number of modes to be controlled to get the desired

shape. The open loop results from the methodology have been quite useful for this

research and are suggested as one of the possible starting points. Burke et.al [19] worked

on developing a closed loop transfer function to get the response for a simply supported

beam with patch actuators. A nonlinear active vibration damper was developed using

Lyapunovs Methodology by Plump et.al [20] in 1987.

Halim and Meheimani [21] developed a spatialH control to suppress the vibration of a

piezoelectric-laminated plate. G. Washington and L. Silverberg [22] worked on

controlling the damping and stiffness of structures with distributed actuators. They

developed a mathematical formulation to calculate the proportional, integral and

derivative gains in a feedback control law.

In 2003, Sumali, et.al [11] made a major contribution in the area of calculating the

deflection results of a quasi-static piezoelectric laminated plate. They developed an

algorithm for shape control of a simply supported isotropic PVDF laminate. The

electrode was divided into grids and the charge was distributed over the PVDF film for

active control of the reflectors shape. A relation between the applied electric field and

the structure shape coefficients was developed by minimizing the strain energy

expression. To get the required electric field for the desired shape, singular value

decomposition was used.


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In order to increase the deflection of actuated film, Sumali et.al [12] investigated active

control of a corner supported isotropic laminated plate. The actuation methodology of

distributing the charge over the PVDF film for active shape control was the same as in

previous work. The corner supported laminated plates have much larger deflections

compared to the other boundary conditions for the same exciting electric field. This

makes the corner supported case very interesting as the number of achievable shapes are

higher so it has more practical applications.

This research improves the existing quasi-static work by including the dynamic term in

the mathematical formulation of the vibration of a three layer PVDF bimorph and also

developing a control methodology which is capable of rejecting external disturbances.

This will not only help in controlling the shape but also will allow the structure to

maintain it under inclement environment.

1.3 Smart Materials

Smart Materials are materials that have one or more properties that can be significantly

altered in a controlled fashion by external stimuli; such has electrical fields, magnetic

fields, stress, moisture etc [23]. Smart Materials convert one form of energy to another,

so it can be said that they are a kind of transducers.

Some of the most common and popular types of smart materials are:

Piezoelectric materials are one of the most popular smart materials. When apiezoelectric material is deformed, it gives a small but measurable electrical

discharge. On the other hand, when an electric field is applied across the material,


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the material deforms and strain is produced. So, piezoelectric materials convert

mechanical energy into electrical energy and vice versa. Quartz is one of the few

naturally found piezoelectric materials. Lead Zirconium Titanate (PZT) and

Polyvinylidene Fluoride (PVDF) are the two most popular man-made

piezoelectric materials.

Shape Memory Alloys (SMA) [24]: These materials undergo a phasetransformation at a specific temperature. They can be plastically deformed at

relatively low temperature. The materials can then recover to their original

unreformed condition if their temperature is raised above a certain transformation

temperature. The process is repeatable with great accuracy. The most common

SMA material is Nickel Titanium Alloy, or Nitinol, which is available in the form

of wires and films.

Electrostrictive [24] materials have similar properties to piezoelectric materialsbut have better strain capability and more sensitivity to temperature. One of the

most common materials is Lead-Magnesium-Niobate or PMN.

Magnetostrictive [24] materials elongate when exposed to a magnetic field astheir magnetic domains align themselves with the external field. One of the most

popular magnetostrictive materials is Terfenol-D (Tb0.3Dy0.7Fe1.9), which

produces relatively low strains and moderate forces over a wide frequency range.

Electrorheological (ER) fluids are suspensions of extremely fine non-conducting

particles (upto 50 diameter) in an electrically insulating fluid [23]. With the

application of an electric field the rheological properties of the fluid (esp.

viscosity) change significantly.


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Magnetorheological (MR) fluids are similar to ER fluids with extremely fineiron particles in a base medium, with a significant change in viscosity on the

application of any external magnetic field.

PVDF, the material used in this research is a piezoelectric material. The next section

explains in detail about the material and piezoelectricity.

1.3.1 Piezoelectr ici ty

Piezoelectricity is the property of some materials to generate electric charge with the

application of an external mechanical stress. It was first discovered in 1880 by Pierre and

Jacque Curie, when surface charges were measured on tourmaline with the application of

mechanical stress. They later extended their study to other materials like quartz and

Rochelle salt. The converse effect was mathematically derived by Gabriel Lippman in

1881 [23] using fundamental thermodynamics principles and later confirmed by the Curie

brothers. Thus, the phenomenon of generating electric charge from applied stress is

known as the Direct Effect, while the deformation due to an applied electric field is

known as the Converse Effect.

Quartz, Rochelle salt and Tourmaline are some of the naturally occurring piezoelectric

materials. Two of the most popular man-made piezoelectric materials are PZT (Lead

Zirconate Titanate), a ceramic, and PVDF (Polyvinylidine Fluoride) a polymer.


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Any piezoelectric material will produce electrical charge on the application of an external

stress, but to really induce the piezoelectric effect, the material needs to be polarized.

This can be done by applying high DC voltage (>2000 V/mm) across the heated material,

with its temperature above the Curie point and then slowly cooling it down while the

electric field is maintained. The Curie temperature is the temperature above which the

material looses its ferroelectric properties [25]. This also sets the maximum operating

temperature limit for the material.

Figure 1.2 shows the three stages before, during and after polarization. During

polarization dipoles are formed and similarly oriented dipoles start grouping together,

which are known as Weiss domains [24]. Application of a high electric field and a high

temperature causes these dipoles to align. These dipoles remain roughly aligned even

after cooling the material to room temperature and in the absence of the electric field.

Once the material is polarized, an external electric field can be applied through the

electrodes plated on the material surface. If an electric field, larger than the one used for

initial polarization, is applied in the opposite direction of the polarization, the material

gets depolarized. Depoling is also possible by heating the material above its Curie point

or by the application of extremely large external stress. Once the material is depolarized,

it can be polarized again.


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Figure 1.2: Three stages of a piezoelectric material (a) Unpolarized, (b) During

Polarization, and (c) After Polarization.

The piezoelectric materials are insulators; hence the charge generated can be calculated

by using the expression,

Charge = Capacitance x Voltage

The properties of a piezoelectric material change with time after the original polarization

of the material. Some time after the initial poling, the material becomes quite stable.

Unless the stress level is very high, the properties of a piezoelectric material are

independent of stress [24].

Polyvinylidene Fluoride or PVDF is highly non-reactive and pure thermoplastic

fluoropolymer [23].


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Figure 1.3: PVDF Chemical Formula [23]

The IUPAC name for PVDF is polyvinylidene difluoride and its molecular formula

is ( )2 2 nCH CF . It is also known as KYNAR or HYLAR. Being a thermoplastic

fluoropolymer, PVDF has a high resistance to solvents, acids and bases. It is also

recyclable and has a glass transition temperature of -35o C.

To induce the piezoelectric properties, the material is polarized under tension and high

temperature depending upon the thickness. General characteristics of a PVDF film are

low mass, flexibility, low elastic stiffness (or high compliance) and relatively high

voltage output. Its piezoelectric constant is approximately 10 to 20 times higher [24] than

other piezoceramic materials. This makes it an ideal material for distributed sensors.

Other advantage over the brittle ceramic materials is its low elastic stiffness, which

makes it easier to mold in customizable shapes.


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The electric charge generated due to any mechanical stress can only be collected if there

is some surface electrode to collect the charge. The amount of charge collected is directly

proportional to the size of electrode i.e. if there are more electrodes, more charge is

collected. Spatial Aperture Shading uses this characteristic to have a special weighting on

the sensor. This selective charge collecting phenomenon is equivalent to performing

signal processing through integration in the spatial domain [24]. This phenomenon can

also be applied when the film is being used as an actuator. Only the portion of the film

covered by the electrode will get actuated.

Some of the different ways to achieve spatial aperture shading are:

Etching or removing the desired electrode Cutting the film in desired shape or Variable polarization.

1.4 Research Objectives

The goal of this research is to develop a methodology for creating a self-shaping

laminated plate, which could counter all the external disturbances while maintaining its

desired shape. This plate or laminate can be shaped to function as a satellite antenna,

optical reflector or a solar reflector. Typically these are the type of applications where a

shape change is difficult once the system in installed. In order to realize the usage of

piezoelectric film for vibration and shape control the following research objectives

should be met: 1) Conceptually develop the model of the laminated plate including the

piezoelectric effect in relation to its electrodes; 2) Perform some preliminary simulated


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experiments by applying different electric fields to get the deflections; 3) Develop and

implement control strategies, which increase the performance of the laminated system.

To meet the stated desired objectives the research is divided into three major categories:

Conceptual Development, Design and Modeling and Control Methodology Development

and Implementation.

1.4.1 Conceptual Development

This stage of research conceptually develops the idea of a self shaping structure, which

could handle external disturbances while maintaining its shape. The term external

disturbances refers to the disturbances directly interacting with the structure as well as

disturbances interacting with the output of the structure. For example significant weather

changes affect the radio frequency (RF) output of an antenna without affecting its shape.

This stage will also study the different applications where this new methodology could be

really useful like satellite antennas and solar reflectors etc., and also identify the real

advantages over earlier work.

1.4.2 Design and Modeling

This category primarily involves the modeling of the piezoelectric laminated plate and

developing all the necessary equations. The stage starts with formulating the stress-strain

relationships of the laminate, which are later used to develop the equations of motion.

Finally a displacement function of the plate with simply supported boundary conditions is

developed. Some preliminary testing is also performed in this section to get an idea of the

deflection to be controlled.


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1.4.3 Control Methodology Development and Implementation

This stage develops the control strategy suitable for shape control of the laminated plate.

The merits and demerits involved in varying different parameters of the system are also

discussed. Previous work done in this area is also mentioned to compare with the current

system and its advantages.

1.5 Thesis Organization

Chapter 1 (current) introduces the problem statement. It also talks about the motivation

behind pursuing this area of work by detailing the previous work done. A small section

on Smart Materials is also covered, which talks about the availability of different kinds of

materials. Polyvinylidene Fluoride (material used) and its properties are described in

detail.

Chapter 2 develops the equations of motion of a multilayer piezoelectric laminated plate

from first principles. It starts out by explaining tensors and developing the compliance

matrix for an orthotropic laminated plate under plane stress. The piezoelectric effects of

the material are incorporated in the equations of motion.

Chapter 3 uses the equations of motion to generate an expression for deflection of a

simply supported PVDF bimorph with respect to time. The chapter also elaborates on the

input function generated to incorporate the piezoelectric effect of the PVDF film. This

dynamic expression is the first attempt in this area of research for the modeling of a

laminated plate. The chapter also discusses the advantages of corner supported boundary


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conditions. An approximate eigenfunction is suggested and used in the Rayleigh-Ritz

method to develop the frequency equation of the laminate.

Chapter 4 develops a control methodology for the simply supported PVDF bimorph. It

starts out by discussing the open loop control methodology and its disadvantages. The

idea is to develop the control methodology for a simply supported beam and then extend

it for a plate. The open loop control methodology for a pin-pin beam by Burke and

Hubbard [18] is discussed with an example. Closed Loop Control and its advantages are

discussed. The dynamic model of a simply supported beam is developed, which is used

for state space modeling of the beam. This state space model was then further used to

design a controller by pole placement technique. A new Observer is developed to

simulate the experiments and is termed an Empirical Observer. This technique is further

used to show the results for simple examples, which proved the methodology is capable

of dynamically rejecting the disturbances. The dynamic model for the plate is simplified

and reduced to a form such that the same control methodology can be used.

Chapter 5 summarizes the whole research and talks about the possible future work in this

area.


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CHAPTER 2

LAMINATED PLATE: EQUATIONS OF MOTION

This chapter describes in detail the dynamic modeling of a laminated plate and also

develops an analytical solution of the simply supported laminate used in the research.

Hookes law defines the relationship between stress and strain as,

C = (2.1)

Where, is the stress applied, is the strain generated and C is the stiffness of the

material. Equation (2.1) shows a single dimensional stressed state; hence the stress, strain

and stiffness are all scalar values. For a three dimensional stress state, equation (2.1) will

be a matrix equation and the three quantities will be tensors. Although the physical

quantities represented are independent of any coordinate system, choosing a coordinate

system will be convenient to describe them.

Tensors are used to mathematically represent these quantities.

Tensors:

Tensors can be defined as a simple array of numbers or functions that transform

according to certain rules under a change of coordinates and are used to define physical

quantities. Tensors are independent of any coordinate system, yet they are specified in a


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particular coordinate system by a certain set of quantities known as components. Once

the components of a tensor in one coordinate system are specified, the components of that

tensor in all other coordinates are defined as well. They are converted from one

coordinate system to another by the use of transformations. Generally, tensors are noted

in terms of orders, like a scalar number is a zero order tensor, a three dimensional vector

is a first order tensor. The number of components in a tensor can be defined as3n , where

n is the order of the tensor. So a 2nd

order tensor will have 23 = 9 components.

Figure 2.1: An Element showing the coordinate system

Figure 2.1 shows the Cartesian Coordinates for a plate element. A fully three

dimensional stress state is represented by the stress tensor as,


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11 12 13

21 22 23

31 32 33

;ij

=

2nd

Order Tensor - 23 = 9 Components. (2.2)

Similarly the strain tensor is,

11 12 13

21 22 23

31 32 33

;ij

= 2nd

Order Tensor - 23 = 9 Components (2.3)

and stiffness is a 4th

order tensor with 43 = 81 with Components,ijklC

So equation (2.1) can be written as,

ij ijkl kl

kl

C = (2.4)

or ij ijkl klkl

S = (2.5)

Where, ijklS is the compliance (inverse of stiffness) of the material.

An alternative to the tensor notation in equation (2.4) is,

[ ] [ ][ ]C = (2.6)

Where, [ ]C is a 9x9 stiffness matrix of the material and [ ] and [ ] are the vectors

containing all the components of the stress and strain tensors from equations (2.2) & (2.3)

Assuming no net moment is acting on the element,

ij ji = (2.7)

i.e.

[ ]

11 12 13

12 22 23

13 23 33

=

(2.8)

thus reducing the number of elements in the stress vector to 6 and in the stiffness matrix

to 54.


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20

Similarly, for strain there are only six independent components,

ij ji = (2.9)

i.e.

11 12 13

12 22 23

13 23 33

= (2.10)

The number of elements in the stiffness matrix is now reduced to 36.

Reducing the stress and strain notations to,

11 1 22 2 33 3 23 4 13 5 12 6; ; ; ; ; = = = = = = (2.11)

and 11 1 22 2 33 3 23 4 13 5 12 6; ; ; ; ; = = = = = = (2.12)

Using the notations of equations (2.11) & (2.12), generalized Hookes law can be

rewritten as,

1 11 12 13 14 15 16 1

2 21 22 23 24 25 26 2

3 31 32 33 34 35 36 3

4 41 42 43 44 45 46 4

5 51 52 53 54 55 56 5

6 61 62 63 64 65 66 6

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

=

(2.13)

If W is the strain energy then by using Chain Rule the strain energy differential can be

expressed as,

i j

i j

W WdW d d

= + (2.14)

Assuming,i

W

andj

W

are continuous then,

j

W

i

=

i

W

j

(2.15)


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21

Strain Energy per unit volume for an element can be written as,

iW = (2.16)

From equation (2.13),

i ij jC = (2.17)

iijj

C

=

(2.18)

From equation (2.17),2 2

ij ji

i j j i

W WC C

= = =

(2.19)

Thus the number of elements in the stiffness matrix of equation (2.13) is reduced to 21

and the equation can be expressed as,

1 11 12 13 14 15 16 1

2 12 22 23 24 25 26 2

3 13 23 33 34 35 36 3

4 14 24 34 44 45 46 4

5 15 25 35 45 55 56 5

6 16 26 36 46 56 66 6

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

=

(2.20)

Symmetry: If a quantity f(x) is symmetric across one plane then mathematically we can

say that f(x) = f(-x) or suppose if the plane of symmetry is 1-2 then looking in the +3

direction is the same as looking in the -3 direction.

So if the material has symmetry with respect to one of the planes then the number of

constants is reduced to 13. This type of material is called Monoclinic. Considering the

above stated example, if the axis 3 is perpendicular to the plane of symmetry (plane 1-2)

then,

14 15 24 25 34 35 46 56 0C C C C C C C C = = = = = = = = (2.21)


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22

and the stiffness matrix reduces to,

1 11 12 13 16 1

2 12 22 23 26 2

3 13 23 33 36 3

4 44 45 4

5 45 55 5

6 16 26 36 66 6

0 0

0 0

0 0

0 0 0 0

0 0 0 0

0 0

C C C C

C C C C

C C C C

C C

C C

C C C C

=

(2.22)

Similarly if there are two planes of symmetry (if a material is symmetrical across two

planes then it is also symmetrical across the third plane) then the number of constants

reduce to 9.

16 26 36 45 0C C C C = = = = (2.23)

and the stiffness equation reduces to,

1 11 12 13 1

2 12 22 23 2

3 13 23 33 3

4 44 4

5 55 5

6 66 6

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

C C C

C C C

C C C

C

C

C

=

(2.24)

A material with two planes of symmetry (if a material is symmetric across two planes,

then is also symmetric across the third plane) and 9 constants is known as Orthotropic.

Some of the naturally occurring orthotropic materials are Barytes and Wood.

Isotropy: This should not be confused with symmetry. Mathematically isotropy could be

termed as f(x1) = f(x2) = f(x3) = i.e. if a material is isotropic in a certain direction then

no matter where we look in that direction, the material properties are the same.


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23

Now if a material has three planes of symmetry and one of the planes is isotropic then the

number of constants is reduced to 5 and the material is known as Transversely Isotropic.

Considering the case when plane 2-3 is isotropic,

( )

22 33 12 13 55 66

44 22 23

; ;

1and

2

C C C C C C

C C C

= = =

= (2.25)

and equation (2.24) will reduce to,

( )

11 12 121 1

12 22 232 2

12 23 22

3 3

4 422 23

5 566

6 666

0 0 0

0 0 0

0 0 0

10 0 0 0 0

2

0 0 0 0 0

0 0 0 0 0

C C C

C C C

C C C

C C

C

C

=

(2.26)

If the material is completely Isotropic,

( )

( )

( )

11 12 12

12 11 121 1

12 12 112 2

3 311 12

4 4

5 511 12

6 6

11 12

0 0 0

0 0 0

0 0 01

0 0 0 0 02

10 0 0 0 0

2

10 0 0 0 0

2

C C C

C C C

C C C

C C

C C

C C

=

(2.27)

and equation (2.27) shows that the stiffness matrix is configured of only 2 unknowns.

This research deals with an orthotropic material, so getting back to equation (2.24), the 9

constants defining the stiffness matrix of the material are the Youngs modulii,

11 22 33, ,E E E


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24

Shear modulii,

12 13 23, ,G G G

and Poissons Ratios,

12 13 23, ,

Considering a plane stress case relative to the plane 1-2,

3 4 5 0 = = = (2.28)

Stress-Strain relationship for an orthotropic element can be written as,

( )11 12 22

2212 22 1112 22 11

1 1

12 22 222 22 2

12 22 11 12 22 11

6 6

12

0(1 / )1 /

0(1 / ) (1 / )

0 0

E E

E EE E

E E

E E E E

G

=

(2.29)

Equation (2.29) is the fundamental equation for dynamic modeling of a thin laminated

plate under plane stress.

Rewriting equation (2.29) in a simplified form as,

1 11 12 1

2 12 22 2

6 66 6

0

0

0 0

Q Q

Q Q

Q

=

(2.30)

Where,

11 12 2211 122 2

12 22 11 12 22 11

2222 66 122

12 22 11

;

(1 / ) (1 / )

;(1 / )

E EQ Q

E E E EE

Q Q GE E

= =

= =

(2.31)

Equation (2.30) can also be written as,


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25

[ ] [ ][ ]Q = (2.32)

Basic Assumptions:

The following basic assumptions [26] are considered for the laminated plate modeling:

1. The laminate is constructed of multiple orthotropic PVDF layers bonded byisotropic epoxy.

2. The plate (laminate) is thin and the thickness is much smaller as compared toother dimensions of the laminate.

3. The laminated plate has constant thickness.4. u, v and w are the displacements in the three directions (1, 2 & 3) of the

coordinate system and are much smaller as compared to the thickness of the plate.

5. The tangential displacements u and v are a linear function of the coordinate in3(z)-direction.

6. The strains 1 2 6, & are very small compared to unity.7. Transverse shear strains 4 5& are negligible.8. Transverse normal strain 3 is negligible.9. Rotary inertia terms are negligible.10.Each lamina obeys Hookes law.11.There are no body forces.


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26

2.1 Piezoelectr ic Effect

As the laminated plate in the research consists of PVDF films, hence the piezoelectric

effect of the smart material should be included in the model. For any piezoelectric

material the charge developed due to strain in the material is known as Direct Effect and

the deflection due to applied electric field is known as Converse Effect. The constitutive

equations for the converse and direct effects can be written as,

Converse Effect: i ij j mi ms d E = + (2.33)

Direct Effect: Tm mi i mk k D d E = + (2.34)

Where, [s] is the compliance matrix, [d] is the piezoelectric strain constant matrix, [ ] is

the permittivity matrix, [E] is the applied electric field vector and [D] is the electric

displacement vector.

The piezoelectric constant mid indicates the strain in i-direction due to the electric field

applied in m-direction. m also indicates the direction of initial polarization of the

material.

The Converse Effect equation (2.33) can be written in its full form as,

1 11 12 13 14 15 16 1 11 21

2 21 22 23 24 25 26 2

3 31 32 33 34 35 36 3

4 41 42 43 44 45 46 4

5 51 52 53 54 55 56 5

6 61 62 63 64 65 66 6

s s s s s s d d

s s s s s s

s s s s s s

s s s s s s

s s s s s s

s s s s s s

= +

31

12 22 32

1

13 23 33

2

14 24 34

3

15 25 35

16 26 36

d

d d d

Ed d dE

d d dE

d d d

d d d

(2.35)


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Plane Stress: A thin structure subjected to in-plane loading such that the stresses with

respect to the thin surface are zero is said to be under Plane Stress.

For a thin plate under a plane stress condition,

3 4 5 0 = = =

So equation (2.35) can be written as,

1 11 12 16 1 11 21 31 1

2 21 22 26 2 12 22 32 2

6 61 62 66 6 16 26 36 3

s s s d d d E

s s s d d d E

s s s d d d E

= +

(2.36)

If the electric field is applied only in 3-direction then 1 2 0E E= = and 1 2 0i id d= = .

Equation (2.36) becomes,

[ ]1 11 12 16 1 31

2 21 22 26 2 32 3

6 61 62 66 6 36

s s s d

s s s d E

s s s d

= +

(2.37)

As compliance of any material is the inverse of its stiffness then for an orthotropic

material from equation (2.32),

[ ] [ ]1

s Q

= (2.38)

Using equation (2.38) in equation (2.37),

[ ] [ ]1 1 31

1

2 2 32 3

6 6 36

d

Q d E

d

= +

(2.39)

or [ ] [ ] [ ]1 1 31

2 2 32 3

6 6 36

d

Q Q d E

d

=

(2.40)


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28

Similarly, Direct Effect equation (2.34) can be written in expanded form as,

11 21 31 1

12 22 32 2

1 11 11 11 1

13 23 33 3

2 21 22 23 2

14 24 34 4

3 31 32 33 3

15 25 35 5

16 26 36 6

Td d d

d d dD E

d d dD E

d d dD E

d d d

d d d

= +

(2.41)

Considering the conditions for plane stress and applying the electric field in the direction

of axis-3, we can write equation (2.41) as,

3 31 1 32 2 36 6 33 3D d d d E = + + + (2.42)

Till now it is assumed that the orthotropic coordinates of each element are aligned with

the Cartesian coordinates of the laminate. To write equations (2.40) & (2.42) in a more

generalized form, let us assume that the local coordinates of each element are

' ' '1 ,2 &3 and the angle of rotation from global coordinates 1 & 2 is , which is the Skew

Angle.

Equations (2.40) and (2.42) can be written as,

[ ] [ ] 363

23

13

6

2

1

6

2

1

=

E

d

d

d

QQ

(2.43)

3336632231133 +++= EdddD (2.44)

Transformation Matrix:


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29

Transformation matrices can be used to convert the equations from local to global

coordinate system.

Transformation matrices for the stress and strain matrices are,

Stress Transformation Matrix:

[ ]

=

22

22

22

sincossincossincos

sincos2cossin

sincos2sincos

T (2.45)

Strain Transformation Matrix:

[ ]

=

22

22

22

sincossincos2sincos2

sincoscossin

sincossincos

T (2.46)

Where, is the angle of rotation from the cartesian coordinates. So if the element is

aligned with the coordinate axis, then the angle of rotation =0. Substituting this value in

either of the transformation matrices will give us an identity matrix (reminder sin (0) = 0

and cos (0) = 1).

Using equations (2.45) and (2.46) to substitute [ ]T

& [ ]T

for stress and strain

(local coordinates) respectively in equation (2.40),

[ ] [ ][ ] [ ]1 1 3 1

2 2 3 2 3

6 6 3 6

d

T Q T Q d E

d

=

(2.47)


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30

[ ] [ ][ ] [ ] [ ] 363

23

13

1

6

2

1

1

6

2

1

=

E

d

d

d

QTTQT

(2.48)

or [ ] [ ] [ ] 363

23

13

1

6

2

1

6

2

1

=

E

d

d

d

QTQ

(2.49)

where, [ ] [ ][ ]1

Q T Q T

= (2.50)

Basic assumption # 6 states that the displacements in the direction 1(x) and 2(y) are linear

functions of the 3(z) coordinate.

Let u, and w be the displacements in 1(x) and 3(z) directions. If ou is the displacement of

the middle plane in x direction then according to Figure 2.2, the net displacement of any

point q can be written as,

ou u z= (2.51)

or ow

u u zx

=

(2.52)

Similarly, if v is the displacement in 2(y) direction and ov is the corresponding middle

plane deflection then net displacement of any point can be written as,

o wv v z

y

=

(2.53)


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31

Figure 2.2: Deformation of a cross-section due to bending

Using equations (2.52) & (2.53), the strain expressions can be written as,

2

1 1 12( )

oo ou w u w

u z z zx x x x x

= = = = +

(2.54)

2

2 2 22( )

oo ov w v wv z z z

y y y y y

= = = = +

(2.55)


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32

2

6 6 62o o

o o ow w u v wu z v z z z

y x x y y x x y

= + = + = +

(2.56)

Where,2

1 2

w

x

=

;

2

2 2

w

y

=

;

2

6 2w

x y

=

(2.57)

Using equations (2.54), (2.55) and (2.56), the stress equation (2.49) can be rewritten as,

[ ] [ ]1 1 1 3 1

1

2 2 2 3 2 3

6 6 6 3 6

o

o

o

z d

Q z T Q d E

z d

+ = + +

(2.58)

or [ ] [ ]1 1 1 3 1

1

2 2 2 3 2 3

6 6 6 3 6

o

o

o

d

Q Q z T Q d E

d

= +

(2.59)

Stress Resultants:

Stress resultants are the total load acting per unit length at the mid plane. They have the

dimensions of force per unit length and are defined as,

dzdzN

dzdzN

dzdzN

h

h

h

hxyxy

h

h

h

hyy

h

h

h

hxx

==

==

==

2/

2/6

2/

2/

2/

2/2

2/

2/

2/

2/1

2/

2/

(2.60)

Where, h is the height of each lamina.

The three forces in equation (2.60) can be combined in a vector form for thk lamina as,


43/129

33

1

/ 2 / 2

2/ 2 / 2

6

x xh h

y yh h

xy xy kk k

N

N dz dz

N

= =

(2.61)

Extending equation (2.61) for an n layer laminate,

1/ 2

2/ 2

1

6

x n h

yh

k

xy k

N

N dz

N

=

=

(2.62)

Substituting the stress equation (2.59) in equation (2.62),

[ ] [ ]1 1

1 1 3 1_

1

2 2 3 2 3

1 1

6 6 3 6

k k

k k

ox h hn n

o

y

k kk oh h

xy

N d

N Q z dz T Q d E dz

N d

= =

= +

(2.63)

or

[ ] [ ]

=

=

+

=

n

k

h

h

n

k

h

h

k

h

h o

o

o

k

xy

y

x

k

k

k

k

k

k

dzE

d

d

d

QT

zdz

QQQ

QQQ

QQQ

dz

QQQ

QQQ

QQQ

N

N

N

1

3

63

23

13

1

1

6

2

1

66

_

26

_

16

_26

_

22

_

12

_16

_

12

_

11

_

6

2

1

66

_

26

_

16

_26

_

22

_

12

_16

_

12

_

11

_

1

11

(2.64)

If ( ) = =

=

=

n

k

n

k

kk

k

ij

h

hk

ijijhhQdzQA

k

k1 1

1

__

1

(2.65)

and ( ) = =

=

=

n

k

n

k

kk

k

ij

h

hk

ijijhhQzdzQB

k

k1 1

2

1

2__

2

1

1

(2.66)

Then using equations (2.65) and (2.66), equation (2.64) can be written as,


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34

[ ] [ ] =

+

=

n

k

h

h

k

o

o

o

kxy

y

x

k

k

dzE

d

d

d

QT

BBB

BBB

BBB

AAA

AAA

AAA

N

N

N

1

3

63

23

13

1

6

2

1

662616

262212

161211

6

2

1

662616

262212

161211

1

(2.67)

and if,

+

=

6

2

1

662616

262212

161211

6

2

1

662616

262212

161211

k

o

o

o

kxy

y

x

BBB

BBB

BBB

AAA

AAA

AAA

N

N

N

(2.68)

Then equation (2.67) can be written as,

[ ] [ ] =

=

n

k

h

h

xy

y

x

xy

y

x k

k

dzE

d

d

d

QT

N

N

N

N

N

N

1

3

63

23

13

1

1

(2.69)

Moment Resultants:

For the Total Equivalent Load on the laminate, moments must also be applied (in

addition to the resultant stresses) at the mid plane. These moments are equivalent to the

moments by the resultant stresses at the mid plane. So the dimensions are length times the

force per unit length (unit of the resultant stress).

Similar to Stress Resultants, the Moment Resultants for each lamina can be written as,


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35

1/ 2 / 2

2

/ 2 / 2

6

x xh h

y y

h h

xy xy kk k

M

M zdz zdz

M

= =

(2.70)

Extending equation (2.70) for an n layer laminate,

[ ] [ ]1 1

1 1 3 1_

1

2 2 3 2 3

1 1

6 6 3 6

k k

k k

o

x h hn no

y

k kk oh h

xy

M d

M Q z zdz T Q d E zdz

M d

= =

= +

(2.71)

or

[ ] [ ]

=

=

+

=

n

k

h

h

n

k

h

h

k

h

h o

o

o

k

xy

y

x

k

k

k

k

k

k

zdzE

d

d

d

QT

dzz

QQQ

QQQ

QQQ

dzz

QQQ

QQQ

QQQ

M

M

M

1

3

63

23

13

1

1

2

6

2

1

66

_

26

_

16

_26

_

22

_

12

_

16

_

12

_

11

_

6

2

1

66

_

26

_

16

_26

_

22

_

12

_

16

_

12

_

11

_

1

11

(2.72)

If ( ) = =

=

=

n

k

n

k

kk

k

ij

h

hk

ijijhhQdzzQD

k

k1 1

3

1

3_

2_

3

1

1

(2.73)

Substituting equations (2.66) and (2.73) in (2.72) will give,

[ ] [ ] =

+

=

n

k

h

h

k

o

o

o

kxy

y

x

k

k

zdzE

d

d

d

QT

DDD

DDD

DDD

BBB

BBB

BBB

M

M

M

1

3

63

23

13

1

6

2

1

662616

262212

161211

6

2

1

662616

262212

161211

1

(2.74)


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36

and if,

+

=

6

2

1

662616

262212

161211

6

2

1

662616

262212

161211

k

o

o

o

kxy

y

x

DDD

DDD

DDD

BBB

BBB

BBB

M

M

M

(2.75)

then, [ ] [ ] =

=

n

k

h

h

xy

y

x

xy

y

x k

k

zdzE

d

d

d

QT

M

M

M

M

M

M

1

3

63

23

13

1

1

(2.76)

2.2 Energy Formulation

The governing equations and natural boundary conditions for the laminate will be

formulated by using the variation in energy principle. This method utilizes the

piezoelectric properties of the PVDF material and could also be used for other

approximate methods like Galerkin and Ritz methods.

2.2.1 Kinetic Energy: If o is the mass density of a layer then kinetic energy for each

lamina can be expressed as,

2 2 21

2o

u v wT dxdydz

t t t

= + +

(2.77)

Substituting the displacement expressions from equations (2.52) and (2.53) in (2.77), the

kinetic energy per lamina can be written as,

2 2 22 21

2

o o

o

u w v w wT z z dxdydz

t t x t t x t

= + +

(2.78)


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37

If ko is theth

k layer density then the mass density of the laminate can be written as,

1

nk

okk dz == (2.79)

Carrying out the integration over z in equation (2.78), using equation (2.79) and

neglecting time derivatives of plate slopes2w

t x

and

2w

t y

, kinetic energy for a laminate

can be expressed as,

dxdyt

w

t

v

t

uT

oo

+

+

=

222

2

1 (2.80)

2.2.2 Strain Energy: Strain Energy for any elastic body under plane stress can be

expressed as,

( )1 1 2 2 6 61

2U dxdydz = + + (2.81)

Substituting equations (2.54), (2.55), (2.56), (2.67) and (2.74) into equation (2.81) and

performing the integration over z yields,


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38

dxdy

yx

wR

x

v

y

uR

y

wR

y

vR

x

wR

x

uR

dxdy

yx

wD

y

wD

x

wD

yx

w

y

wD

y

w

x

wD

x

wD

x

v

y

u

yx

wB

yx

w

y

v

x

v

y

u

y

wB

yx

w

x

u

x

v

y

u

x

wB

y

w

y

vB

y

w

x

u

x

w

y

vB

x

w

x

uB

x

v

y

uA

x

v

y

u

y

vA

x

uA

y

vA

y

v

x

uA

x

uA

U

oooo

oo

oooooo

oooooo

oooooooo

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

=

2

332

2

222

2

11

22

662

2

262

2

16

2

2

2

2

222

2

2

2

12

2

2

2

11

2

66

2

2

2

26

2

2

2

16

2

2

222

2

2

2

122

2

11

2

66

2616

2

2212

2

11

2

2

1

44

24

2222

222

22

2

1

(2.82)

Where, [ ] [ ] [ ]1

1 3 11

2 3 2 3

1

3 3 6

k

k

hn

k h

R d

R R T Q d E dz

R d

=

= =

(2.83)

and [ ] [ ]1

1 3 11

2 3 2 3

1

3 63

k

k

hn

k h

R d

R R T Q d E zdz

dR

=

= =

(2.84)

2.2.3 Potential Energy: Considering a plane stress case, transverse loads are absent.

So the potential energy due to the inplane loads ( iN ) can be expressed as,

dxdyy

w

x

wN

y

wN

x

wNV ixy

i

y

i

x

+

+

= 2

2

122

(2.85)


49/129

39

2.3 Equations of Motion

Hamiltons Principle: states that the development in time for a mechanical system is

such that the integral of the difference between the kinetic and the potential energy is

stationary. More specifically it can expressed as,

( )1

0

0

t

t

U V T dt + = (2.86)

( )1

0

0

t

t

U V T dt + = (2.87)

To find the variation in the strain energy, taking the first term from equation (2.82)

( )2

11 112o o

ou uA A u

x x x

=

(2.88)

and the second term,

( ) ( )12 12 122 2 2o o o o

o ou v u vA A v A ux y x y y x

= +

(2.89)

Similarly, taking the variation of each term in the strain energy expression,


50/129

40

( )

( )

( ) ( )

( )

( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( )

dxdy

wyx

R

vx

uy

Rwy

Rvy

Rwx

Rux

R

dxdy

wyxyx

wD

y

wD

x

wD

x

v

y

uB

y

vB

x

uB

w

yyx

wD

y

wD

x

wD

x

v

y

uB

y

vB

x

uB

wxyx

wD

y

wD

x

wD

x

v

y

uB

y

vB

x

uB

vx

uy

yx

wB

y

wB

x

wB

x

v

y

uA

y

vA

x

uA

vyyx

wB

y

wB

x

wB

x

v

y

uA

y

vA

x

uA

uxyx

wB

y

wB

x

wB

x

v

y

uA

y

vA

x

uA

U

oooo

oooo

oooo

oooo

oo

oooo

ooooo

ooooo

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

=

2

3

32

2

222

2

11

22

662

2

262

2

16662616

2

22

262

2

222

2

12262212

2

22

162

2

122

2

11161211

2

662

2

262

2

16662616

2

262

2

222

2

12262212

2

162

2

122

2

11161211

2

22

2

2

2

2

2

(2.90)

Using equation (2.67) and (2.72),

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )

dxdy

wyx

RM

vxuyRNwyRM

vy

RNwx

RMux

RN

U

xy

oo

xyy

o

yx

o

x

+

+

+

=

2

3

32

2

2

22

2

11

2

(2.91)


51/129

41

Using integration by parts for each term in equation (2.91),

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )

( )( ) ( )( ) ( ) ( )

( )( )

( )( )

( )( ) ( )( ) ( ) ( )

( )( )

( )( )

+

++

+

+

+

+

+

+

+

=

x

y

Sxyx

x

o

xy

o

x

Sxyy

y

o

y

o

xy

yxyx

oxyyoxyx

dy

wy

RMw

x

RM

wx

RMvRNuRN

dx

wx

RMw

y

RM

wy

RMvRNuRN

dxdy

wy

RM

yx

RM

x

RM

vx

RN

y

RNu

y

RN

x

RN

U

31

131

32

223

2

2

2

3

2

2

1

2

3231

2

2

2

(2.92)

Similarly, from equation (2.85), the variation inpotential energy can be written as,

( )

( ) ( )dywy

wN

x

wNdxw

y

wN

x

wN

dxdywy

wN

yx

wN

x

wN

V

xy S

i

xy

i

xS

i

y

i

xy

i

y

i

xy

i

x

+

+

+

+

+

+

=

2

22

2

2

2 (2.93)

Forkinetic energy variation, using equation (2.80),

( ) ( ) ( ) dxdywtt

wv

tt

vu

tt

uT o

oo

o

+

+

= (2.94)


52/129

42

Adding and subtracting ( )oo

ut

u

2

2

, ( )o

o

vt

v

2

2

and ( )w

t

w

2

2

from equation (2.94)

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

dxdy

wtt

ww

t

w

vtt

vv

t

vu

tt

uu

t

u

wt

wv

t

vu

t

u

T oo

oo

oo

oo

oo

oo

+

+

+

+

+

=

2

2

2

2

2

2

2

2

2

2

2

2

(2.95)

( ) ( ) ( )

( ) ( ) ( )dxdy

wt

wv

t

vu

t

u

t

wt

wv

t

vu

t

u

To

oo

o

oo

oo

+

+

+

+

=

2

2

2

2

2

2

(2.96)

Integrating equation (2.96) in the time interval 1ttto and assuming,

0)()()()()()( 111 ====== twtwtvtvtutu oo

o

oo

o

o (2.97)

Total Variation in kinetic energy can be written as,

( ) ( ) ( )1 1 2 2 2

2 2 2

o o

t t o o

o o

t t

u v wTdt u v w dxdyt t t

= + + (2.98)

Substituting equations (2.92), (2.93) and (2.98) in equation (2.87),


53/129

43

( ) ( ) ( )

( ) ( )( )

( ) ( ) ( )

( )

( )( ) ( )( ) ( ) ( )

( )( )

( )( ) ( ) ( )

( )( ) ( )( ) ( ) ( )

( )( )

( )( ) ( )

( )

0

2

2

2

2

1

31

131

32

223

2

2

2

22

2

2

2

2

2

3

2

2

1

2

2

232

2

231

=

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

dt

dy

wy

wN

wx

wNw

y

RMw

x

RM

wx

RMvRNuRN

dx

wy

wNw

x

wNw

x

RMw

y

RM

wy

RMvRNuRN

dxdy

w

t

w

y

wN

yx

wN

x

wN

y

RM

yx

RM

x

RM

vt

v

x

RN

y

RN

u

t

u

y

RN

x

RN

t

t

S

i

xy

i

x

xyx

x

o

xy

o

x

Si

y

i

xy

xyy

yo

yo

xy

i

y

i

xy

i

x

yxyx

oo

xyy

oo

xyx

o

x

y

(2.99)

In equation (2.99) looking at the dynamic terms, it can be said that the surface integral

(double integral) will be zero if the following equations hold good,

( )2

231

t

u

y

RN

x

RN oxyx

=

+

(2.100)

2

232

t

v

x

RN

y

RN oxyy

=

+

(2.101)


54/129

44

( ) ( ) ( )

2

2

2

22

2

2

2

2

2

3

2

2

1

2

2

2

t

w

y

wN

yx

wN

x

wN

y

RM

yx

RM

x

RM iy

i

xy

i

x

yxyx

=

+

+

+

+

+

(2.102)

or,y

R

x

R

t

u

y

N

x

N oxyx

+

+

=

+

312

2

(2.103)

x

R

y

R

t

v

x

N

y

N oxyy

+

+

=

+

32

2

2

(2.104)

2

2

2

3

2

2

1

2

2

2

2

22

2

2

2

22

2

2 2

22

y

R

yx

R

x

R

t

w

y

wN

yx

wN

x

wN

y

M

yx

M

x

M iy

i

xy

i

x

yxyx

+

+

+

=

+

+

+

+

+

(2.105)

Equations (2.103), (2.104) and (2.105) are the equations of motion in x, y and z direction

respectively.


55/129

45

CHAPTER 3

LAMINATED PLATE: DYNAMIC MODELING

This chapter describes in detail the dynamic modeling of a laminated plate.

3.1 PVDF Laminate (Bimorph)

This laminated plate or laminate consists of three layers. The top and bottom layers are

made of an orthotropic piezoelectric material PVDF, and the middle layer consists of

isotropic epoxy.

Bimorph: A piezoelectric laminate with two active layers is called a Bimorph.

Figure 3.1 shows a bimorph with top and bottom active layer of a piezoelectric material.

The x-axis of the laminate is aligned with the direction of stretching of PVDF. The two

active layers have a silver electrode coating. The bottom layer of the laminate is the

reflecting side and acts as the ground. The top layer is generally divided into grids of

electrodes. Each grid can be individually excited by applying an electric field. The

electrodes are assumed to be extremely thin and light. Their effect on the stiffness and

mass of the laminate is neglected and not taken in consideration during the formulation.

PVDF layers are polarized in the opposite directions such that when an electric field is

applied across the laminate the top layer extends in the x-y (or 1-2) plane and the bottom


56/129

46

layer contracts in the x-y plane, thus creating a moment bending and deflecting the plate

in the z direction.

If the thickness of the PVDF and epoxy are ph and eh respectively then the total thickness

of the laminate can be written as 2 p eh h h= + .

Figure 3.1: A three layer laminate with PVDF and epoxy

3.2 Dynamic EquationsAssuming the mid-plane in plane displacements to be zero,

0== oo vu (3.1)


57/129

47

the dynamic terms in equations (2.103) and (2.104) are zero. Therefore, considering

equation (2.105) for transverse deflection in the absence of external

load( 0)i i ix y xyN N N= = = ,

2 22 22 22

31 2

2 2 2 2 2

2 2

xy yxM MM RR Rw

x x y y t x x y y

+ + = + + +

(3.2)

Rewriting equation (2.75) for moment resultants,

+

=

6

2

1

662616

262212

161211

6

2

1

662616

262212

161211

k

o

o

o

kxy

y

x

DDD

DDD

DDD

BBB

BBB

BBB

M

M

M

From equation (3.1), mid-plane in plane displacements are zeros, so from equations

(2.54), (2.55) and (2.56)

1 2 6 0o o o

= = = (3.3)

So, equation (2.75) can be written as,

11 12 16 1

12 22 26 2

16 26 66 6

x

y

xy k

M D D D

M D D D

M D D D

=

(3.4)

Using equations (2.57) and (3.4),

2 2 2

11 12 162 22

x

w w wM D D D

x y x y

=

(3.5)


58/129


59/129

49

Equation (3.11) is a 4th

order differential equation of motion and ( , , )p x y t is the input to

the system.

3.3 Input Function

Rewriting equation (3.10) for the input function,

22 2

31 2

2 2

( , , ) 2

RR Rp x y t

x x y y

= + +

Where, R from equation (2.84) is,

[ ] [ ] =

=n

k

h

h

k

k

zdzE

d

d

d

QTR1

3

63

23

13

1

1

The applied electric field 3E can be expressed as,

3 ( , , ) ( , ) ( )o o oE x y t E P x y G t = (3.12)

Where, 1o = if the direction of polarization is the same as the applied electric field,

1o

= if the direction of polarization is opposite to the applied electric field,

oE is the magnitude of the applied electric field,

G(t) is the time variable function of the applied electric field and

),( yxPo is the spatial distribution of the applied electric field.

The input can be altered by varying the electric field oE and also by changing the shape

of the electrode ),( yxPo .


60/129

50

Figure 3.2: PVDF Bimorph with rectangular patch of electrode for input signal

Figure 3.2 shows the laminate with a rectangular shaped electrode. The spatial function

for this shape can be expressed as,

[ ] [ ])()()()(),( 2121 yyHyyHxxHxxHyxPo = (3.13)

Where, H is the Heaviside Step Function. It is a discontinuous function whose value is

zero for negative argument and one for positive argument. It is also known as Unit Step

Function.

Substituting equations (3.12) and (2.84) in equation (3.10),


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51

[ ] [ ]1

3 1 2 2 21 3 3 3

3 2 2 21

3 6

( , , ) 2k

k

hn

k h

dE E E

p x y t T Q d zdzx x y y

d

=

= + +

(3.14)

[ ] [ ]1

3 1 2 2 21

3 2 2 21

3 6

( , , ) ( , ) 2 ( , ) ( , ) ( )k

k

hn

o o o o o

k h

d

p x y t T Q d E P x y P x y P x y G t zdzx x y y

d

=

= + +

(3.15)

The partial derivatives of the spatial function defined in equation (3.13) are,

[ ] [ ])()()()(),(

2121 yyHyyHxxxxx

yxPo =

(3.16)

[ ] [ ])()()()(),(

21212

2

yyHyyHxxxxx

yxPo =

(3.17)

[ ] [ ])()()()(),(

2121 yyyyxxHxxHy

yxPo =

(3.18)

[ ] [ ])()()()(),(

21212

2

yyyyxxHxxHy

yxPo =

(3.19)

[ ] [ ]2

1 2 1 2

( , )( ) ( ) ( ) ( )o

P x yx x x x y y y y

x y

=

(3.20)

Where,

is the dirac delta function and represents an impulse of a force or a point force, and

is a first derivative of the dirac delta function and represents a point moment.

Substituting equations (3.17), (3.19) and (3.20) in (3.15) will provide the input function.

shape and vibration control of smart laminated plates_phd thesis

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